**Measurement in Quantum Mechanics: Decoherence and the Pointer Basis**

Anu Venugopalan

*University School of Basic and Applied Sciences, GGS Indraprastha University, Sector 16C, Dwarka, New Delhi-110075 India* 

#### **1. Introduction**

Quantum Mechanics represents one of the greatest triumphs of the scientific enterprise of the twentieth century. The stunning success of quantum theory has led to many revolutionary inventions and its extraordinary concepts describe the heart of several important real world applications like transistors and lasers. The theory makes accurate predictions about a wide range of physical phenomena and has historically withstood the tests and scrutiny of every experimental investigation. However, inspite of the fact that quantum theory is widely regarded by the scientific establishment as the fundamental theory of nature and is immensly successful and useful, its conceptual framework makes many predictions which are difficult to comprehend "classically". The theory is, pardoxically, powerful and confusing at the same time. Quantum theory's unusual predictions originate from its basic formailsm which involves concepts like probability amplitudes and the linear superposition principle(Dirac, 1947). The quantum view appears abstract and counterintuitive and at odds with classical perceptions. Many of the conceptual problems of quantum mechanics are encompassed in what is known as the *quantum measurement problem*(Peres, 1986; von Neumann, 1932; Wheeler & Zurek, 1983). Conventionally, the measurement paradox is supposedly 'resolved' by forcing a notion of a sudden collapse of the state vector of the system being measured. However, the nature of this mechanism is at odds with the basic tenets of quantum mechanics and hence may lie outside its realm thus questioning the validity of the theory it self. Closely related with the problem of measurement in quantum mechanics is the question of its connection with the emergence of classicality and the elusive boundary between quantum and classical worlds. What is the connection between the 'classical' and the 'quantum' worlds? Is there a definite relationship? Are classical mechanics and quantum mechanics two mutually exclusive incompatible theories or are they two aspects of the same underlying philosophy? Classical objects are eventually composed of elements of the microworld which can be described quantum mechanically. So, how and where can there be a boundary between the two worlds? In the following section we begin by introducing the quantum measurement problem. In the next section we will discuss attempts to understand and explain away the underlying paradoxes in quantum theory as highlighted in the quantum measurement problem and the question of the quantum-classical connection. From among the various explanations that seek a resolution to the conceptual problems of quantum mechanics, we focus on the 'environment induced decoherence theory' - an approach that employs the methods developed by several authors to analyse the quantum mechanics of

A basic postulate of quantum mechanics regarding *measurement* is that any measurement of the quantity A can only yield *one* of the eigenvalues, *ai*s, but the result is not definite in the sense that different measurements for the quantum state |*ψ*� can yield different eigenvalues. However, quantum theory predicts only that the *probability* of obtaining eigenvalue *ai* is |*ci*|

Measurement in Quantum Mechanics: Decoherence and the Pointer Basis 3

An additional postulate of quantum mechanics is that the measurement of an observable A,

or *collapse* of the state vector |*ψ*� to the eigenstate |*αi*�. This means that every term in the linear superposition vanishes, except one. This reduction is a *non unitary process* and hence in complete contrast to the unitary dynamics of quantum mechanics predicted by the Schrödinger equation and this is where the crux of the conceptual difficulties encountered in quantum theory lies. These two stages of quantum measurement are captured in the well-know von Neumann model through two distinct processes - first, where the system and apparatus interact through linear unitary Schrödinger evolution via an appropriate interaction Hamiltonian, and second - the nonlinear, indeterministic collapse (von Neumann, 1932). In this sense, the idea of measurement is very different from what we understand for classical systems. Classical systems are independent from measurements - the act of measurement does not disturb the state of the system or its 'properties'. In the language of quantum mechanical wave functions, the von Neumann measurement scheme can be

Measurements are described by treating both the system and the measuring apparatus as quantum objects. Let the quantum system be in the superposition state |*ψS*� = ∑*<sup>n</sup> cn*|*ψSn*�, where |*ψSn*� are the eigenstates of the operator that needs to be measured. For a measurement to be affected, the measured system described by |*ψS*� needs to interact with the measuring apparatus described by |*φA*� , so that the total wave function before the interaction is |*ψS*�|*φA*�. During the interaction of the system and the apparatus, the unitary evolution realizes the

Here |*φAn*� are orthonormal states of the measuring apparatus. This unitary evolution is


is often referred to as *the wave function collapse*. The final density operator corresponding to

system states, which, after the measurement will be found in the state |*ψSn*� with probability

corresponds to an additional selection of a subensemble by means of observation. In measurements of the second kind, the unitary evolution during the interaction of the system

in which the states |*χSn*� of the sytsem are determined by the nature of the interaction between system and measuring apparatus. As in the case of measurements of the first kind, the

*cn*|*ψSn*�|*φAn*� *(measurement of the first kind)*. (3)

<sup>2</sup>|*ψSn*��*ψSn*|. This density operator describes an *ensemble* of

<sup>2</sup>|*ψSn*��*ψSn*<sup>|</sup> (4)

<sup>2</sup>|*ψSn*��*ψSn*|→|*ψSn*�, (5)

*cn*|*χSn*�|*φAn*�, (6)

which yields one of the eigenvalues *ai* ( with probability |*ci*|

following transition from the initial to the final total wave function:

<sup>|</sup>*ψS*� → <sup>∑</sup>*<sup>n</sup>*

<sup>|</sup>*ψS*� → <sup>∑</sup>*<sup>n</sup>*


<sup>|</sup>*ψS*�|*φA*� → <sup>∑</sup>*<sup>n</sup>*

<sup>|</sup>*ψS*�|*φA*� → <sup>∑</sup>*<sup>n</sup>*

referred to as *premeasurement*. The transition

the system is calculated as ∑*<sup>n</sup>* |*cn*|

and measuring apparatus is described as:

2. The transition


illustrated as follows:

2.

2) culminates with the *reduction*

a system in interaction with its environment(Zeh, 1970). The central idea of this approach is that classicality is an emergent property triggered in open systems by their environments and it is the influence of environmental interactions that explains the perceived outcomes of quantum measurements(Zurek, 1981). We illustrate this approach through some specific system-apparatus models and highlight some key results of other researchers and ours. Following this, the next section will address a specific aspect of the decoherence theory, i.e., the notion of a 'preferred basis' or a 'pointer basis'. Our experience of the classical world suggests that unlike quantum systems, which are allowed to exist in all possible states, classical systems only exist in a few select states. The decoherence approach demonstrates that such states are singled out by the environment from a larger quantum menu. These special states are the 'preferred basis', also referred to as the 'pointer states' in a quantum-measurement-like scenario. What is the 'preferred' or 'pointer' basis? This question is examined via specific system-apparatus models and answered through some key results of our work and that of other researchers. Some of these results show that the 'pointer states' could emerge independent of the initial state of the apparatus. In the light of several advances in technology and high precision experiments, many of the questions relating to the conceptual problems of quantum mechanics are no longer merely 'academic' in nature. These questions and theories like environment-induced decoherence now offer themselves to experimental tests. Many recent experiments have provided important insights into the role of the environment in bringing about classicality. The decoherence theory is strengthened by these spectacular observations and there is no doubt that this approach has provided many important insights into the actual mechanism of the loss of quantum coherence. However, many researchers believe that many of the conceptual problems of quantum mechanics are still unresolved and the decoherence explanation is not adequate. In the concluding section of this chapter we summarize the main ideas that are presented in it and also highlight some of the difficulties and unresolved issues and their implications.

#### **2. Measurement in quantum mechanics**

Though quantum theory is widely accepted as the fundamental theory of nature and is immensely successful and satisfying, its conceptual framework makes predictions which are difficult to comprehend "classically". Many of these conceptual problems are encompassed in what is known as the quantum measurement problem. While the basic formalism of quantum mechanics was developed between 1925 and 1927, the standard interpretation of quantum measurement is attributed to von Neumann's theory presented in his book in 1932 (von Neumann, 1932). The quantum mechanical description of a system is contained in its wave function or state vector |*ψ*� which lives in an abstract "Hilbert space". The dynamics of the wavefunction is governed by the Schrödinger equation

$$i\hbar\frac{d}{dt}|\psi\rangle = H|\psi\rangle. \tag{1}$$

Here *H* is the Hamiltonian of the system and the equation is linear, deterministic and the time evolution governed by it is unitary. Dynamic variables or *observables* are represented in quantum mechanics by *linear Hermitian operators* which act on the state vector. An operator *A*ˆ, corresponding to the dynamical quantity *A* is associated with *eigenvalues ai* and corresponding *eigenvectors* {*αis*} which form a *complete orthonormal set*. Any arbitrary state vector, |*ψ*� can, in general, be represented as a linear superposition of these eigenvectors:

$$|\psi\rangle = \Sigma c\_i |\alpha\_i\rangle. \tag{2}$$

2 Will-be-set-by-IN-TECH

a system in interaction with its environment(Zeh, 1970). The central idea of this approach is that classicality is an emergent property triggered in open systems by their environments and it is the influence of environmental interactions that explains the perceived outcomes of quantum measurements(Zurek, 1981). We illustrate this approach through some specific system-apparatus models and highlight some key results of other researchers and ours. Following this, the next section will address a specific aspect of the decoherence theory, i.e., the notion of a 'preferred basis' or a 'pointer basis'. Our experience of the classical world suggests that unlike quantum systems, which are allowed to exist in all possible states, classical systems only exist in a few select states. The decoherence approach demonstrates that such states are singled out by the environment from a larger quantum menu. These special states are the 'preferred basis', also referred to as the 'pointer states' in a quantum-measurement-like scenario. What is the 'preferred' or 'pointer' basis? This question is examined via specific system-apparatus models and answered through some key results of our work and that of other researchers. Some of these results show that the 'pointer states' could emerge independent of the initial state of the apparatus. In the light of several advances in technology and high precision experiments, many of the questions relating to the conceptual problems of quantum mechanics are no longer merely 'academic' in nature. These questions and theories like environment-induced decoherence now offer themselves to experimental tests. Many recent experiments have provided important insights into the role of the environment in bringing about classicality. The decoherence theory is strengthened by these spectacular observations and there is no doubt that this approach has provided many important insights into the actual mechanism of the loss of quantum coherence. However, many researchers believe that many of the conceptual problems of quantum mechanics are still unresolved and the decoherence explanation is not adequate. In the concluding section of this chapter we summarize the main ideas that are presented in it and also highlight some of the difficulties

Though quantum theory is widely accepted as the fundamental theory of nature and is immensely successful and satisfying, its conceptual framework makes predictions which are difficult to comprehend "classically". Many of these conceptual problems are encompassed in what is known as the quantum measurement problem. While the basic formalism of quantum mechanics was developed between 1925 and 1927, the standard interpretation of quantum measurement is attributed to von Neumann's theory presented in his book in 1932 (von Neumann, 1932). The quantum mechanical description of a system is contained in its wave function or state vector |*ψ*� which lives in an abstract "Hilbert space". The dynamics of

Here *H* is the Hamiltonian of the system and the equation is linear, deterministic and the time evolution governed by it is unitary. Dynamic variables or *observables* are represented in quantum mechanics by *linear Hermitian operators* which act on the state vector. An operator *A*ˆ, corresponding to the dynamical quantity *A* is associated with *eigenvalues ai* and corresponding *eigenvectors* {*αis*} which form a *complete orthonormal set*. Any arbitrary state vector, |*ψ*� can, in

*dt*|*ψ*� <sup>=</sup> *<sup>H</sup>*|*ψ*�. (1)


and unresolved issues and their implications.

**2. Measurement in quantum mechanics**

the wavefunction is governed by the Schrödinger equation

*ih*¯ *d*

general, be represented as a linear superposition of these eigenvectors:

A basic postulate of quantum mechanics regarding *measurement* is that any measurement of the quantity A can only yield *one* of the eigenvalues, *ai*s, but the result is not definite in the sense that different measurements for the quantum state |*ψ*� can yield different eigenvalues. However, quantum theory predicts only that the *probability* of obtaining eigenvalue *ai* is |*ci*| 2. An additional postulate of quantum mechanics is that the measurement of an observable A, which yields one of the eigenvalues *ai* ( with probability |*ci*| 2) culminates with the *reduction* or *collapse* of the state vector |*ψ*� to the eigenstate |*αi*�. This means that every term in the linear superposition vanishes, except one. This reduction is a *non unitary process* and hence in complete contrast to the unitary dynamics of quantum mechanics predicted by the Schrödinger equation and this is where the crux of the conceptual difficulties encountered in quantum theory lies. These two stages of quantum measurement are captured in the well-know von Neumann model through two distinct processes - first, where the system and apparatus interact through linear unitary Schrödinger evolution via an appropriate interaction Hamiltonian, and second - the nonlinear, indeterministic collapse (von Neumann, 1932). In this sense, the idea of measurement is very different from what we understand for classical systems. Classical systems are independent from measurements - the act of measurement does not disturb the state of the system or its 'properties'. In the language of quantum mechanical wave functions, the von Neumann measurement scheme can be illustrated as follows:

Measurements are described by treating both the system and the measuring apparatus as quantum objects. Let the quantum system be in the superposition state |*ψS*� = ∑*<sup>n</sup> cn*|*ψSn*�, where |*ψSn*� are the eigenstates of the operator that needs to be measured. For a measurement to be affected, the measured system described by |*ψS*� needs to interact with the measuring apparatus described by |*φA*� , so that the total wave function before the interaction is |*ψS*�|*φA*�. During the interaction of the system and the apparatus, the unitary evolution realizes the following transition from the initial to the final total wave function:

$$<\langle \psi\_{\sf S} \rangle | \phi\_{\sf A} \rangle \to \sum\_{\sf n} c\_{\sf n} | \psi\_{\sf S n} \rangle | \phi\_{\sf A n} \rangle \qquad (measurement \ of \ the \sf first \ kind). \tag{3}$$

Here |*φAn*� are orthonormal states of the measuring apparatus. This unitary evolution is referred to as *premeasurement*. The transition

$$
\langle \psi\_{\rm S} \rangle \rightarrow \sum\_{n} |c\_{n}|^{2} |\psi\_{\rm Sn}\rangle \langle \psi\_{\rm Sn}| \tag{4}
$$

is often referred to as *the wave function collapse*. The final density operator corresponding to the system is calculated as ∑*<sup>n</sup>* |*cn*| <sup>2</sup>|*ψSn*��*ψSn*|. This density operator describes an *ensemble* of system states, which, after the measurement will be found in the state |*ψSn*� with probability |*cn*| 2. The transition

$$<\langle \psi\_{\rm S} \rangle \to \sum\_{n} |c\_{n}|^{2} |\psi\_{\rm Sn}\rangle\langle\psi\_{\rm Sn}| \to |\psi\_{\rm Sn}\rangle,\tag{5}$$

corresponds to an additional selection of a subensemble by means of observation. In measurements of the second kind, the unitary evolution during the interaction of the system and measuring apparatus is described as:

$$
\langle |\psi\_S\rangle | \phi\_A \rangle \to \sum\_n c\_n |\chi\_{Sn}\rangle |\phi\_{An}\rangle\_\prime \tag{6}
$$

in which the states |*χSn*� of the sytsem are determined by the nature of the interaction between system and measuring apparatus. As in the case of measurements of the first kind, the

elements ( the first and second terms) can be easily interpreted as probabilities corresponding to the system being in state |*ψS*1� or |*ψS*2� (with the corresponding correlations with the apparatus states |*φA*1� and |*φA*2�, respectively), the off-diagonal elements represented by the third and fourth terms are difficult to interpret classically in terms of probabilities. In order to make 'classical' sense, the density matrix corresponding to the pure state ensemble described by the entangled state (8) must reduce to a *statistical mixture* which is *diagonal* in some basis with appropriate system-apparatus correlations. Such a *mixed* density matrix would look like

Measurement in Quantum Mechanics: Decoherence and the Pointer Basis 5

Several interpretations of quantum mechanics seek to explain this *ρpure* → *ρmixed* transition (von Neumann's irreversible 'reduction' process) and a resolution to the mechanism for the apparently nonunitary 'collapse' in a quantum measurement(Wheeler & Zurek, 1983; Zurek, 1991). In recent years, the decoherence approach(Joos et al., 2003) has been widely discussed and accepted as the mechanism responsible for this transition. The central idea of this approach has been that 'classicality' is an emergent property of systems interacting with an environment. The theory also predicts that in a quantum measurement, the apparatus will have correlations with the system in a set of 'preferred states'(Joos et al., 2003; Zurek, 1981; 1991) selected by the environment. In the next two sections we describe the progress made in adopting this approach to explain the mechanism for the perceived outcomes of a quantum measurement as well as the emergence of classicality from an underlying quantum world. The strength of this approach lies in the fact that it provides a reasonably satisfying explanation

In the previous section, we have seen that there is a serious interpretational problem with the way quantum mechanics deals with the act of measurement. In particular, the problem lies in von Neumann's postulate of an irreversible reduction process which takes the quantum superposition to a statistical mixture which is supposedly classically interpretable and meaningful. However, the non-unitary nature of this reduction is at odds with the inherent unitary nature of the Schrodinger equation, implying, somehow that the mechanism seems to lie *outside* the realm of quantum mechanics. From among the various explanations that seek a resolution to the conceptual problems of quantum mechanics, in this section we focus on the 'environment induced decoherence theory'. As pointed out in the previous section, the problem lies with the *off-diagonal* elements of the density matrix describing the entangled state of the system-apparatus composite, (8). These off-diagonal elements are the signatures of *quantum correlations*. In quantum mechanics, wave functions evolve according to the Schrödinger equation which is linear and deterministic and this evolution is unitary in nature. Unitary evolutions ensure that eigenvalues are preserved. There is no way that some terms of the density matrix can vanish in the course of a unitary evolution. How, then, can 'classical behaviour' (as discussed above) ever emerge from this substrate of the quantum world where entanglements and coherences are ubiquitous and inevitable? How can a pure state density matrix become a "classically interpretable" statistical mixture? The decoherence approach seeks to answer this problem by providing a mechanism which leads to the loss of quantum coherence, thus bringing about the much desired *ρpure* → *ρmixed* transition, and hence, classicality. The answer lies in realizing the fact that the Schrödinger equation driving unitary evolutions is strictly applicable only to *completely isolated systems*. In reality, we know that macroscopic systems are *almost never isolated* from their surroundings but are known to

*within* the realm of quantum mechanics.

**3. Decoherence**

*ρmixed* ∼ |*ψS*1��*ψS*1||*ψA*1��*ψA*1| + |*ψS*2��*ψS*2||*ψA*2��*ψA*2|. (9)

final state of the system will be |*χSn*� with probability |*cn*| 2. The concept that quantum mechanics does not yield an objective description of microscopic reality but deals only with probabilities (as illustrated in the measurement process) is an essential part of the Copenhagen interpretation of quantum mechanics which is regarded as the "standard" interpretation of quantum mechanics. The von Neumann measurement scheme is in tune with the Copenhagen interpretation of quantum mechanics which was one of the first attempts to understand quantum mechanics, initiated by Niels Bohr, and supported by Werner Heisenberg, Max Born and others. The von Neumann scheme described above would typically involve a coupling between the microscopic system and a 'macroscopic' apparatus (meter), resulting in states like (3) and (6), called *entangled states* which are uniquely quantum mechanical states for the composite. The term *entanglement* was coined by Schrödinger and describes a correlated state that is "not separable"(Schrödinger, 1935). Today, entanglement is considered one of the most defining concepts in quantum mechanics - a uniquely quantum mechanical possibility with no classical analouge. In quantum information and quantum computation, entanglement is viewed as a resource for computing tasks that can be performed faster or in a more secure way than is classically possible and there are intensive experimental efforts to create entangled states in the labarotory. The entangled state describing the system-apparatus, as in (3) above should contain one-to-one correlations between the states of the system, {|*ψSn*�}, and the states of the apparatus {|*φAn*�}, so that a read out of the apparatus or 'meter' states gives information about the states of the system. Consider a simple example of a two-level system for which the entangled system-apparatus state after the measurement interaction should look like

$$|\psi\_S\rangle|\phi\_A\rangle \to |\psi\_{S1}\rangle|\phi\_{A1}\rangle + |\psi\_{S2}\rangle|\phi\_{A2}\rangle. \tag{7}$$

Such an entangled state is like a two-particle *superposition state*. The problem with such an entangled state is that it seems to allow the 'meter' (apparatus) to exist in a coherent superposition of the two states |*φA*1� and |*φA*2� which could be macroscopically distinct - a situation hard to reconcile with classical intuition. Historically, it was the Einstein, Podolsky, Rosen (EPR) paper(Einstein et al., 1935) which first highlighted the problem of quantum entanglement. In response to the EPR work, Schrödinger posed a thought experiment which is now famously known as Schrödinger's Cat paradox(Schrödinger, 1935). Schrödinger's cat is the unfortunate victim of a nasty contraption where the decay of a radioactive atom triggers a device which kills the cat . The quantum mechanical description of this scenario demands that a superposition state of 'decayed' and 'not decayed' for the atom lead to an entangled state of the kind (7) for the atom-cat composite with the cat being in a superposition state of 'dead' and 'alive'. This amounts to interpreting the quantum state of the cat as being in a coherent superposition of 'dead' and 'alive' states - a situation which is completely at odds with our familair classical perceptions. Schroödinger's Cat paradox is often presented as an illustration of the conceptual problems of quantum mechanics. It is worth mentioning at this point that the density matrix is a convenient formal tool to compare and contrast quantum and classical systems in terms of probabilities. Some of the conceptual problems of quantum measurement become more transparent when analyzed in this language. It can be easily seen that the density matrix corresponding to the entangled sate (7) is

$$
\begin{split}
\hat{\rho}\_{\text{S}+A} &= |\psi\_{\text{S1}}\rangle\langle\psi\_{\text{S1}}||\psi\_{A1}\rangle\langle\psi\_{A1}| + |\psi\_{\text{S2}}\rangle\langle\psi\_{\text{S2}}||\psi\_{A2}\rangle\langle\psi\_{A2}| \\
&+ - |\psi\_{\text{S1}}\rangle\langle\psi\_{\text{S2}}||\psi\_{A1}\rangle\langle\psi\_{A2}| + |\psi\_{\text{S2}}\rangle\langle\psi\_{\text{S1}}||\psi\_{A2}\rangle\langle\psi\_{A1}|.
\end{split}
\tag{8}
$$

While (8) represents a perfectly legitimate solution of the Schrödinger equation, the physical interpretation in the usual language of probabilities leads to difficulties. While the diagonal elements ( the first and second terms) can be easily interpreted as probabilities corresponding to the system being in state |*ψS*1� or |*ψS*2� (with the corresponding correlations with the apparatus states |*φA*1� and |*φA*2�, respectively), the off-diagonal elements represented by the third and fourth terms are difficult to interpret classically in terms of probabilities. In order to make 'classical' sense, the density matrix corresponding to the pure state ensemble described by the entangled state (8) must reduce to a *statistical mixture* which is *diagonal* in some basis with appropriate system-apparatus correlations. Such a *mixed* density matrix would look like

$$
\rho\_{\rm mixed} \sim |\psi\_{\rm S1}\rangle\langle\psi\_{\rm S1}| |\psi\_{A1}\rangle\langle\psi\_{A1}| + |\psi\_{\rm S2}\rangle\langle\psi\_{\rm S2}| |\psi\_{A2}\rangle\langle\psi\_{A2}|.\tag{9}
$$

Several interpretations of quantum mechanics seek to explain this *ρpure* → *ρmixed* transition (von Neumann's irreversible 'reduction' process) and a resolution to the mechanism for the apparently nonunitary 'collapse' in a quantum measurement(Wheeler & Zurek, 1983; Zurek, 1991). In recent years, the decoherence approach(Joos et al., 2003) has been widely discussed and accepted as the mechanism responsible for this transition. The central idea of this approach has been that 'classicality' is an emergent property of systems interacting with an environment. The theory also predicts that in a quantum measurement, the apparatus will have correlations with the system in a set of 'preferred states'(Joos et al., 2003; Zurek, 1981; 1991) selected by the environment. In the next two sections we describe the progress made in adopting this approach to explain the mechanism for the perceived outcomes of a quantum measurement as well as the emergence of classicality from an underlying quantum world. The strength of this approach lies in the fact that it provides a reasonably satisfying explanation *within* the realm of quantum mechanics.

#### **3. Decoherence**

4 Will-be-set-by-IN-TECH

mechanics does not yield an objective description of microscopic reality but deals only with probabilities (as illustrated in the measurement process) is an essential part of the Copenhagen interpretation of quantum mechanics which is regarded as the "standard" interpretation of quantum mechanics. The von Neumann measurement scheme is in tune with the Copenhagen interpretation of quantum mechanics which was one of the first attempts to understand quantum mechanics, initiated by Niels Bohr, and supported by Werner Heisenberg, Max Born and others. The von Neumann scheme described above would typically involve a coupling between the microscopic system and a 'macroscopic' apparatus (meter), resulting in states like (3) and (6), called *entangled states* which are uniquely quantum mechanical states for the composite. The term *entanglement* was coined by Schrödinger and describes a correlated state that is "not separable"(Schrödinger, 1935). Today, entanglement is considered one of the most defining concepts in quantum mechanics - a uniquely quantum mechanical possibility with no classical analouge. In quantum information and quantum computation, entanglement is viewed as a resource for computing tasks that can be performed faster or in a more secure way than is classically possible and there are intensive experimental efforts to create entangled states in the labarotory. The entangled state describing the system-apparatus, as in (3) above should contain one-to-one correlations between the states of the system, {|*ψSn*�}, and the states of the apparatus {|*φAn*�}, so that a read out of the apparatus or 'meter' states gives information about the states of the system. Consider a simple example of a two-level system for which the entangled system-apparatus state after the measurement interaction should look

Such an entangled state is like a two-particle *superposition state*. The problem with such an entangled state is that it seems to allow the 'meter' (apparatus) to exist in a coherent superposition of the two states |*φA*1� and |*φA*2� which could be macroscopically distinct - a situation hard to reconcile with classical intuition. Historically, it was the Einstein, Podolsky, Rosen (EPR) paper(Einstein et al., 1935) which first highlighted the problem of quantum entanglement. In response to the EPR work, Schrödinger posed a thought experiment which is now famously known as Schrödinger's Cat paradox(Schrödinger, 1935). Schrödinger's cat is the unfortunate victim of a nasty contraption where the decay of a radioactive atom triggers a device which kills the cat . The quantum mechanical description of this scenario demands that a superposition state of 'decayed' and 'not decayed' for the atom lead to an entangled state of the kind (7) for the atom-cat composite with the cat being in a superposition state of 'dead' and 'alive'. This amounts to interpreting the quantum state of the cat as being in a coherent superposition of 'dead' and 'alive' states - a situation which is completely at odds with our familair classical perceptions. Schroödinger's Cat paradox is often presented as an illustration of the conceptual problems of quantum mechanics. It is worth mentioning at this point that the density matrix is a convenient formal tool to compare and contrast quantum and classical systems in terms of probabilities. Some of the conceptual problems of quantum measurement become more transparent when analyzed in this language. It can be easily seen

that the density matrix corresponding to the entangled sate (7) is


*ρ*ˆ*S*+*<sup>A</sup>* = |*ψS*1��*ψS*1||*ψA*1��*ψA*1| + |*ψS*2��*ψS*2||*ψA*2��*ψA*2| (8)

+ |*ψS*1��*ψS*2||*ψA*1��*ψA*2| + |*ψS*2��*ψS*1||*ψA*2��*ψA*1|. While (8) represents a perfectly legitimate solution of the Schrödinger equation, the physical interpretation in the usual language of probabilities leads to difficulties. While the diagonal

2. The concept that quantum

final state of the system will be |*χSn*� with probability |*cn*|

like

In the previous section, we have seen that there is a serious interpretational problem with the way quantum mechanics deals with the act of measurement. In particular, the problem lies in von Neumann's postulate of an irreversible reduction process which takes the quantum superposition to a statistical mixture which is supposedly classically interpretable and meaningful. However, the non-unitary nature of this reduction is at odds with the inherent unitary nature of the Schrodinger equation, implying, somehow that the mechanism seems to lie *outside* the realm of quantum mechanics. From among the various explanations that seek a resolution to the conceptual problems of quantum mechanics, in this section we focus on the 'environment induced decoherence theory'. As pointed out in the previous section, the problem lies with the *off-diagonal* elements of the density matrix describing the entangled state of the system-apparatus composite, (8). These off-diagonal elements are the signatures of *quantum correlations*. In quantum mechanics, wave functions evolve according to the Schrödinger equation which is linear and deterministic and this evolution is unitary in nature. Unitary evolutions ensure that eigenvalues are preserved. There is no way that some terms of the density matrix can vanish in the course of a unitary evolution. How, then, can 'classical behaviour' (as discussed above) ever emerge from this substrate of the quantum world where entanglements and coherences are ubiquitous and inevitable? How can a pure state density matrix become a "classically interpretable" statistical mixture? The decoherence approach seeks to answer this problem by providing a mechanism which leads to the loss of quantum coherence, thus bringing about the much desired *ρpure* → *ρmixed* transition, and hence, classicality. The answer lies in realizing the fact that the Schrödinger equation driving unitary evolutions is strictly applicable only to *completely isolated systems*. In reality, we know that macroscopic systems are *almost never isolated* from their surroundings but are known to

emergence of classicality and the perceived outcomes of quantum measurements(Zurek, 1991). In the following subsections we highlight studies done on two measurement models where the outcome of the system-apparatus interaction is explained by the decoherence

Measurement in Quantum Mechanics: Decoherence and the Pointer Basis 7

X' X

(a)

(b)

after the off-daigonal elements have been partially washed away by decoherence

degrees) correlates with the spin states. The Hamiltonian describing this model is:

*<sup>H</sup>* <sup>=</sup> *λσ<sup>z</sup>* <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

2*m*

While the first two terms represent the self Hamiltonians of the system and apparatus, respectively, the last term is the interaction Hamiltonian. *z* and *p* denote the position and momentum of the particle of mass *m*, *λσ<sup>z</sup>* the Hamiltonian of the system and *�* the product of the field gradient and the magnetic moment of the particle. The most general initial state for

+ *�zσz*. (11)

*ψ* = {*a*| ↑� + *b*| ↓�} ⊗ *φ*(*z*). (12)

**3.1 Decoherence and the Stern-Gerlach measurement**

the system-apparatus combine can be written as

<sup>X</sup> X'

Fig. 1. (a)Density matrix of a superposition of two Gaussian wave packets (b) Density matrix

Venugopalan et al(Venugopalan et al., 1995a;b) first anaylsed a Stern-Gerlach-like measurement model through the decoherence approach. Their analysis shows that decoherence would bring out the desired 'classical-like' outcome in such a measurement scenario. Consider a simple model of measurement consisting of a free particle with spin (for simplicity, consider a two-state system or spin-1/2 which could represent a qubit). Here the spin degrees of freedom represent the system and the position and momentum degrees represent the apparatus. Let us first look at the bare system without the inclusion of additional environment degrees of freedom. The system and apparatus are coupled by a Stern-Gerlach measurementlike interaction such that the trajectory of the particle (position and momentum

approach.

be constantly interacting with a complex environment. In a measurement like situation, the apparatus is almost always a macroscopic object from which one reads out the measured property of the system. In fact, the apparatus is not only considered macroscopic, it is also regarded as 'classical' in its dynamics. How does the apparatus (which starts off as a quantum object) end up appearing 'classical'?

The 'decoherence' explanation is that it is the influence of the environment that makes a quantum system appear 'classical'. The environment 'washes away' quantum coherence ('deoherence'), leaving behind a system which looks and behaves like a classical object of our cherished commonsense world. The system no longer constitutes a closed system but an 'open system' which is coupled to a large number degrees of freedom which constitute the environment. However, one is always monitoring only a few degrees of freedom, which are of relevance. Technically, this amounts to 'tracing' over all other degress of freedom. This tracing over has the effect of causing the transition:

$$
\rho\_{\text{pure}} \rightarrow \rho\_{\text{mixed}}.\tag{10}
$$

An illuminating and popular paradigm for understanding decoherence is the phenomenon of Brownian motion which describes the motion of a particle suspended in a liquid. Such a suspended particle, when examined closely, is seen to bounce around in a random, irregular, 'zig-zag' fashion. Einstein showed that this behaviour is exactly what should be expected if the suspended particle is being repeatedly 'kicked' by other unseen smaller particles. The random motion of the suspended particle can be statistically explained by taking into account its interaction with a large number of particles which constitute the reservoir of liquid molecules or the 'environment'. When we see Brownian motion, we are only focussing on the dynamics of the suspended particle and do not monitor each and every particle of the environment. Mathematically, we trace over all the degrees of freedom of the environment and look only at the reduced system -the suspended particle. As a consequence, the tagged particle is found to show a dynamics that contains dissipation (a steady loss of energy or relaxation) and diffusion (the random zig-zag motion). *Quantum Brownian motion* describes a similar situation at the quantum mechanical level(Agarwal, 1971; Caldeira & Leggett, 1983). A simple example by Zurek(Zurek, 1991) illustrates this point (see Figure 1). Here an initial pure state constructed as a coherent superposition of two spatially separated Gaussian wavepackets decoheres into a statistical mixture (diagonal density matrix) when its dynamics incorporates the coupling to a large number of environmental degrees of freedom. While the pure state density matrix of the system (Fig. 1(a)) has both diagonal and off-diagonal elements, it can be seen that after a certain time, impacted by the environmental influence, the off-diagonal elements of the reduced system are diminished to give us a statistical mixture (Fig1(b)).

Let us now look at the quantum measurement situation through this approach. The microscopic system couples to a macroscopic apparatus, which in turn is interacting with a large number of degrees of freedom which conststitutes the environment . Schrödinger's equation is applied to the entire closed universe of system-apparatus-environment. Hamiltonian evolution drives this closed system from an initial uncoupled state into a gigantic entangled state containing all the degrees of freedom. A tracing over all the environmental degrees of freedom salvages the reduced system-apparatus combine from this mess. After a characteristic time, the apparatus, impacted by the environment, appears classical in its dynamics. Thus, the environment causes a general quantum state to decay into a statistical mixture of "pointer states" which can be understood and interpreted as classical probability distributions. This, in essence, is the approach of the decoherence theory to explain the 6 Will-be-set-by-IN-TECH

be constantly interacting with a complex environment. In a measurement like situation, the apparatus is almost always a macroscopic object from which one reads out the measured property of the system. In fact, the apparatus is not only considered macroscopic, it is also regarded as 'classical' in its dynamics. How does the apparatus (which starts off as a quantum

The 'decoherence' explanation is that it is the influence of the environment that makes a quantum system appear 'classical'. The environment 'washes away' quantum coherence ('deoherence'), leaving behind a system which looks and behaves like a classical object of our cherished commonsense world. The system no longer constitutes a closed system but an 'open system' which is coupled to a large number degrees of freedom which constitute the environment. However, one is always monitoring only a few degrees of freedom, which are of relevance. Technically, this amounts to 'tracing' over all other degress of freedom. This

An illuminating and popular paradigm for understanding decoherence is the phenomenon of Brownian motion which describes the motion of a particle suspended in a liquid. Such a suspended particle, when examined closely, is seen to bounce around in a random, irregular, 'zig-zag' fashion. Einstein showed that this behaviour is exactly what should be expected if the suspended particle is being repeatedly 'kicked' by other unseen smaller particles. The random motion of the suspended particle can be statistically explained by taking into account its interaction with a large number of particles which constitute the reservoir of liquid molecules or the 'environment'. When we see Brownian motion, we are only focussing on the dynamics of the suspended particle and do not monitor each and every particle of the environment. Mathematically, we trace over all the degrees of freedom of the environment and look only at the reduced system -the suspended particle. As a consequence, the tagged particle is found to show a dynamics that contains dissipation (a steady loss of energy or relaxation) and diffusion (the random zig-zag motion). *Quantum Brownian motion* describes a similar situation at the quantum mechanical level(Agarwal, 1971; Caldeira & Leggett, 1983). A simple example by Zurek(Zurek, 1991) illustrates this point (see Figure 1). Here an initial pure state constructed as a coherent superposition of two spatially separated Gaussian wavepackets decoheres into a statistical mixture (diagonal density matrix) when its dynamics incorporates the coupling to a large number of environmental degrees of freedom. While the pure state density matrix of the system (Fig. 1(a)) has both diagonal and off-diagonal elements, it can be seen that after a certain time, impacted by the environmental influence, the off-diagonal elements of the

reduced system are diminished to give us a statistical mixture (Fig1(b)).

Let us now look at the quantum measurement situation through this approach. The microscopic system couples to a macroscopic apparatus, which in turn is interacting with a large number of degrees of freedom which conststitutes the environment . Schrödinger's equation is applied to the entire closed universe of system-apparatus-environment. Hamiltonian evolution drives this closed system from an initial uncoupled state into a gigantic entangled state containing all the degrees of freedom. A tracing over all the environmental degrees of freedom salvages the reduced system-apparatus combine from this mess. After a characteristic time, the apparatus, impacted by the environment, appears classical in its dynamics. Thus, the environment causes a general quantum state to decay into a statistical mixture of "pointer states" which can be understood and interpreted as classical probability distributions. This, in essence, is the approach of the decoherence theory to explain the

*ρpure* → *ρmixed*. (10)

object) end up appearing 'classical'?

tracing over has the effect of causing the transition:

emergence of classicality and the perceived outcomes of quantum measurements(Zurek, 1991). In the following subsections we highlight studies done on two measurement models where the outcome of the system-apparatus interaction is explained by the decoherence approach.

Fig. 1. (a)Density matrix of a superposition of two Gaussian wave packets (b) Density matrix after the off-daigonal elements have been partially washed away by decoherence

#### **3.1 Decoherence and the Stern-Gerlach measurement**

Venugopalan et al(Venugopalan et al., 1995a;b) first anaylsed a Stern-Gerlach-like measurement model through the decoherence approach. Their analysis shows that decoherence would bring out the desired 'classical-like' outcome in such a measurement scenario. Consider a simple model of measurement consisting of a free particle with spin (for simplicity, consider a two-state system or spin-1/2 which could represent a qubit). Here the spin degrees of freedom represent the system and the position and momentum degrees represent the apparatus. Let us first look at the bare system without the inclusion of additional environment degrees of freedom. The system and apparatus are coupled by a Stern-Gerlach measurementlike interaction such that the trajectory of the particle (position and momentum degrees) correlates with the spin states. The Hamiltonian describing this model is:

$$H = \lambda \sigma\_z + \frac{p^2}{2m} + \epsilon z \sigma\_z. \tag{11}$$

While the first two terms represent the self Hamiltonians of the system and apparatus, respectively, the last term is the interaction Hamiltonian. *z* and *p* denote the position and momentum of the particle of mass *m*, *λσ<sup>z</sup>* the Hamiltonian of the system and *�* the product of the field gradient and the magnetic moment of the particle. The most general initial state for the system-apparatus combine can be written as

$$
\psi = \{a \vert \uparrow \rangle + b \vert \downarrow \rangle\} \otimes \phi(z). \tag{12}
$$

dissipation/decoherence involved, and the state remains a pure entangled state with all its non-diagonal elements (*ρ*↑↓, *<sup>ρ</sup>*↓↑) as well as the spatial nonlocality. The establishment of system-apparatus correlations, therefore, *is not enough to affect a measurement as the off diagonal elements of the density matrix have not vanished* and we do not have the desired *mixed state* density

Measurement in Quantum Mechanics: Decoherence and the Pointer Basis 9

Now let us look at the situation where we include the interaction with the environment. A commonly used model to describe the environment is to consider it as a reservoir of quantum oscillators, each of which interacts with the apparatus in our case and is describd by the

> *P*2 *j* 2*Mj* + *M*<sup>2</sup> *<sup>j</sup>* <sup>Ω</sup><sup>2</sup> *j* 2

Here *z* and *p* denote the position and momentum of the particle (apparatus) of mass *m*. *λσ<sup>z</sup>* is the Hamiltonian of the system and *�* is the strength of the system-apparatus coupling as mentioned earlier. The last term represents the Hamiltonian for the bath of oscillators (environment) and the apparatus-environment interaction. *Zj* and *Pj* are the position and momentum coordinates of the jth harmonic oscillator of the bath, *Cj*s are the coupling strengths and Ω*j*s are the frequencies of the oscillators comprising the bath. Note that the coupling of the apparatus with the many environmental degrees of freedom is a coordinate-coordinate coupling. The dynamics for the closed universe of the system, apparatus and environment is governed by the Hamiltonian evolution via (17) and the Schrödinger equation. In the decoherence approach, a tracing over all the degrees of freedom of the environment results in an equation describing the dynamics of the *reduced density matrix* of the system-apparatus combine. The reduced density matrix evolves according to a master equation which is obtained by solving the Schröinger equation for the entire universe of the system, apparatus and the environment and then tracing over the environment degrees of freedom. Several authors like Zeh and Zurek, among others, have worked extensively on the decoherence approach using the master equation for the reduced density matrix. The master equation for this kind of model of the environment was first derived separately by Caldeira and Leggett, Agarwal, Dekker(Dekker, 1977) and others in the context of quantum Brownian motion and is a popular equation for the study of open quantum systems. For our purpose, we deal with the mater equation for the system-apparatus-environment composite described by (17). The master equation for the density matrix, corresponding to the four elements of

*Zj* <sup>−</sup> *Cjz Mj*Ω<sup>2</sup> *j*

2

. (17)

(18)

matrix.

Hamiltonian

*<sup>H</sup>* <sup>=</sup> *λσ<sup>z</sup>* <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

spin space (↑↑, ↑↓, ↓↑, ↓↓) (see (13)) is

, *t*) *<sup>∂</sup><sup>t</sup>* <sup>=</sup>  <sup>−</sup> *<sup>h</sup>*¯ 2*im*

<sup>−</sup> *<sup>D</sup>*

 *∂*<sup>2</sup> *<sup>∂</sup>z*<sup>2</sup> <sup>−</sup> *<sup>∂</sup>*<sup>2</sup> *∂z*�<sup>2</sup> 

)<sup>2</sup> +

4¯*h*<sup>2</sup> (*<sup>z</sup>* <sup>−</sup> *<sup>z</sup>*�

− *γ*(*z* − *z*�

*s*� ) *<sup>h</sup>*¯ <sup>+</sup>

*i�*(*zs* − *z*�

where *s*,*s*� = +1 (for ↑) or −1 (for ↓). Here *γ* is the Langevin friction coefficient and *D* has the usual interpretation of the diffusion coefficient. *γ* and *D* are related to the parameters of the Hamiltonin of the total system. Without going into further details of the master equation, it suffices to point out that the equation naturally separates into three distinct terms, namely (i) a term describing the von Neumann equation which can be derived from the Schrödinger equation and thus represents the pure quantum evolution, (ii) a term that causes dissipation, which can be understood as a steady loss in energy or a relaxation process, and (iii) a term causing diffusion, which can be understood as the fluctuctions or zig-zag

) *∂ <sup>∂</sup><sup>z</sup>* <sup>−</sup> *<sup>∂</sup> ∂z*� 

*iλ*(*s* − *s*�

*h*¯

)

 *ρss*�(*z*, *z*�

, *t*),

*∂ρss*�(*z*, *z*�

2*m*

+ *�zσ<sup>z</sup>* + ∑

*j*

This is a product state of the most general spin state for the system and an arbitrary state, *φ*(*z*) for the apparatus (free particle). | ↑� and | ↓� are the eigenstates of *σz*. Following the measurement interaction governed by Hamiltonian evolution, this initial state becomes an *entangled state* between the system and apparatus. The density matrix of the entangled state, when the initial state, (12), undergoes a Hamiltonian evolution via (11) will be of the form

$$\begin{split} \rho\_{\mathcal{S}+A} &= |a|^2 |\uparrow\rangle \langle \uparrow| \rho\_{\uparrow\uparrow}(z, z', t) + |b|^2 |\downarrow\rangle \langle \downarrow| \rho\_{\downarrow\downarrow}(z, z', t) \\ &+ |ab^\*| \uparrow\rangle \langle \downarrow| \rho\_{\uparrow\downarrow}(z, z', t) + a^\*b |\downarrow\rangle \langle \uparrow| \rho\_{\downarrow\uparrow}(z, z', t). \end{split} \tag{13}$$

Here *<sup>ρ</sup>*↑↑ and *<sup>ρ</sup>*↓↓ correspond to the diagonal elements (in spin) of the density matrix (*ρd*) for the apparatus which could correlate with up and down spin states of the system and *<sup>ρ</sup>*↑↓ and *<sup>ρ</sup>*↓↑ correspond to the off-diagonal elements (*ρod*), and *<sup>ρ</sup>*(*z*, *<sup>z</sup>*� , *t*) = �*z*|*ρ*|*z*� �. The specific form of *ρ<sup>d</sup>* and *ρod* and the system correlations they would (or would not) contain depends on the initial state of the apparatus. When they contain one-to-one system-apparatus correlations, the states corresponding to *ρ<sup>d</sup>* would be candidate *pointer states*. For the purpose of illustration we look at the situation when the apparatus starts off in an initial state which is a Gaussian wave packet as was first analyzed by Venugopalan et al. The initial system-apparatus state is thus given as

$$\{\left|a\right|\uparrow\rangle + b\left|\downarrow\right>\} \otimes \Phi(z) = \{a\left|\uparrow\right> + b\left|\downarrow\right>\} \otimes \frac{1}{\sqrt{\sigma\sqrt{\pi}}} \exp\{\frac{-z^2}{2\sigma^2}\}.\tag{14}$$

Here *σ* is the width of the wavepacket. The wavepacket in quantum mechanics is often viewed as the most "nearly classical" state and is known to exhibit many striking classical properties and hence is a reasonably good choice for the initial state of the apparatus. Following Hamiltonian evolution via (11), the system-apparatus composite ends up in an entangled state whose density matrix is of the form (13). One examining the detailed form of the complete density matrix representing this entangled state, one can identify the parts of the "apparatus", i.e., *<sup>ρ</sup>*↑↑ and *<sup>ρ</sup>*↓↓ that correlate with the up and down spins, and it can be shown that the diagonal elements of *<sup>ρ</sup>*↑↑ and *<sup>ρ</sup>*↓↓ are

$$\rho\_{\uparrow\uparrow}\rho\_{\downarrow\downarrow} \rightarrow \rho\_d(z,t) = \frac{2}{\sigma} \sqrt{\frac{\pi}{N(t)}} \exp\{-\frac{4}{\sigma\sqrt{N(t)}} \left(z \mp \frac{\epsilon t^2}{2m}\right)^2\},\tag{15}$$

in the position representation and

$$\rho\_{\uparrow\uparrow}, \rho\_{\downarrow\downarrow} \rightarrow \rho\_d(p, t)) = 2\sigma \sqrt{\pi} \exp\{-4\sigma^2 \left(p \mp \frac{\epsilon t}{\hbar}\right)^2\},\tag{16}$$

in the momentum representation. It must be kept in mind, though, that the density matrix represents a pure state which has 'nonlocal' quantum correlations both in the spin space and the position and momentum space. In (15) & (16) above, we are looking at the *diagonal* elements of the position and momentum space density matrix and these show system-apparatus correlations. The up and down spin states of the system correlate with a Gaussian wavepackets centered round <sup>−</sup> *�<sup>t</sup>* 2 <sup>2</sup>*<sup>m</sup>* and <sup>+</sup> *�<sup>t</sup>* 2 <sup>2</sup>*<sup>m</sup>* , in the position space, respectively, and around <sup>−</sup>*�<sup>t</sup> <sup>h</sup>*¯ and <sup>+</sup>*�<sup>t</sup> <sup>h</sup>*¯ in the momentum space. This pure entangled state of the system and the apparatus is akin to a 'Schroödinger cat state' which contains one-to-one correlations between the system and 'macroscopic' apparatus states with all quantum coherences intact. Since the dynamics, as governed by the Hamiltonian (11), is purely unitary, there is no 8 Will-be-set-by-IN-TECH

This is a product state of the most general spin state for the system and an arbitrary state, *φ*(*z*) for the apparatus (free particle). | ↑� and | ↓� are the eigenstates of *σz*. Following the measurement interaction governed by Hamiltonian evolution, this initial state becomes an *entangled state* between the system and apparatus. The density matrix of the entangled state, when the initial state, (12), undergoes a Hamiltonian evolution via (11) will be of the form

Here *<sup>ρ</sup>*↑↑ and *<sup>ρ</sup>*↓↓ correspond to the diagonal elements (in spin) of the density matrix (*ρd*) for the apparatus which could correlate with up and down spin states of the system and *<sup>ρ</sup>*↑↓ and

of *ρ<sup>d</sup>* and *ρod* and the system correlations they would (or would not) contain depends on the initial state of the apparatus. When they contain one-to-one system-apparatus correlations, the states corresponding to *ρ<sup>d</sup>* would be candidate *pointer states*. For the purpose of illustration we look at the situation when the apparatus starts off in an initial state which is a Gaussian wave packet as was first analyzed by Venugopalan et al. The initial system-apparatus state is

Here *σ* is the width of the wavepacket. The wavepacket in quantum mechanics is often viewed as the most "nearly classical" state and is known to exhibit many striking classical properties and hence is a reasonably good choice for the initial state of the apparatus. Following Hamiltonian evolution via (11), the system-apparatus composite ends up in an entangled state whose density matrix is of the form (13). One examining the detailed form of the complete density matrix representing this entangled state, one can identify the parts of the "apparatus", i.e., *<sup>ρ</sup>*↑↑ and *<sup>ρ</sup>*↓↓ that correlate with the up and down spins, and it can be shown that the

, *t*) + |*b*|

<sup>2</sup>| ↓��↓ |*ρ*↓↓(*z*, *<sup>z</sup>*�

, *t*) = �*z*|*ρ*|*z*�

exp{−*z*<sup>2</sup>

, *<sup>t</sup>*) + *<sup>a</sup>*∗*b*| ↓��↑ |*ρ*↓↑(*z*, *<sup>z</sup>*�

*σ* <sup>√</sup>*<sup>π</sup>* , *t*)

, *t*). (13)

�. The specific form

<sup>2</sup>*σ*<sup>2</sup> }. (14)

}, (15)

}, (16)

<sup>2</sup>*<sup>m</sup>* , in the position space, respectively,

<sup>2</sup>| ↑��↑ |*ρ*↑↑(*z*, *<sup>z</sup>*�

{*a*| ↑� <sup>+</sup> *<sup>b</sup>*| ↓�} ⊗ *<sup>φ</sup>*(*z*) = {*a*| ↑� <sup>+</sup> *<sup>b</sup>*| ↓�} ⊗ <sup>1</sup>

*σ*

*<sup>ρ</sup>*↑↑, *<sup>ρ</sup>*↓↓ → *<sup>ρ</sup>d*(*p*, *<sup>t</sup>*)) = <sup>2</sup>*<sup>σ</sup>*

*π*

*<sup>N</sup>*(*t*) exp{− <sup>4</sup>

<sup>√</sup>*<sup>π</sup>* exp{−4*σ*<sup>2</sup>

in the momentum representation. It must be kept in mind, though, that the density matrix represents a pure state which has 'nonlocal' quantum correlations both in the spin space and the position and momentum space. In (15) & (16) above, we are looking at the *diagonal* elements of the position and momentum space density matrix and these show system-apparatus correlations. The up and down spin states of the system correlate with

> 2 <sup>2</sup>*<sup>m</sup>* and <sup>+</sup> *�<sup>t</sup>*

and the apparatus is akin to a 'Schroödinger cat state' which contains one-to-one correlations between the system and 'macroscopic' apparatus states with all quantum coherences intact. Since the dynamics, as governed by the Hamiltonian (11), is purely unitary, there is no

*σ N*(*t*)

2

*<sup>h</sup>*¯ in the momentum space. This pure entangled state of the system

 *z* ∓ *�t* 2 2*m* 2

 *p* ∓ *�t h*¯ 2

+ *ab*∗| ↑��↓ |*ρ*↑↓(*z*, *<sup>z</sup>*�

*<sup>ρ</sup>*↓↑ correspond to the off-diagonal elements (*ρod*), and *<sup>ρ</sup>*(*z*, *<sup>z</sup>*�

*ρS*+*<sup>A</sup>* = |*a*|

thus given as

diagonal elements of *<sup>ρ</sup>*↑↑ and *<sup>ρ</sup>*↓↓ are

in the position representation and

and around <sup>−</sup>*�<sup>t</sup>*

a Gaussian wavepackets centered round <sup>−</sup> *�<sup>t</sup>*

*<sup>h</sup>*¯ and <sup>+</sup>*�<sup>t</sup>*

*<sup>ρ</sup>*↑↑, *<sup>ρ</sup>*↓↓ <sup>→</sup> *<sup>ρ</sup>d*(*z*, *<sup>t</sup>*) = <sup>2</sup>

dissipation/decoherence involved, and the state remains a pure entangled state with all its non-diagonal elements (*ρ*↑↓, *<sup>ρ</sup>*↓↑) as well as the spatial nonlocality. The establishment of system-apparatus correlations, therefore, *is not enough to affect a measurement as the off diagonal elements of the density matrix have not vanished* and we do not have the desired *mixed state* density matrix.

Now let us look at the situation where we include the interaction with the environment. A commonly used model to describe the environment is to consider it as a reservoir of quantum oscillators, each of which interacts with the apparatus in our case and is describd by the Hamiltonian

$$H = \lambda \sigma\_z + \frac{p^2}{2m} + \epsilon z \sigma\_z + \sum\_j \frac{P\_j^2}{2M\_j} + \frac{M\_j^2 \Omega\_j^2}{2} \left( Z\_{\dot{\gamma}} - \frac{\mathbb{C}\_{\dot{\gamma}} z}{M\_{\dot{\gamma}} \Omega\_{\dot{\gamma}}^2} \right)^2. \tag{17}$$

Here *z* and *p* denote the position and momentum of the particle (apparatus) of mass *m*. *λσ<sup>z</sup>* is the Hamiltonian of the system and *�* is the strength of the system-apparatus coupling as mentioned earlier. The last term represents the Hamiltonian for the bath of oscillators (environment) and the apparatus-environment interaction. *Zj* and *Pj* are the position and momentum coordinates of the jth harmonic oscillator of the bath, *Cj*s are the coupling strengths and Ω*j*s are the frequencies of the oscillators comprising the bath. Note that the coupling of the apparatus with the many environmental degrees of freedom is a coordinate-coordinate coupling. The dynamics for the closed universe of the system, apparatus and environment is governed by the Hamiltonian evolution via (17) and the Schrödinger equation. In the decoherence approach, a tracing over all the degrees of freedom of the environment results in an equation describing the dynamics of the *reduced density matrix* of the system-apparatus combine. The reduced density matrix evolves according to a master equation which is obtained by solving the Schröinger equation for the entire universe of the system, apparatus and the environment and then tracing over the environment degrees of freedom. Several authors like Zeh and Zurek, among others, have worked extensively on the decoherence approach using the master equation for the reduced density matrix. The master equation for this kind of model of the environment was first derived separately by Caldeira and Leggett, Agarwal, Dekker(Dekker, 1977) and others in the context of quantum Brownian motion and is a popular equation for the study of open quantum systems. For our purpose, we deal with the mater equation for the system-apparatus-environment composite described by (17). The master equation for the density matrix, corresponding to the four elements of spin space (↑↑, ↑↓, ↓↑, ↓↓) (see (13)) is

$$\begin{split} \frac{\partial \rho\_{\rm s\rm s\rm \prime}(z, z', t)}{\partial t} &= \left[ -\frac{\hbar}{2im} \left( \frac{\partial^2}{\partial z^2} - \frac{\partial^2}{\partial z'^2} \right) - \gamma (z - z') \left( \frac{\partial}{\partial z} - \frac{\partial}{\partial z'} \right) \\ & - \frac{D}{4\hbar^2} (z - z')^2 + \frac{i\epsilon (zs - z's')}{\hbar} + \frac{i\lambda (s - s')}{\hbar} \right] \rho\_{\rm s\rm s\rm \prime}(z, z', t), \end{split} \tag{18}$$

where *s*,*s*� = +1 (for ↑) or −1 (for ↓). Here *γ* is the Langevin friction coefficient and *D* has the usual interpretation of the diffusion coefficient. *γ* and *D* are related to the parameters of the Hamiltonin of the total system. Without going into further details of the master equation, it suffices to point out that the equation naturally separates into three distinct terms, namely (i) a term describing the von Neumann equation which can be derived from the Schrödinger equation and thus represents the pure quantum evolution, (ii) a term that causes dissipation, which can be understood as a steady loss in energy or a relaxation process, and (iii) a term causing diffusion, which can be understood as the fluctuctions or zig-zag

with a macroscopic quantum apparatus (harmonic oscillator) coupled to a bath of harmonic

Measurement in Quantum Mechanics: Decoherence and the Pointer Basis 11

*j*

Here *ω* is the frequency of the oscillator and all the other symbols have the same meaning as explained in the previous section. The master equation for this system can be written exactly in the same way as for the free particle in the previous section. The initial system-apparatus state is considered to be a product state of any arbitrary state of the apparatus and a general

Exact solutions of the master equation show that the reduced density matrix of the system-apparatus combine decoheres to a statistical mixture where up and down spins eventually correlate with pointer states of the apparatus. The strength of this analysis is that unlike in the previous section where the initial state of the apparatus was considered to be a Gaussian wavepacket, no particular initial state of the harmonic oscillator was chosen. Venugopalan shows that for the zero temperature bath the system-apparatus combine ends up with spin-apparatus correlations in the *coherent states of the harmonic oscillator* for arbitrary initial states of the apparatus. For a high temperature bath, pointer states are Gaussian distributions or *generalized coherent states*(Venugopalan, 2000). For both cases, the off-diagonal elements in spin-space decohere over a time scale which goes inversely as the square of the "separation" between the "pointers". The "statistical mixture" into which the density matrix

, *t*) + |*b*|

*� mω*<sup>2</sup> 2

2

*P*2 *j* 2*Mj* + *M*<sup>2</sup> *<sup>j</sup>* <sup>Ω</sup><sup>2</sup> *j* 2

*ψ* = {*a*| ↑� + *b*| ↓�} ⊗ *φ*(*z*). (21)

<sup>2</sup>| ↓��↓ |*ρ*↓↓(*z*, *<sup>z</sup>*�

<sup>−</sup> *<sup>m</sup>ωr*<sup>2</sup> 4¯*h*

<sup>−</sup> *<sup>m</sup>ωr*<sup>2</sup> 4¯*h*

). This is nothing but the density matrix corresponding

. (25)

*Zj* <sup>−</sup> *Cjz Mj*Ω<sup>2</sup> *j*

2

. (20)

, *t*) (22)

, (23)

. (24)

oscillators was analysed. The Hamiltonian for this system would be

superposition state of the spin-1/2 state of the system, of the form

*<sup>m</sup>ω*2*z*<sup>2</sup> <sup>+</sup> *�zσ<sup>z</sup>* <sup>+</sup> ∑

*H* = *λσ<sup>z</sup>* +

decoheres looks like

with *R* = (*z* + *z*�

positions=±*�*/*mω*2, and

where the apparatus states are

1 2

*ρS*+*<sup>A</sup>* ∼ |*a*|

*<sup>ρ</sup>*↑↑ =

*<sup>ρ</sup>*↓↓ =

)/2, and *r* = (*z* − *z*�

*mω <sup>π</sup>h*¯ *exp*

*mω <sup>π</sup>h*¯ *exp*

<sup>2</sup>| ↑��↑ |*ρ*↑↑(*z*, *<sup>z</sup>*�


<sup>−</sup> *<sup>m</sup><sup>ω</sup> h*¯ *R* +

<sup>−</sup> *<sup>m</sup><sup>ω</sup> h*¯ *<sup>R</sup>* <sup>−</sup> *� mω*<sup>2</sup>

<sup>2</sup> <sup>=</sup> *<sup>m</sup><sup>ω</sup>* 2¯*h*

to a *coherent state*, |*α*�, of a harmonic oscillator with zero mean momentum and mean

The above result is for a zero temperature bath. For a high temperature bath, *generalized coherent states* constitute the pointer states. Exact results also demonstrate in an unambiguous way that the pointer states in this measurement model emerge independent of the initial state of the apparatus. For details of the analysis and exact solutions, the reader is referred to (Venugopalan, 2000). For both the zero temperature and high temperature cases analysed, the exact solutions for this model demonstrate the two main signatures of the decoherence mechanism in a quantum measurement, namely, (a) the decoherence time is much smaller than the thermal relaxation time, and (b) the decoherence time is inversely proportional to the

 *� mω*<sup>2</sup> 

movement seen in Brownian motion. The initial work of Zeh, Zurek and subsequent work where the decoherence approach has been applied to typical quantum-measurement-like situations like the Stern Gerlach experiment have shown that in the dynamics governed by the master equation, coherent quantum superpositions persist for a very short time. They are rapidly destroyed by the action of the environment. In fact, results show that the larger the quantum superposition, the faster the decoherence. For truly macroscopic superpositions such as Schrödinger's cat, decoherence occurs on such a short time scale that it is impossible to observe these quantum coherences. As mentioned above, the simple example by Zurek (Fig.1) illustrates this point. Several other authors have studied decoherence in other systems and their calculations are seen to contain two main features which can be seen as signatures of decoherence: (a) the decoherence time, over which the superpositions decay is much much shorter than any characteristic time scale of the system (the thermal relaxation time, *γ*<sup>−</sup>1, and (b) the decoherence time varies versely as the square of a quantity that indicates the 'size' of the quantum superposition (e.g. *�<sup>t</sup>* 2 *<sup>m</sup>* in (15) is the "separation" in position space of the two Gaussians correlating with up and down spin states).

Venugopalan et al(Venugopalan et al., 1995a;b) have shown that if the evolution of the initial state (14) is via the mater equation, (18), there is a one-to-one correlation established between the system states (spin) and the apparatus states (position and momentum degrees of freedom). Further, the coupling with the environmental degrees of freedom causes decoherence of the pure density matrix of the entangled state into a statistical mixture. For more details of the exact solutions, the interested reader may see(Kumar, 1998; Venugopalan et al., 1995a;b; Venugopalan, 1997). The density matrix for an initial system-apparatus state described by (14), evolving via the master equation (18) has the form

$$\begin{split} \rho\_{S+A} &= |a|^2 |\uparrow\rangle\langle\uparrow| \rho\_{\uparrow\uparrow}(z, z', t) + |b|^2 |\downarrow\rangle\langle\downarrow| \rho\_{\downarrow\downarrow}(z, z', t) \\ &+ \left[ a b^\* |\uparrow\rangle\langle\downarrow| \rho\_{\uparrow\downarrow}(z, z', t) + a^\* b |\downarrow\rangle\langle\uparrow| \rho\_{\downarrow\uparrow}(z, z', t) . \right] e^{-at^3}. \end{split} \tag{19}$$

Here the off-diagonal elements of the density matrix (last two terms) contain a multiplicative factor of the form *e*−*α<sup>t</sup>* 3 which causes the decay of these terms to zero over a characteristic time making the density matrix diagonal in spin space. Thus, we are left with a mixed state density matrix which can be interpreted in terms of classical probbilities. It also turns out that the spatial nonlocality of the diagonal components *<sup>ρ</sup>*↑↑ and *<sup>ρ</sup>*↓↓ disappear over time and these are rendered completely diagonal in momentum space. The density matrix, thus ends up being completely diagonal with perfect correlations between the spin component value and the average momentum of the particle(Kumar, 1998; Venugopalan et al., 1995a). Thus, the inclusion of environmental interaction has destroyed the quantum corelations (signified by the off-diagonal elements of the density matrix) and rendered the reduced system-apparatus combine into a statistical mixture. This, in effect, explains the mechanism leading to the "collapse" in a quantum measurement, as well as the description of the way classicality emerges from an underlying quantum substrate.

#### **3.2 Decoherence and the harmonic oscillator apparatus**

In the previous section we have seen a measurement-like scenario where the apparatus was represented by the continuous position and momentum degrees of freedom of the particle who's spin was measured. Venugopalan(Venugopalan, 2000) has looked at the same problem where the apparatus is a harmonic osillator. The interaction of a quantum system (spin-1/2) with a macroscopic quantum apparatus (harmonic oscillator) coupled to a bath of harmonic oscillators was analysed. The Hamiltonian for this system would be

$$H = \lambda \sigma\_z + \frac{1}{2} m \omega^2 z^2 + \epsilon z \sigma\_z + \sum\_j \frac{P\_j^2}{2M\_j} + \frac{M\_j^2 \Omega\_j^2}{2} \left( Z\_j - \frac{\mathcal{C}\_j z}{M\_j \Omega\_j^2} \right)^2. \tag{20}$$

Here *ω* is the frequency of the oscillator and all the other symbols have the same meaning as explained in the previous section. The master equation for this system can be written exactly in the same way as for the free particle in the previous section. The initial system-apparatus state is considered to be a product state of any arbitrary state of the apparatus and a general superposition state of the spin-1/2 state of the system, of the form

$$
\psi = \{a \mid \uparrow \rangle + b \mid \downarrow \} \otimes \phi(z). \tag{21}
$$

Exact solutions of the master equation show that the reduced density matrix of the system-apparatus combine decoheres to a statistical mixture where up and down spins eventually correlate with pointer states of the apparatus. The strength of this analysis is that unlike in the previous section where the initial state of the apparatus was considered to be a Gaussian wavepacket, no particular initial state of the harmonic oscillator was chosen. Venugopalan shows that for the zero temperature bath the system-apparatus combine ends up with spin-apparatus correlations in the *coherent states of the harmonic oscillator* for arbitrary initial states of the apparatus. For a high temperature bath, pointer states are Gaussian distributions or *generalized coherent states*(Venugopalan, 2000). For both cases, the off-diagonal elements in spin-space decohere over a time scale which goes inversely as the square of the "separation" between the "pointers". The "statistical mixture" into which the density matrix decoheres looks like

$$
\rho\_{S+A} \sim |a|^2 |\uparrow\rangle\langle\uparrow| \rho\_{\uparrow\uparrow}(z, z', t) + |b|^2 |\downarrow\rangle\langle\downarrow| \rho\_{\downarrow\downarrow}(z, z', t) \tag{22}
$$

where the apparatus states are

10 Will-be-set-by-IN-TECH

movement seen in Brownian motion. The initial work of Zeh, Zurek and subsequent work where the decoherence approach has been applied to typical quantum-measurement-like situations like the Stern Gerlach experiment have shown that in the dynamics governed by the master equation, coherent quantum superpositions persist for a very short time. They are rapidly destroyed by the action of the environment. In fact, results show that the larger the quantum superposition, the faster the decoherence. For truly macroscopic superpositions such as Schrödinger's cat, decoherence occurs on such a short time scale that it is impossible to observe these quantum coherences. As mentioned above, the simple example by Zurek (Fig.1) illustrates this point. Several other authors have studied decoherence in other systems and their calculations are seen to contain two main features which can be seen as signatures of decoherence: (a) the decoherence time, over which the superpositions decay is much much shorter than any characteristic time scale of the system (the thermal relaxation time, *γ*<sup>−</sup>1, and (b) the decoherence time varies versely as the square of a quantity that indicates the 'size' of

Venugopalan et al(Venugopalan et al., 1995a;b) have shown that if the evolution of the initial state (14) is via the mater equation, (18), there is a one-to-one correlation established between the system states (spin) and the apparatus states (position and momentum degrees of freedom). Further, the coupling with the environmental degrees of freedom causes decoherence of the pure density matrix of the entangled state into a statistical mixture. For more details of the exact solutions, the interested reader may see(Kumar, 1998; Venugopalan et al., 1995a;b; Venugopalan, 1997). The density matrix for an initial system-apparatus state

, *t*) + |*b*|

Here the off-diagonal elements of the density matrix (last two terms) contain a multiplicative

time making the density matrix diagonal in spin space. Thus, we are left with a mixed state density matrix which can be interpreted in terms of classical probbilities. It also turns out that the spatial nonlocality of the diagonal components *<sup>ρ</sup>*↑↑ and *<sup>ρ</sup>*↓↓ disappear over time and these are rendered completely diagonal in momentum space. The density matrix, thus ends up being completely diagonal with perfect correlations between the spin component value and the average momentum of the particle(Kumar, 1998; Venugopalan et al., 1995a). Thus, the inclusion of environmental interaction has destroyed the quantum corelations (signified by the off-diagonal elements of the density matrix) and rendered the reduced system-apparatus combine into a statistical mixture. This, in effect, explains the mechanism leading to the "collapse" in a quantum measurement, as well as the description of the way classicality

In the previous section we have seen a measurement-like scenario where the apparatus was represented by the continuous position and momentum degrees of freedom of the particle who's spin was measured. Venugopalan(Venugopalan, 2000) has looked at the same problem where the apparatus is a harmonic osillator. The interaction of a quantum system (spin-1/2)

*<sup>m</sup>* in (15) is the "separation" in position space of the two

<sup>2</sup>| ↓��↓ |*ρ*↓↓(*z*, *<sup>z</sup>*�

which causes the decay of these terms to zero over a characteristic

, *<sup>t</sup>*) + *<sup>a</sup>*∗*b*| ↓��↑ |*ρ*↓↑(*z*, *<sup>z</sup>*�

, *t*)

, *t*). *e* −*αt* 3

. (19)

2

described by (14), evolving via the master equation (18) has the form

<sup>2</sup>| ↑��↑ |*ρ*↑↑(*z*, *<sup>z</sup>*�

*ab*∗| ↑��↓ |*ρ*↑↓(*z*, *<sup>z</sup>*�

the quantum superposition (e.g. *�<sup>t</sup>*

*ρS*+*<sup>A</sup>* = |*a*|

3

emerges from an underlying quantum substrate.

**3.2 Decoherence and the harmonic oscillator apparatus**

+

factor of the form *e*−*α<sup>t</sup>*

Gaussians correlating with up and down spin states).

$$\rho\_{\uparrow\uparrow} = \sqrt{\frac{m\omega}{\pi\hbar}} \exp\left\{-\frac{m\omega}{\hbar} \left(\mathcal{R} + \frac{\varepsilon}{m\omega^2}\right)^2 - \frac{m\omega r^2}{4\hbar}\right\},\tag{23}$$

$$\rho\_{\downarrow\downarrow} = \sqrt{\frac{m\omega}{\pi\hbar}} \exp\left\{-\frac{m\omega}{\hbar} \left(R - \frac{\varepsilon}{m\omega^2}\right)^2 - \frac{m\omega r^2}{4\hbar}\right\}.\tag{24}$$

with *R* = (*z* + *z*� )/2, and *r* = (*z* − *z*� ). This is nothing but the density matrix corresponding to a *coherent state*, |*α*�, of a harmonic oscillator with zero mean momentum and mean positions=±*�*/*mω*2, and

$$\left|a\right|^2 = \frac{m\omega}{2\hbar} \left(\frac{\varepsilon}{m\omega^2}\right). \tag{25}$$

The above result is for a zero temperature bath. For a high temperature bath, *generalized coherent states* constitute the pointer states. Exact results also demonstrate in an unambiguous way that the pointer states in this measurement model emerge independent of the initial state of the apparatus. For details of the analysis and exact solutions, the reader is referred to (Venugopalan, 2000). For both the zero temperature and high temperature cases analysed, the exact solutions for this model demonstrate the two main signatures of the decoherence mechanism in a quantum measurement, namely, (a) the decoherence time is much smaller than the thermal relaxation time, and (b) the decoherence time is inversely proportional to the

one looks at the reduced Wigner distributions of infinite systems of coupled oscillatirs as *t* → ∞. Paz and Zurek(Paz & Zurek, 1999) have investigated decoherence in the limit of weak interaction with the environment and show that the eigenstates of energy emerge as pointer states. Roy and Venugopalan have also obtained the exact solutions of the Master equation for a harmonic oscillator and a free particle in a compact factorizable form and have shown that the density matrix diagonalizes in the energy basis which is number states for the oscillator and momentum states for the free particle for arbitrary initial conditions(Roy & Venugopalan, 1999). It is intuitive that the pointer states should naturally be a consequence of the interplay between the various components of the total Hamiltonian and one should also expect them to

Measurement in Quantum Mechanics: Decoherence and the Pointer Basis 13

The studies mentioned above do shed light on the nature of the preferred basis but are inadequate and there is a need to analyze more systems. In particular, it is important to look at systems like the harmonic oscillator apparatus model which is fairly generic and exact solutions make it an interesting candidate to explore experimentally in the context of decoherence and quantum measurements. Also, this example indicates that it seems pertinent to look at a system-apparatus-environment like scenario for measurement to analyse the issue

Finally, we would like to refer to a recent work of Venugopalan(Venugopalan, 2011) which indicates that there is a need to look at the bare system-apparatus interaction in a measurement like scenaorio more carefully to get insights into what states would eventually end up as pointer states and be selected in the event of environmental influence. Venugopalan has looked at a simple one dimensional model for the system-apparatus interaction where the system is a spin-1/2 particle, and its position and momentum degrees constitutes the apparatus, like the Stern-Gerlach model discussed above. An analysis involving only unitary Schrödinger dynamics illustrates the nature of the correlations established in the system-apparatus entangled state. It is shown that even in the absence of any environment-induced decoherence, or any other measurement model, certain initial states of the apparatus -like localized Gaussian wavepackets - are preferred over others, in terms of the establishment of measurementlike one-to-one correlations in the pure system-apparatus entangled state. This result indicates that perhaps there already exist special states of the apparatus within the quantum menu, and it is these that end up being ultimately selected by

The central theme of the decoherence approach has been to explain, within the realm of quantum theory, the appearance of classicality in the macrosopic, familiar physical world. The strength of the theory has been often claimed to be the fact that it is within the realm of quantum mechanics, and uses the rules of quantum theory to explain the emergence of classicality. In the case of quantum measurement, decoherence is believed to "mimic" wave function collapse. This is achieved via the transition from a pure state density matrix to a statistical mixture with 'classically' meaningful terms. In a sense, decoherence explains the washing away of quantum coherences and the emergence of a state which makes classical sense. In this chapter we have seen this happen in the two measurement models considered above - one with free particle as an apparatus and one with a harmonic oscillator as the apparatus. We have seen that the environmental influence is crucial in not only destroying the quantum coherences, but also is selecting a special state or a preferred basis. A criticism levelled against the decoherence approach is often that it does not explain the fact that *only*

be independent of the initial state of the system/apparatus.

of the pointer basis and the states singled out by the environment.

the environment as the preferred states.

**5. Conclusions**

square of the "separation" between the two "pointers" that correlate with the system states. Thus, one again, one can see that the inclusion of the environmental degrees of freedom takes a pure entangled system-apparatus state to a statistical mixture which lets us interpret the measurement in terms of classical probabilities as well as lets us explain the emergence of classical correlations from an underlying quantum substrate.

#### **4. Decoherence and the pointer basis**

Finally, we focus on a specific aspect of the decoherence theory, i.e., the notion of a 'preferred basis' or a 'pointer basis'. Our experience of the classical world suggests that unlike quantum systems, which are allowed to exist in all possible states, classical systems only exist in a few select states which are singled out by the environment from a larger quantum menu(Joos et al., 2003; Zurek, 1981). These special states are the *preferred basis*, also referred to as the "pointer states" in a quantum-measurement-like scenario. In a measurement-like scenario the pointer basis should also be understood as those states of the apparatus in which correlations with the system states are eventually established (irrespective of the initial states of the apparatus). Inspite of many theoretical studies on the decoherence approach, not many systems have been analysed with the aim of predicting what the pointer basis should be in a given situation. Let us look at the issue of the pointer basis a little more closely by once again referring to the two quantum measurement examples discussed in the previous sections: (i)the spin-1/2 system with the free particle as an apparatus in the Stern-Gerlach-like interaction described in section 3.1 (ii) the spin-1/2 system with the harmonic oscillator as an apparatus described in section 3.2.

What is the preferred basis in a given scenario, and what decides in which pointer basis the system-apparatus will be finally established? For simplified models where the self Hamiltonian of the system has either been ignored or considered co-diagonal with the interaction Hamiltonian, the pointer variable has been shown to be the one which commutes with the interaction Hamiltonian(Zurek, 1981). However, in more general situations where all terms are included, as in the two examples discussed here, the various parts of the Hamiltonian may not commute. In such situations it is not obvious what decides the preferred basis. For example, in the case of the spin system and free particle apparatus with Stern-Gerlach like interaction discussed above, the coordinate-coordinate coupling indicates that the position basis is intuitively expected to emerge as the preferred basis. However, as we have seen, this is contrary to the conclusion of Venugopalan et al(Kumar, 1998; Venugopalan, 1994; Venugopalan et al., 1995a) in their analysis of the Stern-Gerlach measurement model where the spin components eventually correlate with distributions which are completely diagonal in the momentum basis and only approximately diagonal in the position basis(Kumar, 1998).

In the literature, the preferred basis has been variously described as the one in which the final state density matrix becomes diagonal or that set of basis states which are characterized by maximum stability or a minimum increase in linear or statistical entropy, decided by a *predictability sieve*(Zurek et al. , 1993). Using the Markovian Master equation for a harmonic oscillator coupled to a heat bath and the criterion of the predictability sieve Zurek argues that coherent states emerge as the preferred basis. This is in tune with the results for the harmonic oscillator apparatus model where we have seen that pointer states end up being coherent states or generalized coherent states , *irrespective* of the initial state in which the apparatus starts off. This result also agrees with a study by Tegmark and Shapiro(Tegmark & Shapiro, 1999) where they show that generalized coherent states tend to be produced naturally when one looks at the reduced Wigner distributions of infinite systems of coupled oscillatirs as *t* → ∞. Paz and Zurek(Paz & Zurek, 1999) have investigated decoherence in the limit of weak interaction with the environment and show that the eigenstates of energy emerge as pointer states. Roy and Venugopalan have also obtained the exact solutions of the Master equation for a harmonic oscillator and a free particle in a compact factorizable form and have shown that the density matrix diagonalizes in the energy basis which is number states for the oscillator and momentum states for the free particle for arbitrary initial conditions(Roy & Venugopalan, 1999). It is intuitive that the pointer states should naturally be a consequence of the interplay between the various components of the total Hamiltonian and one should also expect them to be independent of the initial state of the system/apparatus.

The studies mentioned above do shed light on the nature of the preferred basis but are inadequate and there is a need to analyze more systems. In particular, it is important to look at systems like the harmonic oscillator apparatus model which is fairly generic and exact solutions make it an interesting candidate to explore experimentally in the context of decoherence and quantum measurements. Also, this example indicates that it seems pertinent to look at a system-apparatus-environment like scenario for measurement to analyse the issue of the pointer basis and the states singled out by the environment.

Finally, we would like to refer to a recent work of Venugopalan(Venugopalan, 2011) which indicates that there is a need to look at the bare system-apparatus interaction in a measurement like scenaorio more carefully to get insights into what states would eventually end up as pointer states and be selected in the event of environmental influence. Venugopalan has looked at a simple one dimensional model for the system-apparatus interaction where the system is a spin-1/2 particle, and its position and momentum degrees constitutes the apparatus, like the Stern-Gerlach model discussed above. An analysis involving only unitary Schrödinger dynamics illustrates the nature of the correlations established in the system-apparatus entangled state. It is shown that even in the absence of any environment-induced decoherence, or any other measurement model, certain initial states of the apparatus -like localized Gaussian wavepackets - are preferred over others, in terms of the establishment of measurementlike one-to-one correlations in the pure system-apparatus entangled state. This result indicates that perhaps there already exist special states of the apparatus within the quantum menu, and it is these that end up being ultimately selected by the environment as the preferred states.

#### **5. Conclusions**

12 Will-be-set-by-IN-TECH

square of the "separation" between the two "pointers" that correlate with the system states. Thus, one again, one can see that the inclusion of the environmental degrees of freedom takes a pure entangled system-apparatus state to a statistical mixture which lets us interpret the measurement in terms of classical probabilities as well as lets us explain the emergence of

Finally, we focus on a specific aspect of the decoherence theory, i.e., the notion of a 'preferred basis' or a 'pointer basis'. Our experience of the classical world suggests that unlike quantum systems, which are allowed to exist in all possible states, classical systems only exist in a few select states which are singled out by the environment from a larger quantum menu(Joos et al., 2003; Zurek, 1981). These special states are the *preferred basis*, also referred to as the "pointer states" in a quantum-measurement-like scenario. In a measurement-like scenario the pointer basis should also be understood as those states of the apparatus in which correlations with the system states are eventually established (irrespective of the initial states of the apparatus). Inspite of many theoretical studies on the decoherence approach, not many systems have been analysed with the aim of predicting what the pointer basis should be in a given situation. Let us look at the issue of the pointer basis a little more closely by once again referring to the two quantum measurement examples discussed in the previous sections: (i)the spin-1/2 system with the free particle as an apparatus in the Stern-Gerlach-like interaction described in section 3.1 (ii) the spin-1/2 system with the harmonic oscillator as an apparatus described in section

What is the preferred basis in a given scenario, and what decides in which pointer basis the system-apparatus will be finally established? For simplified models where the self Hamiltonian of the system has either been ignored or considered co-diagonal with the interaction Hamiltonian, the pointer variable has been shown to be the one which commutes with the interaction Hamiltonian(Zurek, 1981). However, in more general situations where all terms are included, as in the two examples discussed here, the various parts of the Hamiltonian may not commute. In such situations it is not obvious what decides the preferred basis. For example, in the case of the spin system and free particle apparatus with Stern-Gerlach like interaction discussed above, the coordinate-coordinate coupling indicates that the position basis is intuitively expected to emerge as the preferred basis. However, as we have seen, this is contrary to the conclusion of Venugopalan et al(Kumar, 1998; Venugopalan, 1994; Venugopalan et al., 1995a) in their analysis of the Stern-Gerlach measurement model where the spin components eventually correlate with distributions which are completely diagonal in the momentum basis and only approximately diagonal in

In the literature, the preferred basis has been variously described as the one in which the final state density matrix becomes diagonal or that set of basis states which are characterized by maximum stability or a minimum increase in linear or statistical entropy, decided by a *predictability sieve*(Zurek et al. , 1993). Using the Markovian Master equation for a harmonic oscillator coupled to a heat bath and the criterion of the predictability sieve Zurek argues that coherent states emerge as the preferred basis. This is in tune with the results for the harmonic oscillator apparatus model where we have seen that pointer states end up being coherent states or generalized coherent states , *irrespective* of the initial state in which the apparatus starts off. This result also agrees with a study by Tegmark and Shapiro(Tegmark & Shapiro, 1999) where they show that generalized coherent states tend to be produced naturally when

classical correlations from an underlying quantum substrate.

**4. Decoherence and the pointer basis**

the position basis(Kumar, 1998).

3.2.

The central theme of the decoherence approach has been to explain, within the realm of quantum theory, the appearance of classicality in the macrosopic, familiar physical world. The strength of the theory has been often claimed to be the fact that it is within the realm of quantum mechanics, and uses the rules of quantum theory to explain the emergence of classicality. In the case of quantum measurement, decoherence is believed to "mimic" wave function collapse. This is achieved via the transition from a pure state density matrix to a statistical mixture with 'classically' meaningful terms. In a sense, decoherence explains the washing away of quantum coherences and the emergence of a state which makes classical sense. In this chapter we have seen this happen in the two measurement models considered above - one with free particle as an apparatus and one with a harmonic oscillator as the apparatus. We have seen that the environmental influence is crucial in not only destroying the quantum coherences, but also is selecting a special state or a preferred basis. A criticism levelled against the decoherence approach is often that it does not explain the fact that *only*

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Measurement, *Phys. Rev. Lett.* 77, 4887-4890 (1996)

Haroche, S., Observing the Progressive Decoherence of the "Meter" in a Quantum

*one* of the mixed outcomes is actually observed, nor does it allow us to predict exactly which one will be observed. Inspite of these criticism, it is generally accepted that the decoherence explanation has certainly provided valuable insights into the actual mechanism of the loss of quantum coherences. In recent years, the predictions of the decoherence theory have been tested in several spectacular experiments which put the theory on a firm footing. Of these, two experiments are particularly noteworthy as they have succeeded in monitoring the decoherence mechanism, i.e., the actual transition from a pure entangled state to a statistical mixture. Moreover, the experiments also give a quantitative estimate of the decoherence time. These experiments have done no less than create Schrödinger-cat-like entangled states in the laboratory and seen them transform into classically recognizable objects under the influence of environmental coupling. Among the first successful attempts is an experiment by Brune et al(Brune et al., 1996) at the Ecole Normale Superieure in Paris. Using Rubidium atoms and high technology superconducting microwave cavities, Brune et al created a superposition of quantum states involving radiation fields. The superposition was the equivalent of a "system+ measuring apparatus" situation in which the "meter" was pointing simultaneously towards two different directions. This is a Schrönger-Cat-like entangled state. Through a series of ingenious "atom-interferometry" experiments, Brune et al. managed to not only "read" this pure state but also monitored the decoherence phenomenon as it unfolded, transforming this superposition state to a statistical mixture. Besides providing a direct insight into the role of the environment in a quantum measurement process, their experiment also confirmed the basic tenets of the decoherence theory. The two main signatures of the decoherence theory were clearly observed in this classic experiment. The environment in this experiment are the "modes" of the electromagnetic field in the cavity. At the National Institute of Standards and Technology, Boulder, Colorado, the group headed by Wineland(Monroe et al., 1996) created a Schrödinger-Cat-like state using a series of laser pulses to entangle the internal (electronic) and the external (motional) states of a Beryllium ion in a "Paul trap". The motion of this trapped ion couples to an electric field which changes randomly, thus simulating an environment . Monroe et al call this environment an engineered reservoir whose state and coupling can be controlled. Through their measurements, Wineland et al. have successfully demonstrated the two important signatures of the decoherence mechanism. Thus one can see that the qualitative and quantitative predictions of the decoherence theory are experimentally tested. These tests would be highly relevant to all experimental implementations of the novel ideas of quantum information and computation as decoherence would ruin the functioning of devices which use uniquely quantum mechanical effects for information processing. The above two experiments, along with others have provided important insights into the role of the environment in bringing about classicality and thus decoherence theory is strengthened by these spectacular observations. What, then, is the final verdict of the decoherence theory? Has it resolved the conceptual problems of quantum mechanics? There are many who believe that the conceptual problems of quantum mechanics are still unresolved and decoherence does not answer many issues. At the end of the day we can say that the decoherence explanation takes away some of the mystery from the idea of 'wave function collapse' and provides a conventional mechanism to explain the appearance of a classical world. Many physicists find this a satisfactory explanation and there is no doubt that the experiments discussed clearly show how decoherence washes away quantum coherences providing a fairly convincing evidence for explaining the absence of Schödinger's Cats in the real world. For all practical purposes, the decoherence explanation finds favour as a satisfactory settlement of the quantum measurement problem.

#### **6. References**

14 Will-be-set-by-IN-TECH

*one* of the mixed outcomes is actually observed, nor does it allow us to predict exactly which one will be observed. Inspite of these criticism, it is generally accepted that the decoherence explanation has certainly provided valuable insights into the actual mechanism of the loss of quantum coherences. In recent years, the predictions of the decoherence theory have been tested in several spectacular experiments which put the theory on a firm footing. Of these, two experiments are particularly noteworthy as they have succeeded in monitoring the decoherence mechanism, i.e., the actual transition from a pure entangled state to a statistical mixture. Moreover, the experiments also give a quantitative estimate of the decoherence time. These experiments have done no less than create Schrödinger-cat-like entangled states in the laboratory and seen them transform into classically recognizable objects under the influence of environmental coupling. Among the first successful attempts is an experiment by Brune et al(Brune et al., 1996) at the Ecole Normale Superieure in Paris. Using Rubidium atoms and high technology superconducting microwave cavities, Brune et al created a superposition of quantum states involving radiation fields. The superposition was the equivalent of a "system+ measuring apparatus" situation in which the "meter" was pointing simultaneously towards two different directions. This is a Schrönger-Cat-like entangled state. Through a series of ingenious "atom-interferometry" experiments, Brune et al. managed to not only "read" this pure state but also monitored the decoherence phenomenon as it unfolded, transforming this superposition state to a statistical mixture. Besides providing a direct insight into the role of the environment in a quantum measurement process, their experiment also confirmed the basic tenets of the decoherence theory. The two main signatures of the decoherence theory were clearly observed in this classic experiment. The environment in this experiment are the "modes" of the electromagnetic field in the cavity. At the National Institute of Standards and Technology, Boulder, Colorado, the group headed by Wineland(Monroe et al., 1996) created a Schrödinger-Cat-like state using a series of laser pulses to entangle the internal (electronic) and the external (motional) states of a Beryllium ion in a "Paul trap". The motion of this trapped ion couples to an electric field which changes randomly, thus simulating an environment . Monroe et al call this environment an engineered reservoir whose state and coupling can be controlled. Through their measurements, Wineland et al. have successfully demonstrated the two important signatures of the decoherence mechanism. Thus one can see that the qualitative and quantitative predictions of the decoherence theory are experimentally tested. These tests would be highly relevant to all experimental implementations of the novel ideas of quantum information and computation as decoherence would ruin the functioning of devices which use uniquely quantum mechanical effects for information processing. The above two experiments, along with others have provided important insights into the role of the environment in bringing about classicality and thus decoherence theory is strengthened by these spectacular observations. What, then, is the final verdict of the decoherence theory? Has it resolved the conceptual problems of quantum mechanics? There are many who believe that the conceptual problems of quantum mechanics are still unresolved and decoherence does not answer many issues. At the end of the day we can say that the decoherence explanation takes away some of the mystery from the idea of 'wave function collapse' and provides a conventional mechanism to explain the appearance of a classical world. Many physicists find this a satisfactory explanation and there is no doubt that the experiments discussed clearly show how decoherence washes away quantum coherences providing a fairly convincing evidence for explaining the absence of Schödinger's Cats in the real world. For all practical purposes, the decoherence explanation finds favour as a satisfactory settlement of

the quantum measurement problem.

Agarwal, G. S., Brownian Motion of a Quantum Oscillator, *Phys. Rev. A* 4, 739-747 (1971)


**0**

**2**

1,3*Ukraine* <sup>2</sup>*Italy*

**Time as Quantum Observable, Canonical**

<sup>1</sup>*Institute for Nuclear Research, National Academy of Sciences* <sup>2</sup>*Facoltà di Ingegneria, Università statale di Bergamo, Bergamo*

<sup>3</sup>*Institute for Nuclear Research, National Academy of Sciences*

Vladislav S. Olkhovsky1, Erasmo Recami2 and Sergei P. Maydanyuk<sup>3</sup>

Time, as well as 3-position, sometimes is a parameter, but sometimes is an observable that in quantum theory would be expected to be associated with an operator. However, almost from the birth of quantum mechanics (cf., e.g., Ref.(Pauli, 1926; 1980)), it is known that time cannot be represented by a selfadjoint operator, except in the case of special systems (such as an electrically charged particle in an infinite uniform electric field)1. The list of papers devoted to the problem of time in quantum mechanics is extremely large (see, for instance, Refs. (Aharonov et al., 1998; Atmanspacher & Amann, 1998; Blanchard P & Jadczyk, 1996; Busch et al., 1994; Delgado, 1999; Egusquiza & Muga, 1999; Giannitrapani, 1997; Gó´zd ´zA& De¸bicki, 2007; Grot et al., 1996; Holevo, 1978; 1982; Kijowski, 1997; Kobe et al., 1994; Kocha'nski & Wo'dkievicz, 1999; Leo'n, 1997; Muga et al., 1999; Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; 1990; 1992; Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky, 1998; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007; Olkhovsky, 2009; 2011; Recami, 1976; 1977; Srinivas & Vijayalakshmi, 1981; Toller, 1999; Wang & Xiong, 2007), and references therein). The same situation had to be faced also in quantum electrodynamics and, more in general, in relativistic quantum field theory (see, for instance, Refs.(Olkhovsky & Recami, 1968; 1969; Olkhovsky et

As to quantum mechanics, the very first relevant articles are probably Refs. (Holevo, 1978; 1982; Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; 1990; 1992; 1998; Recami, 1976; 1977), and refs. therein. A second set of papers on time in quantum physics (Aharonov et al., 1998; Atmanspacher & Amann, 1998; Blanchard P & Jadczyk, 1996; Busch et al., 1994; Delgado, 1999; Egusquiza & Muga, 1999; Giannitrapani, 1997; Gó´zd ´z A & De¸bicki, 2007; Grot et al., 1996; Kijowski, 1997; Kobe et al.,

<sup>1</sup> This is a consequence of the semi-boundedness of the continuous energy spectra from below (usually from zero). Only for an electrically charged particle in an infinite uniform electric field, and other very rare special systems, the continuous energy spectrum is not bounded and extends over the whole axis from −∞ to +∞. It is curious that for systems with continuous energy spectra bounded from above and from below, the time operator is however selfadjoint and yields a discrete time spectrum.

**1. Introduction**

al., 2004; Olkhovsky & Recami, 2007)).

**Conjugated to Energy**

*and INFN-Sezione di Milano, Milan*


## **Time as Quantum Observable, Canonical Conjugated to Energy**

Vladislav S. Olkhovsky1, Erasmo Recami2 and Sergei P. Maydanyuk<sup>3</sup>

<sup>1</sup>*Institute for Nuclear Research, National Academy of Sciences* <sup>2</sup>*Facoltà di Ingegneria, Università statale di Bergamo, Bergamo and INFN-Sezione di Milano, Milan* <sup>3</sup>*Institute for Nuclear Research, National Academy of Sciences* 1,3*Ukraine* <sup>2</sup>*Italy*

#### **1. Introduction**

16 Will-be-set-by-IN-TECH

16 Measurements in Quantum Mechanics

Wheeler, J. A. and Zurek, W. H., eds. *Quantum Theory and Measurement*, Princeton University

Zeh, H. D., On the interpretation of measurement in quantum theory, *Foundations of Physics*,

Zurek, W. H., Pointer basis of quantum apparatus: Into what mixture does the wave packet

Zurek, W.H., Decoherence and the Transition from the Quantum to the Classical *Physics Today*,

Zurek, W. H., Habib,S. & Paz, J. P. Coherent states via decoherence, *Phys. Rev. Lett.*70,

Press (1983)

36-44 (October 1991).

1187-1190 (1993).

Volume 1, Issue 1, pp.69-76, (1970)

collapse? *Phys. Rev. D* 24, 1516 - 1525 (1981)

Time, as well as 3-position, sometimes is a parameter, but sometimes is an observable that in quantum theory would be expected to be associated with an operator. However, almost from the birth of quantum mechanics (cf., e.g., Ref.(Pauli, 1926; 1980)), it is known that time cannot be represented by a selfadjoint operator, except in the case of special systems (such as an electrically charged particle in an infinite uniform electric field)1. The list of papers devoted to the problem of time in quantum mechanics is extremely large (see, for instance, Refs. (Aharonov et al., 1998; Atmanspacher & Amann, 1998; Blanchard P & Jadczyk, 1996; Busch et al., 1994; Delgado, 1999; Egusquiza & Muga, 1999; Giannitrapani, 1997; Gó´zd ´zA& De¸bicki, 2007; Grot et al., 1996; Holevo, 1978; 1982; Kijowski, 1997; Kobe et al., 1994; Kocha'nski & Wo'dkievicz, 1999; Leo'n, 1997; Muga et al., 1999; Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; 1990; 1992; Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky, 1998; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007; Olkhovsky, 2009; 2011; Recami, 1976; 1977; Srinivas & Vijayalakshmi, 1981; Toller, 1999; Wang & Xiong, 2007), and references therein). The same situation had to be faced also in quantum electrodynamics and, more in general, in relativistic quantum field theory (see, for instance, Refs.(Olkhovsky & Recami, 1968; 1969; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007)).

As to quantum mechanics, the very first relevant articles are probably Refs. (Holevo, 1978; 1982; Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; 1990; 1992; 1998; Recami, 1976; 1977), and refs. therein. A second set of papers on time in quantum physics (Aharonov et al., 1998; Atmanspacher & Amann, 1998; Blanchard P & Jadczyk, 1996; Busch et al., 1994; Delgado, 1999; Egusquiza & Muga, 1999; Giannitrapani, 1997; Gó´zd ´z A & De¸bicki, 2007; Grot et al., 1996; Kijowski, 1997; Kobe et al.,

<sup>1</sup> This is a consequence of the semi-boundedness of the continuous energy spectra from below (usually from zero). Only for an electrically charged particle in an infinite uniform electric field, and other very rare special systems, the continuous energy spectrum is not bounded and extends over the whole axis from −∞ to +∞. It is curious that for systems with continuous energy spectra bounded from above and from below, the time operator is however selfadjoint and yields a discrete time spectrum.

time. As remarked, those papers did not refer to the Naimark theorem2 (Naimark, 1940) which had mathematically supported, on the contrary, the results in (Holevo, 1978; 1982; Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; 1990; 1992; 1998; Recami, 1976; 1977), and afterwards in (Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007). Indeed, already in the seventies (in Refs. (Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Recami, 1976; 1977) while more detailed presentations and reviews can be found in (Olkhovsky, 1984; 1990; 1992; 1998) and independently in (Holevo, 1978; 1982)), it was proven that, for systems with continuous energy spectra, Time **is** a quantum-mechanical observable, canonically conjugate to energy. Namely, it had been shown

Time as Quantum Observable, Canonical Conjugated to Energy 19

*t*, in the time (*t*-)representation, (a)

to be not selfadjoint, but hermitian, and to act on square-integrable space-time wave packets in the representation (1a), and on their Fourier-transforms in (1b), once point *E* = 0 is eliminated (i. e., once one deals only with moving packets, excluding any *non-moving* rear tails and the cases with zero fluxes)<sup>3</sup> In Refs.(Olkhovsky, 1984; 1990; 1992; 1998) and (Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007), the operator ˆ*t* (in the *t*-representation) had the property that any averages over time, in the one-dimensional (1D) scalar case, were to be obtained by use of the following

> *<sup>W</sup>* (*t*, *<sup>x</sup>*) *dt* <sup>=</sup> *<sup>j</sup>*(*x*, *<sup>t</sup>*) *dt* + ∞ −∞

where the the flux density *j*(*x*, *t*) corresponds to the (temporal) probability for a particle to pass through point *x* during the unit time centered at *t*, when traveling in the positive *x*-direction. Such a measure is not postulated, but is a direct consequence of the well-known probabilistic *spatial* interpretation of *ρ* (*x*, *t*) and of the continuity relation *∂ρ* (*x*, *t*)/*∂ t* + div*j*(*x*, *t*) = 0. Quantity *ρ*(*x*, *t*) is, as usual, the probability of finding the considered moving

Quantities *ρ*(*x*, *t*) and *j*(*x*, *t*) are related to the wave function Ψ (*x*, *t*) by the ordinary

density *j*(*x*, *t*) changes its sign, quantity *W* (*x*, *t*) *dt* is no longer positive-definite and, as in Refs.(Olkhovsky, 1984; Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007), it acquires the physical meaning of a probability density *only* during those partial time-intervals in which the flux

<sup>2</sup> The Naimark theorem states in particular the following(Naimark, 1940): The non-orthogonal spectral decomposition of a maximal hermitian operator can be approximated by an orthogonal spectral function (which corresponds to a selfadjoint operator), in a weak convergence, with any desired

<sup>3</sup> Such a condition is enough for operator (1a,b) to be a *hermitian*, or more precisely a *maximal hermitian*[2–8] *operator* (see also (Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007)); but it can be dispensed with by recourse to bilinear forms (see, e.g., Refs.(Recami, 1976; 1977; Recami et al., 1983) and refs. therein), as

*j*(*x*, *t*) *dt*

*<sup>∂</sup>E*, in the energy (*E*-)representation (b) (1)

<sup>2</sup> and *<sup>j</sup>*(*x*, *<sup>t</sup>*) = �[Ψ∗(*x*, *<sup>t</sup>*) (*h*¯ /*iμ*) <sup>Ψ</sup> (*x*, *<sup>t</sup>*))]). When the flux

, (2)

the time operator

*measure* (or weight):

definitions *ρ* (*x*, *t*) = |Ψ (*x*, *t*)|

accuracy.

we shall see below.

ˆ*t* =

<sup>−</sup>*ih*¯ *<sup>∂</sup>*

particle inside a unit space interval, centered at point *x*, at time *t*.

1994; Kocha'nski & Wo'dkievicz, 1999; Leo'n, 1997; Muga et al., 1999; Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007; Srinivas & Vijayalakshmi, 1981; Toller, 1999; Wang & Xiong, 2007) appeared in the nineties, stimulated partially by the need of a consistent definition for the tunneling time. It is noticeable, and let us stress it right now, that this second set of papers seems however to have ignored Naimark's theorem(Naimark, 1940), which had previously constituted (directly or indirectly) an important basis for the results in Refs. (Holevo, 1978; 1982; Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; 1990; 1992; 1998; Recami, 1976; 1977), moreover, all the papers (Aharonov et al., 1998; Atmanspacher & Amann, 1998; Blanchard P & Jadczyk, 1996; Busch et al., 1994; Grot et al., 1996; Kobe et al., 1994; Leo'n, 1997; Srinivas & Vijayalakshmi, 1981) attempted at solving the problem of time as a quantum observable by means of formal mathematical operations performed *outside* the usual Hilbert space of conventional quantum mechanics. Let us recall that Naimark's theorem states(Naimark, 1940) that the non-orthogonal spectral decomposition of a hermitian operator *can be approximated* by an orthogonal spectral function (which corresponds to a selfadjoint operator), in a weak convergence, *with any desired accuracy*.

The main goal of the first part of the present paper is to justify the use of time as a quantum observable, basing ourselves on the properties of the hermitian (or, rather, maximal hermitian) operators for the case of continuous energy spectra: cf., e.g., the Refs. (Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; 2004; Olkhovsky & Recami, 2007)).

The question of time as a quantum-theoretical observable is conceptually connected with the much more general problem of the four-position operator and of the canonically conjugate four-momentum operator, both endowed with an hermitian and an anti-hermitian part, for relativistic spin-zero particles: This problem is analyzed in the second part of the present paper.

In the third part of this work, it is shown how non-hermitian operators can be meaningfully and extensively used, for instance, for describing *unstable states* (decaying resonances). Brief mentions are added of the cases of quantum dissipation, and of the nuclear optical potential.

#### **2. Time operator in non-relativistic quantum mechanics and in quantum electrodynamics 2. Time operator in non-relativistic quantum mechanics and in quantum**

#### **2.1 On Time as an observable in non-relativistic quantum mechanics for systems with continuous energy spectra electrodynamics 2.1 On** *Time* **as an observable in non-relativistic quantum mechanics for systems with continuous energy spectra**

The last part of the above-mentioned list (Aharonov et al., 1998; Atmanspacher & Amann, 1998; Blanchard P & Jadczyk, 1996; Busch et al., 1994; Delgado, 1999; Egusquiza & Muga, 1999; Giannitrapani, 1997; Gó´zd ´z A & De¸bicki, 2007; Grot et al., 1996; Kijowski, 1997; Kobe et al., 1994; Kocha'nski & Wo'dkievicz, 1999; Leo'n, 1997; Muga et al., 1999; Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007; Toller, 1999; Wang & Xiong, 2007), of papers, in particular Refs. (Aharonov et al., 1998; Atmanspacher & Amann, 1998; Blanchard P & Jadczyk, 1996; Delgado, 1999; Egusquiza & Muga, 1999; Giannitrapani, 1997; Gó´zd ´z A & De¸bicki, 2007; Grot et al., 1996; Kijowski, 1997; Kobe et al., 1994; Kocha'nski & Wo'dkievicz, 1999; Leo'n, 1997; Muga et al., 1999; Toller, 1999; Wang & Xiong, 2007), appeared in the nineties, devoted to the problem of Time in non-relativistic quantum mechanics, essentially because of the need to define the tunnelling 2 Will-be-set-by-IN-TECH

1994; Kocha'nski & Wo'dkievicz, 1999; Leo'n, 1997; Muga et al., 1999; Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007; Srinivas & Vijayalakshmi, 1981; Toller, 1999; Wang & Xiong, 2007) appeared in the nineties, stimulated partially by the need of a consistent definition for the tunneling time. It is noticeable, and let us stress it right now, that this second set of papers seems however to have ignored Naimark's theorem(Naimark, 1940), which had previously constituted (directly or indirectly) an important basis for the results in Refs. (Holevo, 1978; 1982; Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; 1990; 1992; 1998; Recami, 1976; 1977), moreover, all the papers (Aharonov et al., 1998; Atmanspacher & Amann, 1998; Blanchard P & Jadczyk, 1996; Busch et al., 1994; Grot et al., 1996; Kobe et al., 1994; Leo'n, 1997; Srinivas & Vijayalakshmi, 1981) attempted at solving the problem of time as a quantum observable by means of formal mathematical operations performed *outside* the usual Hilbert space of conventional quantum mechanics. Let us recall that Naimark's theorem states(Naimark, 1940) that the non-orthogonal spectral decomposition of a hermitian operator *can be approximated* by an orthogonal spectral function (which corresponds to a selfadjoint

The main goal of the first part of the present paper is to justify the use of time as a quantum observable, basing ourselves on the properties of the hermitian (or, rather, maximal hermitian) operators for the case of continuous energy spectra: cf., e.g., the Refs. (Olkhovsky & Recami,

The question of time as a quantum-theoretical observable is conceptually connected with the much more general problem of the four-position operator and of the canonically conjugate four-momentum operator, both endowed with an hermitian and an anti-hermitian part, for relativistic spin-zero particles: This problem is analyzed in the second part of the present

In the third part of this work, it is shown how non-hermitian operators can be meaningfully and extensively used, for instance, for describing *unstable states* (decaying resonances). Brief mentions are added of the cases of quantum dissipation, and of the nuclear optical potential.

**2. Time operator in non-relativistic quantum mechanics and in quantum**

**2.1 On Time as an observable in non-relativistic quantum mechanics for systems with**

**2. Time operator in non-relativistic quantum mechanics and in quantum** 

The last part of the above-mentioned list (Aharonov et al., 1998; Atmanspacher & Amann, 1998; Blanchard P & Jadczyk, 1996; Busch et al., 1994; Delgado, 1999; Egusquiza & Muga, 1999; Giannitrapani, 1997; Gó´zd ´z A & De¸bicki, 2007; Grot et al., 1996; Kijowski, 1997; Kobe et al., 1994; Kocha'nski & Wo'dkievicz, 1999; Leo'n, 1997; Muga et al., 1999; Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007; Toller, 1999; Wang & Xiong, 2007), of papers, in particular Refs. (Aharonov et al., 1998; Atmanspacher & Amann, 1998; Blanchard P & Jadczyk, 1996; Delgado, 1999; Egusquiza & Muga, 1999; Giannitrapani, 1997; Gó´zd ´z A & De¸bicki, 2007; Grot et al., 1996; Kijowski, 1997; Kobe et al., 1994; Kocha'nski & Wo'dkievicz, 1999; Leo'n, 1997; Muga et al., 1999; Toller, 1999; Wang & Xiong, 2007), appeared in the nineties, devoted to the problem of Time in non-relativistic quantum mechanics, essentially because of the need to define the tunnelling

**electrodynamics 2.1 On** *Time* **as an observable in non-relativistic quantum mechanics for systems with** 

operator), in a weak convergence, *with any desired accuracy*.

paper.

**electrodynamics**

**continuous energy spectra**

**continuous energy spectra** 

1992; Olkhovsky et al., 1995; 2004; Olkhovsky & Recami, 2007)).

time. As remarked, those papers did not refer to the Naimark theorem2 (Naimark, 1940) which had mathematically supported, on the contrary, the results in (Holevo, 1978; 1982; Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; 1990; 1992; 1998; Recami, 1976; 1977), and afterwards in (Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007). Indeed, already in the seventies (in Refs. (Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Recami, 1976; 1977) while more detailed presentations and reviews can be found in (Olkhovsky, 1984; 1990; 1992; 1998) and independently in (Holevo, 1978; 1982)), it was proven that, for systems with continuous energy spectra, Time **is** a quantum-mechanical observable, canonically conjugate to energy. Namely, it had been shown the time operator

$$f = \begin{cases} \displaystyle t\_\prime & \text{in the time (t-)representation,} \\ \displaystyle -i\hbar \frac{\partial}{\partial E} \prime \text{ in the energy (E-)representation (b)} \end{cases} \tag{1}$$

to be not selfadjoint, but hermitian, and to act on square-integrable space-time wave packets in the representation (1a), and on their Fourier-transforms in (1b), once point *E* = 0 is eliminated (i. e., once one deals only with moving packets, excluding any *non-moving* rear tails and the cases with zero fluxes)<sup>3</sup> In Refs.(Olkhovsky, 1984; 1990; 1992; 1998) and (Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007), the operator ˆ*t* (in the *t*-representation) had the property that any averages over time, in the one-dimensional (1D) scalar case, were to be obtained by use of the following *measure* (or weight):

$$W(t, \mathbf{x}) \, dt = \frac{\int \mathbf{(x, t)} \, dt}{\int\_{-\infty}^{\infty} j(\mathbf{x}, t) \, dt} \, \mathbf{} \tag{2}$$

where the the flux density *j*(*x*, *t*) corresponds to the (temporal) probability for a particle to pass through point *x* during the unit time centered at *t*, when traveling in the positive *x*-direction. Such a measure is not postulated, but is a direct consequence of the well-known probabilistic *spatial* interpretation of *ρ* (*x*, *t*) and of the continuity relation *∂ρ* (*x*, *t*)/*∂ t* + div*j*(*x*, *t*) = 0. Quantity *ρ*(*x*, *t*) is, as usual, the probability of finding the considered moving particle inside a unit space interval, centered at point *x*, at time *t*.

Quantities *ρ*(*x*, *t*) and *j*(*x*, *t*) are related to the wave function Ψ (*x*, *t*) by the ordinary definitions *ρ* (*x*, *t*) = |Ψ (*x*, *t*)| <sup>2</sup> and *<sup>j</sup>*(*x*, *<sup>t</sup>*) = �[Ψ∗(*x*, *<sup>t</sup>*) (*h*¯ /*iμ*) <sup>Ψ</sup> (*x*, *<sup>t</sup>*))]). When the flux density *j*(*x*, *t*) changes its sign, quantity *W* (*x*, *t*) *dt* is no longer positive-definite and, as in Refs.(Olkhovsky, 1984; Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007), it acquires the physical meaning of a probability density *only* during those partial time-intervals in which the flux

<sup>2</sup> The Naimark theorem states in particular the following(Naimark, 1940): The non-orthogonal spectral decomposition of a maximal hermitian operator can be approximated by an orthogonal spectral function (which corresponds to a selfadjoint operator), in a weak convergence, with any desired accuracy.

<sup>3</sup> Such a condition is enough for operator (1a,b) to be a *hermitian*, or more precisely a *maximal hermitian*[2–8] *operator* (see also (Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007)); but it can be dispensed with by recourse to bilinear forms (see, e.g., Refs.(Recami, 1976; 1977; Recami et al., 1983) and refs. therein), as we shall see below.

do clearly satisfy the commutation relation(Olkhovsky & Recami, 2007; 2008; Recami, 1976;

Time as Quantum Observable, Canonical Conjugated to Energy 21

The Stone and von Neumann theorem(Stone, 1930), has been always interpreted as establishing a commutation relation like (8) for the pair of the canonically conjugate operators (1) and (7), in both representations, for selfadjoint operators only. However, it can be generalized for (maximal) hermitian operators, once one introduces ˆ*t* by means of the *single-valued* Fourier transformation from the *t*-axis (−∞ < *t* < ∞) to the *E*-semiaxis (0 < *E* < ∞), and utilizes the properties(Akhiezer & Glazman, 1981; D ter Haar, 1971) of the "(maximal) hermitian" operators: This has been shown, e.g., in the last one of Refs.(Olkhovsky & Recami,

(where the standard deviations are <sup>Δ</sup>*<sup>a</sup>* <sup>=</sup> <sup>√</sup>*Da*, quantity *Da* being the variance *Da* <sup>=</sup> �*a*2�−�*a*�2, and *<sup>a</sup>* <sup>=</sup> *<sup>E</sup>*, *<sup>t</sup>*, while �...� denotes the average over *<sup>t</sup>* with the measures *<sup>W</sup>* (*x*, *<sup>t</sup>*) *dt* or *W*<sup>±</sup> (*x*, *t*) *dt* in the *t*-representation) can be derived also for operators which are simply hermitian, by a straightforward generalization of the procedures which are common in the case of *selfadjoint* (canonically conjugate) quantities, like coordinate *x*ˆ and momentum *p*ˆ*x*. Moreover, relation (8) satisfies(Olkhovsky & Recami, 2007; 2008) the Dirac "correspondence" principle, since the classical Poisson brackets {*q*0, *p*0}, with *q*<sup>0</sup> = *t* and *p*<sup>0</sup> = −*E*, are equal to 1. In Refs. (Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; Recami, 1976; 1977), and (Olkhovsky & Recami, 2007; 2008), it was also shown that *the differences*, between the mean times at which a wave-packet passes through a *pair* of points, obey the Ehrenfest

As a consequence, one can state that, for systems with continuous energy spectra, the mathematical properties of (maximal) hermitian operators, like ˆ*t* in eq. (1), are sufficient for considering them as quantum observables. Namely, the *uniqueness*(Akhiezer & Glazman,

guarantees the "equivalence" of the mean values of any analytical function of time when evaluated in the *t* and in the *E*-representations. In other words, such an expansion is equivalent to a completeness relation, for the (approximate) eigenfunctions of ˆ*t<sup>n</sup>* (*n* > 1), which *with any accuracy* can be regarded as orthogonal, and corresponds to the actual eigenvalues for the continuous spectrum. These approximate eigenfunctions belong to the space of the square-integrable functions of the energy *E* (cf., for instance, see, for instance Refs. (Olkhovsky, 1984; 1990; 1992; 1998; Olkhovsky & Recami, 2007; Recami, 1976; 1977) and

From this point of view, there is no *practical* difference between selfadjoint and maximal hermitian operators for systems with continuous energy spectra. Let us repeat that the

mechanical observable (like energy, momentum, space coordinates, etc.) *without having to*

It is remarkable that von Neumann himself(Von Neumann, 1955), before confining himself for simplicity to selfajoint operators, stressed that operators like our time ˆ*t* may represent physical observables, even if they are not selfadjoint. Namely, he explicitly considered the example of the operator − *ih*¯ *∂*/*∂x* associated with a particle living in the right semi-space bounded by a rigid wall located at *x* = 0; that operator is not selfadjoint (acting on wave packets defined

*<sup>n</sup>* (*n* > 1) are enough for considering time as a quantum

1981) of the spectral decomposition (although not orthogonal) for operators ˆ*t*, and ˆ*t*

1968; 1969) as well as in Refs.(Olkhovsky & Recami, 2007; 2008).

Indeed, from eq. (8) the uncertainty relation

correspondence principle.

refs. therein).

mathematical properties of ˆ*t*

*introduce any new physical postulates*.

[*E*ˆ, ˆ*t*] = *ih*¯. (8)

Δ*E* Δ*t* ≥ *h*¯ /2 (9)

*<sup>n</sup>* (*n* > 1),

1977)

density *j*(*x*, *t*) does keep its sign. Therefore, let us introduce the *two* measures(Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; 2004; Olkhovsky & Recami, 2007) by separating the positive and the negative flux-direction values (that is, the flux signs)

$$\mathcal{W}\_{\pm}\left(t,\mathbf{x}\right)dt = \frac{j\_{\pm}\left(\mathbf{x},t\right)dt}{\int\_{-\infty}^{\infty} j\_{\pm}\left(\mathbf{x},t\right)dt} \tag{3}$$

with *j*<sup>±</sup> (*x*, *t*) = *j*(*x*, *t*) *θ*(±*j*).

Then, the mean value �*t*±(*x*)� of the time *t* at which the particle passes through position *x*, *when traveling in the positive or negative direction*, is, respectively,

$$\langle t\_{\pm}(\mathbf{x})\rangle = \frac{\int\_{-\infty}^{+\infty} t \, j\_{\pm}(\mathbf{x}, t) \, dt}{\int\_{-\infty}^{+\infty}} = \frac{\int\_{0}^{+\infty} \frac{1}{2} \left[ \mathbf{G}^{\*}(\mathbf{x}, E) \, \hat{\mathbf{f}} \, \mathbf{v} \, \mathbf{G} \left(\mathbf{x}, E\right) + \mathbf{v} \, \mathbf{G}^{\*}(\mathbf{x}, E) \, \hat{\mathbf{f}} \, \mathbf{G} \left(\mathbf{x}, E\right) \right] \, dE}{\int\_{0}^{+\infty}}, \qquad \text{(4)}$$

where *G* (*x*, *E*) is the Fourier-transform of the moving 1D wave-packet

$$\begin{aligned} \Psi(\mathbf{x},t) &= \int\_0^{+\infty} \mathbf{G}\left(\mathbf{x},E\right) \exp(-iEt/\hbar) \, dE = \\ &= \int\_0^{+\infty} \mathbf{g}\left(E\right) \varrho(\mathbf{x},E) \exp(-iEt/\hbar) \, dE \end{aligned}$$

when going on from the time to the energy representation. For free motion, one has *G*(*x*, *E*) = *g*(*E*) exp(*ikx*), and *ϕ*(*x*, *E*) = exp(*ikx*), while *E* = *μ h*¯ <sup>2</sup>*k*2/ 2 = *μ v*2/ 2. In Refs. (Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; 2004; Olkhovsky & Recami, 2007), there were defined the mean time *durations* for the particle 1D transmission from *xi* to *x <sup>f</sup>* > *xi*, and reflection from the region (*xi*, +∞) back to the interval *x <sup>f</sup>* ≤ *xi*. Namely

$$<\langle \pi\_T(\mathbf{x}\_i, \mathbf{x}\_f) \rangle = \langle t\_+(\mathbf{x}\_f) \rangle - \langle t\_+(\mathbf{x}\_i) \rangle \tag{5}$$

and

$$
\langle \langle \pi\_{\mathcal{R}}(\mathbf{x}\_{i\prime} \mathbf{x}\_{f}) \rangle \rangle = \langle t\_{-}(\mathbf{x}\_{f}) \rangle - \langle t\_{+}(\mathbf{x}\_{i}) \rangle\_{\prime} \tag{6}
$$

respectively. The 3D generalization for the mean durations of quantum collisions and nuclear reactions appeared in (Olkhovsky, 1984; 1990; 1992; 1998). Finally, suitable definitions of the averages �*tn*� on time of *<sup>t</sup>n*, with *<sup>n</sup>* <sup>=</sup> 1, 2 . . ., and of �*f*(*t*)�, quantity *<sup>f</sup>*(*t*) being any analytical function of time, can be found in (Olkhovsky & Recami, 2007; 2008), where single-valued expressions have been explicitly written down.

The two canonically conjugate operators, the time operator (1) and the energy operator

$$\hat{E} = \begin{cases} E\_{\prime} & \text{in the energy (E-) representation, (a)} \\ i\hbar \frac{\partial}{\partial t^{\prime}} & \text{in the time (t-) representation} \end{cases} \tag{7}$$

4 Will-be-set-by-IN-TECH

density *j*(*x*, *t*) does keep its sign. Therefore, let us introduce the *two* measures(Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; 2004; Olkhovsky & Recami, 2007) by separating the

> *<sup>W</sup>*<sup>±</sup> (*t*, *<sup>x</sup>*) *dt* <sup>=</sup> *<sup>j</sup>*<sup>±</sup> (*x*, *<sup>t</sup>*) *dt* + ∞ −∞

Then, the mean value �*t*±(*x*)� of the time *t* at which the particle passes through position *x*,

+ ∞

0 *v G* (*x*, *E*)

*G* (*x*, *E*) exp(−*iEt*/¯*h*) *dE* =

*g*(*E*) *ϕ*(*x*, *E*) exp(−*iEt*/¯*h*) *dE*

when going on from the time to the energy representation. For free motion, one has *G*(*x*, *E*) = *g*(*E*) exp(*ikx*), and *ϕ*(*x*, *E*) = exp(*ikx*), while *E* = *μ h*¯ <sup>2</sup>*k*2/ 2 = *μ v*2/ 2. In Refs. (Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; 2004; Olkhovsky & Recami, 2007), there were defined the mean time *durations* for the particle 1D transmission from *xi* to *x <sup>f</sup>* > *xi*, and reflection from

respectively. The 3D generalization for the mean durations of quantum collisions and nuclear reactions appeared in (Olkhovsky, 1984; 1990; 1992; 1998). Finally, suitable definitions of the averages �*tn*� on time of *<sup>t</sup>n*, with *<sup>n</sup>* <sup>=</sup> 1, 2 . . ., and of �*f*(*t*)�, quantity *<sup>f</sup>*(*t*) being any analytical function of time, can be found in (Olkhovsky & Recami, 2007; 2008), where single-valued

*E*, in the energy (*E*-) representation, (a)

The two canonically conjugate operators, the time operator (1) and the energy operator

*j*<sup>±</sup> (*x*, *t*) *dt*

*G*∗(*x*, *E*) ˆ*tvG* (*x*, *E*) + *v G*∗(*x*, *E*) ˆ*t G* (*x*, *E*)

�*τT*(*xi*, *x <sup>f</sup>*)� = �*t*+(*x <sup>f</sup>*)�−�*t*+(*xi*)� (5)

�*τR*(*xi*, *x <sup>f</sup>*)� = �*t*−(*x <sup>f</sup>*)�−�*t*+(*xi*)�, (6)

, in the time (*t*-) representation (b) (7)

 <sup>2</sup> *dE* (3)

, (4)

 *dE*

positive and the negative flux-direction values (that is, the flux signs)

*when traveling in the positive or negative direction*, is, respectively,

+ ∞

1 2 

0

where *G* (*x*, *E*) is the Fourier-transform of the moving 1D wave-packet

+ ∞

0

0

= + ∞

=

Ψ (*x*, *t*) =

the region (*xi*, +∞) back to the interval *x <sup>f</sup>* ≤ *xi*. Namely

expressions have been explicitly written down.

*E*ˆ =

*ih*¯ *<sup>∂</sup> ∂t*

with *j*<sup>±</sup> (*x*, *t*) = *j*(*x*, *t*) *θ*(±*j*).

+ ∞

*t j*<sup>±</sup> (*x*, *t*) *dt*

*j*<sup>±</sup> (*x*, *t*) *dt*

−∞

+ ∞

−∞

�*t*±(*x*)� =

and

do clearly satisfy the commutation relation(Olkhovsky & Recami, 2007; 2008; Recami, 1976; 1977)

$$\left[\mathbf{f}, \hat{\mathbf{f}}\right] = i\hbar. \tag{8}$$

The Stone and von Neumann theorem(Stone, 1930), has been always interpreted as establishing a commutation relation like (8) for the pair of the canonically conjugate operators (1) and (7), in both representations, for selfadjoint operators only. However, it can be generalized for (maximal) hermitian operators, once one introduces ˆ*t* by means of the *single-valued* Fourier transformation from the *t*-axis (−∞ < *t* < ∞) to the *E*-semiaxis (0 < *E* < ∞), and utilizes the properties(Akhiezer & Glazman, 1981; D ter Haar, 1971) of the "(maximal) hermitian" operators: This has been shown, e.g., in the last one of Refs.(Olkhovsky & Recami, 1968; 1969) as well as in Refs.(Olkhovsky & Recami, 2007; 2008). Indeed, from eq. (8) the uncertainty relation

Δ*E* Δ*t* ≥ *h*¯ /2 (9)

(where the standard deviations are <sup>Δ</sup>*<sup>a</sup>* <sup>=</sup> <sup>√</sup>*Da*, quantity *Da* being the variance *Da* <sup>=</sup> �*a*2�−�*a*�2, and *<sup>a</sup>* <sup>=</sup> *<sup>E</sup>*, *<sup>t</sup>*, while �...� denotes the average over *<sup>t</sup>* with the measures *<sup>W</sup>* (*x*, *<sup>t</sup>*) *dt* or *W*<sup>±</sup> (*x*, *t*) *dt* in the *t*-representation) can be derived also for operators which are simply hermitian, by a straightforward generalization of the procedures which are common in the case of *selfadjoint* (canonically conjugate) quantities, like coordinate *x*ˆ and momentum *p*ˆ*x*. Moreover, relation (8) satisfies(Olkhovsky & Recami, 2007; 2008) the Dirac "correspondence" principle, since the classical Poisson brackets {*q*0, *p*0}, with *q*<sup>0</sup> = *t* and *p*<sup>0</sup> = −*E*, are equal to 1. In Refs. (Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; Recami, 1976; 1977), and (Olkhovsky & Recami, 2007; 2008), it was also shown that *the differences*, between the mean times at which a wave-packet passes through a *pair* of points, obey the Ehrenfest correspondence principle.

As a consequence, one can state that, for systems with continuous energy spectra, the mathematical properties of (maximal) hermitian operators, like ˆ*t* in eq. (1), are sufficient for considering them as quantum observables. Namely, the *uniqueness*(Akhiezer & Glazman, 1981) of the spectral decomposition (although not orthogonal) for operators ˆ*t*, and ˆ*t <sup>n</sup>* (*n* > 1), guarantees the "equivalence" of the mean values of any analytical function of time when evaluated in the *t* and in the *E*-representations. In other words, such an expansion is equivalent to a completeness relation, for the (approximate) eigenfunctions of ˆ*t<sup>n</sup>* (*n* > 1), which *with any accuracy* can be regarded as orthogonal, and corresponds to the actual eigenvalues for the continuous spectrum. These approximate eigenfunctions belong to the space of the square-integrable functions of the energy *E* (cf., for instance, see, for instance Refs. (Olkhovsky, 1984; 1990; 1992; 1998; Olkhovsky & Recami, 2007; Recami, 1976; 1977) and refs. therein).

From this point of view, there is no *practical* difference between selfadjoint and maximal hermitian operators for systems with continuous energy spectra. Let us repeat that the mathematical properties of ˆ*t <sup>n</sup>* (*n* > 1) are enough for considering time as a quantum mechanical observable (like energy, momentum, space coordinates, etc.) *without having to introduce any new physical postulates*.

It is remarkable that von Neumann himself(Von Neumann, 1955), before confining himself for simplicity to selfajoint operators, stressed that operators like our time ˆ*t* may represent physical observables, even if they are not selfadjoint. Namely, he explicitly considered the example of the operator − *ih*¯ *∂*/*∂x* associated with a particle living in the right semi-space bounded by a rigid wall located at *x* = 0; that operator is not selfadjoint (acting on wave packets defined

space (for instance, adding the negative values of the energy, too), by exploiting the Naimark dilation-theorem(Naimark, 1943): But such a program has been realized till now only in the

Time as Quantum Observable, Canonical Conjugated to Energy 23

By contrast, our approach is based on a different Naimark's theorem(Naimark, 1940), which, as already mentioned above, allows a much more direct, simple and general –and at the same time non less rigorous– introduction of a quantum operator for Time. More precisely, our approach is based on the so-called *Carleman theorem*(Carleman, 1923), utilized in Ref.(Naimark, 1940), about approximating a hermitian operator by suitable successions of "bounded" selfadjoint operators: That is, of selfadjoint operators whose spectral functions do weakly converge to the non-orthogonal spectral function of the considered hemitian operator. And our approach is applicable to a large family of three-dimensional (3D) particle collisions, with all possible Hamiltonians. Actually, our approach was proposed in the early Refs. (Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; Recami, 1976; 1977) and in the first one of Ref. (Olkhovsky & Recami, 1992; Olkhovsky et al., 1995), and applied therein for the time analyzis of quantum collisions,

In the continuous spectrum case, instead of the *E*-representation, with 0 < *E* < +∞, in eqs.(1)–(4) one can also use the *k*-representation(Holevo, 1978; 1982), with the advantage that

**2.3 An alternative weight for time averages (in the cases of particle dwelling inside a certain**

**2.3 An alternative weight for time averages (in the cases of particle** *dwelling* **inside a** 

We recall that the weight (2) [as well as its modifications (3)] has the meaning of a probability for the considered particle to pass through point *x* during the time interval (*t*, *t* + *dt*). Let us follow the procedure presented in Refs. (Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007) and

> + ∞

 Ψ(*x*, *t*) 

> Ψ(*x*, *t*) <sup>2</sup> *dx*

> > Ψ(*x*, *t*) <sup>2</sup> *dx*

+ ∞

−∞

−∞

obtained from the 1D continuity equation. One can easily realize that a second, alternative

*g*(*k*) *ϕ*(*x*, *k*) exp(−*iEt*/¯*h*) *dk* (11)

*<sup>n</sup>*�, with *<sup>n</sup>* <sup>&</sup>gt; 1, we confine

<sup>2</sup> *dx* (12)

(13)

simple cases of one-dimensional particle free motion.

nuclear reactions and tunnelling processes.

−∞ < *k* < +∞:

with *<sup>E</sup>* <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup>*k*2/ 2*μ*, and *<sup>k</sup>* �<sup>=</sup> 0.

**spatial region)**

**certain spatial region)** 

weight can be adopted:

**2.2 On the momentum representation of the Time operator**

Ψ (*x*, *t*) =

refs. therein, and analyze the consequences of the equality

+ ∞

−∞

For the extension of the momentum representation to the case of �*t*

+ ∞

−∞

ourselves here to refer the reader to the papers (Olkhovsky & Recami, 2007; 2008).

*j*(*x*, *t*) *dt* =

*d P*(*x*, *t*) ≡ *Z* (*x*, *t*) *dx* =

on the positive *x*-axis) only, nevertheless it obviously corresponds to the *x*-component of the observable *momentum* for that particle: See Fig.1.

Fig. 1. For a particle *Q* free to move in a semi-space, bounded by a rigid wall located at *x* = 0, the operator −*i∂*/*∂x* has the clear physical meaning of the particle momentum *x*-component even if it is *not* selfadjoint (cf. von Neumann(Von Neumann, 1955), and Ref. (Recami, 1976; 1977)): See the text.

At this point, let us emphasize that our previously assumed boundary condition *E* �= 0 can be dispensed with, by having recourse (Olkhovsky & Recami, 1968; 1969; Recami, 1976; 1977) to the *bi-linear* hermitian operator

$$
\hat{t} = \frac{-i\hbar}{2} \frac{\stackrel{\leftrightarrow}{\partial}}{\partial E} \tag{10}
$$

where the meaning of the sign ↔ is clear from the accompanying definition

$$(f, \hat{t} \, g) = \left(f, -\frac{i\hbar}{2}\frac{\partial}{\partial E} \, g\right) + \left(-\frac{i\hbar}{2}\frac{\partial}{\partial E} \, f, \, g\right) \dots$$

By adopting this expression for the time operator, the algebraic sum of the two terms in the r.h.s. of the last relation results to be automatically zero at point *E* = 0. This question will be exploited below, in Sect. 3 (when dealing with the more general case of the four-position operator). Incidentally, such an "elimination" (Olkhovsky & Recami, 1968; 1969; Recami, 1976; 1977) of point *E* = 0 is not only simpler, but also more physical, than other kinds of elimination obtained much later in papers like (Egusquiza & Muga, 1999; Muga et al., 1999).

In connection with the last quotation, leu us for briefly comment on the so-called *positive-operator-value-measure* (POVM) approach, often used or discussed in the second set of papers on time in quantum physics mentioned in our Introduction. Actually, an analogous procedure had been proposed, since the sixties (Aharonov & Bohm, 1961), in some approaches to the quantum theory of measurements. Afterwards, and much later, the POVM approach has been applied, in a simplified and shortened form, to the time-operator problem in the case of one-dimensional free motion: for instance, in Refs. (Delgado, 1999; Egusquiza & Muga, 1999; Giannitrapani, 1997; Gó´zd ´z A & De¸bicki, 2007; Kijowski, 1997; Kobe et al., 1994; Kocha'nski & Wo'dkievicz, 1999; Leo'n, 1997; Muga et al., 1999; Srinivas & Vijayalakshmi, 1981; Toller, 1999; Wang & Xiong, 2007) and especially in (Egusquiza & Muga, 1999; Muga et al., 1999). These papers stated that a generalized decomposition of unity (or "POV measure") could be obtained from selfadjoint extensions of the time operator inside an extended Hilbert 6 Will-be-set-by-IN-TECH

on the positive *x*-axis) only, nevertheless it obviously corresponds to the *x*-component of the

Fig. 1. For a particle *Q* free to move in a semi-space, bounded by a rigid wall located at *x* = 0, the operator −*i∂*/*∂x* has the clear physical meaning of the particle momentum *x*-component even if it is *not* selfadjoint (cf. von Neumann(Von Neumann, 1955), and Ref. (Recami, 1976;

At this point, let us emphasize that our previously assumed boundary condition *E* �= 0 can be dispensed with, by having recourse (Olkhovsky & Recami, 1968; 1969; Recami, 1976; 1977) to

> ↔ *∂*

*<sup>∂</sup><sup>E</sup>* (10)

 .

<sup>ˆ</sup>*<sup>t</sup>* <sup>=</sup> <sup>−</sup>*ih*¯ 2

By adopting this expression for the time operator, the algebraic sum of the two terms in the r.h.s. of the last relation results to be automatically zero at point *E* = 0. This question will be exploited below, in Sect. 3 (when dealing with the more general case of the four-position operator). Incidentally, such an "elimination" (Olkhovsky & Recami, 1968; 1969; Recami, 1976; 1977) of point *E* = 0 is not only simpler, but also more physical, than other kinds of elimination

In connection with the last quotation, leu us for briefly comment on the so-called *positive-operator-value-measure* (POVM) approach, often used or discussed in the second set of papers on time in quantum physics mentioned in our Introduction. Actually, an analogous procedure had been proposed, since the sixties (Aharonov & Bohm, 1961), in some approaches to the quantum theory of measurements. Afterwards, and much later, the POVM approach has been applied, in a simplified and shortened form, to the time-operator problem in the case of one-dimensional free motion: for instance, in Refs. (Delgado, 1999; Egusquiza & Muga, 1999; Giannitrapani, 1997; Gó´zd ´z A & De¸bicki, 2007; Kijowski, 1997; Kobe et al., 1994; Kocha'nski & Wo'dkievicz, 1999; Leo'n, 1997; Muga et al., 1999; Srinivas & Vijayalakshmi, 1981; Toller, 1999; Wang & Xiong, 2007) and especially in (Egusquiza & Muga, 1999; Muga et al., 1999). These papers stated that a generalized decomposition of unity (or "POV measure") could be obtained from selfadjoint extensions of the time operator inside an extended Hilbert

where the meaning of the sign ↔ is clear from the accompanying definition

obtained much later in papers like (Egusquiza & Muga, 1999; Muga et al., 1999).

 *<sup>f</sup>* , <sup>−</sup>*ih* 2 *∂ <sup>∂</sup><sup>E</sup> <sup>g</sup>* + −*ih* 2 *∂ <sup>∂</sup><sup>E</sup> <sup>f</sup>* , *<sup>g</sup>*

(*f* , ˆ*t g*) =

observable *momentum* for that particle: See Fig.1.

1977)): See the text.

the *bi-linear* hermitian operator

space (for instance, adding the negative values of the energy, too), by exploiting the Naimark dilation-theorem(Naimark, 1943): But such a program has been realized till now only in the simple cases of one-dimensional particle free motion.

By contrast, our approach is based on a different Naimark's theorem(Naimark, 1940), which, as already mentioned above, allows a much more direct, simple and general –and at the same time non less rigorous– introduction of a quantum operator for Time. More precisely, our approach is based on the so-called *Carleman theorem*(Carleman, 1923), utilized in Ref.(Naimark, 1940), about approximating a hermitian operator by suitable successions of "bounded" selfadjoint operators: That is, of selfadjoint operators whose spectral functions do weakly converge to the non-orthogonal spectral function of the considered hemitian operator. And our approach is applicable to a large family of three-dimensional (3D) particle collisions, with all possible Hamiltonians. Actually, our approach was proposed in the early Refs. (Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; Recami, 1976; 1977) and in the first one of Ref. (Olkhovsky & Recami, 1992; Olkhovsky et al., 1995), and applied therein for the time analyzis of quantum collisions, nuclear reactions and tunnelling processes.

#### **2.2 On the momentum representation of the Time operator**

In the continuous spectrum case, instead of the *E*-representation, with 0 < *E* < +∞, in eqs.(1)–(4) one can also use the *k*-representation(Holevo, 1978; 1982), with the advantage that −∞ < *k* < +∞:

$$\Psi(\mathbf{x},t) = \int\_{-\infty}^{+\infty} g(k) \,\varrho(\mathbf{x},k) \, \exp(-iEt/\hbar) \, dk \tag{11}$$

with *<sup>E</sup>* <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup>*k*2/ 2*μ*, and *<sup>k</sup>* �<sup>=</sup> 0.

For the extension of the momentum representation to the case of �*t <sup>n</sup>*�, with *<sup>n</sup>* <sup>&</sup>gt; 1, we confine ourselves here to refer the reader to the papers (Olkhovsky & Recami, 2007; 2008).

#### **2.3 An alternative weight for time averages (in the cases of particle dwelling inside a certain spatial region) 2.3 An alternative weight for time averages (in the cases of particle** *dwelling* **inside a certain spatial region)**

We recall that the weight (2) [as well as its modifications (3)] has the meaning of a probability for the considered particle to pass through point *x* during the time interval (*t*, *t* + *dt*). Let us follow the procedure presented in Refs. (Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007) and refs. therein, and analyze the consequences of the equality

$$\int\_{-\infty}^{+\infty} j(\mathbf{x}, t) \, dt = \int\_{-\infty}^{+\infty} \left| \Psi(\mathbf{x}, t) \right|^2 \, d\mathbf{x} \tag{12}$$

obtained from the 1D continuity equation. One can easily realize that a second, alternative weight can be adopted:

$$d\,P(\mathbf{x},t) \equiv Z\left(\mathbf{x},t\right)d\mathbf{x} = \frac{\left|\Psi(\mathbf{x},t)\right|^2 d\mathbf{x}}{\int \left|\Psi(\mathbf{x},t)\right|^2 d\mathbf{x}}\tag{13}$$

where, as usual, *A*(*r*, *t*) is the electromagnetic vector potential, while *r* = {*x*, *y*, *z*}, *k* = {*kx*, *ky*, *kz*}, *k*<sup>0</sup> ≡ *w*/*c* = *ε*/ ¯*hc*, and *k* ≡ |*k*| = *k*0. The axis *x* has been chosen as the

Time as Quantum Observable, Canonical Conjugated to Energy 25

*xi*, *xj* = *y*, *z*, while *χi*(*k*) is the probability amplitude for the photon to have momentum *k* and polarization *e<sup>j</sup>* along *xj*. Moreover, it is *ϕ*(*k*,*r*) = exp(*ikxx*) in the case of plane waves, while *ϕ*(*k*,*r*) is a linear combination of evanescent (decreasing) and anti-evanescent (increasing) waves in the case of "photon barriers" (i.e., band-gap filters, or even undersized segments of waveguides for microwaves, or frustrated total-internal-reflection regions for light, and so on). Although it is not easy to localize a photon in the direction of its polarization(Akhiezer & Berestezky, 1959; Schweber, 1961), nevertheless for 1D propagations it is possible to use the

*<sup>S</sup>*<sup>0</sup> *dx* , *<sup>S</sup>*<sup>0</sup> <sup>=</sup>

(*s*<sup>0</sup> = [*E*<sup>∗</sup> · *E* + *H*<sup>∗</sup> · *H*]/ 4*π* being the energy density, with the electromagnetic field *H* = rot *A*, and *E* = −1/*c ∂A*/*∂t*), which represents the probability density *of a photon to be found (localized) in the spatial interval (x, x* + *dx) along the x-axis at the instant t*; and the quantity

(*sx* = *c* �[*E*<sup>∗</sup> ∧ *H*]*<sup>x</sup>* / 8*π* being the energy flux density), which represents *the flux probability density of a photon to pass through point x in the time interval (t, t* + *dt)*: in full analogy with the probabilistic quantities for non-relativistic particles. The justification and convenience of such definitions is self-evident, when the wave-packet group velocity coincides with the velocity of the energy transport; in particular: (i) the wave-packet (17) is quite similar to wave-packets for non-relativistic particles, and (ii) in analogy with conventional non-relativistic quantum mechanics, one can define the "mean time instant" for a photon (i.e., an electromagnetic

*i*=*y*,*z*

*χi*(*k*) *ei*(*k*), with *eie<sup>j</sup>* = *δij*, and

*<sup>s</sup>*<sup>0</sup> *dy dz* (18)

*Sx*(*x*, *<sup>t</sup>*) *dt*, *Sx*(*x*, *<sup>t</sup>*) = *sx dy dz* (19)

propagation direction. Let us notice that *χ*(*k*) = ∑

space-time probabilistic interpretation of eq. (17), and define the quantity

*<sup>ρ</sup>*em(*x*, *<sup>t</sup>*) *dx* <sup>=</sup> *<sup>S</sup>*<sup>0</sup> *dx*

*<sup>j</sup>*em(*x*, *<sup>t</sup>*) *dt* <sup>=</sup> *Sx dt*

wave-packet) to pass through point *x*, as follows

Recami, 2007) processes.

�*t*(*x*)� =

+ ∞

*t J*em, *<sup>x</sup> dt* =

As a consequence [in the same way as in the case of equations (1)–(2)], the form (1) for the time operator in the energy representation is valid also for photons, with the same boundary conditions adopted in the case of particles, that is, with *χ<sup>i</sup>* (0) = *χ<sup>i</sup>* (∞) and with *E* = *hck* ¯ 0. The energy density *s*<sup>0</sup> and energy flux density *sx* satisfy the relevant continuity equation

*∂sx*

which is Lorentz-invariant for 1D spatial propagation(Olkhovsky et al., 2004; Olkhovsky &

*∂s*<sup>0</sup> *<sup>∂</sup><sup>t</sup>* <sup>+</sup> + ∞

*t Sx*(*x*, *t*) *dt*

.

*<sup>∂</sup><sup>x</sup>* <sup>=</sup> <sup>0</sup> (20)

*Sx*(*x*, *t*) *dt*

−∞

+ ∞

−∞

−∞

which possesses the meaning of probability for the particle to be located (or to sojourn, i. e., to *dwell*) inside the infinitesimal space region (*x*, *x* + *dx*) at the instant *t*, independently of its motion properties. Then, the quantity

$$P(\mathbf{x}\_1, \mathbf{x}\_2, t) = \frac{\int\_{\mathbf{x}\_1}^{\mathbf{x}\_2} \left| \Psi(\mathbf{x}, t) \right|^2 d\mathbf{x}}{\int\_{-\infty}^{\infty} \left| \Psi(\mathbf{x}, t) \right|^2 d\mathbf{x}} \tag{14}$$

will have the meaning of probability for the particle to dwell inside the spatial interval (*x*1, *x*2) at the instant *t*.

As it is known (see, for instance, Refs.(Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; 2004; Olkhovsky & Recami, 2007) and refs. therein), the *mean dwell time* can be written in the *two* equivalent forms:

$$\langle \pi(\mathbf{x}\_{i\prime}\mathbf{x}\_f) \rangle = \frac{\int\_{-\infty}^{+\infty} dt \int\_{\mathbf{x}\_i}^{\mathbf{x}\_f} |\Psi(\mathbf{x}\_i t)|^2 \, d\mathbf{x}}{\int\_{-\infty}^{+\infty} \dot{\jmath}\_{\text{in}}(\mathbf{x}\_i, t) \, dt} \tag{15}$$

and

$$
\langle \pi(\mathbf{x}\_{i}, \mathbf{x}\_{f}) \rangle = \frac{\int\_{-\infty}^{+\infty} t \, j(\mathbf{x}\_{f}, t) \, dt - \int\_{-\infty}^{+\infty} t \, j(\mathbf{x}\_{i}, t) \, dt}{\int\_{-\infty}^{+\infty} j\_{\text{in}}(\mathbf{x}\_{i}, t) \, dt} \, \, . \tag{16}
$$

where it has been taken account, in particular, of relation (12), which follows — as already said — from the continuity equation.

Thus, in correspondence with the two measures (2) and (13), when integrating over time one gets *two* different kinds of time distributions (mean values, variances...), which refer to the particle traversal time in the case of measure (2), and to the particle dwelling in the case of measure (13). Some examples for 1D tunneling are contained in Refs.(Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; 2004; Olkhovsky & Recami, 2007).

#### **2.4 Extension of the notion of Time as a quantum-theoretical observable for the case of photons 2.4 Extension of the notion of Time as a quantum-theoretical observable for the case of photons**

As is known (see, for instance, Refs. (Akhiezer & Berestezky, 1959; Olkhovsky et al., 2004; Schweber, 1961)), in first quantization the single-photon wave function can be probabilistically described in the 1D case by the wave-packet4

*<sup>A</sup>*(*r*, *<sup>t</sup>*) = *k*0 *d*3*k k*0 *χ*(*k*) *ϕ*(*k*,*r*) exp(−*ik*0*t*) , (17)

<sup>4</sup> The gauge condition div*A* = 0 is assumed.

8 Will-be-set-by-IN-TECH

which possesses the meaning of probability for the particle to be located (or to sojourn, i. e., to *dwell*) inside the infinitesimal space region (*x*, *x* + *dx*) at the instant *t*, independently of its

*x*2

*x*1 Ψ(*x*, *t*) <sup>2</sup> *dx*

+ ∞

 Ψ(*x*, *t*) <sup>2</sup> *dx* (14)

(15)

, (16)

−∞

will have the meaning of probability for the particle to dwell inside the spatial interval (*x*1, *x*2)

As it is known (see, for instance, Refs.(Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; 2004; Olkhovsky & Recami, 2007) and refs. therein), the *mean dwell time* can be written in the

*dt*

+ ∞

−∞

*t j*(*x <sup>f</sup>* , *t*) *dt* −

+ ∞

−∞

where it has been taken account, in particular, of relation (12), which follows — as already

Thus, in correspondence with the two measures (2) and (13), when integrating over time one gets *two* different kinds of time distributions (mean values, variances...), which refer to the particle traversal time in the case of measure (2), and to the particle dwelling in the case of measure (13). Some examples for 1D tunneling are contained in Refs.(Olkhovsky & Recami,

**2.4 Extension of the notion of Time as a quantum-theoretical observable for the case of**

As is known (see, for instance, Refs. (Akhiezer & Berestezky, 1959; Olkhovsky et al., 2004; Schweber, 1961)), in first quantization the single-photon wave function can be probabilistically

**2.4 Extension of the notion of Time as a quantum-theoretical observable for the case** 

*xf* 


*j*in(*xi*, *t*) *dt*

+ ∞

−∞

*j*in(*xi*, *t*) *dt*

<sup>2</sup> *dx*

*t j*(*xi*, *t*) *dt*

*χ*(*k*) *ϕ*(*k*,*r*) exp(−*ik*0*t*) , (17)

*xi*

+ ∞

−∞

*P*(*x*1, *x*2, *t*) =

�*τ*(*xi*, *x <sup>f</sup>*)� =

+ ∞

−∞

�*τ*(*xi*, *x <sup>f</sup>*)� =

1992; Olkhovsky et al., 1995; 2004; Olkhovsky & Recami, 2007).

*A*(*r*, *t*) =

*d*3*k k*0

*k*0

said — from the continuity equation.

described in the 1D case by the wave-packet4

<sup>4</sup> The gauge condition div*A* = 0 is assumed.

motion properties. Then, the quantity

at the instant *t*.

and

**photons**

**of photons** 

*two* equivalent forms:

where, as usual, *A*(*r*, *t*) is the electromagnetic vector potential, while *r* = {*x*, *y*, *z*}, *k* = {*kx*, *ky*, *kz*}, *k*<sup>0</sup> ≡ *w*/*c* = *ε*/ ¯*hc*, and *k* ≡ |*k*| = *k*0. The axis *x* has been chosen as the propagation direction. Let us notice that *χ*(*k*) = ∑ *i*=*y*,*z χi*(*k*) *ei*(*k*), with *eie<sup>j</sup>* = *δij*, and

*xi*, *xj* = *y*, *z*, while *χi*(*k*) is the probability amplitude for the photon to have momentum *k* and polarization *e<sup>j</sup>* along *xj*. Moreover, it is *ϕ*(*k*,*r*) = exp(*ikxx*) in the case of plane waves, while *ϕ*(*k*,*r*) is a linear combination of evanescent (decreasing) and anti-evanescent (increasing) waves in the case of "photon barriers" (i.e., band-gap filters, or even undersized segments of waveguides for microwaves, or frustrated total-internal-reflection regions for light, and so on). Although it is not easy to localize a photon in the direction of its polarization(Akhiezer & Berestezky, 1959; Schweber, 1961), nevertheless for 1D propagations it is possible to use the space-time probabilistic interpretation of eq. (17), and define the quantity

$$
\rho\_{\text{em}}(\mathbf{x},t) \, d\mathbf{x} = \frac{S\_0 \, d\mathbf{x}}{\int S\_0 \, d\mathbf{x}}, \, S\_0 = \int \int s\_0 \, dy \, dz \tag{18}
$$

(*s*<sup>0</sup> = [*E*<sup>∗</sup> · *E* + *H*<sup>∗</sup> · *H*]/ 4*π* being the energy density, with the electromagnetic field *H* = rot *A*, and *E* = −1/*c ∂A*/*∂t*), which represents the probability density *of a photon to be found (localized) in the spatial interval (x, x* + *dx) along the x-axis at the instant t*; and the quantity

$$j\_{\rm em}(\mathbf{x},t) \, dt = \frac{\mathbf{S}\_{\mathbf{x}} \, dt}{\int \mathbf{S}\_{\mathbf{x}}(\mathbf{x},t) \, dt'}, \mathbf{S}\_{\mathbf{x}}(\mathbf{x},t) = \int \int \mathbf{s}\_{\mathbf{x}} \, dy \, dz \tag{19}$$

(*sx* = *c* �[*E*<sup>∗</sup> ∧ *H*]*<sup>x</sup>* / 8*π* being the energy flux density), which represents *the flux probability density of a photon to pass through point x in the time interval (t, t* + *dt)*: in full analogy with the probabilistic quantities for non-relativistic particles. The justification and convenience of such definitions is self-evident, when the wave-packet group velocity coincides with the velocity of the energy transport; in particular: (i) the wave-packet (17) is quite similar to wave-packets for non-relativistic particles, and (ii) in analogy with conventional non-relativistic quantum mechanics, one can define the "mean time instant" for a photon (i.e., an electromagnetic wave-packet) to pass through point *x*, as follows

$$\langle t(\mathbf{x})\rangle = \int\_{-\infty}^{+\infty} t \, f\_{\mathbf{em},\mathbf{x}} \, dt = \frac{\int\_{-\infty}^{+\infty} t \, S\_{\mathbf{x}}(\mathbf{x}, t) \, dt}{\int\_{-\infty}^{+\infty} S\_{\mathbf{x}}(\mathbf{x}, t) \, dt}.$$

As a consequence [in the same way as in the case of equations (1)–(2)], the form (1) for the time operator in the energy representation is valid also for photons, with the same boundary conditions adopted in the case of particles, that is, with *χ<sup>i</sup>* (0) = *χ<sup>i</sup>* (∞) and with *E* = *hck* ¯ 0. The energy density *s*<sup>0</sup> and energy flux density *sx* satisfy the relevant continuity equation

$$\frac{\partial \mathbf{s}\_0}{\partial t} + \frac{\partial \mathbf{s}\_x}{\partial x} = \mathbf{0} \tag{20}$$

which is Lorentz-invariant for 1D spatial propagation(Olkhovsky et al., 2004; Olkhovsky & Recami, 2007) processes.

quantity *x*/*v* being the free-motion time (for a particle with velocity *v* ) for traveling the

Time as Quantum Observable, Canonical Conjugated to Energy 27

On the basis of what precedes, it is possible to show that the wave function Ψ(*x*, *t*) of a

In the energy representation, and in the stationary case, we obtain again *two* (dual) equations

and *T*ˆ Ψ = *t* Ψ . (26)

Ψ(*x*, *t*) *eiεt*/¯*<sup>h</sup> dt* . (28)

*gn ϕn*(*x*) exp[−*i*(*ε<sup>n</sup>* − *ε*0)*t*/¯*h*] , (29)

Θ(−*t* − [2*n* + 1]*T*/2 . (30)

<sup>2</sup> <sup>=</sup> 1. We omitted the non-significant phase factor exp(−*iε*0*t*/¯*h*)

∞ ∑ *n*=0 *∂ε* , (27)

*∂ϕε*

*∂*Ψ *∂t*

*<sup>H</sup>*<sup>ˆ</sup> *ϕε* <sup>=</sup> *ε ϕε* and *<sup>T</sup>*<sup>ˆ</sup> *ϕε* <sup>=</sup> <sup>−</sup>*ih*¯

+ ∞

−∞

**2.6 Time as an observable (and the time-energy uncertainty relation), for quantum-**

It might be interesting to apply the two pairs of the last dual equations also for investigating tunnelling processes through the quantum gravitational barrier, which appears during inflation, or at the beginning of the big-bang expansion, whenever a quasi-linear

For describing the time evolution of non-relativistic quantum systems endowed with a purely *discrete* (or a continuous *and discrete*) spectrum, let us now introduce wave-packets of the

where *ϕn*(*x*) are orthogonal and normalized bound states which satisfy the equation *H*ˆ *ϕn*(*x*) = *ε<sup>n</sup> ϕn*(*x*), quantity *H*ˆ being the Hamiltonian of the system; while the coefficients

Let us first consider the systems whose energy levels are separated by intervals admitting a maximum common divisor *D* (for ex., harmonic oscillator, particle in a rigid box, and spherical spinning top), so that the wave packet (29) is a periodic function of time possessing as period the Poincaré cycle time *T* = 2*πh*¯ /*D*. For such systems it is possible (Olkhovsky, 1990; 1992; 1998; Olkhovsky & Recami, 2007; 2008) to construct a *selfadjoint* time operator with the form (in the time representation) of a saw-function of *t*, choosing *t* = 0 as the initial time instant:

This periodic function for the time operator is a linear (increasing) function of time *t* within

Θ(*t* − [2*n* + 1]*T*/2) + *T*

*H*ˆ Ψ = *ih*¯

*ϕε* <sup>=</sup> <sup>1</sup> 2*πh*¯

**2.6 Time as an observable (and the time-energy uncertainty relation), for quantum-mechanical systems with discrete energy spectra**

form (Olkhovsky, 1990; 1992; 1998; Olkhovsky & Recami, 2007; 2008):

*n*=0

*ψ* (*x*, *t*) = ∑

*gn* are normalized: ∑

of the fundamental state.

*n*=0 |*gn*|

<sup>ˆ</sup>*<sup>t</sup>* <sup>=</sup> *<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*

each Poincarè cycle: see Fig.2.

∞ ∑ *n*=0

distance *x*.

quantum system satisfies the two (dual) equations

quantity *ϕε* being the Fourier-transform of Ψ:

Schrödinger-type equation does approximately show up.

**mechanical systems with discrete energy spectra** 

#### **2.5 Introducing the analogue of the "Hamiltonian" for the case of the Time operator: A new hamiltonian approach 2.5 Introducing the analogue of the "Hamiltonian" for the case of the Time operator: A new hamiltonian approach**

In non-relativistic quantum theory, the Energy operator acquires (cf., e.g., Refs. (Olkhovsky, 1990; 1992; 1998; Olkhovsky & Recami, 2007)) the *two* forms: (i) *ih*¯ *<sup>∂</sup> ∂t* in the *t*-representation, and (ii) *H*ˆ (*p*ˆ*x*, *x*ˆ,...) in the hamiltonianian formalism. The "duality" of these two forms can be easily inferred from the Schröedinger equation itself, *H*ˆ Ψ = *ih*¯ *∂*Ψ *∂t* . One can introduce in quantum mechanics a similar duality for the case of *Time*: Besides the general form (1) for the Time operator in the energy representation, which is valid for any physical systems in the region of continuous energy spectra, one can *express the time operator also in a "hamiltonian form"*, i.e., in terms of the coordinate and momentum operators, by having recourse to their commutation relations. Thus, by the replacements

$$\begin{aligned} \hat{\mathsf{E}} & \rightarrow \hat{\mathsf{H}} \left( \mathfrak{p}\_{\mathsf{X}}, \mathfrak{k}\_{\prime}, \ldots \right), \\\\ \hat{\mathsf{H}} & \rightarrow \hat{\mathsf{T}} \left( \mathfrak{p}\_{\mathsf{X}}, \mathfrak{k}\_{\prime}, \ldots \right), \end{aligned} \tag{21}$$

and on using the commutation relation [similar to eq. (3)]

$$[\hat{H}, \hat{T}] = i\hbar \,\,,\tag{22}$$

one can obtain(Rosenbaum, 1969), given a specific ordinary Hamiltonian, the corresponding explicit expression for *T*ˆ (*p*ˆ*x*, *x*ˆ,...).

Indeed, this procedure can be adopted for any physical system with a known Hamiltonian *H*ˆ (*p*ˆ*x*, *x*ˆ,...), and we are going to see a concrete example. By going on from the coordinate to the momentum representation, one realizes that the *formal* expressions of *both* the hamiltonian-type operators *H*ˆ (*p*ˆ*x*, *x*ˆ,...) and *T*ˆ (*p*ˆ*x*, *x*ˆ,...) *do not change*, except for an obvious change of sign in the case of operator *T*ˆ (*p*ˆ*x*, *x*ˆ,...).

As an explicit example, let us address the simple case of a free particle whose Hamiltonian is

$$\hat{H} = \begin{cases} \not p\_{\text{x}}^2 / 2\mu \,, \not p\_{\text{x}} = -i\hbar \frac{\partial}{\partial \mathbf{x}^{\prime}} & \text{in the coordinate representation} \\ p\_{\text{x}}^2 / 2\mu \,. & \text{in the momentum representation (a)} \end{cases} \tag{23}$$

Correspondingly, the Hamilton-type *time operator,* in its symmetrized form, will write

$$\hat{T} = \begin{cases} \frac{\mu}{2} \left( \hat{p}\_{\text{x}}^{-1} \mathbf{x} + x \hat{p}\_{\text{x}}^{-1} + i\hbar \right) \text{, in the coordinate representation} & \text{(a)}\\ -\frac{\mu}{2} \left( p\_{\text{x}}^{-1} \mathbf{\hat{x}} + \mathbf{\hat{x}} p\_{\text{x}}^{-1} + i\hbar / p\_{\text{x}}^{2} \right) \text{, in the momentum representation (b)} & \end{cases} \tag{24}$$

where

$$
\hat{p}\_{\mathbf{x}}^{-1} = \frac{\dot{\mathbf{t}}}{\hbar} \int d\mathbf{x} \dots \qquad \qquad \hat{\mathbf{x}} = i\hbar \frac{\partial}{\partial p\_{\mathbf{x}}} \dots
$$

Incidentally, operator (24b) is equivalent to <sup>−</sup>*ih*¯ *<sup>∂</sup> <sup>∂</sup><sup>E</sup>* , since *<sup>E</sup>* <sup>=</sup> *<sup>p</sup>*<sup>2</sup> *<sup>x</sup>*/ 2*μ*; and therefore it is also a (maximal) *hermitian* operator. Indeed, by applying the operator *T*ˆ (*p*ˆ*x*, *x*ˆ,...), for instance, to a plane-wave of the type exp(*ikx*), we obtain the same result in both the coordinate and the momentum representations:

$$
\hat{T}\,\exp(ik\mathbf{x}) = \frac{\mathbf{x}}{v}\,\exp(ik\mathbf{x})\tag{25}
$$

10 Will-be-set-by-IN-TECH

**2.5 Introducing the analogue of the "Hamiltonian" for the case of the Time operator: A new**

**2.5 Introducing the analogue of the "Hamiltonian" for the case of the Time operator: A new** 

In non-relativistic quantum theory, the Energy operator acquires (cf., e.g., Refs. (Olkhovsky,

and (ii) *H*ˆ (*p*ˆ*x*, *x*ˆ,...) in the hamiltonianian formalism. The "duality" of these two forms can

quantum mechanics a similar duality for the case of *Time*: Besides the general form (1) for the Time operator in the energy representation, which is valid for any physical systems in the region of continuous energy spectra, one can *express the time operator also in a "hamiltonian form"*, i.e., in terms of the coordinate and momentum operators, by having recourse to their

*<sup>E</sup>*<sup>ˆ</sup> <sup>→</sup> *<sup>H</sup>*<sup>ˆ</sup> (*p*ˆ*x*, *<sup>x</sup>*ˆ,...),

<sup>ˆ</sup>*<sup>t</sup>* <sup>→</sup> *<sup>T</sup>*<sup>ˆ</sup> (*p*ˆ*x*, *<sup>x</sup>*ˆ,...),

one can obtain(Rosenbaum, 1969), given a specific ordinary Hamiltonian, the corresponding

Indeed, this procedure can be adopted for any physical system with a known Hamiltonian *H*ˆ (*p*ˆ*x*, *x*ˆ,...), and we are going to see a concrete example. By going on from the coordinate to the momentum representation, one realizes that the *formal* expressions of *both* the hamiltonian-type operators *H*ˆ (*p*ˆ*x*, *x*ˆ,...) and *T*ˆ (*p*ˆ*x*, *x*ˆ,...) *do not change*, except for an obvious

As an explicit example, let us address the simple case of a free particle whose Hamiltonian is

Correspondingly, the Hamilton-type *time operator,* in its symmetrized form, will write

−2 *x* �

> *x* �

*<sup>T</sup>*<sup>ˆ</sup> exp(*ikx*) = *<sup>x</sup>*

*<sup>x</sup>* + *ih*¯ /*p*<sup>2</sup>

*<sup>x</sup>*/ 2*μ* . in the momentum representation (a)

*dx* ..., *<sup>x</sup>*<sup>ˆ</sup> <sup>=</sup> *ih*¯ *<sup>∂</sup>*

a (maximal) *hermitian* operator. Indeed, by applying the operator *T*ˆ (*p*ˆ*x*, *x*ˆ,...), for instance, to a plane-wave of the type exp(*ikx*), we obtain the same result in both the coordinate and the

*∂t*

*∂*Ψ *∂t*

[*H*ˆ , *T*ˆ] = *ih*¯ , (22)

*<sup>∂</sup><sup>x</sup>* , in the coordinate representation (a)

, in the coordinate representation (a)

, in the momentum representation (b)

*∂px* .

*<sup>∂</sup><sup>E</sup>* , since *<sup>E</sup>* <sup>=</sup> *<sup>p</sup>*<sup>2</sup>

in the *t*-representation,

. One can introduce in

(21)

(23)

(24)

*<sup>x</sup>*/ 2*μ*; and therefore it is also

*<sup>v</sup>* exp(*ikx*) (25)

1990; 1992; 1998; Olkhovsky & Recami, 2007)) the *two* forms: (i) *ih*¯ *<sup>∂</sup>*

be easily inferred from the Schröedinger equation itself, *H*ˆ Ψ = *ih*¯

commutation relations. Thus, by the replacements

change of sign in the case of operator *T*ˆ (*p*ˆ*x*, *x*ˆ,...).

*<sup>x</sup>*/ 2*μ*, *<sup>p</sup>*ˆ*<sup>x</sup>* <sup>=</sup> <sup>−</sup>*ih*¯ *<sup>∂</sup>*

−1 *<sup>x</sup>* + *ih*¯ ; *p*ˆ

*<sup>p</sup>*ˆ−<sup>1</sup> *<sup>x</sup>* <sup>=</sup> *<sup>i</sup> h*¯ �

*<sup>x</sup> <sup>x</sup>*<sup>ˆ</sup> + *xp*<sup>ˆ</sup> <sup>−</sup><sup>1</sup>

Incidentally, operator (24b) is equivalent to <sup>−</sup>*ih*¯ *<sup>∂</sup>*

explicit expression for *T*ˆ (*p*ˆ*x*, *x*ˆ,...).

⎧ ⎨ ⎩

*p*ˆ2

*p*2

*H*ˆ =

⎧ ⎪⎨ *μ* 2 � *p*ˆ −1 *<sup>x</sup> x* + *xp*ˆ

− *μ* 2 � *p*−<sup>1</sup>

⎪⎩

momentum representations:

*T*ˆ =

where

and on using the commutation relation [similar to eq. (3)]

**hamiltonian approach**

**hamiltonian approach** 

quantity *x*/*v* being the free-motion time (for a particle with velocity *v* ) for traveling the distance *x*.

On the basis of what precedes, it is possible to show that the wave function Ψ(*x*, *t*) of a quantum system satisfies the two (dual) equations

$$
\hat{H}\,\Psi = i\hbar \frac{\partial \Psi}{\partial t} \text{ and } \hat{T}\,\Psi = t\,\Psi\,. \tag{26}
$$

In the energy representation, and in the stationary case, we obtain again *two* (dual) equations

$$
\hat{H}\,\varphi\_{\varepsilon} = \varepsilon\,\varphi\_{\varepsilon} \text{ and } \hat{\Upsilon}\,\varphi\_{\varepsilon} = -i\hbar \frac{\partial \varphi\_{\varepsilon}}{\partial \varepsilon} \,, \tag{27}
$$

quantity *ϕε* being the Fourier-transform of Ψ:

$$\varphi\_{\varepsilon} = \frac{1}{2\pi\hbar} \int\_{-\infty}^{+\infty} \Psi(\mathbf{x}, t) \, e^{i\varepsilon t/\hbar} \, dt \, \,. \tag{28}$$

It might be interesting to apply the two pairs of the last dual equations also for investigating tunnelling processes through the quantum gravitational barrier, which appears during inflation, or at the beginning of the big-bang expansion, whenever a quasi-linear Schrödinger-type equation does approximately show up.

#### **2.6 Time as an observable (and the time-energy uncertainty relation), for quantum-mechanical systems with discrete energy spectra 2.6 Time as an observable (and the time-energy uncertainty relation), for quantummechanical systems with discrete energy spectra**

For describing the time evolution of non-relativistic quantum systems endowed with a purely *discrete* (or a continuous *and discrete*) spectrum, let us now introduce wave-packets of the form (Olkhovsky, 1990; 1992; 1998; Olkhovsky & Recami, 2007; 2008):

$$\psi\left(\mathbf{x},t\right) = \sum\_{n=0} g\_{\mathbb{H}} \,\varphi\_{\mathbb{H}}\left(\mathbf{x}\right) \,\exp\left[-i(\varepsilon\_{\mathbb{H}} - \varepsilon\_{0})t/\hbar\right] \,\tag{29}$$

where *ϕn*(*x*) are orthogonal and normalized bound states which satisfy the equation *H*ˆ *ϕn*(*x*) = *ε<sup>n</sup> ϕn*(*x*), quantity *H*ˆ being the Hamiltonian of the system; while the coefficients *gn* are normalized: ∑ *n*=0 |*gn*| <sup>2</sup> <sup>=</sup> 1. We omitted the non-significant phase factor exp(−*iε*0*t*/¯*h*) of the fundamental state.

Let us first consider the systems whose energy levels are separated by intervals admitting a maximum common divisor *D* (for ex., harmonic oscillator, particle in a rigid box, and spherical spinning top), so that the wave packet (29) is a periodic function of time possessing as period the Poincaré cycle time *T* = 2*πh*¯ /*D*. For such systems it is possible (Olkhovsky, 1990; 1992; 1998; Olkhovsky & Recami, 2007; 2008) to construct a *selfadjoint* time operator with the form (in the time representation) of a saw-function of *t*, choosing *t* = 0 as the initial time instant:

$$\hat{f} = \ t - T \sum\_{n=0}^{\infty} \Theta(t - [2n+1]T/2) + T \sum\_{n=0}^{\infty} \Theta(-t - [2n+1]T/2) \,. \tag{30}$$

This periodic function for the time operator is a linear (increasing) function of time *t* within each Poincarè cycle: see Fig.2.

accuracy. For them, when choosing an approximate Poincaré-cycle time, one can include in one cycle as many quasi-cycles as it is necessary for the demanded accuracy. Then, with the

Time as Quantum Observable, Canonical Conjugated to Energy 29

**3.1 Tunneling in consideration of multiple internal reflections of waves between internal**

**3.1 Tunneling in consideration of multiple internal reflections of waves between** 

An approach for description of one-dimensional motion of a non-relativistic particle above a barrier on the basis of multiple internal reflections of stationary waves relatively boundaries has been studied in number of papers and is known (see (Anderson, 1989; Fermor, 1966; McVoy et al., 1967) and references therein). Tunneling of the particle under the barrier was described successfully on the basis of multiple internal reflections of the wave packets relatively boundaries (approach was called as *method of multiple internal reflections* or *method MIR*, see Refs. (Maydanyuk et al., 2002a;b; Maydanyuk, 2003; Olkhovsky, 2000)). In such approach it succeeded in connecting: 1) continuous transition of solutions for packets after each reflection, total packets between the above-barrier motion and the under-barrier tunneling; 2) coincidence of transmitted and reflected amplitudes of stationary wave function in each spatial region obtained by approach MIR with the corresponding amplitudes obtained by standard method of quantum mechanics; 3) all non-stationary fluxes in each step, are non-zero that confirms propagation of packets under the barrier (i. e. their "tunneling"). In frameworks of such a method, non-stationary tunneling obtained own interpretation, allowing to study this process at interesting time moment or space point. In calculation of phase times this method turns out to be enough simple and convenient (Cardone et al., 2006). It has been adapted for scattering of the particle on nucleus and *α*-decay in the spherically symmetric approximation with the simplest radial barriers (Maydanyuk et al., 2002a; Maydanyuk, 2003; Olkhovsky, 2000) and for tunneling of photons (Cardone et al., 2006; Maydanyuk et al., 2002a). However, further realization of the MIR approach meets with three questions. 1) *Question on effectiveness.* The multiple reflections have been proved for the motion above one rectangular barrier and for tunneling under it (Anderson, 1989; Cardone et al., 2006; Maydanyuk et al., 2002a). However, after addition of the second step it becomes unclear how to separate the needed reflected waves from all their variety in calculation of all needed amplitudes. After obtaining exact solutions of the stationary amplitudes for two arbitrary rectangular barriers (Maydanyuk, 2003; Olkhovsky, 2000), it becomes unclear how to generalize such approach for barriers with arbitrary complicate shape. So, *we come to a serious unresolved problem of realization of the approach of multiple reflections in real quantum systems with complicated barriers*, and clear algorithms of calculation of amplitudes should be constructed. 2) *Question on correctness.* Whether is interference between packets formed relatively different boundaries appeared? Whether does this come to principally different results of the approach of multiple internal reflections and direct methods of quantum mechanics? Note that such interference cannot be appeared in tunneling through one rectangular barrier and, therefore,

3) *Question on uncertainty in radial problem.* Calculations of half-lives of different types of decays based on the semiclassical approach are prevailing today. For example, in Ref. (Buck et al., 1993) agreement between experimental data of *α*-decay half-lives and ones calculated by theory is demonstrated in a wide region of nuclei from 106Te up to nucleus with *Ad* = 266 and *Zd* = 109 (see Ref. (Denisov & Ikezoe, 2005) for some improved approaches). In review (Sobiczewski & Pomorski, 2007) methodology of calculation of half-lives for

chosen accuracy, a *quasi-selfadjoint time operator* can be introduced.

**boundaries**

**internal boundaries** 

it could not visible in the previous papers.

**3. Multiple internal reflections approach in description of tunneling**

**3. Multiple internal reflections approach in description of tunneling** 

Fig. 2. The periodic saw-tooth function for the time operator in the case of quantum mechanical systems with *discrete* energy spectra: Namely, for the case of eq. (30).

The commutation relations of the Energy and Time operators, now both selfajoint, acquires in the case of discrete energies and of a periodic Time operator the form

$$\left[\left[\hat{E},\hat{t}\right]\right] = i\hbar \left\{ 1 - T \sum\_{n=0}^{\infty} \delta(t - \left[2n+1\right]T) \right\}\tag{31}$$

wherefrom the uncertainty relation follows in the new form

$$\left(\left(\Delta E\right)^{2}\left(\Delta t\right)^{2} = \hbar^{2}\left[1 - \frac{T|\psi(T/2+\gamma)|^{2}}{\int\_{-T/2}^{T/2} |\psi(t)|^{2}dt}\right],\tag{32}$$

where it has been introduced a parameter *γ*, with −*T*/2 < *γ* < *T*/2, in order to assure that the r.h.s. integral is single-valued(Olkhovsky & Recami, 2007; 2008).

When Δ*E* → 0 (that is, when |*gn*| → *δnn*�), the r.h.s. of eq. (32) tends to zero too, since |*ψ*(*t*)| <sup>2</sup> tends to a constant value. In such a case, the distribution of the time instants at which the wave-packet passes through point *x* becomes flat within each Poincaré cycle. When, by contrast, Δ*E* >> *D* and |*ψ*(*T* + *γ*)| <sup>2</sup> << ( � *<sup>T</sup>*/2 <sup>−</sup>*T*/2 <sup>|</sup>*ψ*(*t*)<sup>|</sup> <sup>2</sup>*dt*)/*T*, the periodicity condition may become inessential whenever Δ*t* << *t*. In other words, our uncertainty relation (32) transforms into the ordinary uncertainty relation for systems with continuous spectra.

In more general cases, for excited states of nuclei, atoms and molecules, the *energy-level intervals, for discrete and quasi-discrete (resonance) spectra, are not multiples of a maximum common divisor,* and hence the Poincaré cycle is not well-defined for such systems. Nevertheless, even for those systems one can introduce an approximate description (sometimes, with any desired degree of accuracy) in terms of Poincaré quasi-cycles and a quasi-periodical evolution; so that for sufficiently long time intervals the behavior of the wave-packets can be associated with a *a periodical motion (oscillation)*, sometimes — e.g., for very narrow resonances — with any desired 12 Will-be-set-by-IN-TECH

Fig. 2. The periodic saw-tooth function for the time operator in the case of quantum mechanical systems with *discrete* energy spectra: Namely, for the case of eq. (30).

the case of discrete energies and of a periodic Time operator the form

� 1 − *T*

<sup>2</sup> (Δ*t*)<sup>2</sup> = *h*¯ <sup>2</sup>

the r.h.s. integral is single-valued(Olkhovsky & Recami, 2007; 2008).

[*E*ˆ, ˆ*t*] = *ih*¯

wherefrom the uncertainty relation follows in the new form

(Δ*E*)

contrast, Δ*E* >> *D* and |*ψ*(*T* + *γ*)|


The commutation relations of the Energy and Time operators, now both selfajoint, acquires in

∞ ∑ *n*=0

⎡

where it has been introduced a parameter *γ*, with −*T*/2 < *γ* < *T*/2, in order to assure that

When Δ*E* → 0 (that is, when |*gn*| → *δnn*�), the r.h.s. of eq. (32) tends to zero too, since

may become inessential whenever Δ*t* << *t*. In other words, our uncertainty relation (32) transforms into the ordinary uncertainty relation for systems with continuous spectra. In more general cases, for excited states of nuclei, atoms and molecules, the *energy-level intervals, for discrete and quasi-discrete (resonance) spectra, are not multiples of a maximum common divisor,* and hence the Poincaré cycle is not well-defined for such systems. Nevertheless, even for those systems one can introduce an approximate description (sometimes, with any desired degree of accuracy) in terms of Poincaré quasi-cycles and a quasi-periodical evolution; so that for sufficiently long time intervals the behavior of the wave-packets can be associated with a *a periodical motion (oscillation)*, sometimes — e.g., for very narrow resonances — with any desired

<sup>2</sup> << (

<sup>2</sup> tends to a constant value. In such a case, the distribution of the time instants at which the wave-packet passes through point *x* becomes flat within each Poincaré cycle. When, by

> � *<sup>T</sup>*/2 <sup>−</sup>*T*/2 <sup>|</sup>*ψ*(*t*)<sup>|</sup>

*δ*(*t* − [2*n* + 1]*T*)

<sup>⎣</sup><sup>1</sup> <sup>−</sup> *<sup>T</sup>*|*ψ*(*T*/2 <sup>+</sup> *<sup>γ</sup>*)<sup>|</sup>

<sup>−</sup>*T*/2 <sup>|</sup>*ψ*(*t*)|2*dt*

� *<sup>T</sup>*/2

�

2

⎤

, (31)

⎦ , (32)

<sup>2</sup>*dt*)/*T*, the periodicity condition

accuracy. For them, when choosing an approximate Poincaré-cycle time, one can include in one cycle as many quasi-cycles as it is necessary for the demanded accuracy. Then, with the chosen accuracy, a *quasi-selfadjoint time operator* can be introduced.

#### **3. Multiple internal reflections approach in description of tunneling 3. Multiple internal reflections approach in description of tunneling**

#### **3.1 Tunneling in consideration of multiple internal reflections of waves between internal boundaries 3.1 Tunneling in consideration of multiple internal reflections of waves between internal boundaries**

An approach for description of one-dimensional motion of a non-relativistic particle above a barrier on the basis of multiple internal reflections of stationary waves relatively boundaries has been studied in number of papers and is known (see (Anderson, 1989; Fermor, 1966; McVoy et al., 1967) and references therein). Tunneling of the particle under the barrier was described successfully on the basis of multiple internal reflections of the wave packets relatively boundaries (approach was called as *method of multiple internal reflections* or *method MIR*, see Refs. (Maydanyuk et al., 2002a;b; Maydanyuk, 2003; Olkhovsky, 2000)). In such approach it succeeded in connecting: 1) continuous transition of solutions for packets after each reflection, total packets between the above-barrier motion and the under-barrier tunneling; 2) coincidence of transmitted and reflected amplitudes of stationary wave function in each spatial region obtained by approach MIR with the corresponding amplitudes obtained by standard method of quantum mechanics; 3) all non-stationary fluxes in each step, are non-zero that confirms propagation of packets under the barrier (i. e. their "tunneling"). In frameworks of such a method, non-stationary tunneling obtained own interpretation, allowing to study this process at interesting time moment or space point. In calculation of phase times this method turns out to be enough simple and convenient (Cardone et al., 2006). It has been adapted for scattering of the particle on nucleus and *α*-decay in the spherically symmetric approximation with the simplest radial barriers (Maydanyuk et al., 2002a; Maydanyuk, 2003; Olkhovsky, 2000) and for tunneling of photons (Cardone et al., 2006; Maydanyuk et al., 2002a). However, further realization of the MIR approach meets with three questions. 1) *Question on effectiveness.* The multiple reflections have been proved for the motion above one rectangular barrier and for tunneling under it (Anderson, 1989; Cardone et al., 2006; Maydanyuk et al., 2002a). However, after addition of the second step it becomes unclear how to separate the needed reflected waves from all their variety in calculation of all needed amplitudes. After obtaining exact solutions of the stationary amplitudes for two arbitrary rectangular barriers (Maydanyuk, 2003; Olkhovsky, 2000), it becomes unclear how to generalize such approach for barriers with arbitrary complicate shape. So, *we come to a serious unresolved problem of realization of the approach of multiple reflections in real quantum systems with complicated barriers*, and clear algorithms of calculation of amplitudes should be constructed.

2) *Question on correctness.* Whether is interference between packets formed relatively different boundaries appeared? Whether does this come to principally different results of the approach of multiple internal reflections and direct methods of quantum mechanics? Note that such interference cannot be appeared in tunneling through one rectangular barrier and, therefore, it could not visible in the previous papers.

3) *Question on uncertainty in radial problem.* Calculations of half-lives of different types of decays based on the semiclassical approach are prevailing today. For example, in Ref. (Buck et al., 1993) agreement between experimental data of *α*-decay half-lives and ones calculated by theory is demonstrated in a wide region of nuclei from 106Te up to nucleus with *Ad* = 266 and *Zd* = 109 (see Ref. (Denisov & Ikezoe, 2005) for some improved approaches). In review (Sobiczewski & Pomorski, 2007) methodology of calculation of half-lives for

where stationary WF is:

and *k* = <sup>1</sup>

accordingly.

II we obtain:

*ψ*1 *tr*(*x*, *t*) = + � ∞

0

*h*¯

<sup>√</sup>2*mE*, *<sup>ξ</sup>* <sup>=</sup> <sup>1</sup>

*h*¯

where *θ*-function satisfies to the requirement

to a requirement of the normalization � <sup>|</sup>*g*(*<sup>E</sup>* <sup>−</sup> *<sup>E</sup>*¯)<sup>|</sup>

*ϕ*(*x*) =

⎧ ⎨ ⎩

for above-barrier energies, having included the additional transformation

*eikx* + *ARe*−*ikx*, for *x* < 0; *αeξ<sup>x</sup>* + *βe*−*ξx*, for 0 < *x* < *a*; *ATeikx*, for *x* > *a*;

Time as Quantum Observable, Canonical Conjugated to Energy 31

accordingly. The weight amplitude *<sup>g</sup>*(*<sup>E</sup>* <sup>−</sup> *<sup>E</sup>*¯) can be written in a form of gaussian and satisfies

the particle. One can calculate coefficients *AT*, *AR*, *α* and *β* analytically, using a requirements of a continuity of WF *ϕ*(*x*) and its derivative on each boundary of the barrier. Substituting in eq. (33) instead of *ϕ*(*k*, *x*) the incident *ϕinc*(*k*, *x*), transmitted *ϕtr*(*k*, *x*) or reflected part of WF *ϕref*(*k*, *x*), defined by eq. (34), we receive the incident, transmitted or reflected WP,

We assume, that a time, for which the WP tunnels through the barrier, is enough small. So, the time necessary for a tunneling of proton through a barrier of decay in proton-decay of a nucleus, is about 10−<sup>21</sup> seconds. We consider, that one can neglect a spreading of the WP for this time. And a breadth of the WP appears essentially more narrow on a comparison with a barrier breadth. Considering only sub-barrier processes, we exclude a component of waves

*<sup>θ</sup>*(*η*) = � 0, for *<sup>η</sup>* <sup>&</sup>lt; 0;

The method of multiple internal reflections considers the propagation process of the WP describing a motion of the particle, sequentially on steps of its penetration in relation to each boundary of the barrier (Anderson, 1989; Fermor, 1966; McVoy et al., 1967). Using this method, we find expressions for the transmitted and reflected WP in relation to the barrier. At the first step we consider the WP in the region I, which is incident upon the first (initial) boundary of the barrier. Let us assume, that this package transforms into the WP, transmitted through this boundary and tunneling further in the region II, and into the WP, reflected from the boundary and propagating back in the region I. Thus we consider, that the WP, tunneling in the region II, is not reached the second (final) boundary of the barrier because of a terminating velocity of its propagation, and consequently at this step we consider only two regions I and II. Because of physical reasons to construct an expression for this packet, we consider, that its amplitude should decrease in a positive *x*-direction. We use only one item *β* exp(−*ξx*) in eq. (34), throwing the second increasing item *α* exp(*ξx*) (in an opposite case we break a requirement of a finiteness of the WF for an indefinitely wide barrier). In result, in the region

Thus the WF in the barrier region constructed by such way, is an analytic continuation of a relevant expression for the WF, corresponding to a similar problem with above-barrier

1, for *η* > 0.

�2*m*(*V*<sup>1</sup> <sup>−</sup> *<sup>E</sup>*), *<sup>E</sup>* and *<sup>m</sup>* are the total energy and mass of the particle,

*<sup>g</sup>*(*<sup>E</sup>* <sup>−</sup> *<sup>E</sup>*¯) <sup>→</sup> *<sup>g</sup>*(*<sup>E</sup>* <sup>−</sup> *<sup>E</sup>*¯)*θ*(*V*<sup>1</sup> <sup>−</sup> *<sup>E</sup>*), (35)

*<sup>g</sup>*(*<sup>E</sup>* <sup>−</sup> *<sup>E</sup>*¯)*θ*(*V*<sup>1</sup> <sup>−</sup> *<sup>E</sup>*)*β*0*e*−*ξx*−*iEt*/¯*hdE*, for 0 <sup>&</sup>lt; *<sup>x</sup>* <sup>&</sup>lt; *<sup>a</sup>*. (36)

<sup>2</sup>*dE* = 1, value *E*¯ is an average energy of

(34)

spontaneous-fission is presented (see eqs. (21)–(24) in p. 321). Let us consider proton-decay of nucleus where proton penetrates from the internal region outside with its tunneling through the barrier. At the same boundary condition, reflected and incident waves turn out to be defined with uncertainty. How to determine them? The semiclassical approach gives such answer: *according to theory, in construction of well known formula for probability we neglect completely by the second (increasing) item of the wave function inside tunneling region* (see Ref. (Landau & Lifshitz, 1989), eq. (50.2), p. 221). In result, equality *T*<sup>2</sup> + *R*<sup>2</sup> = 1 has no any sense (where *T* and *R* are coefficients of penetrability and reflection). Condition of continuity for the wave function and for total flux is broken at turning point. So, we do not find reflection *R*. We do not suppose on possible interference between incident and reflected waves which can be non zero. The penetrability is determined by the barrier shape inside tunneling region, while internal and external parts do not take influence on it. The penetrability does not dependent on depth of the internal well (while the simplest rectangular well and barrier give another exact result). But, the semiclassical approach is so prevailing that one can suppose that it has enough well approximation of the penetrability estimated. It turns out that if in fully quantum approach to determine the penetrability through the barrier (constructed on the basis of realistic potential of interaction between proton and daughter nucleus) then one can obtain answer "no". Fully quantum penetrability is a function of new additional independent parameters, it can achieve essential difference from semiclassical one (at the same boundary condition imposed on the wave function). This will be demonstrated below.

#### **3.2 Tunneling of packet through one-dimensional rectangular step**

Let us consider a problem of tunneling of a particle in a positive *x*-direction through an one-dimensional rectangular potential barrier (see Fig. 3). Let us label a region I for *x* < 0, a region II for 0 < *x* < *a* and a region III for *x* > *a*, accordingly. In standard approach, with

Fig. 3. Tunneling of the particle through one-dimensional rectangular barrier

energy less than the barrier height the tunneling evolution of the particle is described using a non-stationary propagation of WP

$$\Psi(\mathbf{x},t) = \int\_0^{+\infty} \mathbf{g}(E-\vec{E})\varphi(k,\mathbf{x})e^{-iEt/\hbar}d\mathbf{E},\tag{33}$$

where stationary WF is:

14 Will-be-set-by-IN-TECH

spontaneous-fission is presented (see eqs. (21)–(24) in p. 321). Let us consider proton-decay of nucleus where proton penetrates from the internal region outside with its tunneling through the barrier. At the same boundary condition, reflected and incident waves turn out to be defined with uncertainty. How to determine them? The semiclassical approach gives such answer: *according to theory, in construction of well known formula for probability we neglect completely by the second (increasing) item of the wave function inside tunneling region* (see Ref. (Landau & Lifshitz, 1989), eq. (50.2), p. 221). In result, equality *T*<sup>2</sup> + *R*<sup>2</sup> = 1 has no any sense (where *T* and *R* are coefficients of penetrability and reflection). Condition of continuity for the wave function and for total flux is broken at turning point. So, we do not find reflection *R*. We do not suppose on possible interference between incident and reflected waves which can be non zero. The penetrability is determined by the barrier shape inside tunneling region, while internal and external parts do not take influence on it. The penetrability does not dependent on depth of the internal well (while the simplest rectangular well and barrier give another exact result). But, the semiclassical approach is so prevailing that one can suppose that it has enough well approximation of the penetrability estimated. It turns out that if in fully quantum approach to determine the penetrability through the barrier (constructed on the basis of realistic potential of interaction between proton and daughter nucleus) then one can obtain answer "no". Fully quantum penetrability is a function of new additional independent parameters, it can achieve essential difference from semiclassical one (at the same boundary

condition imposed on the wave function). This will be demonstrated below.

Let us consider a problem of tunneling of a particle in a positive *x*-direction through an one-dimensional rectangular potential barrier (see Fig. 3). Let us label a region I for *x* < 0, a region II for 0 < *x* < *a* and a region III for *x* > *a*, accordingly. In standard approach, with

energy less than the barrier height the tunneling evolution of the particle is described using a

*<sup>g</sup>*(*<sup>E</sup>* <sup>−</sup> *<sup>E</sup>*¯)*ϕ*(*k*, *<sup>x</sup>*)*e*−*iEt*/¯*hdE*, (33)

**3.2 Tunneling of packet through one-dimensional rectangular step**

Fig. 3. Tunneling of the particle through one-dimensional rectangular barrier

+ ∞

0

*ψ*(*x*,*t*) =

non-stationary propagation of WP

$$\varphi(\mathbf{x}) = \begin{cases} e^{i\mathbf{k}\mathbf{x}} + A\_R e^{-i\mathbf{k}\mathbf{x}}, \text{ for } \mathbf{x} < 0; \\ a e^{\mathbf{j}\mathbf{x}} + \beta e^{-\mathbf{j}\mathbf{x}}, \quad \text{for } 0 < \mathbf{x} < a; \\ A\_T e^{i\mathbf{k}\mathbf{x}}, \qquad \text{for } \mathbf{x} > a; \end{cases} \tag{34}$$

and *k* = <sup>1</sup> *h*¯ <sup>√</sup>2*mE*, *<sup>ξ</sup>* <sup>=</sup> <sup>1</sup> *h*¯ �2*m*(*V*<sup>1</sup> <sup>−</sup> *<sup>E</sup>*), *<sup>E</sup>* and *<sup>m</sup>* are the total energy and mass of the particle, accordingly. The weight amplitude *<sup>g</sup>*(*<sup>E</sup>* <sup>−</sup> *<sup>E</sup>*¯) can be written in a form of gaussian and satisfies to a requirement of the normalization � <sup>|</sup>*g*(*<sup>E</sup>* <sup>−</sup> *<sup>E</sup>*¯)<sup>|</sup> <sup>2</sup>*dE* = 1, value *E*¯ is an average energy of the particle. One can calculate coefficients *AT*, *AR*, *α* and *β* analytically, using a requirements of a continuity of WF *ϕ*(*x*) and its derivative on each boundary of the barrier. Substituting in eq. (33) instead of *ϕ*(*k*, *x*) the incident *ϕinc*(*k*, *x*), transmitted *ϕtr*(*k*, *x*) or reflected part of WF *ϕref*(*k*, *x*), defined by eq. (34), we receive the incident, transmitted or reflected WP, accordingly.

We assume, that a time, for which the WP tunnels through the barrier, is enough small. So, the time necessary for a tunneling of proton through a barrier of decay in proton-decay of a nucleus, is about 10−<sup>21</sup> seconds. We consider, that one can neglect a spreading of the WP for this time. And a breadth of the WP appears essentially more narrow on a comparison with a barrier breadth. Considering only sub-barrier processes, we exclude a component of waves for above-barrier energies, having included the additional transformation

$$g(E-\bar{E}) \to g(E-\bar{E})\theta(V\_1 - E),\tag{35}$$

where *θ*-function satisfies to the requirement

$$\theta(\eta) = \begin{cases} 0, \text{ for } \eta < 0; \\ 1, \text{ for } \eta > 0. \end{cases}$$

The method of multiple internal reflections considers the propagation process of the WP describing a motion of the particle, sequentially on steps of its penetration in relation to each boundary of the barrier (Anderson, 1989; Fermor, 1966; McVoy et al., 1967). Using this method, we find expressions for the transmitted and reflected WP in relation to the barrier. At the first step we consider the WP in the region I, which is incident upon the first (initial) boundary of the barrier. Let us assume, that this package transforms into the WP, transmitted through this boundary and tunneling further in the region II, and into the WP, reflected from the boundary and propagating back in the region I. Thus we consider, that the WP, tunneling in the region II, is not reached the second (final) boundary of the barrier because of a terminating velocity of its propagation, and consequently at this step we consider only two regions I and II. Because of physical reasons to construct an expression for this packet, we consider, that its amplitude should decrease in a positive *x*-direction. We use only one item *β* exp(−*ξx*) in eq. (34), throwing the second increasing item *α* exp(*ξx*) (in an opposite case we break a requirement of a finiteness of the WF for an indefinitely wide barrier). In result, in the region II we obtain:

$$\psi\_{tr}^{1}(\mathbf{x},t) = \int\_{0}^{+\infty} g(E-\bar{E})\theta(V\_{1}-E)\mathfrak{f}^{0}e^{-\mathfrak{f}\mathbf{x}-i\mathcal{E}t/\hbar}d\mathbf{E},\text{for } 0<\mathbf{x}$$

Thus the WF in the barrier region constructed by such way, is an analytic continuation of a relevant expression for the WF, corresponding to a similar problem with above-barrier

boundaries, one can obtain the recurrence relations:

*<sup>R</sup>* <sup>=</sup> *<sup>k</sup>* <sup>−</sup> *<sup>i</sup><sup>ξ</sup> k* + *iξ*

, *<sup>α</sup><sup>n</sup>* <sup>=</sup> *<sup>β</sup><sup>n</sup> <sup>i</sup><sup>ξ</sup>* <sup>−</sup> *<sup>k</sup>*

*<sup>T</sup>* <sup>=</sup> *<sup>β</sup><sup>n</sup>* <sup>2</sup>*i<sup>ξ</sup>*

, *A<sup>n</sup>*

*ϕtr*(*k*, *x*) =

*ϕref*(*k*, *x*) =

1

2

1

*<sup>i</sup>* can be obtained.

<sup>1</sup> <sup>=</sup> *<sup>β</sup>*0, *<sup>T</sup>*<sup>+</sup>

Using the recurrence relations, one can find series of coefficients *αn*, *βn*, *A<sup>n</sup>*

*T*<sup>+</sup>

*R*<sup>+</sup> <sup>1</sup> <sup>=</sup> *<sup>A</sup>*<sup>0</sup>

these series can be calculated easier, using coefficients *T*±

*ϕ*1 *inc* exp(*ikx*)

*ϕ*2 *inc* exp(−*ξx*)

> *ϕ*1 *inc* exp(*ξx*)

negative) *x*-direction and incident upon the boundary with number *i*. Using *T*±

*<sup>R</sup>*, *<sup>R</sup>*<sup>+</sup>

(*i* = 2). In result, this WP transforms into the WP *ψ<sup>i</sup>*

on *x* parts of the stationary WF one can write:

*ϕ*1 *tr* exp(−*ξx*) <sup>=</sup> *<sup>T</sup>*<sup>+</sup>

*ϕ*2 *tr* exp(*ikx*) <sup>=</sup> *<sup>T</sup>*<sup>+</sup>

*ϕ*1 *tr* exp(−*ikx*) <sup>=</sup> *<sup>T</sup>*<sup>−</sup>

*<sup>i</sup>* and *R*<sup>±</sup>

with number *i*, and into the WP *ψ<sup>i</sup>*

the coefficients *T*±

*iξ* + *k e* −2*ξa*

*iξ* + *k e*

Considering the propagation of the WP by such way, we obtain expressions for the WF on each region which can be written through series of multiple WP. Using eq. (33) with account eq. (35), we determine resultant expressions for the incident, transmitted and reflected WP in relation to the barrier, where one can need to use following expressions for the stationary WF:

Time as Quantum Observable, Canonical Conjugated to Energy 33

*ϕinc*(*k*, *x*) = *eikx*, for *x* < 0,

Now we consider the WP formed in result of sequential *n* reflections from the boundaries of the barrier and incident upon one of these boundaries at point *x* = 0 (*i* = 1) or at point *x* = *a*

> , *<sup>ϕ</sup>*<sup>1</sup> *ref* exp(−*ikx*) <sup>=</sup> *<sup>R</sup>*<sup>+</sup>

, *<sup>ϕ</sup>*<sup>2</sup> *ref* exp(*ξx*) <sup>=</sup> *<sup>R</sup>*<sup>+</sup>

, *<sup>ϕ</sup>*<sup>1</sup> *ref* exp(−*ξx*) <sup>=</sup> *<sup>R</sup>*<sup>−</sup>

where the sign "+" (or "-") corresponds to the WP, tunneling (or propagating) in a positive (or

can precisely describe an arbitrary WP which has formed in result of *n*-multiple reflections, if to know a "path" of its propagation along the barrier. Using the recurrence relations eq. (40),

> <sup>2</sup> <sup>=</sup> *<sup>A</sup><sup>n</sup> T <sup>β</sup><sup>n</sup>* , *<sup>T</sup>*<sup>−</sup>

<sup>2</sup> <sup>=</sup> *<sup>α</sup><sup>n</sup>*

*<sup>β</sup><sup>n</sup>* , *<sup>R</sup>*<sup>−</sup>

+∞ ∑ *n*=0 *An*

+∞ ∑ *n*=0 *An* , *<sup>β</sup>n*+<sup>1</sup> <sup>=</sup> *<sup>α</sup><sup>n</sup> <sup>i</sup><sup>ξ</sup>* <sup>−</sup> *<sup>k</sup>*

*<sup>R</sup>* <sup>=</sup> *<sup>α</sup><sup>n</sup>* <sup>2</sup>*i<sup>ξ</sup>*

<sup>−</sup>*ξa*−*ika*, *An*+<sup>1</sup>

*<sup>T</sup>eikx*, for *<sup>x</sup>* <sup>&</sup>gt; *<sup>a</sup>*,

*<sup>R</sup>e*−*ikx*, for *<sup>x</sup>* <sup>&</sup>lt; 0.

*iξ* + *k* ,

(40)

(41)

(42)

*<sup>i</sup>* , one

(43)

*<sup>R</sup>*. However,

*<sup>i</sup>* and *R*<sup>±</sup>

*<sup>T</sup>* and *<sup>A</sup><sup>n</sup>*

*<sup>i</sup>* . Analyzing all possible

*iξ* + *k* .

*tr*(*x*, *t*), transmitted through boundary

,

,

,

*ref*(*x*, *t*), reflected from this boundary. For an independent

1

2

<sup>1</sup> <sup>=</sup> *<sup>A</sup>n*+<sup>1</sup> *R <sup>α</sup><sup>n</sup>* ,

<sup>1</sup> <sup>=</sup> *<sup>β</sup>n*+<sup>1</sup> *<sup>α</sup><sup>n</sup>* .

*<sup>i</sup>* and *R*<sup>±</sup>

1

*ϕ*1 *inc* exp(*ikx*)

*ϕ*2 *inc* exp(−*ξx*)

> *ϕ*1 *inc* exp(*ξx*)

*<sup>β</sup>*<sup>0</sup> <sup>=</sup> <sup>2</sup>*<sup>k</sup> k* + *iξ*

*A*0

energies, where as a stationary expression we select the wave exp(*ik*2*x*), propagated to the right.

Let us consider the first step further. One can write expressions for the incident and the reflected WP in relation to the first boundary as follows

$$\begin{aligned} \psi\_{\rm inc}(\mathbf{x},t) &= \int\_0^{+\infty} \mathfrak{g}(E-\bar{E})\theta(V\_1-E)e^{i\mathbf{k}\cdot\mathbf{x}-i\mathbf{E}t/\hbar}d\mathbf{E}, \qquad \text{for } \mathbf{x}<0, \\\psi\_{\rm ref}^1(\mathbf{x},t) &= \int\_0^{+\infty} \mathfrak{g}(E-\bar{E})\theta(V\_1-E)A\_\mathcal{R}^0e^{-i\mathbf{k}\cdot\mathbf{x}-i\mathbf{E}t/\hbar}d\mathbf{E}, \text{ for } \mathbf{x}<0. \end{aligned} \tag{37}$$

A sum of these expressions represents the complete WF in the region I, which is dependent on a time. Let us require, that this WF and its derivative continuously transform into the WF (36) and its derivative at point *<sup>x</sup>* <sup>=</sup> 0 (we assume, that the weight amplitude *<sup>g</sup>*(*<sup>E</sup>* <sup>−</sup> *<sup>E</sup>*¯) differs weakly at transmitting and reflecting of the WP in relation to the barrier boundaries). In result, we obtain two equations, in which one can pass from the time-dependent WP to the corresponding stationary WF and obtain the unknown coefficients *β*<sup>0</sup> and *A*<sup>0</sup> *R*.

At the second step we consider the WP, tunneling in the region II and incident upon the second boundary of the barrier at point *x* = *a*. It transforms into the WP, transmitted through this boundary and propagated in the region III, and into the WP, reflected from the boundary and tunneled back in the region II. For a determination of these packets one can use eq. (33) with account eq. (35), where as the stationary WF we use:

$$\begin{array}{ll} \mathfrak{gl}\_{\text{juc}}^{2}(k,\mathbf{x}) = \mathfrak{gl}\_{\text{tr}}^{1}(k,\mathbf{x}) = \mathfrak{gl}^{0}e^{-\mathfrak{f}\mathbf{x}}, \text{ for } 0 < \mathbf{x} < a, \\\mathfrak{gl}\_{\text{tr}}^{2}(k,\mathbf{x}) = A\_{\text{T}}^{0}e^{i\mathbf{x}}, & \text{ for } \mathbf{x} > a, \\\mathfrak{gl}\_{\text{ref}}^{2}(k,\mathbf{x}) = a^{0}e^{\mathfrak{f}\mathbf{x}}, & \text{ for } 0 < \mathbf{x} < a. \end{array} \tag{38}$$

Here, for forming an expression for the WP reflected from the boundary, we select an increasing part of the stationary solution *α*<sup>0</sup> exp(*ξx*) only. Imposing a condition of continuity on the time-dependent WF and its derivative at point *x* = *a*, we obtain 2 new equations, from which we find the unknowns coefficients *A*<sup>0</sup> *<sup>T</sup>* and *<sup>α</sup>*0.

At the third step the WP, tunneling in the region II, is incident upon the first boundary of the barrier. Then it transforms into the WP, transmitted through this boundary and propagated further in the region I, and into the WP, reflected from boundary and tunneled back in the region II. For a determination of these packets one can use eq. (33) with account eq. (35), where as the stationary WF we use:

$$\begin{array}{ll}\mathfrak{q}\_{\text{inc}}^{\mathfrak{T}}(k,\mathbf{x}) = \mathfrak{q}\_{ref}^{2}(k,\mathbf{x}), \text{ for } 0 < \mathbf{x} < a, \\\mathfrak{q}\_{\text{ff}}^{\mathfrak{T}}(k,\mathbf{x}) = A\_{R}^{1}e^{-i\mathbf{k}\mathbf{x}}, \text{ for } \mathbf{x} < 0, \\\mathfrak{q}\_{ref}^{\mathfrak{T}}(k,\mathbf{x}) = \mathfrak{beta}^{1}e^{-\mathfrak{T}\mathbf{x}}, \quad \text{ for } 0 < \mathbf{x} < a. \end{array} \tag{39}$$

Using a conditions of continuity for the time-dependent WF and its derivative at point *x* = 0, we obtain the unknowns coefficients *A*<sup>1</sup> *<sup>R</sup>* and *<sup>β</sup>*1.

Analyzing further possible processes of the transmission (and the reflection) of the WP through the boundaries of the barrier, we come to a deduction, that any of following steps can be reduced to one of 2 considered above. For the unknown coefficients *αn*, *βn*,*A<sup>n</sup> <sup>T</sup>* and *An <sup>R</sup>*, used in expressions for the WP, forming in result of some internal reflections from the boundaries, one can obtain the recurrence relations:

16 Will-be-set-by-IN-TECH

energies, where as a stationary expression we select the wave exp(*ik*2*x*), propagated to the

Let us consider the first step further. One can write expressions for the incident and the

A sum of these expressions represents the complete WF in the region I, which is dependent on a time. Let us require, that this WF and its derivative continuously transform into the WF (36) and its derivative at point *<sup>x</sup>* <sup>=</sup> 0 (we assume, that the weight amplitude *<sup>g</sup>*(*<sup>E</sup>* <sup>−</sup> *<sup>E</sup>*¯) differs weakly at transmitting and reflecting of the WP in relation to the barrier boundaries). In result, we obtain two equations, in which one can pass from the time-dependent WP to the

At the second step we consider the WP, tunneling in the region II and incident upon the second boundary of the barrier at point *x* = *a*. It transforms into the WP, transmitted through this boundary and propagated in the region III, and into the WP, reflected from the boundary and tunneled back in the region II. For a determination of these packets one can use eq. (33) with

*<sup>g</sup>*(*<sup>E</sup>* <sup>−</sup> *<sup>E</sup>*¯)*θ*(*V*<sup>1</sup> <sup>−</sup> *<sup>E</sup>*)*A*<sup>0</sup>

corresponding stationary WF and obtain the unknown coefficients *β*<sup>0</sup> and *A*<sup>0</sup>

*<sup>g</sup>*(*<sup>E</sup>* <sup>−</sup> *<sup>E</sup>*¯)*θ*(*V*<sup>1</sup> <sup>−</sup> *<sup>E</sup>*)*eikx*−*iEt*/¯*hdE*, for *<sup>x</sup>* <sup>&</sup>lt; 0,

*tr*(*k*, *<sup>x</sup>*) = *<sup>β</sup>*0*e*−*ξx*, for 0 < *<sup>x</sup>* < *<sup>a</sup>*,

*ref*(*k*, *x*), for 0 < *x* < *a*,

*<sup>R</sup>e*−*ikx*, for *<sup>x</sup>* <sup>&</sup>lt; 0,

*ref*(*k*, *<sup>x</sup>*) = *<sup>β</sup>*1*e*−*ξx*, for 0 <sup>&</sup>lt; *<sup>x</sup>* <sup>&</sup>lt; *<sup>a</sup>*.

Using a conditions of continuity for the time-dependent WF and its derivative at point *x* = 0,

*<sup>R</sup>*, used in expressions for the WP, forming in result of some internal reflections from the

*<sup>R</sup>* and *<sup>β</sup>*1. Analyzing further possible processes of the transmission (and the reflection) of the WP through the boundaries of the barrier, we come to a deduction, that any of following steps can be reduced to one of 2 considered above. For the unknown coefficients *αn*, *βn*,*A<sup>n</sup>*

*<sup>T</sup>eikx*, for *<sup>x</sup>* <sup>&</sup>gt; *<sup>a</sup>*,

*ref*(*k*, *<sup>x</sup>*) = *<sup>α</sup>*0*eξx*, for <sup>0</sup> <sup>&</sup>lt; *<sup>x</sup>* <sup>&</sup>lt; *<sup>a</sup>*.

Here, for forming an expression for the WP reflected from the boundary, we select an increasing part of the stationary solution *α*<sup>0</sup> exp(*ξx*) only. Imposing a condition of continuity on the time-dependent WF and its derivative at point *x* = *a*, we obtain 2 new equations, from

*<sup>T</sup>* and *<sup>α</sup>*0. At the third step the WP, tunneling in the region II, is incident upon the first boundary of the barrier. Then it transforms into the WP, transmitted through this boundary and propagated further in the region I, and into the WP, reflected from boundary and tunneled back in the region II. For a determination of these packets one can use eq. (33) with account eq. (35),

*<sup>R</sup>e*−*ikx*−*iEt*/¯*hdE*, for *<sup>x</sup>* <sup>&</sup>lt; 0.

*R*.

(37)

(38)

(39)

*<sup>T</sup>* and

reflected WP in relation to the first boundary as follows

+ ∞ 0

+ ∞ 0

account eq. (35), where as the stationary WF we use:

*inc*(*k*, *<sup>x</sup>*) = *<sup>ϕ</sup>*<sup>1</sup>

*tr*(*k*, *<sup>x</sup>*) = *<sup>A</sup>*<sup>0</sup>

*ϕ*3

*ϕ*3

*ϕ*3

*inc*(*k*, *<sup>x</sup>*) = *<sup>ϕ</sup>*<sup>2</sup>

*tr*(*k*, *<sup>x</sup>*) = *<sup>A</sup>*<sup>1</sup>

*ϕ*2

*ϕ*2

*ϕ*2

which we find the unknowns coefficients *A*<sup>0</sup>

where as the stationary WF we use:

we obtain the unknowns coefficients *A*<sup>1</sup>

*An*

*ψinc*(*x*, *t*) =

*ref*(*x*, *t*) =

*ψ*1

right.

$$\begin{split} \beta^{0} &= \frac{2k}{k + i\xi'}, \quad a^{n} = \beta^{n} \frac{i\overline{\xi} - k}{i\overline{\xi} + k} e^{-2\overline{\xi}a}, \quad \beta^{n+1} = a^{n} \frac{i\overline{\xi} - k}{i\overline{\xi} + k'} \\\ A\_{R}^{0} &= \frac{k - i\overline{\xi}}{k + i\overline{\xi}}, \ A\_{T}^{n} = \beta^{n} \frac{2i\overline{\xi}}{i\overline{\xi} + k} e^{-\overline{\xi}a - ika}, \ A\_{R}^{n+1} = a^{n} \frac{2i\overline{\xi}}{i\overline{\xi} + k}. \end{split} \tag{40}$$

Considering the propagation of the WP by such way, we obtain expressions for the WF on each region which can be written through series of multiple WP. Using eq. (33) with account eq. (35), we determine resultant expressions for the incident, transmitted and reflected WP in relation to the barrier, where one can need to use following expressions for the stationary WF:

$$\begin{aligned} \varrho\_{\rm inc}(k, \mathbf{x}) &= e^{ik\mathbf{x}}, \quad &\text{ for } \mathbf{x} < \mathbf{0}, \\ \varrho\_{\rm tr}(k, \mathbf{x}) &= \sum\_{n=0}^{+\infty} A\_T^n e^{ik\mathbf{x}}, \quad \text{ for } \mathbf{x} > a, \\ \varrho\_{\rm ref}(k, \mathbf{x}) &= \sum\_{n=0}^{+\infty} A\_R^n e^{-ik\mathbf{x}}, \text{ for } \mathbf{x} < \mathbf{0}. \end{aligned} \tag{41}$$

Now we consider the WP formed in result of sequential *n* reflections from the boundaries of the barrier and incident upon one of these boundaries at point *x* = 0 (*i* = 1) or at point *x* = *a* (*i* = 2). In result, this WP transforms into the WP *ψ<sup>i</sup> tr*(*x*, *t*), transmitted through boundary with number *i*, and into the WP *ψ<sup>i</sup> ref*(*x*, *t*), reflected from this boundary. For an independent on *x* parts of the stationary WF one can write:

$$\begin{split} \frac{\varrho\_{tr}^{1}}{\exp(-\mathfrak{f}\boldsymbol{x})} &= T\_{1}^{+} \frac{\varrho\_{inc}^{1}}{\exp(ik\boldsymbol{x})}, \frac{\varrho\_{ref}^{1}}{\exp(-ik\boldsymbol{x})} = R\_{1}^{+} \frac{\varrho\_{inc}^{1}}{\exp(ik\boldsymbol{x})},\\ \frac{\varrho\_{tr}^{2}}{\exp(ik\boldsymbol{x})} &= T\_{2}^{+} \frac{\varrho\_{inc}^{2}}{\exp(-\mathfrak{f}\boldsymbol{x})}, \frac{\varrho\_{ref}^{2}}{\exp(\mathfrak{f}\boldsymbol{x})} = R\_{2}^{+} \frac{\varrho\_{inc}^{2}}{\exp(-\mathfrak{f}\boldsymbol{x})},\\ \frac{\varrho\_{tr}^{1}}{\exp(-ik\boldsymbol{x})} &= T\_{1}^{-} \frac{\varrho\_{inc}^{1}}{\exp(\mathfrak{f}\boldsymbol{x})}, \frac{\varrho\_{ref}^{1}}{\exp(-\mathfrak{f}\boldsymbol{x})} = R\_{1}^{-} \frac{\varrho\_{inc}^{1}}{\exp(\mathfrak{f}\boldsymbol{x})} \end{split} \tag{42}$$

where the sign "+" (or "-") corresponds to the WP, tunneling (or propagating) in a positive (or negative) *x*-direction and incident upon the boundary with number *i*. Using *T*± *<sup>i</sup>* and *R*<sup>±</sup> *<sup>i</sup>* , one can precisely describe an arbitrary WP which has formed in result of *n*-multiple reflections, if to know a "path" of its propagation along the barrier. Using the recurrence relations eq. (40), the coefficients *T*± *<sup>i</sup>* and *R*<sup>±</sup> *<sup>i</sup>* can be obtained.

$$\begin{aligned} T\_1^+ &= \beta^0, \ T\_2^+ = \frac{A\_T^n}{\beta^n}, T\_1^- = \frac{A\_R^{n+1}}{a^n}, \\ R\_1^+ &= A\_{R'}^0, R\_2^+ = \frac{a^n}{\beta^n}, \ R\_1^- = \frac{\beta^{n+1}}{a^n}. \end{aligned} \tag{43}$$

Using the recurrence relations, one can find series of coefficients *αn*, *βn*, *A<sup>n</sup> <sup>T</sup>* and *<sup>A</sup><sup>n</sup> <sup>R</sup>*. However, these series can be calculated easier, using coefficients *T*± *<sup>i</sup>* and *R*<sup>±</sup> *<sup>i</sup>* . Analyzing all possible

in solutions for above barrier energies while the solution for tunneling could be obtained after by change *i ξ<sup>i</sup>* → *ki*. A general solution of the wave function (up to its normalization) has the

Time as Quantum Observable, Canonical Conjugated to Energy 35

*r*

where *α<sup>j</sup>* and *β<sup>j</sup>* are unknown amplitudes, *AT* and *AR* are unknown amplitudes of

wave numbers. We shall be looking for solution for such problem in approach of multiple internal reflections (we restrict ourselves by a case of orbital moment *l* = 0 while its non-zero generalization changes the barrier shape which was used as arbitrary before in development

According to the method of multiple internal reflections, scattering of the particle on the barrier is considered on the basis of wave packet consequently by steps of its propagation relatively to each boundary of the barrier (the most clearly idea of such approach can be understood in the problem of tunneling through the simplest rectangular barrier, see (Cardone et al., 2006; Maydanyuk et al., 2002a; Maydanyuk, 2003) where one can find proof of this fully quantum exactly solvable method, one can analyze its properties). Each step in such consideration of propagation of the packet will be similar to one from the first 2*N* − 1 steps, independent between themselves. From analysis of these steps recurrent relations are

*n*, summation of these amplitudes are calculated. We shall be looking for the unknown amplitudes, requiring wave function and its derivative to be continuous at each boundary.

amplitudes *e*±*ikx*. Here, bottom index denotes number of the boundary, upper (top) signs "+" and "−" denote directions of the wave to the right or to the left, correspondingly. At the

<sup>2</sup> ... *R*<sup>±</sup>

, *T*−

, *R*−

<sup>1</sup> , *R*<sup>±</sup>

*i*(*kj*−*kj*<sup>+</sup>1)*rj*

<sup>3</sup> . . . and *R*<sup>±</sup>

*<sup>N</sup>*−1:

*<sup>j</sup>* <sup>=</sup> <sup>2</sup>*kj*+<sup>1</sup> *kj* + *kj*+<sup>1</sup>

*<sup>j</sup>* <sup>=</sup> *kj*<sup>+</sup><sup>1</sup> <sup>−</sup> *kj kj* + *kj*<sup>+</sup><sup>1</sup>

<sup>=</sup> *<sup>T</sup>*˜ <sup>+</sup>

<sup>1</sup> <sup>−</sup> *<sup>R</sup>*<sup>+</sup>

*<sup>j</sup> <sup>T</sup>*<sup>+</sup> *j*+1

*j*+1*R*˜ <sup>−</sup> *j* ,

<sup>1</sup> , *T*<sup>±</sup> <sup>2</sup> , *T*<sup>±</sup>

*<sup>N</sup>*−<sup>1</sup> and *<sup>R</sup>*<sup>±</sup>

*e*

*e* 2*ikjrj* , *S*(*n*)

<sup>1</sup> , *R*<sup>±</sup> <sup>2</sup> , *R*<sup>±</sup>

*e*

*e* −2*ikj*<sup>+</sup>1*rj* .

*i*(*kj*−*kj*<sup>+</sup>1)*rj*

,

*T*<sup>+</sup> *<sup>j</sup>*−1*R*˜ <sup>+</sup> *<sup>j</sup> T*<sup>−</sup> *j*−1

*T*− *j*+1*R*˜ <sup>−</sup> *<sup>j</sup> <sup>T</sup>*<sup>+</sup> *j*+1

<sup>1</sup> <sup>−</sup> *<sup>R</sup>*˜ <sup>+</sup> *<sup>j</sup> R*<sup>−</sup> *j*−1 ,

<sup>1</sup> <sup>−</sup> *<sup>R</sup>*<sup>+</sup>

*j*+1*R*˜ <sup>−</sup> *j* ,

*<sup>e</sup>ik*1*<sup>r</sup>* <sup>+</sup> *AR <sup>e</sup>*−*ik*1*r*, at *<sup>R</sup>*min <sup>&</sup>lt; *<sup>r</sup>* <sup>≤</sup> *<sup>r</sup>*<sup>1</sup> (region 1), *<sup>α</sup>*<sup>2</sup> *<sup>e</sup>ik*2*<sup>r</sup>* <sup>+</sup> *<sup>β</sup>*<sup>2</sup> *<sup>e</sup>*−*ik*2*r*, at *<sup>r</sup>*<sup>1</sup> <sup>≤</sup> *<sup>r</sup>* <sup>≤</sup> *<sup>r</sup>*<sup>2</sup> (region 2), ... ... ...

*<sup>α</sup>n*−<sup>1</sup> *<sup>e</sup>ikN*−1*<sup>r</sup>* <sup>+</sup> *<sup>β</sup>N*−<sup>1</sup> *<sup>e</sup>*−*ikN*−1*r*, at *rN*−<sup>2</sup> <sup>≤</sup> *<sup>r</sup>* <sup>≤</sup> *rN*−<sup>1</sup> (region *<sup>N</sup>* <sup>−</sup> 1), *AT <sup>e</sup>ikNr*, at *rN*−<sup>1</sup> <sup>≤</sup> *<sup>r</sup>* <sup>≤</sup> *<sup>R</sup>*max (region *<sup>N</sup>*),

*Ylm*(*θ*, *ϕ*), (49)

�2*m*(*<sup>E</sup>* <sup>−</sup> *Vi*) are complex

, *α*(*n*) and *β*(*n*) for arbitrary step

<sup>3</sup> . . . as additional factors to

*h*¯

(50)

(51)

(52)

*<sup>ψ</sup>*(*r*, *<sup>θ</sup>*, *<sup>ϕ</sup>*) = *<sup>χ</sup>*(*r*)

transmission and reflection, *Ylm*(*θ*, *ϕ*) is spherical function, *ki* = <sup>1</sup>

of formalism MIR and, so, is absolutely non principal).

found for calculation of unknown amplitudes *A*(*n*)

<sup>2</sup> ... *T*<sup>±</sup>

*<sup>j</sup>* <sup>=</sup> *kj* <sup>−</sup> *kj*<sup>+</sup><sup>1</sup> *kj* + *kj*<sup>+</sup><sup>1</sup>

> *<sup>j</sup>*−1*R*˜ <sup>+</sup> *<sup>j</sup> T*<sup>−</sup> *j*−1 � 1 + +∞ ∑ *m*=1 (*R*˜ <sup>+</sup> *<sup>j</sup> R*<sup>−</sup> *<sup>j</sup>*−1)*<sup>m</sup>* � = *R*<sup>+</sup> *<sup>j</sup>*−<sup>1</sup> <sup>+</sup>

> *j*+1*R*˜ <sup>−</sup> *<sup>j</sup> <sup>T</sup>*<sup>+</sup> *j*+1 � 1 + +∞ ∑ *m*=1 (*R*<sup>+</sup> *j*+1*R*˜ <sup>−</sup> *<sup>j</sup>* )*<sup>m</sup>* � = *R*− *<sup>j</sup>*+<sup>1</sup> +

*<sup>j</sup>* <sup>=</sup> <sup>2</sup>*kj kj* + *kj*+<sup>1</sup>

We shall consider the coefficients *T*±

<sup>1</sup> , *T*<sup>±</sup>

*T*<sup>+</sup>

*R*<sup>+</sup>

*<sup>j</sup>*−<sup>1</sup> <sup>+</sup> *<sup>T</sup>*<sup>+</sup>

*<sup>j</sup>*+<sup>1</sup> + *T*<sup>−</sup>

*<sup>j</sup> <sup>T</sup>*<sup>+</sup> *j*+1 � 1 + +∞ ∑ *m*=1 (*R*<sup>+</sup> *j*+1*R*˜ <sup>−</sup> *<sup>j</sup>* )*<sup>m</sup>* �

first, we calculate *T*±

Using recurrent relations:

*R*˜ <sup>+</sup>

*R*˜ <sup>−</sup>

*T*˜<sup>+</sup> *<sup>j</sup>*+<sup>1</sup> <sup>=</sup> *<sup>T</sup>*˜<sup>+</sup>

*<sup>j</sup>*−<sup>1</sup> <sup>=</sup> *<sup>R</sup>*<sup>+</sup>

*<sup>j</sup>*+<sup>1</sup> = *R*<sup>−</sup>

following form:

*χ*(*r*) =

⎧ ⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

"paths" of the WP propagations along the barrier, we receive:

$$\begin{aligned} \sum\_{n=0}^{+\infty} A\_T^n &= T\_2^+ T\_1^- \left( 1 + \sum\_{n=1}^{+\infty} (R\_2^+ R\_1^-)^n \right) = \frac{i4 k \xi e^{-\xi a - ik a}}{F\_{sub}}, \\ \sum\_{n=0}^{+\infty} A\_R^n &= R\_1^+ + T\_1^+ R\_2^+ T\_1^- \left( 1 + \sum\_{n=1}^{+\infty} (R\_2^+ R\_1^-)^n \right) = \frac{k\_0^2 D\_-}{F\_{sub}}, \\ \sum\_{n=0}^{+\infty} a^n &= a^0 \left( 1 + \sum\_{n=1}^{+\infty} (R\_2^+ R\_1^-)^n \right) = \frac{2 k (i\xi - k) e^{-2\xi a}}{F\_{sub}}, \\ \sum\_{n=0}^{+\infty} \beta^n &= \beta^0 \left( 1 + \sum\_{i=1}^{+\infty} (R\_2^+ R\_1^-)^n \right) = \frac{2 k (i\xi + k)}{F\_{sub}}, \end{aligned} \tag{44}$$

where

$$\begin{array}{l} F\_{sub} = (k^2 - \tilde{\xi}^2)D\_- + 2ik\tilde{\xi}D\_+, \\ D\_\pm = 1 \pm e^{-2\tilde{\xi}a}, \\ k\_0^2 = k^2 + \tilde{\xi}^2 = \frac{2mV\_1}{\hbar^2}. \end{array} \tag{45}$$

All series ∑ *αn*, ∑ *βn*, ∑ *A<sup>n</sup> <sup>T</sup>* and <sup>∑</sup> *<sup>A</sup><sup>n</sup> <sup>R</sup>*, obtained using the method of multiple internal reflections, coincide with the corresponding coefficients *α*, *β*, *AT* and *AR* of the eq. (34), calculated by a stationary methods. Using the following substitution

$$i\mathfrak{F} \to k\_{2\prime} \tag{46}$$

where *k*<sup>2</sup> = <sup>1</sup> *h*¯ �2*m*(*<sup>E</sup>* <sup>−</sup> *<sup>V</sup>*1) is a wave number for a case of above-barrier energies, expression for the coefficients *αn*, *βn*, *A<sup>n</sup> <sup>T</sup>* and *<sup>A</sup><sup>n</sup> <sup>R</sup>* for each step, expressions for the WF for each step, the total eqs. (44) and (45) transform into the corresponding expressions for a problem of the particle propagation above this barrier. At the transformation of the WP and the time-dependent WF one can need to change a sign of argument at *θ*-function. Besides the following property is fulfilled:

$$\left|\sum\_{n=0}^{+\infty} A\_T^n \right|^2 + \left|\sum\_{n=0}^{+\infty} A\_R^n \right|^2 = 1. \tag{47}$$

#### **3.3 Exact solutions for wave function for tunneling through radial barrier composed from arbitrary number of potential steps 3.3 Exact solutions for wave function for tunneling through radial barrier composed from arbitrary number of potential steps**

Now we shall come to radial problem (Maydanyuk & Belchikov, 2011). Let us assume that starting from some time moment before decay the nucleus could be considered as system composite from daughter nucleus and fragment emitted. Its decay is described by a particle with reduced mass *m* which moves in radial direction inside a radial potential with a barrier. We shall be interesting in the radial barrier of arbitrary shape, which has successfully been approximated by finite number *N* of rectangular steps:

$$V(r) = \begin{cases} V\_{1\prime} \text{ at } R\_{\text{min}} < r \le r\_1 & \text{(region 1)},\\ V\_{2\prime} \text{ at } r\_1 \le r \le r\_2 & \text{(region 2)},\\ \dots \dots & \dots \\ V\_{N\prime} \text{ at } r\_{N-1} \le r \le R\_{\text{max}} \text{ (region N)}, \end{cases} \tag{48}$$

where *Vi* are constants (*i* = 1... *N*). We define the first region 1 starting from point *R*min, assuming that the fragment is formed here and then it moves outside. We shall be interesting

18 Will-be-set-by-IN-TECH

*Fsub* = (*k*<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*2)*D*<sup>−</sup> <sup>+</sup> <sup>2</sup>*ikξD*+,

reflections, coincide with the corresponding coefficients *α*, *β*, *AT* and *AR* of the eq. (34),

the total eqs. (44) and (45) transform into the corresponding expressions for a problem of the particle propagation above this barrier. At the transformation of the WP and the time-dependent WF one can need to change a sign of argument at *θ*-function. Besides the

> +∞ ∑ *n*=0 *An R* � � � �

**3.3 Exact solutions for wave function for tunneling through radial barrier composed from**

**3.3 Exact solutions for wave function for tunneling through radial barrier composed from** 

Now we shall come to radial problem (Maydanyuk & Belchikov, 2011). Let us assume that starting from some time moment before decay the nucleus could be considered as system composite from daughter nucleus and fragment emitted. Its decay is described by a particle with reduced mass *m* which moves in radial direction inside a radial potential with a barrier. We shall be interesting in the radial barrier of arbitrary shape, which has successfully been

> *V*1, at *R*min < *r* ≤ *r*<sup>1</sup> (region 1), *V*2, at *r*<sup>1</sup> ≤ *r* ≤ *r*<sup>2</sup> (region 2), ... ... ...

*VN*, at *rN*−<sup>1</sup> ≤ *r* ≤ *R*max (region N),

where *Vi* are constants (*i* = 1... *N*). We define the first region 1 starting from point *R*min, assuming that the fragment is formed here and then it moves outside. We shall be interesting

*<sup>h</sup>*¯ <sup>2</sup> .

�2*m*(*<sup>E</sup>* <sup>−</sup> *<sup>V</sup>*1) is a wave number for a case of above-barrier energies, expression

2

*<sup>D</sup>*<sup>±</sup> <sup>=</sup> <sup>1</sup> <sup>±</sup> *<sup>e</sup>*−2*ξa*,

<sup>0</sup> <sup>=</sup> *<sup>k</sup>*<sup>2</sup> <sup>+</sup> *<sup>ξ</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup>*mV*<sup>1</sup>

<sup>=</sup> *<sup>i</sup>*4*kξe*−*ξa*−*ika Fsub*

,

*<sup>R</sup>*, obtained using the method of multiple internal

*iξ* → *k*2, (46)

= 1. (47)

*<sup>R</sup>* for each step, expressions for the WF for each step,

<sup>=</sup> <sup>2</sup>*k*(*i<sup>ξ</sup>* <sup>−</sup> *<sup>k</sup>*)*e*−2*ξ<sup>a</sup> Fsub*

<sup>=</sup> <sup>2</sup>*k*(*i<sup>ξ</sup>* <sup>+</sup> *<sup>k</sup>*) *Fsub*

,

,

(44)

(45)

(48)

"paths" of the WP propagations along the barrier, we receive:

<sup>1</sup> <sup>+</sup> *<sup>T</sup>*<sup>+</sup> <sup>1</sup> *<sup>R</sup>*<sup>+</sup> <sup>2</sup> *T*<sup>−</sup> 1 � 1 + +∞ ∑ *n*=1 (*R*<sup>+</sup> <sup>2</sup> *R*<sup>−</sup> <sup>1</sup> )*<sup>n</sup>* � <sup>=</sup> *<sup>k</sup>*<sup>2</sup> <sup>0</sup>*D*<sup>−</sup> *Fsub* ,

� 1 + +∞ ∑ *n*=1 (*R*<sup>+</sup> <sup>2</sup> *R*<sup>−</sup> <sup>1</sup> )*<sup>n</sup>* �

� 1 + +∞ ∑ *i*=1 (*R*<sup>+</sup> <sup>2</sup> *R*<sup>−</sup> <sup>1</sup> )*<sup>n</sup>* �

*k*2

*<sup>T</sup>* and <sup>∑</sup> *<sup>A</sup><sup>n</sup>*

calculated by a stationary methods. Using the following substitution

*<sup>T</sup>* and *<sup>A</sup><sup>n</sup>*

� � � �

+∞ ∑ *n*=0 *An T* � � � �

2 + � � � �

+∞ ∑ *n*=0 *An <sup>T</sup>* <sup>=</sup> *<sup>T</sup>*<sup>+</sup> <sup>2</sup> *T*<sup>−</sup> 1 � 1 + +∞ ∑ *n*=1 (*R*<sup>+</sup> <sup>2</sup> *R*<sup>−</sup> <sup>1</sup> )*<sup>n</sup>* �

+∞ ∑ *n*=0 *An <sup>R</sup>* <sup>=</sup> *<sup>R</sup>*<sup>+</sup>

+∞ ∑ *n*=0

+∞ ∑ *n*=0

All series ∑ *αn*, ∑ *βn*, ∑ *A<sup>n</sup>*

*h*¯

for the coefficients *αn*, *βn*, *A<sup>n</sup>*

following property is fulfilled:

**arbitrary number of potential steps**

**arbitrary number of potential steps** 

approximated by finite number *N* of rectangular steps:

⎧ ⎪⎪⎨

⎪⎪⎩

*V*(*r*) =

where

where *k*<sup>2</sup> = <sup>1</sup>

*α<sup>n</sup>* = *α*<sup>0</sup>

*β<sup>n</sup>* = *β*<sup>0</sup>

in solutions for above barrier energies while the solution for tunneling could be obtained after by change *i ξ<sup>i</sup>* → *ki*. A general solution of the wave function (up to its normalization) has the following form:

$$
\psi(r,\theta,\varphi) = \frac{\chi(r)}{r} Y\_{lm}(\theta,\varphi),
\tag{49}
$$

$$\chi(r) = \begin{cases} e^{ik\_1r} + A\_R e^{-ik\_1r} & \text{at } R\_{\text{min}} < r \le r\_1 \quad \text{(region 1)},\\ \alpha\_2 e^{ik\_2r} + \beta\_2 e^{-ik\_2r} & \text{at } r\_1 \le r \le r\_2 \quad \text{(region 2)},\\ \dots & \dots & \dots \\ \alpha\_{n-1} e^{ik\_{N-1}r} + \beta\_{N-1} e^{-ik\_{N-1}r} \text{ at } r\_{N-2} \le r \le r\_{N-1} \text{ (region } N-1),\\ A\_T e^{ik\_Nr} & \text{at } r\_{N-1} \le r \le R\_{\text{max}} \text{ (region N)}. \end{cases} \tag{50}$$

where *α<sup>j</sup>* and *β<sup>j</sup>* are unknown amplitudes, *AT* and *AR* are unknown amplitudes of transmission and reflection, *Ylm*(*θ*, *ϕ*) is spherical function, *ki* = <sup>1</sup> *h*¯ �2*m*(*<sup>E</sup>* <sup>−</sup> *Vi*) are complex wave numbers. We shall be looking for solution for such problem in approach of multiple internal reflections (we restrict ourselves by a case of orbital moment *l* = 0 while its non-zero generalization changes the barrier shape which was used as arbitrary before in development of formalism MIR and, so, is absolutely non principal).

According to the method of multiple internal reflections, scattering of the particle on the barrier is considered on the basis of wave packet consequently by steps of its propagation relatively to each boundary of the barrier (the most clearly idea of such approach can be understood in the problem of tunneling through the simplest rectangular barrier, see (Cardone et al., 2006; Maydanyuk et al., 2002a; Maydanyuk, 2003) where one can find proof of this fully quantum exactly solvable method, one can analyze its properties). Each step in such consideration of propagation of the packet will be similar to one from the first 2*N* − 1 steps, independent between themselves. From analysis of these steps recurrent relations are found for calculation of unknown amplitudes *A*(*n*) , *S*(*n*) , *α*(*n*) and *β*(*n*) for arbitrary step *n*, summation of these amplitudes are calculated. We shall be looking for the unknown amplitudes, requiring wave function and its derivative to be continuous at each boundary. We shall consider the coefficients *T*± <sup>1</sup> , *T*<sup>±</sup> <sup>2</sup> , *T*<sup>±</sup> <sup>3</sup> . . . and *R*<sup>±</sup> <sup>1</sup> , *R*<sup>±</sup> <sup>2</sup> , *R*<sup>±</sup> <sup>3</sup> . . . as additional factors to amplitudes *e*±*ikx*. Here, bottom index denotes number of the boundary, upper (top) signs "+" and "−" denote directions of the wave to the right or to the left, correspondingly. At the first, we calculate *T*± <sup>1</sup> , *T*<sup>±</sup> <sup>2</sup> ... *T*<sup>±</sup> *<sup>N</sup>*−<sup>1</sup> and *<sup>R</sup>*<sup>±</sup> <sup>1</sup> , *R*<sup>±</sup> <sup>2</sup> ... *R*<sup>±</sup> *<sup>N</sup>*−1:

$$\begin{aligned} T\_j^+ &= \frac{2k\_j}{k\_j + k\_{j+1}} e^{i(k\_j - k\_{j+1})r\_j} \ \_I T\_j^- = \frac{2k\_{j+1}}{k\_j + k\_{j+1}} e^{i(k\_j - k\_{j+1})r\_j} \\\ R\_j^+ &= \frac{k\_j - k\_{j+1}}{k\_j + k\_{j+1}} e^{2ik\_j r\_j} \ \_ \end{aligned} \tag{51}$$

Using recurrent relations:

$$\begin{split} \tilde{R}\_{j-1}^{+} &= R\_{j-1}^{+} + T\_{j-1}^{+} \tilde{R}\_{j}^{+} T\_{j-1}^{-} \left( 1 + \sum\_{m=1}^{+\infty} (\tilde{R}\_{j}^{+} R\_{j-1}^{-})^{m} \right) = R\_{j-1}^{+} + \frac{T\_{j-1}^{+} \tilde{R}\_{j}^{+} T\_{j-1}^{-}}{1 - \tilde{R}\_{j}^{+} R\_{j-1}^{-}}, \\ \tilde{R}\_{j+1}^{-} &= R\_{j+1}^{-} + T\_{j+1}^{-} \tilde{R}\_{j}^{-} T\_{j+1}^{+} \left( 1 + \sum\_{m=1}^{+\infty} (\tilde{R}\_{j+1}^{+} \tilde{R}\_{j}^{-})^{m} \right) = R\_{j+1}^{-} + \frac{T\_{j+1}^{-} \tilde{R}\_{j}^{-} T\_{j+1}^{+}}{1 - \tilde{R}\_{j+1}^{+} \tilde{R}\_{j}^{-}}, \\ \tilde{T}\_{j+1}^{+} &= \tilde{T}\_{j}^{+} T\_{j+1}^{+} \left( 1 + \sum\_{m=1}^{+\infty} (\tilde{R}\_{j+1}^{+} \tilde{R}\_{j}^{-})^{m} \right) = \frac{\tilde{T}\_{j}^{+} T\_{j+1}^{+}}{1 - \tilde{R}\_{j+1}^{+} \tilde{R}\_{j}^{-}}, \end{split} \tag{52}$$

the penetrability coefficient *TMIR* by eqs. (56). We check the found amplitudes, coefficients *TMIR* and *RMIR* comparing them with corresponding amplitudes and coefficients calculated by standard approach of quantum mechanics. We restrict ourselves by eq. (50) for *F*<sup>1</sup> and find width Γ by eq. (48) and half-live *τMIR* by eq. (52). We define the penetrability *TWKB* by eq. (49),

Time as Quantum Observable, Canonical Conjugated to Energy 37

The first interesting result which we have obtained is *essential dependence of penetrability on the position of the first region where we localize the wave incidenting on the barrier*. In particular, we have analyzed how much the internal boundary *R*min takes influence on the penetrability. Taking into account that width of each interval is 0.01 fm, we consider left boundary *R*min of the first interval as a *starting point* (with error up to 0.01 fm), from here proton begins to move outside and is incident on the internal part of the barrier in the first stage of the proton decay. In the Fig. 4 [left panel] one can see that half-live of the proton decay of <sup>157</sup>

is changed essentially at displacement of *R*min. So, we establish *essential dependence of the penetrability on the starting point R*start*, where the proton starts to move outside by approach MIR.* At

*τMIR* on the starting point *R*min, the right panel is for dependence of the half-live *τMIR* on *R*max (here, we use *R*form = 7.2127 fm where calculated *τMIR* at *R*max = 250 fm coincides with experimental data *τ*exp for this nucleus). In all calculations factor *F* is the same.

*R*form = 7.2127 fm this dependence allows us to achieve very close coincidence between the

The region of the barrier located between turning points *R*<sup>2</sup> and *R*<sup>3</sup> is main part of the potential used in calculation of the penetrability in the semiclassical approach (up to the second correction), while the internal and external parts of this potential do not take influence on it. Let us analyze whether convergence exists in calculations of the penetrability in the approach MIR if to increase the external boundary *R*max (*R*max > *R*3). Keeping width of each interval (step) to be the same, we shall increase *R*max (through increasing number of intervals in the external region), starting from the external turning point *R*3, and calculate the corresponding penetrability *TMIR*. In Fig. 4 [central panel] one can see how the penetrability

in the next figure 4 [right panel]. One can see that the method MIR gives convergent values for the penetrability and half-life at increasing of *R*max. From such figures we find that *inclusion of the external region into calculations changes the half-life up to 1.5 times* (*τ*min = 0.20 sec is the

<sup>73</sup> Ta with increasing *R*max. Dependence of the half-life *τMIR* on *R*max is shown

half-live calculated by the approach MIR and experimental data.

**3.4.2 Dependence of the penetrability on the external region**

<sup>73</sup> Ta nucleus: the left panel is for dependence of the half-life

<sup>73</sup> Ta

calculate Γ-width and half-live *τWKB* by eqs. (48) and (52).

**3.4.1 Dependence of the penetrability on the starting point**

Fig. 4. Proton-decay for the <sup>157</sup>

is changed for <sup>157</sup>

and selecting as starting the following values:

$$
\tilde{\mathcal{R}}\_{N-1}^{+} = \mathcal{R}\_{N-1'}^{+} \, \tilde{\mathcal{R}}\_{1}^{-} = \mathcal{R}\_{1}^{-} \, \tilde{T}\_{1}^{+} = T\_{1}^{+} \, \tag{53}
$$

we calculate successively coefficients *R*˜ <sup>+</sup> *<sup>N</sup>*−<sup>2</sup> ... *<sup>R</sup>*˜ <sup>+</sup> <sup>1</sup> , *<sup>R</sup>*˜ <sup>−</sup> <sup>2</sup> ... *<sup>R</sup>*˜ <sup>−</sup> *<sup>N</sup>*−<sup>1</sup> and *<sup>T</sup>*˜<sup>+</sup> <sup>2</sup> ... *<sup>T</sup>*˜ <sup>+</sup> *<sup>N</sup>*−1. At finishing, we determine coefficients *βj*:

$$\beta\_{\vec{j}} = \tilde{T}\_{\vec{j}-1}^{+} \left( 1 + \sum\_{m=1}^{+\infty} (\tilde{\mathcal{R}}\_{\vec{j}}^{+} \tilde{\mathcal{R}}\_{\vec{j}-1}^{-})^{m} \right) = \frac{\tilde{T}\_{\vec{j}-1}^{+}}{1 - \tilde{\mathcal{R}}\_{\vec{j}}^{+} \tilde{\mathcal{R}}\_{\vec{j}-1}^{-}} \,\tag{54}$$

the amplitudes of transmission and reflection:

$$A\_T = \tilde{T}\_{N-1'}^+ A\_R = \tilde{R}\_1^+ \tag{55}$$

and corresponding coefficients of penetrability *T* and reflection *R*:

$$T\_{MIR} = \frac{k\_{\rm II}}{k\_1} \left| A\_T \right|^2 \text{ } \mathcal{R}\_{MIR} = \left| A\_R \right|^2. \tag{56}$$

We check the property:

$$\frac{k\_n}{k\_1} \left| A\_T \right|^2 + \left| A\_R \right|^2 = 1 \quad \text{or} \quad T\_{MIR} + R\_{MIR} = 1,\tag{57}$$

which should be the test, whether the method MIR gives us proper solution for wave function. Now if energy of the particle is located below then height of one step with number *m*, then for description of transition of this particle through such barrier with its tunneling it shall need to use the following change:

$$k\_m \to i\,\mathfrak{J}\_m.\tag{58}$$

For the potential from two rectangular steps (with different choice of their sizes) after comparison between the all amplitudes obtained by method of MIR and the corresponding amplitudes obtained by standard approach of quantum mechanics, we obtain coincidence up to first 15 digits. Increasing of number of steps up to some thousands keeps such accuracy and fulfillment of the property (57). This is important test which confirms reliability of the method MIR. So, we have obtained full coincidence between all amplitudes, calculated by method MIR and by standard approach of quantum mechanics, and that is way we generalize the method MIR for description of tunneling of the particle through potential, consisting from arbitrary number of rectangular barriers and wells of arbitrary shape.

#### **3.4 Analysis of the proton-decay for** <sup>157</sup> <sup>73</sup> Ta**,** <sup>161</sup> <sup>75</sup> Re**,** <sup>167</sup> <sup>77</sup> Ir **and** <sup>185</sup> <sup>83</sup> Bi

Today, there are a lot of modern methods able to calculate half-lives, which have been studied experimentally well. So, we have a rich theoretical and experimental material for analysis. We shall use these nuclei: <sup>157</sup> <sup>73</sup> Ta, <sup>161</sup> <sup>75</sup> Re, <sup>167</sup> <sup>77</sup> Ir for *<sup>l</sup>* <sup>=</sup> 0, and <sup>109</sup> <sup>53</sup> I, <sup>112</sup> <sup>55</sup> Cm, <sup>147</sup> <sup>69</sup> Tm for *l* �= 0. Such a choice we explain by that they have small coefficient of quadruple deformation *β*<sup>2</sup> and at good approximation can be considered as spherical. We shall study proton-decay on the basis of leaving of the particle with reduced mass from the internal region outside with its tunneling through the barrier. This particle is supposed to start from *R*min ≤ *r* ≤ *r*<sup>1</sup> and move outside (*r*<sup>1</sup> is defined in eq. (1)). Using technique of the *T*<sup>±</sup> *<sup>j</sup>* and *R*<sup>±</sup> *<sup>j</sup>* coefficients in eqs. (51)–(53), we calculate total amplitudes of transmission *AT* and reflection *AR* by eqs. (55), 20 Will-be-set-by-IN-TECH

<sup>1</sup> = *R*<sup>−</sup>

<sup>1</sup> , *<sup>R</sup>*˜ <sup>−</sup>

*<sup>N</sup>*−1, *AR* <sup>=</sup> *<sup>R</sup>*˜ <sup>+</sup>

which should be the test, whether the method MIR gives us proper solution for wave function. Now if energy of the particle is located below then height of one step with number *m*, then for description of transition of this particle through such barrier with its tunneling it shall need to

For the potential from two rectangular steps (with different choice of their sizes) after comparison between the all amplitudes obtained by method of MIR and the corresponding amplitudes obtained by standard approach of quantum mechanics, we obtain coincidence up to first 15 digits. Increasing of number of steps up to some thousands keeps such accuracy and fulfillment of the property (57). This is important test which confirms reliability of the method MIR. So, we have obtained full coincidence between all amplitudes, calculated by method MIR and by standard approach of quantum mechanics, and that is way we generalize the method MIR for description of tunneling of the particle through potential, consisting from

, *RMIR* =

<sup>1</sup> , *<sup>T</sup>*˜<sup>+</sup>

<sup>2</sup> ... *<sup>R</sup>*˜ <sup>−</sup>

<sup>1</sup> <sup>=</sup> *<sup>T</sup>*<sup>+</sup>

<sup>=</sup> *<sup>T</sup>*˜ <sup>+</sup> *j*−1 <sup>1</sup> <sup>−</sup> *<sup>R</sup>*˜ <sup>+</sup> *<sup>j</sup> <sup>R</sup>*˜ <sup>−</sup> *j*−1

> *AR* 2

<sup>2</sup> = 1 or *TMIR* + *RMIR* = 1, (57)

*km* → *i ξm*. (58)

*<sup>N</sup>*−<sup>1</sup> and *<sup>T</sup>*˜<sup>+</sup>

<sup>1</sup> , (53)

*<sup>N</sup>*−1. At finishing,

, (54)

<sup>2</sup> ... *<sup>T</sup>*˜ <sup>+</sup>

<sup>1</sup> (55)

. (56)

*<sup>N</sup>*−1, *<sup>R</sup>*˜ <sup>−</sup>

*AT* = *T*˜<sup>+</sup>

*k*1 *AT* 2

and corresponding coefficients of penetrability *T* and reflection *R*:

*TMIR* <sup>=</sup> *kn*

<sup>2</sup> <sup>+</sup> <sup>|</sup>*AR*<sup>|</sup>

arbitrary number of rectangular barriers and wells of arbitrary shape.

<sup>73</sup> Ta, <sup>161</sup>

move outside (*r*<sup>1</sup> is defined in eq. (1)). Using technique of the *T*<sup>±</sup>

<sup>73</sup> Ta**,** <sup>161</sup>

<sup>75</sup> Re, <sup>167</sup>

<sup>75</sup> Re**,** <sup>167</sup>

Today, there are a lot of modern methods able to calculate half-lives, which have been studied experimentally well. So, we have a rich theoretical and experimental material for analysis.

Such a choice we explain by that they have small coefficient of quadruple deformation *β*<sup>2</sup> and at good approximation can be considered as spherical. We shall study proton-decay on the basis of leaving of the particle with reduced mass from the internal region outside with its tunneling through the barrier. This particle is supposed to start from *R*min ≤ *r* ≤ *r*<sup>1</sup> and

eqs. (51)–(53), we calculate total amplitudes of transmission *AT* and reflection *AR* by eqs. (55),

<sup>77</sup> Ir **and** <sup>185</sup>

<sup>77</sup> Ir for *<sup>l</sup>* <sup>=</sup> 0, and <sup>109</sup>

<sup>83</sup> Bi

<sup>53</sup> I, <sup>112</sup>

<sup>55</sup> Cm, <sup>147</sup>

*<sup>j</sup>* and *R*<sup>±</sup>

<sup>69</sup> Tm for *l* �= 0.

*<sup>j</sup>* coefficients in

*<sup>N</sup>*−<sup>2</sup> ... *<sup>R</sup>*˜ <sup>+</sup>

and selecting as starting the following values:

we calculate successively coefficients *R*˜ <sup>+</sup>

we determine coefficients *βj*:

We check the property:

use the following change:

**3.4 Analysis of the proton-decay for** <sup>157</sup>

We shall use these nuclei: <sup>157</sup>

*R*˜ <sup>+</sup>

*<sup>β</sup><sup>j</sup>* = *<sup>T</sup>*˜ <sup>+</sup> *j*−1 1 + +∞ ∑ *m*=1 (*R*˜ <sup>+</sup> *<sup>j</sup> <sup>R</sup>*˜ <sup>−</sup> *<sup>j</sup>*−1)*<sup>m</sup>* 

the amplitudes of transmission and reflection:

*kn k*1 |*AT*|

*<sup>N</sup>*−<sup>1</sup> <sup>=</sup> *<sup>R</sup>*<sup>+</sup>

the penetrability coefficient *TMIR* by eqs. (56). We check the found amplitudes, coefficients *TMIR* and *RMIR* comparing them with corresponding amplitudes and coefficients calculated by standard approach of quantum mechanics. We restrict ourselves by eq. (50) for *F*<sup>1</sup> and find width Γ by eq. (48) and half-live *τMIR* by eq. (52). We define the penetrability *TWKB* by eq. (49), calculate Γ-width and half-live *τWKB* by eqs. (48) and (52).

#### **3.4.1 Dependence of the penetrability on the starting point**

The first interesting result which we have obtained is *essential dependence of penetrability on the position of the first region where we localize the wave incidenting on the barrier*. In particular, we have analyzed how much the internal boundary *R*min takes influence on the penetrability. Taking into account that width of each interval is 0.01 fm, we consider left boundary *R*min of the first interval as a *starting point* (with error up to 0.01 fm), from here proton begins to move outside and is incident on the internal part of the barrier in the first stage of the proton decay. In the Fig. 4 [left panel] one can see that half-live of the proton decay of <sup>157</sup> <sup>73</sup> Ta is changed essentially at displacement of *R*min. So, we establish *essential dependence of the penetrability on the starting point R*start*, where the proton starts to move outside by approach MIR.* At

Fig. 4. Proton-decay for the <sup>157</sup> <sup>73</sup> Ta nucleus: the left panel is for dependence of the half-life *τMIR* on the starting point *R*min, the right panel is for dependence of the half-live *τMIR* on *R*max (here, we use *R*form = 7.2127 fm where calculated *τMIR* at *R*max = 250 fm coincides with experimental data *τ*exp for this nucleus). In all calculations factor *F* is the same.

*R*form = 7.2127 fm this dependence allows us to achieve very close coincidence between the half-live calculated by the approach MIR and experimental data.

#### **3.4.2 Dependence of the penetrability on the external region**

The region of the barrier located between turning points *R*<sup>2</sup> and *R*<sup>3</sup> is main part of the potential used in calculation of the penetrability in the semiclassical approach (up to the second correction), while the internal and external parts of this potential do not take influence on it. Let us analyze whether convergence exists in calculations of the penetrability in the approach MIR if to increase the external boundary *R*max (*R*max > *R*3). Keeping width of each interval (step) to be the same, we shall increase *R*max (through increasing number of intervals in the external region), starting from the external turning point *R*3, and calculate the corresponding penetrability *TMIR*. In Fig. 4 [central panel] one can see how the penetrability is changed for <sup>157</sup> <sup>73</sup> Ta with increasing *R*max. Dependence of the half-life *τMIR* on *R*max is shown in the next figure 4 [right panel]. One can see that the method MIR gives convergent values for the penetrability and half-life at increasing of *R*max. From such figures we find that *inclusion of the external region into calculations changes the half-life up to 1.5 times* (*τ*min = 0.20 sec is the

Moreover, while performing it, we shall meet the opportunity of introducing bilinear operators, which will be used even more in the next case of the full 4-position operator. Let us recall that in Sect.2.1 we mentioned that the boundary condition *E* � 0, therein imposed to guarantee (maximal) hermiticity of the time operator, can be dispensed with just by having recourse to bilinear forms. Namely, by considering the bilinear hermitian operator(Recami,

Time as Quantum Observable, Canonical Conjugated to Energy 39

The standard position operators, being hermitian and moreover selfadjoint, are known to possess real eigenvalues: i.e., they yield a *point-like* localization. J. M. Jauch showed, however, that a point-like localization would be in contrast with "unimodularity". In the relativistic case, moreover, phenomena so as the pair production forbid a localization with precision better than one Compton wave-length. The eigenvalues of a realistic position operator ˆ*z* are therefore expected to represent space *regions*, rather than points. This can be obtained only by having recourse to non-hermitian (and therefore non-selfadjoint) position operators ˆ*z* (a priori, one can have recourse either to non-normal operators with commuting components, or to normal operators with non-commuting components). Following, e.g., the ideas in Ref. (Gallardo et al., 1967b;c; Ka'lnay, 1966; Ka'lnay & Toledo, 1967; Olkhovsky et al., 1967), we are going to show that the mean values of the *hermitian (selfadjoint) part* of ˆ*z* will yield a mean (point-like) position (Baldo & Recami, 1969; Recami, 1970), while the mean values of the *anti-hermitian (anti-selfadjoint) part* of ˆ*z* will yield the sizes of the localization region(Olkhovsky

Let us consider, e.g., the case of relativistic spin-zero particles, in natural units and with metric (+ − −−). The position operator *i* ∇*p*, is known to be actually non-hermitian, and may be in itself a good candidate for an extended-type position operator. To show this, we want to split (Gallardo et al., 1967b;c; Ka'lnay, 1966; Ka'lnay & Toledo, 1967; Olkhovsky et al., 1967) it

Consider, then, a vector space *V* of complex differentiable functions on a 3-dimensional

*p*0

where the integral is taken over the surface of a sphere of radius *R*. If *U* : *V* → *V* is a differential operator of degree one, condition (60) allows a definition of the transpose *U<sup>T</sup>* by

<sup>0</sup>. Let the functions in *V* satisfy moreover the condition

(*UT*Ψ, <sup>Φ</sup>)=(Ψ, *<sup>U</sup>* <sup>Φ</sup>) for all <sup>Ψ</sup>, <sup>Φ</sup> <sup>∈</sup> *<sup>V</sup>* , (61)

*∂* /*∂E*)/2, where the sign ↔ is defined through the

Ψ∗(*p*) Φ(*p*) (59)

Ψ∗(*p*) Φ(*p*) = 0 (60)

↔

*<sup>f</sup>* , <sup>−</sup> *ih* 2 *∂ <sup>∂</sup><sup>E</sup> g* + <sup>−</sup> *ih* 2 *∂ <sup>∂</sup><sup>E</sup> f* , *g* .

1976; 1977; Recami et al., 1983) <sup>ˆ</sup>*<sup>t</sup>* = (−*ih*¯

**4.1 The Klein-Gordon case: Three-position operators**

into its hermitian and anti-hermitian (or skew-hermitian) parts.

*p*<sup>2</sup> + *m*<sup>2</sup>

phase-space(Recami et al., 1983) equipped with an inner product defined by

lim *R*→∞ *dS p*0

*SR*

where *U* is changed into *UT*, or vice-versa, by means of integration by parts.

(Ψ, <sup>Φ</sup>) = *<sup>d</sup>*3*<sup>p</sup>*

accompanying equality (*<sup>f</sup>* , <sup>ˆ</sup>*t g*) =

& Recami, 1968; 1969).

quantity *<sup>p</sup>*<sup>0</sup> being


Table 1. Experimental and calculated half-lives of some proton emitters. Here, *S*th *<sup>p</sup>* is theoretical spectroscopic factor, *τWKB* is half-life calculated by in the semiclassical approach, *τMIR* is half-life calculated by in the approach MIR, *τ*˜*WKB* = *τWKB*/ *S*th *<sup>p</sup>* , *τ*˜*MIR* = *τMIR*/*S*th *p* , *τ*exp is experimental data. Values for *S*th *<sup>p</sup>* , *τ*exp are used from Table IV in Ref. (Aberg et al., 1997) (see p. 1770); in calculations for each nucleus we use: *R*min = 0.11 fm, *R*max = 250 fm; number of intervals in region from *R*min to maximum of the barrier is 10000, from maximum of the barrier to *R*max is 10000.

minimal half-life calculated at *R*<sup>3</sup> ≤ *R*max ≤ 250 fm, *τ*as = 0.30 sec is the half-life calculated at *R*max = 250 fm, error = *τ*as/*τ*min ≈ 1.5 or 50 percents). So, *error in determination of the penetrability in the semiclassical approach (if to take the external region into account) is expected to be the same as a minimum on such a basis*.

#### **3.4.3 Results of calculations of half-lives in our and semiclassical approaches**

As we have demonstrated above, the fully quantum calculations of the penetrability of the barrier for the proton decay give us its essential dependence on the starting point. In order to give power of predictions of half-lives calculated by the approach MIR, we need to find recipe able to resolve such uncertainty in calculations of the half-lives. So, we shall introduce the following hypothesis: *we shall assume that in the first stage of the proton decay proton starts to move outside the most probably at the coordinate of minimum of the internal well*. If such a point is located in the minimum of the well, the penetrability turns out to be maximal and half-life minimal. So, as criterion we could use minimum of half-live for the given potential, which has stable basis. We should take into account that the half-lives obtained before are for the proton occupied ground state while it needs to take into account probability that this state is empty in the daughter nucleus. In order to obtain proper values for the half-lives we should divide them on the spectroscopic factor *S* (which we take from (Aberg et al., 1997)), and then to compare them with experimental data. Results of such calculations and experimental data for some proton emitters are presented in Table 1. To complete a picture, we add half-lives calculated by the semiclassical approach to these data.

#### **4. On four-position operators in quantum field theory, in terms of bilinear operators 4. On four-position operators in quantum field theory, in terms of bilinear operators**

In this Section we approach the *relativistic* case, taking into consideration — therefore — the space-time (four-dimensional) "position" operator, starting however with an analysis of the 3-dimensional (spatial) position operator in the simple relativistic case of the Klein-Gordon equation. Actually, this analysis will lead us to tackle already with non-hermitian operators. 22 Will-be-set-by-IN-TECH

Parent nucleus Half-live-values, sec

<sup>73</sup> Ta84 0.947 <sup>2</sup>*s*1/2 0.66 2.813 · <sup>10</sup>−<sup>1</sup> 2.789 · <sup>10</sup>−<sup>1</sup> 3.0 · <sup>10</sup>−<sup>1</sup>

<sup>75</sup> Re84 1.214 <sup>2</sup>*s*1/2 0.59 2.720 · <sup>10</sup>−<sup>4</sup> 2.673 · <sup>10</sup>−<sup>4</sup> 3.7 · <sup>10</sup>−<sup>4</sup>

<sup>77</sup> Ir90 1.086 <sup>2</sup>*s*1/2 0.51 5.85 · <sup>10</sup>−<sup>2</sup> 5.84 · <sup>10</sup>−<sup>2</sup> 1.1 · <sup>10</sup>−<sup>2</sup>

<sup>53</sup> I56 0.829 <sup>1</sup>*d*5/2 0.76 3.937 · <sup>10</sup>−<sup>6</sup> 3.992 · <sup>10</sup>−<sup>6</sup> 1.0 · <sup>10</sup>−<sup>4</sup>

<sup>55</sup> Cm57 0.823 <sup>1</sup>*d*5/2 0.59 3.526 · <sup>10</sup>−<sup>5</sup> 3.539 · <sup>10</sup>−<sup>5</sup> 5.0 · <sup>10</sup>−<sup>4</sup>

<sup>69</sup> Tm78 1.132 <sup>1</sup>*d*5/2 0.79 7.911 · <sup>10</sup>−<sup>5</sup> 7.796 · <sup>10</sup>−<sup>5</sup> 3.6 · <sup>10</sup>−<sup>4</sup>

theoretical spectroscopic factor, *τWKB* is half-life calculated by in the semiclassical approach,

1997) (see p. 1770); in calculations for each nucleus we use: *R*min = 0.11 fm, *R*max = 250 fm; number of intervals in region from *R*min to maximum of the barrier is 10000, from maximum

minimal half-life calculated at *R*<sup>3</sup> ≤ *R*max ≤ 250 fm, *τ*as = 0.30 sec is the half-life calculated at *R*max = 250 fm, error = *τ*as/*τ*min ≈ 1.5 or 50 percents). So, *error in determination of the penetrability in the semiclassical approach (if to take the external region into account) is expected to be*

As we have demonstrated above, the fully quantum calculations of the penetrability of the barrier for the proton decay give us its essential dependence on the starting point. In order to give power of predictions of half-lives calculated by the approach MIR, we need to find recipe able to resolve such uncertainty in calculations of the half-lives. So, we shall introduce the following hypothesis: *we shall assume that in the first stage of the proton decay proton starts to move outside the most probably at the coordinate of minimum of the internal well*. If such a point is located in the minimum of the well, the penetrability turns out to be maximal and half-life minimal. So, as criterion we could use minimum of half-live for the given potential, which has stable basis. We should take into account that the half-lives obtained before are for the proton occupied ground state while it needs to take into account probability that this state is empty in the daughter nucleus. In order to obtain proper values for the half-lives we should divide them on the spectroscopic factor *S* (which we take from (Aberg et al., 1997)), and then to compare them with experimental data. Results of such calculations and experimental data for some proton emitters are presented in Table 1. To complete a picture, we add half-lives

Table 1. Experimental and calculated half-lives of some proton emitters. Here, *S*th

**3.4.3 Results of calculations of half-lives in our and semiclassical approaches**

**4. On four-position operators in quantum field theory, in terms of bilinear**

**4. On four-position operators in quantum field theory, in terms of bilinear** 

In this Section we approach the *relativistic* case, taking into consideration — therefore — the space-time (four-dimensional) "position" operator, starting however with an analysis of the 3-dimensional (spatial) position operator in the simple relativistic case of the Klein-Gordon equation. Actually, this analysis will lead us to tackle already with non-hermitian operators.

*τMIR* is half-life calculated by in the approach MIR, *τ*˜*WKB* = *τWKB*/ *S*th

*<sup>p</sup> τ*˜*WKB τ*˜*MIR τ*exp

*<sup>p</sup>* , *τ*exp are used from Table IV in Ref. (Aberg et al.,

*<sup>p</sup>* is

*p* ,

*<sup>p</sup>* , *τ*˜*MIR* = *τMIR*/*S*th

Nucleus *Q*, MeV Orbit *S*th

157

161

167

109

*τ*exp is experimental data. Values for *S*th

of the barrier to *R*max is 10000.

*the same as a minimum on such a basis*.

calculated by the semiclassical approach to these data.

**operators**

**operators** 

112

147

Moreover, while performing it, we shall meet the opportunity of introducing bilinear operators, which will be used even more in the next case of the full 4-position operator.

Let us recall that in Sect.2.1 we mentioned that the boundary condition *E* � 0, therein imposed to guarantee (maximal) hermiticity of the time operator, can be dispensed with just by having recourse to bilinear forms. Namely, by considering the bilinear hermitian operator(Recami, ↔

1976; 1977; Recami et al., 1983) <sup>ˆ</sup>*<sup>t</sup>* = (−*ih*¯ *∂* /*∂E*)/2, where the sign ↔ is defined through the accompanying equality (*<sup>f</sup>* , <sup>ˆ</sup>*t g*) = *<sup>f</sup>* , <sup>−</sup> *ih* 2 *∂ <sup>∂</sup><sup>E</sup> g* + <sup>−</sup> *ih* 2 *∂ <sup>∂</sup><sup>E</sup> f* , *g* .

#### **4.1 The Klein-Gordon case: Three-position operators**

The standard position operators, being hermitian and moreover selfadjoint, are known to possess real eigenvalues: i.e., they yield a *point-like* localization. J. M. Jauch showed, however, that a point-like localization would be in contrast with "unimodularity". In the relativistic case, moreover, phenomena so as the pair production forbid a localization with precision better than one Compton wave-length. The eigenvalues of a realistic position operator ˆ*z* are therefore expected to represent space *regions*, rather than points. This can be obtained only by having recourse to non-hermitian (and therefore non-selfadjoint) position operators ˆ*z* (a priori, one can have recourse either to non-normal operators with commuting components, or to normal operators with non-commuting components). Following, e.g., the ideas in Ref. (Gallardo et al., 1967b;c; Ka'lnay, 1966; Ka'lnay & Toledo, 1967; Olkhovsky et al., 1967), we are going to show that the mean values of the *hermitian (selfadjoint) part* of ˆ*z* will yield a mean (point-like) position (Baldo & Recami, 1969; Recami, 1970), while the mean values of the *anti-hermitian (anti-selfadjoint) part* of ˆ*z* will yield the sizes of the localization region(Olkhovsky & Recami, 1968; 1969).

Let us consider, e.g., the case of relativistic spin-zero particles, in natural units and with metric (+ − −−). The position operator *i* ∇*p*, is known to be actually non-hermitian, and may be in itself a good candidate for an extended-type position operator. To show this, we want to split (Gallardo et al., 1967b;c; Ka'lnay, 1966; Ka'lnay & Toledo, 1967; Olkhovsky et al., 1967) it into its hermitian and anti-hermitian (or skew-hermitian) parts.

Consider, then, a vector space *V* of complex differentiable functions on a 3-dimensional phase-space(Recami et al., 1983) equipped with an inner product defined by

$$\left(\Psi,\,\Phi\right) = \int \frac{d^3 p}{p\_0} \,\Psi^\*(p)\,\Phi(p) \tag{59}$$

quantity *<sup>p</sup>*<sup>0</sup> being *p*<sup>2</sup> + *m*<sup>2</sup> <sup>0</sup>. Let the functions in *V* satisfy moreover the condition

$$\lim\_{R \to \infty} \int \frac{dS}{p\_0} \left. \Psi^\*(p) \Phi(p) = 0 \right. \tag{60}$$

where the integral is taken over the surface of a sphere of radius *R*. If *U* : *V* → *V* is a differential operator of degree one, condition (60) allows a definition of the transpose *U<sup>T</sup>* by

$$(\mathcal{U}^T \Psi, \Phi) = (\Psi, \mathcal{U} \Phi) \text{ for all } \Psi, \Phi \in V \text{ \textit{\tiny} } \tag{61}$$

where *U* is changed into *UT*, or vice-versa, by means of integration by parts.

where Ψ<sup>∗</sup> <sup>↔</sup>

writing explicitly

localization-region.

expressed by eqs.(59)–(68).

commutators (*i*, *j* = 1, 2, 3):

yield

<sup>∇</sup>*<sup>p</sup>* <sup>Φ</sup> <sup>≡</sup> <sup>Ψ</sup>∗∇*p*<sup>Φ</sup> <sup>−</sup> <sup>Φ</sup>∇*p*Ψ<sup>∗</sup> and <sup>Ψ</sup><sup>∗</sup> <sup>↔</sup>

Ka'lnay, 1966; Ka'lnay & Toledo, 1967; Olkhovsky et al., 1967; Recami, 1970)

(Ψ, *x*ˆ Φ) = *i*

Φ, *x*ˆ Φ = *i*

verify that eq. (70) *is equivalent to the ordinary Newton-Wigner operator:*

<sup>∇</sup>*<sup>p</sup>* <sup>≡</sup> *<sup>i</sup>* <sup>∇</sup>*<sup>p</sup>* <sup>−</sup> *<sup>i</sup>*

*<sup>y</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>i</sup>* 2 ↔ ∇ (+)

�

*<sup>x</sup>*ˆ*<sup>h</sup>* <sup>≡</sup> *<sup>i</sup>* 2 ↔

In general, the extended-type position operator *z*ˆ will yeld

 *i* 2

↔ *∂ ∂pi* , *i* 2

wherefrom the noticeable "uncertainty correlations" follow:

Δ*α<sup>i</sup>* Δ*β<sup>j</sup>* ≥

↔ *∂* (+) *∂pj*

> 1 4 1 *p*2 0

 <sup>=</sup> *<sup>i</sup>* 2 *p*<sup>2</sup> 0

We are left with the (bilinear) anti-hermitian part

∇ (+)

we always referred to a suitable (Baldo & Recami, 1969; Gallardo et al., 1967b;c; Ka'lnay, 1966; Ka'lnay & Toledo, 1967; Olkhovsky et al., 1967; Recami, 1970; 1976; 1977; Recami et al., 1983) space of wave packets. Its hermitian part (Baldo & Recami, 1969; Gallardo et al., 1967b;c;

Time as Quantum Observable, Canonical Conjugated to Energy 41

which was expected to yield an (ordinary) point-like localization, has been derived also by

and imposing hermiticity, i.e., imposing the reality of the diagonal elements. The calculations

suggesting to adopt just the Lorentz-invariant quantity (70) as a bilinear hermitian position operator. Then, on integrating by parts (and due to the vanishing of the surface integral), we

2

whose *average values* over the considered state (wave-packet) can be regarded as yielding (Baldo & Recami, 1969; Gallardo et al., 1967b;c; Ka'lnay, 1966; Ka'lnay & Toledo, 1967; Olkhovsky et al., 1967; Recami, 1970; 1976; 1977; Recami et al., 1983)the sizes of an ellipsoidal

After the digression associated with eqs.(69)–(74), let us go back to the present formalism, as

where Δ*α* and Δ*β* are the mean-errors encountered when measuring the point-like position and the sizes of the localization region, respectively. It is interesting to evaluate the

*<sup>δ</sup>ij* <sup>−</sup> <sup>2</sup> *pi pj p*2 0

<sup>Φ</sup>∗(*p*) <sup>↔</sup>

*p*

 *d*<sup>3</sup> *p p*0

 *d*<sup>3</sup> *p p*0

*<sup>x</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>i</sup>* 2 ↔ *<sup>p</sup>* Φ ≡ Ψ∗∇*p*Φ + Φ∇*p*Ψ<sup>∗</sup> , and where

∇*<sup>p</sup>* , (70)

Ψ∗(*p*) ∇*<sup>p</sup>* Φ(*p*) (71)

*<sup>p</sup>*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*<sup>2</sup> <sup>≡</sup> <sup>N</sup> <sup>−</sup> W . (73)

*<sup>p</sup>* (74)

�Ψ| *z*ˆ |Ψ� = (*α* + Δ*α*) + *i* (*β* + Δ*β*) , (75)

*<sup>δ</sup>ij* <sup>−</sup> <sup>2</sup> *pi pj p*2 0

> 

, (76)

. (77)

∇*<sup>p</sup>* <sup>Φ</sup>(*p*) , (72)

This allows, further, to introduce a *dual representation*(Recami et al., 1983) (*U*1, *U*2) of a *single* operator *U<sup>T</sup>* <sup>1</sup> + *U*<sup>2</sup> by

$$(\mathcal{U}\_1 \Psi, \Phi) + (\Psi, \mathcal{U}\_2 \Phi) = (\Psi, (\mathcal{U}\_1^T + \mathcal{U}\_2)\, \Phi). \tag{62}$$

With such a dual representation, it is easy to split any operator into its hermitian and anti-hermitian parts

$$\left( (\Psi, \,\mathrm{U}\Phi) = \frac{1}{2} \left( (\Psi, \,\mathrm{U}\Phi) + (\mathrm{U}^\*\Psi, \,\Phi) \right) + \frac{1}{2} \left( (\Psi, \,\mathrm{U}\Phi) - (\mathrm{U}^\*\Psi, \,\Phi) \right). \tag{63}$$

Here the pair

$$\frac{1}{2}\begin{pmatrix} \mathcal{U}^\* \ \mathcal{U} \end{pmatrix} \equiv \stackrel{\leftrightarrow}{\mathcal{U}}\_{\mathcal{U}} \tag{64}$$

corresponding to (1/2) (*U* + *U*∗*T*), represents the hermitian part, while

$$\frac{1}{2}(-\mathcal{U}^\*,\mathcal{U}) \equiv \stackrel{\leftrightarrow}{\mathcal{U}\_a} \tag{65}$$

represents the anti-hermitian part.

Let us apply what precedes to the case of the Klein-Gordon position-operator *z*ˆ = *i* ∇*p*. When

$$
\hat{\mathbf{U}} = \hat{\mathbf{i}} \frac{\partial}{\partial p\_j} \tag{66}
$$

we have(Olkhovsky & Recami, 1968; 1969)

$$\begin{aligned} \frac{1}{2} \left( \mathcal{U}^\* \lrcorner \mathcal{U} \right) &= \frac{1}{2} \left( -i \frac{\partial}{\partial p\_j} \lrcorner i \frac{\partial}{\partial p\_j} \right) \equiv \frac{i}{2} \stackrel{\leftrightarrow}{\frac{\partial}{\partial p\_j}} \lrcorner (a) \\\frac{1}{2} \left( -\mathcal{U}^\* \lrcorner \mathcal{U} \right) &= \frac{1}{2} \left( i \frac{\partial}{\partial p\_j} \lrcorner i \frac{\partial}{\partial p\_j} \right) \equiv \frac{i}{2} \stackrel{\leftrightarrow}{\frac{\partial}{\partial p\_j}} \lrcorner (b) \end{aligned} \tag{67}$$

And the corresponding *single* operators turn out to be

$$\begin{aligned} \frac{1}{2} \left( \mathcal{U} + \mathcal{U}^{\*T} \right) &= i \frac{\partial}{\partial p\_j} - \frac{i}{2} \frac{p\_j}{p^2 + m\_0^2} \prime \begin{array}{c} (a) \\\\ \frac{1}{2} \left( \mathcal{U} - \mathcal{U}^{\*T} \right) &= \frac{i}{2} \frac{p\_j}{p^2 + m\_0^2} \,, \qquad (b) \end{array} \tag{68}$$

It is noteworthy(Olkhovsky & Recami, 1968; 1969) that, as we are going to see, operator (68a) is nothing but the usual Newton-Wigner operator, while (68b) can be interpreted (Gallardo et al., 1967b;c; Ka'lnay, 1966; Ka'lnay & Toledo, 1967; Olkhovsky et al., 1967; Olkhovsky & Recami, 1968; 1969; Toller, 1999) as yielding the sizes of the localization-region (an ellipsoid) via its average values over the considered wave-packet.

Let us underline that the previous formalism justifies from the mathematical point of view the treatment presented in papers like (Baldo & Recami, 1969; Gallardo et al., 1967b;c; Ka'lnay, 1966; Ka'lnay & Toledo, 1967; Olkhovsky et al., 1967; Recami, 1970). We can split(Olkhovsky & Recami, 1968; 1969) the operator *z*ˆ into two *bilinear* parts, as follows:

$$\mathcal{Z} = \dot{\mathbf{r}} \nabla\_p = \frac{\dot{\mathbf{r}}}{2} \stackrel{\leftrightarrow}{\nabla}\_p + \frac{\dot{\mathbf{r}}}{2} \stackrel{\leftrightarrow}{\nabla}\_p^{(+)} \tag{69}$$

24 Will-be-set-by-IN-TECH

This allows, further, to introduce a *dual representation*(Recami et al., 1983) (*U*1, *U*2) of a *single*

With such a dual representation, it is easy to split any operator into its hermitian and

<sup>2</sup> (*U*∗, *<sup>U</sup>*) <sup>≡</sup> <sup>↔</sup>

(−*U*∗, *<sup>U</sup>*) <sup>≡</sup> <sup>↔</sup>

Let us apply what precedes to the case of the Klein-Gordon position-operator *z*ˆ = *i* ∇*p*. When

*<sup>U</sup>* <sup>=</sup> *<sup>i</sup> <sup>∂</sup> ∂pj*

> , *<sup>i</sup> <sup>∂</sup> ∂pj* ≡ *i* 2

, *<sup>i</sup> <sup>∂</sup> ∂pj* ≡ *i* 2

*pj p*<sup>2</sup> + *m*<sup>2</sup> 0

*∂pj* − *i* 2

It is noteworthy(Olkhovsky & Recami, 1968; 1969) that, as we are going to see, operator (68a) is nothing but the usual Newton-Wigner operator, while (68b) can be interpreted (Gallardo et al., 1967b;c; Ka'lnay, 1966; Ka'lnay & Toledo, 1967; Olkhovsky et al., 1967; Olkhovsky & Recami, 1968; 1969; Toller, 1999) as yielding the sizes of the localization-region (an ellipsoid)

Let us underline that the previous formalism justifies from the mathematical point of view the treatment presented in papers like (Baldo & Recami, 1969; Gallardo et al., 1967b;c; Ka'lnay, 1966; Ka'lnay & Toledo, 1967; Olkhovsky et al., 1967; Recami, 1970). We can split(Olkhovsky

> 2 ↔ ∇*<sup>p</sup>* + *i* 2 ↔ ∇ (+)

2

↔ *∂ ∂pj* , (*a*)

↔ *∂* + *∂pj* . (*b*)

*pj p*<sup>2</sup> + *m*<sup>2</sup> 0 , (*a*)

. (*b*)

 + 1 2 

<sup>1</sup> + *U*2) Φ). (62)

. (63)

(66)

(67)

(68)

(Ψ, *U*Φ) − (*U*∗Ψ, Φ)

*Uh* , (64)

*Ua* (65)

*<sup>p</sup>* (69)

(*U*1Ψ, Φ)+(Ψ, *U*2Φ)=(Ψ, (*U<sup>T</sup>*

(Ψ, *U*Φ)+(*U*∗Ψ, Φ)

1

1 2

2 <sup>−</sup>*<sup>i</sup> <sup>∂</sup> ∂pj*

<sup>2</sup> (*<sup>U</sup>* <sup>+</sup> *<sup>U</sup>*∗*T*) = *<sup>i</sup> <sup>∂</sup>*

<sup>2</sup> (*<sup>U</sup>* <sup>−</sup> *<sup>U</sup>*∗*T*) = *<sup>i</sup>*

& Recami, 1968; 1969) the operator *z*ˆ into two *bilinear* parts, as follows:

*<sup>z</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>i</sup>* <sup>∇</sup>*<sup>p</sup>* <sup>=</sup> *<sup>i</sup>*

2 *i ∂ ∂pj*

corresponding to (1/2) (*U* + *U*∗*T*), represents the hermitian part, while

operator *U<sup>T</sup>*

Here the pair

anti-hermitian parts

<sup>1</sup> + *U*<sup>2</sup> by

(Ψ, *<sup>U</sup>*Φ) = <sup>1</sup>

represents the anti-hermitian part.

we have(Olkhovsky & Recami, 1968; 1969)

1

1

And the corresponding *single* operators turn out to be 1

1

via its average values over the considered wave-packet.

<sup>2</sup> (*U*∗, *<sup>U</sup>*) = <sup>1</sup>

<sup>2</sup> (−*U*∗, *<sup>U</sup>*) = <sup>1</sup>

2  where Ψ<sup>∗</sup> <sup>↔</sup> <sup>∇</sup>*<sup>p</sup>* <sup>Φ</sup> <sup>≡</sup> <sup>Ψ</sup>∗∇*p*<sup>Φ</sup> <sup>−</sup> <sup>Φ</sup>∇*p*Ψ<sup>∗</sup> and <sup>Ψ</sup><sup>∗</sup> <sup>↔</sup> ∇ (+) *<sup>p</sup>* Φ ≡ Ψ∗∇*p*Φ + Φ∇*p*Ψ<sup>∗</sup> , and where we always referred to a suitable (Baldo & Recami, 1969; Gallardo et al., 1967b;c; Ka'lnay, 1966; Ka'lnay & Toledo, 1967; Olkhovsky et al., 1967; Recami, 1970; 1976; 1977; Recami et al., 1983) space of wave packets. Its hermitian part (Baldo & Recami, 1969; Gallardo et al., 1967b;c; Ka'lnay, 1966; Ka'lnay & Toledo, 1967; Olkhovsky et al., 1967; Recami, 1970)

$$
\mathfrak{X} = \frac{\stackrel{\!\!\! }{\!}}{\! 2} \stackrel{\!\!\! \/}{\stackrel{\!\!\! }{\! \! \! }}{\stackrel{\!\!\! \/} } \stackrel{\!\!\! \/}{\stackrel{\!\!\! \/} }{\! \! \! \/} \tag{70}
$$

which was expected to yield an (ordinary) point-like localization, has been derived also by writing explicitly

$$\mathbf{i}\left(\Psi,\mathbf{\hat{x}}\Phi\right) = \mathbf{i}\int \frac{d^3p}{p\_0} \,\mathbf{\hat{Y}}^\*(p)\,\nabla\_p\Phi(p) \tag{71}$$

and imposing hermiticity, i.e., imposing the reality of the diagonal elements. The calculations yield

$$\mathfrak{R}\left(\Phi,\hat{\mathfrak{x}}\Phi\right) = \mathrm{i}\int \frac{d^3p}{p\_0} \,\Phi^\*(p)\stackrel{\leftrightarrow}{\nabla}\_p\Phi(p)\,. \tag{72}$$

suggesting to adopt just the Lorentz-invariant quantity (70) as a bilinear hermitian position operator. Then, on integrating by parts (and due to the vanishing of the surface integral), we verify that eq. (70) *is equivalent to the ordinary Newton-Wigner operator:*

$$\text{At}\_{\text{h}} \equiv \frac{\dot{\mathbf{i}}}{2} \stackrel{\leftrightarrow}{\nabla}\_{p} \equiv \dot{\mathbf{i}} \,\nabla\_{p} - \frac{\dot{\mathbf{i}}}{2} \,\frac{p}{p^{2} + m^{2}} \,\,\equiv \text{N} - \text{W} \,. \tag{73}$$

We are left with the (bilinear) anti-hermitian part

$$\mathcal{Y} = \frac{i}{2} \stackrel{\leftrightarrow}{\nabla}\_p^{(+)} \tag{74}$$

whose *average values* over the considered state (wave-packet) can be regarded as yielding (Baldo & Recami, 1969; Gallardo et al., 1967b;c; Ka'lnay, 1966; Ka'lnay & Toledo, 1967; Olkhovsky et al., 1967; Recami, 1970; 1976; 1977; Recami et al., 1983)the sizes of an ellipsoidal localization-region.

After the digression associated with eqs.(69)–(74), let us go back to the present formalism, as expressed by eqs.(59)–(68).

In general, the extended-type position operator *z*ˆ will yeld

$$
\langle \Psi | \hat{z} | \Psi \rangle = (\mathfrak{a} + \Delta \mathfrak{a}) + i \left( \mathfrak{f} + \Delta \mathfrak{f} \right) \,, \tag{75}
$$

where Δ*α* and Δ*β* are the mean-errors encountered when measuring the point-like position and the sizes of the localization region, respectively. It is interesting to evaluate the commutators (*i*, *j* = 1, 2, 3):

$$\left(\frac{i}{2}\frac{\stackrel{\leftrightarrow}{\partial}}{\partial p\_i}, \frac{i}{2}\frac{\stackrel{\leftrightarrow}{\partial}}{\partial p\_j}\right) = \frac{i}{2}\frac{i}{p\_0^2}\left(\delta\_{ij} - \frac{2\,p\_i p\_j}{p\_0^2}\right),\tag{76}$$

wherefrom the noticeable "uncertainty correlations" follow:

$$
\Delta \alpha\_i \, \Delta \beta\_j \ge \frac{1}{4} \left| \left\langle \frac{1}{p\_0^2} \left( \delta\_{ij} - \frac{2 \, p\_i p\_j}{p\_0^2} \right) \right\rangle \right| . \tag{77}$$

Schröedinger equation (as well as the Liouville–von Neumann equation) in three different ways: "retarded", "symmetrical", and "advanced". One of such three formulations — the *retarded* one — describes in an elementary way a system which is exchanging (and losing) energy with the environment. In its density-matrix version, indeed, it can be easily shown

Time as Quantum Observable, Canonical Conjugated to Energy 43

Let us refer, in particular, to the theory of the *chronon* as proposed by P. Caldirola. Let us recall that such an interesting "finite difference" theory, forwards — at the classical level a solution for the motion of a particle endowed with a non-negligible charge in an external electromagnetic field, overcoming all the known difficulties met by Abraham-Lorentz's and Dirac's approaches (and even allowing a clear answer to the question whether a free falling charged particle does or does not emit radiation), and — at the quantum level — yields a

It is easy to compare one another the *new* representations of Quantum Mechanics (QM) resulting from it, in the Schröedinger, Heisenberg and densityUoperator (Liouville–von ˝

For each representation, three (*retarded*, *symmetric* and *advanced*) *formulations* are possible, which refer either to times *t* and *t* − *τ*0, or to times *t* − *τ*0/2 and *t* + *τ*0/2, or to times *t* and *t* + *τ*0, respectively. It is interesting to notice that, when the chronon tends to zero, the ordinary QM is obtained as the limiting case of the "symmetric" formulation only; while the "retarded" one does naturally appear to describe QM with friction, i. e., to describe *dissipative* quantum

In this sense, *discretized* QM is much richer than the ordinary one. Here, we want to pay attention to the fact that, when applying the density matrix formalism to the solution of the *measurement problem* in QM, interesting results are met, as, for instance, a natural explication of the "decoherence" due to dissipation: which seem to reveal the power of dicretized (in

Let us approach our eventual application of the discretization procedures for a possible solution of the measurement problem in Quantum Mechanics, without having to make recourse to the reduction (wave-packet instantaneous collapse) postulate. Namely, let us focus our attention, now, on the consequences for QM of the introduction of a chronon. In QM, time will still be a continuous variable, but the evolution of the system along its world line will be regarded as discontinuous. In analogy with the electron theory in the *non-relativistic* limit, one has to substitute the corresponding finite-difference expression for the time derivatives; e. g.:

*dt* <sup>=</sup> *<sup>f</sup>*(*t*) <sup>−</sup> *<sup>f</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>Δ</sup>*t*)

where proper time is now replaced by the local time *t*. The chronon procedure can then be applied to obtain the finite-difference form of the Schröedinger equation. As for the electron case, there are three different ways to perform the discretization, and three "Schröedinger

<sup>Δ</sup>*<sup>t</sup>* , (82)

*<sup>τ</sup>* [Ψ(*x*, *<sup>t</sup>*) <sup>−</sup> <sup>Ψ</sup>(*x*, *<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)] = *<sup>H</sup>*<sup>ˆ</sup> <sup>Ψ</sup>(*x*, *<sup>t</sup>*), (83)

<sup>2</sup>*<sup>τ</sup>* [Ψ(*x*, *<sup>t</sup>* <sup>+</sup> *<sup>τ</sup>*) <sup>−</sup> <sup>Ψ</sup>(*x*, *<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)] = *<sup>H</sup>*<sup>ˆ</sup> <sup>Ψ</sup>(*x*, *<sup>t</sup>*), (84)

*d f*(*t*)

*i h*¯

*i h*¯

that all non-diagonal terms go to zero very rapidly.

systems (like a particle moving in an absorbing medium).

remarkable mass spectrum for leptons.

Neumann) pictures, respectively.

particular, *retarded*) QM.

equations" can be obtained:

**5.1 On discretized Quantum Mechanics**

#### **4.2 Four-position operators**

It is tempting to propose as *four-position operator* the quantity *z*ˆ*<sup>μ</sup>* = *x*ˆ*<sup>μ</sup>* + *i y*ˆ*μ*, whose hermitian (Lorentz-covariant) part can be written

$$\mathfrak{X}^{\mu} = -\frac{i}{2} \frac{\stackrel{\leftrightarrow}{\partial}}{\partial p\_{\mu}} \,' \,. \tag{78}$$

to be associated with its corresponding "operator" in four-momentum space

$$
\mathfrak{p}\_h^{\mu} = + \frac{i}{2} \frac{\overleftrightarrow{\partial}}{\partial \mathfrak{x}\_{\mu}} \,. \tag{79}
$$

Let us recall the proportionality between the 4-momentum operator and the 4-current density operator in the chronotopical space, and then underline the canonical correspondence (in the 4-position and 4-momentum spaces, respectively) between the "operators" (cf. the previous subsection):

$$\begin{aligned} m\_0 \mathfrak{p} &\equiv \mathfrak{p}\_0 = \frac{i}{2} \stackrel{\leftrightarrow}{\frac{\partial}{\partial t}} & & (a) \\ m\_0 \hat{\mathfrak{p}} &\equiv \hat{\mathfrak{p}} = -\frac{i}{2} \stackrel{\leftrightarrow}{\frac{\partial}{\partial r}} & & (b) \end{aligned} \tag{80}$$

and the operators

$$\begin{aligned} \text{f} & \equiv -\frac{\dot{l}}{2} \frac{\stackrel{\leftrightarrow}{\partial}}{\partial p\_0} \begin{pmatrix} a \\ \end{pmatrix} \\ \text{f} & \equiv \frac{\dot{l}}{2} \frac{\stackrel{\leftrightarrow}{\partial}}{\partial p} \begin{pmatrix} b \\ \end{pmatrix} \end{aligned} \tag{81}$$

where the four-position "operator" (81) can be considered as a 4-current density operator in the energy-impulse space. Analogous considerations can be carried on for the anti-hermitian parts (see the last one of Refs.(Olkhovsky & Recami, 1968; 1969)).

Finally, by recalling the properties of the time operator as a maximal hermitian operator in the non-relativistic case (Sec.2.1), one can see that the relativistic time operator (81a) (for the Klein-Gordon case) is also a selfadjoint bilinear operator for the case of continuous energy spectra, and a (maximal) hermitian linear operator for free particles [due to the presence of the lower limit zero for the kinetic energy, or *m*<sup>0</sup> for the total energy].

#### **5. Decoherence (without instantaneous wave-function collapse**5**)**

In this paper we want to show, within the density matrix formalism, that a simple way to get decoherence is through the introduction of a "quantum" of time (or rather of a *chronon*): thus replacing the differential Liouville–von Neumann equation with a finite-difference version of it. In this way, one is given the possibility of using a very simple quantum equation to describe the decoherence effects due to dissipation, and of partially solving the measurement-problem in quantum mechanics (avoiding any recourse to the wave-function collapse). *Namely*, the mere introduction (not of a "time-lattice", but simply) of the "chronon" allows us to go on from differential to finite-difference equations; and in particular to write down the

<sup>5</sup> This section is developed by Erasmo Recami.

26 Will-be-set-by-IN-TECH

It is tempting to propose as *four-position operator* the quantity *z*ˆ*<sup>μ</sup>* = *x*ˆ*<sup>μ</sup>* + *i y*ˆ*μ*, whose hermitian

*i* 2

Let us recall the proportionality between the 4-momentum operator and the 4-current density operator in the chronotopical space, and then underline the canonical correspondence (in the 4-position and 4-momentum spaces, respectively) between the "operators" (cf. the previous

2

2

↔ *∂ ∂p*<sup>0</sup> (*a*)

↔ *∂ <sup>∂</sup><sup>p</sup>* , (*b*)

where the four-position "operator" (81) can be considered as a 4-current density operator in the energy-impulse space. Analogous considerations can be carried on for the anti-hermitian

Finally, by recalling the properties of the time operator as a maximal hermitian operator in the non-relativistic case (Sec.2.1), one can see that the relativistic time operator (81a) (for the Klein-Gordon case) is also a selfadjoint bilinear operator for the case of continuous energy spectra, and a (maximal) hermitian linear operator for free particles [due to the presence of

In this paper we want to show, within the density matrix formalism, that a simple way to get decoherence is through the introduction of a "quantum" of time (or rather of a *chronon*): thus replacing the differential Liouville–von Neumann equation with a finite-difference version of it. In this way, one is given the possibility of using a very simple quantum equation to describe the decoherence effects due to dissipation, and of partially solving the measurement-problem in quantum mechanics (avoiding any recourse to the wave-function collapse). *Namely*, the mere introduction (not of a "time-lattice", but simply) of the "chronon" allows us to go on from differential to finite-difference equations; and in particular to write down the

↔ *∂ <sup>∂</sup><sup>t</sup>* (*a*)

↔ *∂ <sup>∂</sup><sup>r</sup>* , (*b*)

↔ *∂ ∂p<sup>μ</sup>*

↔ *∂ ∂x<sup>μ</sup>* , (78)

. (79)

(80)

(81)

*x*ˆ *<sup>μ</sup>* <sup>=</sup> <sup>−</sup> *<sup>i</sup>* 2

to be associated with its corresponding "operator" in four-momentum space

*p*ˆ *μ <sup>h</sup>* = +

*<sup>m</sup>*<sup>0</sup> *<sup>ρ</sup>*<sup>ˆ</sup> <sup>≡</sup> *<sup>p</sup>*ˆ0 <sup>=</sup> *<sup>i</sup>*

*<sup>m</sup>*<sup>0</sup> <sup>ˆ</sup>*<sup>j</sup>* <sup>≡</sup> *<sup>p</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>−</sup> *<sup>i</sup>*

<sup>ˆ</sup>*<sup>t</sup>* ≡ − *<sup>i</sup>* 2

*<sup>x</sup>*<sup>ˆ</sup> <sup>≡</sup> *<sup>i</sup>* 2

parts (see the last one of Refs.(Olkhovsky & Recami, 1968; 1969)).

the lower limit zero for the kinetic energy, or *m*<sup>0</sup> for the total energy].

<sup>5</sup> This section is developed by Erasmo Recami.

**5. Decoherence (without instantaneous wave-function collapse**5**)**

**4.2 Four-position operators**

subsection):

and the operators

(Lorentz-covariant) part can be written

Schröedinger equation (as well as the Liouville–von Neumann equation) in three different ways: "retarded", "symmetrical", and "advanced". One of such three formulations — the *retarded* one — describes in an elementary way a system which is exchanging (and losing) energy with the environment. In its density-matrix version, indeed, it can be easily shown that all non-diagonal terms go to zero very rapidly.

Let us refer, in particular, to the theory of the *chronon* as proposed by P. Caldirola. Let us recall that such an interesting "finite difference" theory, forwards — at the classical level a solution for the motion of a particle endowed with a non-negligible charge in an external electromagnetic field, overcoming all the known difficulties met by Abraham-Lorentz's and Dirac's approaches (and even allowing a clear answer to the question whether a free falling charged particle does or does not emit radiation), and — at the quantum level — yields a remarkable mass spectrum for leptons.

It is easy to compare one another the *new* representations of Quantum Mechanics (QM) resulting from it, in the Schröedinger, Heisenberg and densityUoperator (Liouville–von ˝ Neumann) pictures, respectively.

For each representation, three (*retarded*, *symmetric* and *advanced*) *formulations* are possible, which refer either to times *t* and *t* − *τ*0, or to times *t* − *τ*0/2 and *t* + *τ*0/2, or to times *t* and *t* + *τ*0, respectively. It is interesting to notice that, when the chronon tends to zero, the ordinary QM is obtained as the limiting case of the "symmetric" formulation only; while the "retarded" one does naturally appear to describe QM with friction, i. e., to describe *dissipative* quantum systems (like a particle moving in an absorbing medium).

In this sense, *discretized* QM is much richer than the ordinary one. Here, we want to pay attention to the fact that, when applying the density matrix formalism to the solution of the *measurement problem* in QM, interesting results are met, as, for instance, a natural explication of the "decoherence" due to dissipation: which seem to reveal the power of dicretized (in particular, *retarded*) QM.

#### **5.1 On discretized Quantum Mechanics**

Let us approach our eventual application of the discretization procedures for a possible solution of the measurement problem in Quantum Mechanics, without having to make recourse to the reduction (wave-packet instantaneous collapse) postulate. Namely, let us focus our attention, now, on the consequences for QM of the introduction of a chronon. In QM, time will still be a continuous variable, but the evolution of the system along its world line will be regarded as discontinuous. In analogy with the electron theory in the *non-relativistic* limit, one has to substitute the corresponding finite-difference expression for the time derivatives; e. g.:

$$\frac{df(t)}{dt} = \frac{f(t) - f(t - \Delta t)}{\Delta t},\tag{82}$$

where proper time is now replaced by the local time *t*. The chronon procedure can then be applied to obtain the finite-difference form of the Schröedinger equation. As for the electron case, there are three different ways to perform the discretization, and three "Schröedinger equations" can be obtained:

$$i\frac{\hbar}{\tau}[\Psi(\mathbf{x},t) - \Psi(\mathbf{x},t-\tau)] = \hat{H}\,\Psi(\mathbf{x},t),\tag{83}$$

$$i\hbar\frac{\hbar}{2\pi}[\Psi(\mathbf{x},t+\tau)-\Psi(\mathbf{x},t-\tau)]=\hat{H}\,\Psi(\mathbf{x},t),\tag{84}$$

energy: *H*|*n*� = *En*|*n*�. Since the time evolution operator is a function of the Hamiltonian, and commutes with it, the basis of the energy eigenstates will be a basis also for this operator.

Time as Quantum Observable, Canonical Conjugated to Energy 45

which reduces to the LvN equation when *τ* → 0. The essential point is that, following e.g. a procedure similar to Bonifacio's, one gets in this case a *non-unitary* time-evolution operator:

> 1 + *iτ*L *h*¯

which, as all non-unitary operators, does not preserve the probabilities associated with each of the energy eigenstates (that make up the expansion of the initial state in such a basis of

Thus, the time-evolution operator (92) takes the initial density operator *ρ*in to a final state for

*<sup>h</sup>*¯ (*Er* <sup>−</sup> *Es*) <sup>≡</sup> <sup>1</sup>

*ρrs*(*t*) = *ρrs*(0) *e*−*γrst e*−*iνrst*

One can observe, indeed, that the non-diagonal terms tend to zero with time, and that the larger the value of *τ*, the faster the decay becomes. Actually, the chronon *τ* is now an interval of time related no longer to a single electron, but to the whole system O + A. If one imagines the time interval *τ* to be linked to the possibility of distinguishing two successive, different states of the system, then *τ* can be significantly larger than 10−<sup>23</sup> sec, implying an extremely

Thus, the reduction to the diagonal form occurs, provided that *τ* possesses a finite value, no matter how small, and provided that *wnmτ*, for every *n*, *m*, is not much smaller than 1; where

are the transition frequencies between the different energy eigenstates (the last condition being

It is essential to notice that *decoherence has been obtained above, without having recourse to any statistical approach, and in particular without assuming any "coarse graining" of time*. The reduction to the diagonal form illustrated by us is a consequence of the discrete (retarded) LiouvilleUvon ˝

Moreover, the measurement problem is still controversial even with regard to its mathematical approach: In the simplified formalization introduced above, however, we have not included any consideration beyond those common to the quantum formalism, allowing an as clear as

*rs*

1 + *w*<sup>2</sup>

*rsτ*2

<sup>−</sup>*t*/*<sup>τ</sup>*

1 + *iwrsτ*

−*t*/*<sup>τ</sup>*

*<sup>τ</sup>* <sup>=</sup> <sup>−</sup>*i*L*ρ*(*t*), (91)

, (92)

, (93)

*<sup>h</sup>*¯ (Δ*E*)*rs*. (94)

, (95)

, (96)

tan−1(*wrsτ*). (97)

*wnm* = (*En* − *Em*)/¯*h* (98)

*ρ*(*t*) − *ρ*(*t* − *τ*)

*<sup>V</sup>*(*t*, 0) =

eigenstates). We are interested in the time instants *t* = *kτ*, with *k* an integer.

*rs* <sup>=</sup> �*r*|*V*(*t*, 0)|*s*� <sup>=</sup> *<sup>ρ</sup>*in

*<sup>γ</sup>rs* <sup>≡</sup> <sup>1</sup> 2*τ* ln

> *<sup>ν</sup>rs* <sup>≡</sup> <sup>1</sup> *τ*

faster damping of the non-diagonal terms of the density operator.

Neumann equation only, once the inequality *wnmτ* � 1 is *not* verified.

possible recognition of the effects of the introduction of a chronon.

always satisfied, e. g., for non-bounded systems).

which the non-diagonal terms decay exponentially with time; namely, to

*wrs* <sup>≡</sup> <sup>1</sup>

*ρ*fin

where

with

Expression (93) can be written

The discretized (*retarded*) Liouville-von Neumann equation is

$$i\frac{\hbar}{\tau}[\Psi(\mathbf{x},t+\tau)-\Psi(\mathbf{x},t)]=\hat{H}\,\Psi(\mathbf{x},t),\tag{85}$$

which are, respectively, the *retarded*, *symmetric* and *advanced* Schröedinger equations, all of them transforming into the (same) continuous equation when the fundamental interval of time (that can now be called just *τ*) goes to zero.

Since the equations are different, the solutions they provide are also fundamentally different. In the classical theory of the electron the symmetric equation represented a non-radiating motion, providing only an approximate description of the motion (without taking into account the effects due to the self fields of the electron). However, in the quantum theory it plays a fundamental role. In the discrete formalism too, the symmetrical equation constitutes the only way to describe a bound non-radiating particle.

However, the solutions of the *retarded* (and *advanced*) equations show a completely different behaviour. For a Hamiltonian explicitly independent of time, the solutions have a general form given by

$$\Psi(\mathbf{x},t) = \left[1 + i\frac{\tau}{\hbar}\hat{H}\right]^{-t/\tau} f(\mathbf{x})\tag{86}$$

and, expanding *f*(*x*) in terms of the eigenfunctions of *H*ˆ :

$$
\hat{H}\,u\_n(\mathbf{x}) = \mathcal{W}\_n\,u\_n(\mathbf{x})\,.\tag{87}
$$

that is, writing *f*(*x*) = ∑ *n cn un*(*x*), with ∑ *n* |*cn*| <sup>2</sup> = 1, one can obtain that

$$\Psi(\mathbf{x},t) = \sum\_{\boldsymbol{\mathsf{H}}} c\_{\boldsymbol{\mathsf{H}}} \left[ \mathbf{1} + i \, \frac{\boldsymbol{\mathsf{T}}}{\hbar} \, \boldsymbol{\mathsf{W}}\_{\boldsymbol{\mathsf{H}}} \right]^{-t/\tau} u\_{\boldsymbol{\mathsf{H}}}(\mathbf{x}). \tag{88}$$

The norm of this solution is given by

$$|\Psi(\mathbf{x},t)|^2 = \sum\_{\mathbf{n}} |c\_{\mathbf{n}}|^2 \exp\left(-\gamma\_{\mathbf{n}}t\right) \tag{89}$$

with

$$\gamma\_{\boldsymbol{\eta}} = \frac{1}{\tau} \ln \left( 1 + \frac{\tau^2}{\hbar^2} W\_{\boldsymbol{\eta}}^2 \right) = \frac{W\_{\boldsymbol{\eta}}^2}{\hbar^2} \,\mathrm{\boldsymbol{\tau}} + \mathcal{O}\left(\tau^3\right), \tag{90}$$

where it is apparent that the damping factor depends critically on the value *τ* of the chronon. This *dissipative* behaviour originates from the character of the *retarded* equation; in the case of the electron, the retarded equation possesses intrinsically dissipative solutions, representing a radiating system. The Hamiltonian has the same status as in the ordinary (continuous) case: It is an observable, since it is a hermitian operator and its eigenvectors form a basis of the state space. However, as we have seen, the norm of the state vector is not constant any longer, due to the damping factor. An opposite behaviour is observed for the solutions of the advanced equation, in the sense that they increase exponentially.

#### **5.2 Discretized (retarded) Liouville equation, and a solution of the measurement problem: Decoherence from dissipation 5.2 Discretized (retarded) Liouville equation, and a solution of the measurement problem: Decoherence from dissipation**

Suppose we want to measure the dynamical variable *R* of a (microscopic) object O, by utilizing a (macroscopic) measuring apparatus A.

*In the discrete case* the interaction is embedded in the Hamiltonian *H*ˆ , with the following consequences. Let us consider the energy representation, where |*n*� are the states with defined energy: *H*|*n*� = *En*|*n*�. Since the time evolution operator is a function of the Hamiltonian, and commutes with it, the basis of the energy eigenstates will be a basis also for this operator. The discretized (*retarded*) Liouville-von Neumann equation is

$$\frac{\rho(t) - \rho(t-\tau)}{\tau} = -i\mathcal{L}\rho(t),\tag{91}$$

which reduces to the LvN equation when *τ* → 0. The essential point is that, following e.g. a procedure similar to Bonifacio's, one gets in this case a *non-unitary* time-evolution operator:

$$V(t,0) = \left[1 + \frac{i\tau\mathcal{L}}{\hbar}\right]^{-t/\tau} \text{ \AA} \tag{92}$$

which, as all non-unitary operators, does not preserve the probabilities associated with each of the energy eigenstates (that make up the expansion of the initial state in such a basis of eigenstates). We are interested in the time instants *t* = *kτ*, with *k* an integer.

Thus, the time-evolution operator (92) takes the initial density operator *ρ*in to a final state for which the non-diagonal terms decay exponentially with time; namely, to

$$
\rho\_{rs}^{\text{fin}} = \langle r | V(t, 0) | s \rangle = \rho\_{rs}^{\text{in}} \left[ 1 + i w\_{rs} \tau \right]^{-t/\tau}, \tag{93}
$$

where

28 Will-be-set-by-IN-TECH

which are, respectively, the *retarded*, *symmetric* and *advanced* Schröedinger equations, all of them transforming into the (same) continuous equation when the fundamental interval of

Since the equations are different, the solutions they provide are also fundamentally different. In the classical theory of the electron the symmetric equation represented a non-radiating motion, providing only an approximate description of the motion (without taking into account the effects due to the self fields of the electron). However, in the quantum theory it plays a fundamental role. In the discrete formalism too, the symmetrical equation constitutes the only

However, the solutions of the *retarded* (and *advanced*) equations show a completely different behaviour. For a Hamiltonian explicitly independent of time, the solutions have a general

<sup>−</sup>*t*/*<sup>τ</sup>*

*f*(*x*) (86)

*un*(*x*). (88)

<sup>2</sup> exp (−*γnt*) (89)

*<sup>h</sup>*¯ <sup>2</sup> *<sup>τ</sup>* <sup>+</sup> *<sup>O</sup>* (*τ*3), (90)

*H u* ˆ *<sup>n</sup>*(*x*) = *Wn un*(*x*), (87)

<sup>2</sup> = 1, one can obtain that

<sup>−</sup>*t*/*<sup>τ</sup>*

Ψ(*x*, *t*) =

*cn un*(*x*), with ∑

<sup>Ψ</sup>(*x*, *<sup>t</sup>*) = <sup>∑</sup>*<sup>n</sup>*


*<sup>γ</sup><sup>n</sup>* <sup>=</sup> <sup>1</sup> *τ* ln 1 + *τ*2 *<sup>h</sup>*¯ <sup>2</sup> *<sup>W</sup>*<sup>2</sup> *n* <sup>=</sup> *<sup>W</sup>*<sup>2</sup> *n*

equation, in the sense that they increase exponentially.

**Decoherence from dissipation**

a (macroscopic) measuring apparatus A.

**problem: Decoherence from dissipation** 

 1 + *i τ h*¯ *H*ˆ

*n* |*cn*|

*cn* 1 + *i τ <sup>h</sup>*¯ *Wn*

<sup>2</sup> <sup>=</sup> <sup>∑</sup>*<sup>n</sup>*


where it is apparent that the damping factor depends critically on the value *τ* of the chronon. This *dissipative* behaviour originates from the character of the *retarded* equation; in the case of the electron, the retarded equation possesses intrinsically dissipative solutions, representing a radiating system. The Hamiltonian has the same status as in the ordinary (continuous) case: It is an observable, since it is a hermitian operator and its eigenvectors form a basis of the state space. However, as we have seen, the norm of the state vector is not constant any longer, due to the damping factor. An opposite behaviour is observed for the solutions of the advanced

**5.2 Discretized (retarded) Liouville equation, and a solution of the measurement problem:**

**5.2 Discretized (retarded) Liouville equation, and a solution of the measurement** 

Suppose we want to measure the dynamical variable *R* of a (microscopic) object O, by utilizing

*In the discrete case* the interaction is embedded in the Hamiltonian *H*ˆ , with the following consequences. Let us consider the energy representation, where |*n*� are the states with defined

*<sup>τ</sup>* [Ψ(*x*, *<sup>t</sup>* <sup>+</sup> *<sup>τ</sup>*) <sup>−</sup> <sup>Ψ</sup>(*x*, *<sup>t</sup>*)] = *<sup>H</sup>*<sup>ˆ</sup> <sup>Ψ</sup>(*x*, *<sup>t</sup>*), (85)

*i h*¯

time (that can now be called just *τ*) goes to zero.

way to describe a bound non-radiating particle.

*n*

The norm of this solution is given by

and, expanding *f*(*x*) in terms of the eigenfunctions of *H*ˆ :

form given by

with

that is, writing *f*(*x*) = ∑

$$w\_{rs} \equiv \frac{1}{\hbar} \left( E\_r - E\_s \right) \equiv \frac{1}{\hbar} \left( \Delta E \right)\_{rs}.\tag{94}$$

Expression (93) can be written

$$
\rho\_{rs}(t) = \rho\_{rs}(0) \, e^{-\gamma\_{rs}t} \, e^{-i\nu\_{rs}t} \, \prime \tag{95}
$$

with

$$\gamma\_{rs} \equiv \frac{1}{2\pi} \ln\left(1 + w\_{rs}^2 \tau^2\right),\tag{96}$$

$$\nu\_{rs} \equiv \frac{1}{\tau} \tan^{-1}(w\_{rs}\tau). \tag{97}$$

One can observe, indeed, that the non-diagonal terms tend to zero with time, and that the larger the value of *τ*, the faster the decay becomes. Actually, the chronon *τ* is now an interval of time related no longer to a single electron, but to the whole system O + A. If one imagines the time interval *τ* to be linked to the possibility of distinguishing two successive, different states of the system, then *τ* can be significantly larger than 10−<sup>23</sup> sec, implying an extremely faster damping of the non-diagonal terms of the density operator.

Thus, the reduction to the diagonal form occurs, provided that *τ* possesses a finite value, no matter how small, and provided that *wnmτ*, for every *n*, *m*, is not much smaller than 1; where

$$w\_{nm} = (E\_n - E\_m) / \hbar \tag{98}$$

are the transition frequencies between the different energy eigenstates (the last condition being always satisfied, e. g., for non-bounded systems).

It is essential to notice that *decoherence has been obtained above, without having recourse to any statistical approach, and in particular without assuming any "coarse graining" of time*. The reduction to the diagonal form illustrated by us is a consequence of the discrete (retarded) LiouvilleUvon ˝ Neumann equation only, once the inequality *wnmτ* � 1 is *not* verified.

Moreover, the measurement problem is still controversial even with regard to its mathematical approach: In the simplified formalization introduced above, however, we have not included any consideration beyond those common to the quantum formalism, allowing an as clear as possible recognition of the effects of the introduction of a chronon.

2. Albrecht's operator (see Ref.(Albrecht, 1975)):

3. Ref.(Hasse, 1975):

1. Ref.(Gisin, 1982):

2. Ref.(Exner, 1983):

B) *Linear (non-hermitian) Hamiltonians:*

Lie-admissible type(Santilli, 1983).

**7. Some conclusions**

where � � is the averaging produced over |Ψ(*x*)|

*<sup>W</sup>*<sup>ˆ</sup> *<sup>H</sup>*(*x*) = <sup>1</sup>

where [*A*, *B*]+ is the anticommutator: [*A*, *B*]+ = *AB* + *BA*.

1983), even if they are not based on the Schröedinger equation.

as for analyzing a few irreversible phenomena. See the Appendix.

particle collisions, within a wide class of Hamiltonians.

4 

One might recall also the important, so-called "microscopic models"(Caldeira & Leggett,

All such proposals are to be further investigated, and completed, since they have not been apparently exploited enough, till now. Let us remark, just as an example, that it would be desirable to take into deeper consideration other related phenomena, like the ones associated with the "Hartman effect" (and "generalized Hartman effect") (Aharonov, 2002; Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; 2002; 2004; 2005; Recami, 2004), in the case of tunneling with dissipation: a topic faced in few papers, like (Nimtz et al., 1994; Raciti & Salesi, 1994). As a small contribution of ours, in the Appendix we present a scheme of iterations (successive approximations) as a possible tool for explicit calculations of wave-functions in the presence of dissipation, by using as an example the simple Albreht's potential. Our scheme may be useful, in any case, for the investigation of possible violations of the Hartman effect, as well

At last, let us incidentally recall that two generalized Schröedinger equations, introduced by Caldirola (Caldirola, 1941; 1976a;b; 1977) in order to describe two different dissipative processes (behavior of open systems, and the radiation of a charged particle) have been shown — see, e.g., Ref.(Mignani, 1983)) — to possess the same algebraic structure of the

1. We have shown that the Time operator (1), hermitian even if non-selfadjoint, works for any quantum collisions or motions, in the case of a continuum energy spectrum, in non-relativistic quantum mechanics and in one-dimensional quantum electrodynamics. The uniqueness of the (maximal) hermitian time operator (1) directly follows from the uniqueness of the Fourier-transformations from the time to the energy representation. The time operator (1) has been fruitfully used in the case, for instance, of tunnelling times (see Refs. (Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007)), and of nuclear reactions and decays (see Refs. (Olkhovsky, 1984; 1990; 1992; 1998) and also Ref. (Olkhovsky et al., 2006)). We have discussed the advantages of such an approach with respect to POVM's, which is not applicable for three-dimensional

*<sup>W</sup>*<sup>ˆ</sup> *<sup>A</sup>*(*x*) = �*<sup>p</sup>* �(*<sup>x</sup>* − �*x*�) , (103)

<sup>+</sup> , (104)

*<sup>H</sup>*<sup>ˆ</sup> *<sup>G</sup>* = (<sup>1</sup> <sup>−</sup> *<sup>i</sup>γ*) *<sup>H</sup>*<sup>ˆ</sup> <sup>+</sup> *<sup>i</sup>γ*�*H*<sup>ˆ</sup> � ; (105)

*<sup>H</sup>*<sup>ˆ</sup> *<sup>E</sup>* <sup>=</sup> *<sup>H</sup>*<sup>ˆ</sup> <sup>+</sup> *<sup>i</sup> <sup>W</sup>*<sup>ˆ</sup> (*x*) <sup>−</sup> *<sup>i</sup>* �*W*<sup>ˆ</sup> (*x*)� . (106)

2;

Time as Quantum Observable, Canonical Conjugated to Energy 47

*x* − �*x*�, *p* + �*p* �

#### **6. Non-hermitian Hamiltonians and microscopic quantum dissipation**

Various different approached are known, aimed at getting dissipation — and possibly decoherence — within quantum mechanics. First of all, the simple introduction of a "chronon" (see, e.g., Refs. (Caldirola et al., 1978; Caldirola, 1979; Caldirola & Montaldi, 1979; Caldirola & Lugiato, 1982; Caldirola, 1983; Farias & Recami, 2007)) allows one to go on from differential to finite-difference equations, and in particular to write down the quantum theoretical equations (Schrödinger's, Liouville-von Neumann's, etc.) in three different ways: symmetrical, retarded, and advanced. The retarded "Schrödinger" equation describes in a rather simple and natural way a dissipative system, which exchanges (loses) energy with the environment. The corresponding non-unitary time-evolution operator obeys a semigroup law and refers to irreversible processes. The retarded approach furnishes, moreover, an interesting way for solving the "measurement problem" in quantum mechanics, without any need for a wave-function collapse: see Refs.(Bonifacio, 1983; Bonifacio & Caldirola, 1983; Farias & Recami, 2007; Ghirardi & Weber, 1984; Recami & Farias, 2002). The chronon theory can be regarded as a peculiar "coarse grained" description of the time evolution.

Let us stress that it has been shown that the mentioned discrete approach can be replaced with a continuous one, at the price of introducing a *non-hermitian* Hamiltonian: see, e.g., Ref.(Casagrande & Montaldi, 1977).

Further relevant work can be found, for instance, in papers like (Caldirola, 1941; Janussis et al., 1980; 1981a;b; 1982a;b;c; 1984; 1991; 1995; Mignani, 1983) and refs. therein.

Let us add, at this point, that much work is still needed for the description of time irreversibility at the microscopic level. Indeed, various approaches have been proposed, in which new parameters are introduced (regulation or dissipation) into the microscopic dynamics (building a bridge, in a sense, between microscopic structure and macroscopic characteristics). Besides the Caldirola-Kanai(Caldirola, 1941; Kanai, 1948) Hamiltonian

$$
\hat{H}\_{\mathbb{C}K}(t) = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} e^{-\gamma t} + V(\mathfrak{x}) \ e^{\gamma t} \tag{99}
$$

(which has been used, e.g., in Ref.(Angelopoulon et al., 1995)), other rather simple approaches, based of course on the Schrödinger equation

$$i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{x},t) = \hat{H}\Psi(\mathbf{x},t) \,. \tag{100}$$

and adopting a microscopic dissipation defined via a coefficient of extinction *γ*, are for instance the following ones:

A) *Non-linear (non-hermitian) Hamiltonians*

$$
\hat{H}\_{nl} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial \mathbf{x}^2} + V(\mathbf{x}) + \hat{W} \,. \tag{101}
$$

with "potential" operators *W*ˆ of the type:

1. Kostin's operator (see Ref.(Kostin, 1972)):

$$\hat{\mathcal{W}}\_{\mathbf{K}} = -\frac{i\hbar}{2m} \left\{ \frac{\ln \Psi}{\Psi^\*} - \left\langle \frac{\ln \Psi}{\Psi^\*} \right\rangle \right\};\tag{102}$$

2. Albrecht's operator (see Ref.(Albrecht, 1975)):

$$
\hat{W}\_A(\mathbf{x}) = \left< p \right> \left( \mathbf{x} - \left< \mathbf{x} \right> \right) \,, \tag{103}
$$

where � � is the averaging produced over |Ψ(*x*)| 2;

3. Ref.(Hasse, 1975):

30 Will-be-set-by-IN-TECH

Various different approached are known, aimed at getting dissipation — and possibly decoherence — within quantum mechanics. First of all, the simple introduction of a "chronon" (see, e.g., Refs. (Caldirola et al., 1978; Caldirola, 1979; Caldirola & Montaldi, 1979; Caldirola & Lugiato, 1982; Caldirola, 1983; Farias & Recami, 2007)) allows one to go on from differential to finite-difference equations, and in particular to write down the quantum theoretical equations (Schrödinger's, Liouville-von Neumann's, etc.) in three different ways: symmetrical, retarded, and advanced. The retarded "Schrödinger" equation describes in a rather simple and natural way a dissipative system, which exchanges (loses) energy with the environment. The corresponding non-unitary time-evolution operator obeys a semigroup law and refers to irreversible processes. The retarded approach furnishes, moreover, an interesting way for solving the "measurement problem" in quantum mechanics, without any need for a wave-function collapse: see Refs.(Bonifacio, 1983; Bonifacio & Caldirola, 1983; Farias & Recami, 2007; Ghirardi & Weber, 1984; Recami & Farias, 2002). The chronon theory can be

Let us stress that it has been shown that the mentioned discrete approach can be replaced with a continuous one, at the price of introducing a *non-hermitian* Hamiltonian: see, e.g.,

Further relevant work can be found, for instance, in papers like (Caldirola, 1941; Janussis et

Let us add, at this point, that much work is still needed for the description of time irreversibility at the microscopic level. Indeed, various approaches have been proposed, in which new parameters are introduced (regulation or dissipation) into the microscopic dynamics (building a bridge, in a sense, between microscopic structure and macroscopic characteristics). Besides the Caldirola-Kanai(Caldirola, 1941; Kanai, 1948) Hamiltonian

*∂*2

(which has been used, e.g., in Ref.(Angelopoulon et al., 1995)), other rather simple approaches,

and adopting a microscopic dissipation defined via a coefficient of extinction *γ*, are for

*∂*2

ln Ψ Ψ<sup>∗</sup> − ln Ψ Ψ∗

*<sup>∂</sup>x*<sup>2</sup> *<sup>e</sup>*−*γ<sup>t</sup>* <sup>+</sup> *<sup>V</sup>*(*x*) *<sup>e</sup>γ<sup>t</sup>* (99)

Ψ(*x*, *t*) = *H*ˆ Ψ(*x*, *t*), (100)

*<sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *<sup>V</sup>*(*x*) + *<sup>W</sup>*<sup>ˆ</sup> , (101)

; (102)

**6. Non-hermitian Hamiltonians and microscopic quantum dissipation**

regarded as a peculiar "coarse grained" description of the time evolution.

al., 1980; 1981a;b; 1982a;b;c; 1984; 1991; 1995; Mignani, 1983) and refs. therein.

*<sup>H</sup>*<sup>ˆ</sup> *CK*(*t*) = <sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup>

*ih*¯ *<sup>∂</sup> ∂t*

*<sup>H</sup>*<sup>ˆ</sup> *nl* <sup>=</sup> <sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup>

*<sup>W</sup>*<sup>ˆ</sup> *<sup>K</sup>* <sup>=</sup> <sup>−</sup> *ih*¯

2*m*

2*m*

2*m*

Ref.(Casagrande & Montaldi, 1977).

based of course on the Schrödinger equation

A) *Non-linear (non-hermitian) Hamiltonians*

with "potential" operators *W*ˆ of the type:

1. Kostin's operator (see Ref.(Kostin, 1972)):

instance the following ones:

*<sup>W</sup>*<sup>ˆ</sup> *<sup>H</sup>*(*x*) = <sup>1</sup> 4 *x* − �*x*�, *p* + �*p* � <sup>+</sup> , (104)

where [*A*, *B*]+ is the anticommutator: [*A*, *B*]+ = *AB* + *BA*.


$$
\hat{H}\_{\rm G} = \left(1 - i\gamma\right)\hat{H} + i\gamma\langle\hat{H}\rangle\,\,;\tag{105}
$$

2. Ref.(Exner, 1983):

$$
\hat{H}\_E = \hat{H} + i\,\hat{\mathcal{W}}(\mathbf{x}) - i\,\langle \hat{\mathcal{W}}(\mathbf{x}) \rangle \,. \tag{106}
$$

One might recall also the important, so-called "microscopic models"(Caldeira & Leggett, 1983), even if they are not based on the Schröedinger equation.

All such proposals are to be further investigated, and completed, since they have not been apparently exploited enough, till now. Let us remark, just as an example, that it would be desirable to take into deeper consideration other related phenomena, like the ones associated with the "Hartman effect" (and "generalized Hartman effect") (Aharonov, 2002; Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; 2002; 2004; 2005; Recami, 2004), in the case of tunneling with dissipation: a topic faced in few papers, like (Nimtz et al., 1994; Raciti & Salesi, 1994).

As a small contribution of ours, in the Appendix we present a scheme of iterations (successive approximations) as a possible tool for explicit calculations of wave-functions in the presence of dissipation, by using as an example the simple Albreht's potential. Our scheme may be useful, in any case, for the investigation of possible violations of the Hartman effect, as well as for analyzing a few irreversible phenomena. See the Appendix.

At last, let us incidentally recall that two generalized Schröedinger equations, introduced by Caldirola (Caldirola, 1941; 1976a;b; 1977) in order to describe two different dissipative processes (behavior of open systems, and the radiation of a charged particle) have been shown — see, e.g., Ref.(Mignani, 1983)) — to possess the same algebraic structure of the Lie-admissible type(Santilli, 1983).

#### **7. Some conclusions**

1. We have shown that the Time operator (1), hermitian even if non-selfadjoint, works for any quantum collisions or motions, in the case of a continuum energy spectrum, in non-relativistic quantum mechanics and in one-dimensional quantum electrodynamics. The uniqueness of the (maximal) hermitian time operator (1) directly follows from the uniqueness of the Fourier-transformations from the time to the energy representation. The time operator (1) has been fruitfully used in the case, for instance, of tunnelling times (see Refs. (Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007)), and of nuclear reactions and decays (see Refs. (Olkhovsky, 1984; 1990; 1992; 1998) and also Ref. (Olkhovsky et al., 2006)). We have discussed the advantages of such an approach with respect to POVM's, which is not applicable for three-dimensional particle collisions, within a wide class of Hamiltonians.

with the environment (Farias & Recami, 2007; Recami & Farias, 2002). This topic is touched in Section 5; together with questions related with collisions in absorbing media. In particular, in Sec.5 we mention also the case of the optical model in nuclear physics; without forgetting that non-hermitian operators show up even in the case of tunnelling — e. g., in fission phenomena — with quantum dissipation, and of quantum friction. As to the former topic of microscopic quantum dissipation, among the many approaches to quantum irreversibility we have discussed in Sec. 5.2 a possible solution of the quantum measurement problem (via interaction with the environment) by the introduction of finite-difference equations (e. g., in

Time as Quantum Observable, Canonical Conjugated to Energy 49

6. Let us eventually observe that the "dual equations" (26) and (27) seem to be promising also for the study the initial stage of our cosmos, when tunnellings can take place through the barriers which appear in quantum gravity in the limiting case of quasi-Schröedinger

Part of this paper is based on work performed by one of us in cooperation with P. Smrz, and with A. Agodi, M. Baldo and A. Pennisi di Floristella. Thanks are moreover due for stimulating discussions, or kind collaboration, to Y. Aharonov, S. V. Belchikov, G. Battistoni, R. H. A. Farias, L. Fraietta, A. S. Holevo, V. L. Lyuboshitz, R. Mignani, V. Petrillo, P. Pizzochero,

**Time-dependent Schrödinger equation with dissipative terms**

where we put ¯*h* = 1. Let us rewrite the time-dependent wave function (WF), Ψ(*x*, *t*) (which

where *ϕ*(*E*, *x*) is the WF component independent of time, and *g*(*E*) is a weight factor. One can

*<sup>g</sup>*(*E*) = *A e*−*a*<sup>2</sup>(*k*−¯

of the WP. We substitute the Fourier-expansion (108) of WF into eq. (107). Thus, the l.h.s. of

*E*0

0

*k*)<sup>2</sup>

*<sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *<sup>V</sup>*(*x*, *<sup>t</sup>*)

Ψ(*x*, *t*), (107)

. (109)

*g*(*E*)*e*−*iEt ϕ*(*E*, *x*) *dE* , (108)

*g*(*E*) *e*−*iEt ϕ*(*E*, *x*) *EdE* . (110)

*k* is the selected value for the impulse, constituting the center

− *∂*2

terms of a "chronon").

**8. Acknowledgements**

G. Salesi, B. N. Zakhariev, and M. Zamboni-Rached.

Let's consider the time-dependent Schrödinger equation:

<sup>Ψ</sup>(*x*, *<sup>t</sup>*) =

can be considered as a wave-packet (WP)), in the form of a Fourier integral:

*E*0

0

<sup>Ψ</sup>(*x*, *<sup>t</sup>*) =

<sup>Ψ</sup>(*x*, *<sup>t</sup>*) =

*i ∂ ∂t*

choose the function *g*(*E*) to be, e.g., a *Gaussian*:

*i ∂ ∂t*

Here, *A* and *a* are constants, and ¯

this equation transforms into

equations.

**9. Appendix**

**9.1 Introduction**

The mathematical properties of the present Time operator have actually demonstrated — without introducing any new physical postulates — that *time* can be regarded as a quantum-mechanical observable, at the same degree of other physical quantities (energy, momentum, spatial coordinates,...).

The commutation relations (eqs. (8), (22), (31)) here analyzed, and the uncertainty relations (9), result to be analogous to those known for other pairs of canonically conjugate observables (as for coordinate *x*ˆ and momentum *p*ˆ*x*, in the case of eq. (9)). Of course, our new relations do not replace, but merely extend the meaning of the classic time and energy uncertainties, given e.g. in Ref. (Olkhovsky & Recami, 2008). In subsection 2.6, we have studied the properties of Time, as an observable, for quantum-mechanical systems with *discrete* energy spectra.

2. Let us recall that the Time operator (1), and relations (2), (3), (4), (15), (16), have been used for the temporal analysis of nuclear reactions and decays in Refs. (Olkhovsky, 1984; 1990; 1992; 1998); as well as of new phenomena, about time delays-advances in nuclear physics and about time resonances or explosions of highly excited compound nuclei, in Refs. (D'Arrigo et al., 1992; 1993; Olkhovsky & Doroshko, 1992; Olkhovsky et al., 2006). Let us also recall that, besides the time operator, other quantities, to which (maximal) hermitian operators correspond, can be analogously regarded as quantum-physical observables: For example, von Neumann himself (Recami, 1976; 1977; Von Neumann, 1955)) considered the case of the momentum operator −*i∂*/*∂x* in a semi-space with a rigid wall orthogonal to the *x*-axis at *x* = 0, or of the radial momentum −*i∂*/*∂r*, even if both act on packets defined only over the positive *x* or *r* axis, respectively.

Subsection 2.5 has been devoted to a new "hamiltonian approach": namely, to the introduction of the analogue of the "Hamiltonian" for the case of the Time operator.

3. In Section 3, we have proposed a suitable generalization for the Time operator (or, rather, for a Space-Time operator) in relativistic quantum mechanics. For instance, for the Klein-Gordon case, we have shown that the hermitian part of the three-position operator *x*ˆ is nothing but the Newton-Wigner operator, and corresponds to a point-like position; while its anti-hermitian part can be regarded as yielding the sizes of an extended-type (ellipsoidal) localization. When dealing with a 4-position operator, one finds that the Time operator is selfadjoint for unbounded energy spectra, while it is a (maximal) hermitian operator when the kinetic energy, and the total energy, are bounded from below, as for a free particle. We have extensively made recourse, in the latter case, to *bilinear* forms, which dispense with the necessity of eliminating the lower point — corresponding to zero velocity — of the spectra. It would be interesting to proceed to further generalizations of the 3- and 4-position operator for other relativistic cases, and analyze the localization problems associated with Dirac particles, or in 2D and 3D quantum electrodynamics, etc. Work is in progress on time analyses in 2D quantum electrodynamic, for application, e.g., to frustrated (almost total) internal reflections. Further work has still to be done also about the joint consideration of particles and antiparticles.

4. Section 4 has been devoted to the association of unstable states (decaying "resonances") with the eigenvectors of quasi-hermitian (Agodi et al., 1973; Olkhovsky et al., 2006; Recami et al., 1983) Hamiltonians.

5. Non-hermitian Hamiltonians, and non-unitary time-evolution operators, can play an important role also in microscopic quantum dissipation (Bonifacio, 1983; Bonifacio & Caldirola, 1983; Caldirola, 1941; Caldirola et al., 1978; Caldirola, 1979; Caldirola & Montaldi, 1979; Caldirola & Lugiato, 1982; Caldirola, 1983; Casagrande & Montaldi, 1977; Farias & Recami, 2007; Ghirardi & Weber, 1984; Janussis et al., 1980; 1981a;b; 1982a;b;c; 1984; 1991; 1995; Mignani, 1983; Recami & Farias, 2002): namely, in getting decoherence through interaction

with the environment (Farias & Recami, 2007; Recami & Farias, 2002). This topic is touched in Section 5; together with questions related with collisions in absorbing media. In particular, in Sec.5 we mention also the case of the optical model in nuclear physics; without forgetting that non-hermitian operators show up even in the case of tunnelling — e. g., in fission phenomena — with quantum dissipation, and of quantum friction. As to the former topic of microscopic quantum dissipation, among the many approaches to quantum irreversibility we have discussed in Sec. 5.2 a possible solution of the quantum measurement problem (via interaction with the environment) by the introduction of finite-difference equations (e. g., in terms of a "chronon").

6. Let us eventually observe that the "dual equations" (26) and (27) seem to be promising also for the study the initial stage of our cosmos, when tunnellings can take place through the barriers which appear in quantum gravity in the limiting case of quasi-Schröedinger equations.

#### **8. Acknowledgements**

Part of this paper is based on work performed by one of us in cooperation with P. Smrz, and with A. Agodi, M. Baldo and A. Pennisi di Floristella. Thanks are moreover due for stimulating discussions, or kind collaboration, to Y. Aharonov, S. V. Belchikov, G. Battistoni, R. H. A. Farias, L. Fraietta, A. S. Holevo, V. L. Lyuboshitz, R. Mignani, V. Petrillo, P. Pizzochero, G. Salesi, B. N. Zakhariev, and M. Zamboni-Rached.

#### **9. Appendix**

32 Will-be-set-by-IN-TECH

The mathematical properties of the present Time operator have actually demonstrated — without introducing any new physical postulates — that *time* can be regarded as a quantum-mechanical observable, at the same degree of other physical quantities (energy,

The commutation relations (eqs. (8), (22), (31)) here analyzed, and the uncertainty relations (9), result to be analogous to those known for other pairs of canonically conjugate observables (as for coordinate *x*ˆ and momentum *p*ˆ*x*, in the case of eq. (9)). Of course, our new relations do not replace, but merely extend the meaning of the classic time and energy uncertainties, given e.g. in Ref. (Olkhovsky & Recami, 2008). In subsection 2.6, we have studied the properties of Time, as an observable, for quantum-mechanical systems with *discrete* energy spectra.

2. Let us recall that the Time operator (1), and relations (2), (3), (4), (15), (16), have been used for the temporal analysis of nuclear reactions and decays in Refs. (Olkhovsky, 1984; 1990; 1992; 1998); as well as of new phenomena, about time delays-advances in nuclear physics and about time resonances or explosions of highly excited compound nuclei, in Refs. (D'Arrigo et al., 1992; 1993; Olkhovsky & Doroshko, 1992; Olkhovsky et al., 2006). Let us also recall that, besides the time operator, other quantities, to which (maximal) hermitian operators correspond, can be analogously regarded as quantum-physical observables: For example, von Neumann himself (Recami, 1976; 1977; Von Neumann, 1955)) considered the case of the momentum operator −*i∂*/*∂x* in a semi-space with a rigid wall orthogonal to the *x*-axis at *x* = 0, or of the radial momentum −*i∂*/*∂r*, even if both act on packets defined only over the

Subsection 2.5 has been devoted to a new "hamiltonian approach": namely, to the introduction

3. In Section 3, we have proposed a suitable generalization for the Time operator (or, rather, for a Space-Time operator) in relativistic quantum mechanics. For instance, for the Klein-Gordon case, we have shown that the hermitian part of the three-position operator *x*ˆ is nothing but the Newton-Wigner operator, and corresponds to a point-like position; while its anti-hermitian part can be regarded as yielding the sizes of an extended-type (ellipsoidal) localization. When dealing with a 4-position operator, one finds that the Time operator is selfadjoint for unbounded energy spectra, while it is a (maximal) hermitian operator when the kinetic energy, and the total energy, are bounded from below, as for a free particle. We have extensively made recourse, in the latter case, to *bilinear* forms, which dispense with the necessity of eliminating the lower point — corresponding to zero velocity — of the spectra. It would be interesting to proceed to further generalizations of the 3- and 4-position operator for other relativistic cases, and analyze the localization problems associated with Dirac particles, or in 2D and 3D quantum electrodynamics, etc. Work is in progress on time analyses in 2D quantum electrodynamic, for application, e.g., to frustrated (almost total) internal reflections. Further work has still to be done also about the joint consideration of particles and antiparticles. 4. Section 4 has been devoted to the association of unstable states (decaying "resonances") with the eigenvectors of quasi-hermitian (Agodi et al., 1973; Olkhovsky et al., 2006; Recami et

5. Non-hermitian Hamiltonians, and non-unitary time-evolution operators, can play an important role also in microscopic quantum dissipation (Bonifacio, 1983; Bonifacio & Caldirola, 1983; Caldirola, 1941; Caldirola et al., 1978; Caldirola, 1979; Caldirola & Montaldi, 1979; Caldirola & Lugiato, 1982; Caldirola, 1983; Casagrande & Montaldi, 1977; Farias & Recami, 2007; Ghirardi & Weber, 1984; Janussis et al., 1980; 1981a;b; 1982a;b;c; 1984; 1991; 1995; Mignani, 1983; Recami & Farias, 2002): namely, in getting decoherence through interaction

of the analogue of the "Hamiltonian" for the case of the Time operator.

momentum, spatial coordinates,...).

positive *x* or *r* axis, respectively.

al., 1983) Hamiltonians.

#### **Time-dependent Schrödinger equation with dissipative terms**

#### **9.1 Introduction**

Let's consider the time-dependent Schrödinger equation:

$$i\frac{\partial}{\partial t}\Psi(\mathbf{x},t) = \left(-\frac{\partial^2}{\partial \mathbf{x}^2} + V(\mathbf{x},t)\right)\Psi(\mathbf{x},t),\tag{107}$$

where we put ¯*h* = 1. Let us rewrite the time-dependent wave function (WF), Ψ(*x*, *t*) (which can be considered as a wave-packet (WP)), in the form of a Fourier integral:

$$\Psi(\mathbf{x},t) = \int\_0^{E\_0} \mathbf{g}(E) \, e^{-iEt} \, \boldsymbol{\varrho}(E, \mathbf{x}) \, dE,\tag{108}$$

where *ϕ*(*E*, *x*) is the WF component independent of time, and *g*(*E*) is a weight factor. One can choose the function *g*(*E*) to be, e.g., a *Gaussian*:

$$\lg(E) = A \, e^{-a^2(k-\tilde{k})^2} \,. \tag{109}$$

Here, *A* and *a* are constants, and ¯ *k* is the selected value for the impulse, constituting the center of the WP. We substitute the Fourier-expansion (108) of WF into eq. (107). Thus, the l.h.s. of this equation transforms into

$$i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{x},t) = \int\_0^{E\_0} g(E) \, e^{-iEt} \, \boldsymbol{\varrho}(E,\mathbf{x}) \, E d\mathbf{E} \,. \tag{110}$$

where the averages are fulfilled by integrating over *x* by means of the functions Ψ∗(*x*, *t*) and

Time as Quantum Observable, Canonical Conjugated to Energy 51

*dE*<sup>2</sup> *g*(*E*1) *g*(*E*2) *e*

*<sup>i</sup>*(*E*1−*E*<sup>2</sup> )*<sup>t</sup> <sup>ϕ</sup>*∗(*E*1, *<sup>x</sup>*) *∂ϕ*(*E*2, *<sup>x</sup>*)

*dE*<sup>4</sup> *g*(*E*1) *g*(*E*2) *g*(*E*3) *g*(*E*4)

*dE*<sup>4</sup> *g*(*E*1) *g*(*E*2) *g*(*E*3) *g*(*E*4)

*ϕ*∗(*E*3, *x*2) *ϕ*(*E*4, *x*2).

*dE*<sup>4</sup> *g*(*E*3) *g*(*E*4) *ei*(*E*3−*E*4)*<sup>t</sup> x ϕ*∗(*E*3, *x*) *ϕ*(*E*4, *x*);

*<sup>∂</sup><sup>x</sup>* ,

(118)

(119)

(120)

Ψ(*x*, *t*). For the right part of eq. (117) one gets

 *dx E*0

 *dx E*0

so that the total potential *V*(*x*, *t*) becomes

<sup>×</sup> *<sup>e</sup>i*(*E*1−*E*2+*E*3−*E*<sup>4</sup> )*<sup>t</sup>*

*dE g*(*E*) *ϕ*(*E*, *x*)

*dE g*(*E*) *ϕ*(*E*, *x*)

) *ϕ*(*E*�

0

*dE*<sup>3</sup> *E*0

 *x* − *x*<sup>2</sup> 

*V*(*x*, *E*¯, *t*)*e*

0

 *dx*<sup>1</sup> *dx*<sup>2</sup> *E*0

*dE g*(*E*) *ϕ*(*E*, *x*) *V*0(*x*) *δ*(*E*� − *E*) −

*<sup>ϕ</sup>*∗(*E*1, *<sup>x</sup>*1) *∂ϕ*(*E*2, *<sup>x</sup>*1)

*dE*<sup>1</sup> *E*0

*<sup>ϕ</sup>*∗(*E*1, *<sup>x</sup>*1) *∂ϕ*(*E*2, *<sup>x</sup>*1)

) *E*� *ϕ*(*E*�

0

*dE*<sup>1</sup> *E*0

0

, *x*) *V*0(*x*) −

0

*g*(*E*�

× *x* − *x*<sup>2</sup> 

− *iγ dx*<sup>1</sup> *dx*<sup>2</sup> *E*0

 *dx*<sup>1</sup> *dx*<sup>2</sup> *E*0

*∂x*<sup>1</sup>

*dE*<sup>2</sup> *E*0

*∂x*<sup>1</sup>

As a consequence, the whole eq. (115) gets transformed into

0

, *x*) = −*g*(*E*�

*dE*<sup>2</sup> *E*0

*<sup>ϕ</sup>*∗(*E*1, *<sup>x</sup>*1) *∂ϕ*(*E*2, *<sup>x</sup>*1)

0

*∂x*<sup>1</sup>

0

*dE*<sup>1</sup> *E*0

0

0

0

*dE*<sup>1</sup> *E*0

Taking into account this, we find the second term, in the r.h.s. of eq. (115), to be:

*i*(*E*�

0

*dE*<sup>3</sup> *E*0

0

) *∂*2*ϕ*(*E*�

*dE*<sup>3</sup> *E*0

0

*dE*<sup>1</sup> *E*0

0

*dE*<sup>2</sup> *E*0

0

*ϕ*∗(*E*3, *x*2) *ϕ*(*E*4, *x*2) *ϕ*(*E*��, *x*),

, *x*) *<sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *<sup>g</sup>*(*E*�

*dE*<sup>3</sup> *E*0

0

*dE*<sup>4</sup> *g*(*E*1) *g*(*E*2) *g*(*E*3) *g*(*E*4) *g*(*E*��)

*E*�� = *E*� + *E*<sup>1</sup> − *E*<sup>2</sup> + *E*<sup>3</sup> − *E*4. (121)

) *ϕ*(*E*�

*ϕ*∗(*E*3, *x*2) *ϕ*(*E*4, *x*2) *ϕ*(*E*��, *x*)

*dE*<sup>4</sup> *g*(*E*1) *g*(*E*2) *g*(*E*3) *g*(*E*4) *g*(*E*��)

, *x*) *V*0(*x*)

*ϕ*∗(*E*3, *x*2) *ϕ*(*E*4, *x*2) *δ*(*E*� − *E* + *E*<sup>1</sup> − *E*<sup>2</sup> + *E*<sup>3</sup> − *E*4) =

0

*dE*<sup>2</sup> *E*0

<sup>−</sup>*E*)*<sup>t</sup> dt* =

0

*<sup>ϕ</sup>*∗(*E*1, *<sup>x</sup>*1) *∂ϕ*(*E*2, *<sup>x</sup>*1)

*dE*<sup>3</sup> *E*0

*∂x*<sup>1</sup>

0

< *p* > = −*i*

< *x* > =

*V*(*x*, *t*) = *V*0(*x*) − *iγ*

1 2*π E*0

= *E*0

0

0

0

= *g*(*E*�

− *iγ dx*<sup>1</sup> *dx*<sup>2</sup> *E*0

× *x* − *x*<sup>2</sup> 

where

− *iγ E*0

× *x* − *x*<sup>2</sup> 

Afterwards, the r.h.s. of eq. (107) gets transformed into

$$\left(-\frac{\partial^2}{\partial \mathbf{x}^2} + V(\mathbf{x}, t)\right)\Psi(\mathbf{x}, t) = -\int\_0^{\underline{\mathcal{E}\_0}} \mathbf{g}(\mathbf{E}) e^{-i\underline{\mathcal{E}}t} \frac{\partial^2 \varrho(\mathbf{E}, \mathbf{x})}{\partial \mathbf{x}^2} \, d\mathbf{E} + \int\_0^{\underline{\mathcal{E}\_0}} \mathbf{g}(\mathbf{E}) V(\mathbf{x}, \mathbf{\bar{E}}, t) e^{-i\underline{\mathcal{E}}t} \varrho(\mathbf{E}, \mathbf{x}) \, d\mathbf{E}.\tag{111}$$

Therefore, the whole equation (107) has been transformed into

$$\int\_{0}^{E\_{0}} \operatorname{g}(\mathbf{E}) \, e^{-i\mathbf{E}\cdot\mathbf{t}} \, \boldsymbol{\varrho}(\mathbf{E}, \mathbf{x}) \, \mathrm{E}d\mathbf{E} = -\int\_{0}^{E\_{0}} \mathbf{g}(\mathbf{E}) \, e^{-i\mathbf{E}\cdot\mathbf{t}} \frac{\partial^{2} \boldsymbol{\varrho}(\mathbf{E}, \mathbf{x})}{\partial \mathbf{x}^{2}} \, d\mathbf{E} + \int\_{0}^{E\_{0}} \mathbf{g}(\mathbf{E}) \, V(\mathbf{x}, \mathbf{E}, t) \, e^{-i\mathbf{E}\cdot\mathbf{t}} \, \boldsymbol{\varrho}(\mathbf{E}, \mathbf{x}) \, d\mathbf{E}.\tag{112}$$

Let us now apply the inverse Fourier-transformation to this equation. Its left part becomes

$$\begin{split} \frac{1}{2\pi} \int dt \, e^{iE't} \int \limits\_{0}^{E\_0} \oint \left( E \right) e^{-iEt} \, \boldsymbol{\varrho}(E, \mathbf{x}) \, \mathrm{E} \, \mathrm{d}E &= \frac{1}{2\pi} \int \limits\_{0}^{E\_0} \mathrm{d}E \, \mathbb{E} \, \mathbf{g}(\mathcal{E}) \, \boldsymbol{\varrho}(\mathcal{E}, \mathbf{x}) \int e^{i(\mathcal{E}' - \mathcal{E})t} \, \mathrm{d}t = \\ &= \int \limits\_{0} \boldsymbol{\varrho}(\mathcal{E}) \, \boldsymbol{\varrho}(\mathcal{E}, \mathbf{x}) \delta(\mathcal{E}' - \mathcal{E}) \, \mathrm{E} \, \mathrm{d}E = \boldsymbol{\varrho}(\mathcal{E}') \, \mathrm{E}' \, \boldsymbol{\varrho}(\mathcal{E}', \mathbf{x}) \,, \end{split} \tag{113}$$

while its right part becomes

$$\begin{split} & -\frac{1}{2\pi} \int dt \, e^{iE't} \int \limits\_{0}^{E\_{0}} \oint (E \,) \, e^{-iEt} \frac{\partial^{2} \varrho(E, \mathbf{x})}{\partial \mathbf{x}^{2}} \, d\mathbf{E} + \frac{1}{2\pi} \int dt \, e^{iE't} \int \limits\_{0}^{E\_{0}} \varrho(E) \, V(\mathbf{x}, \mathbf{E}, t) \, e^{-iEt} \varrho(E, \mathbf{x}) \, d\mathbf{E} = \\ & - \frac{1}{2\pi} \int \limits\_{0}^{E\_{0}} \mathrm{d} \mathbb{E} \, g(E) \frac{\partial^{2} \varrho(E, \mathbf{x})}{\partial \mathbf{x}^{2}} \int \limits\_{0}^{i(E'-E)t} dt + \frac{1}{2\pi} \int \mathrm{d} \mathbb{E} \, g(E) \, \varrho(E, \mathbf{x}) \int V(\mathbf{x}, \mathbf{E}, t) \, e^{i(E'-E) - t} \, dt = \\ & - \mathcal{g}(E') \frac{\partial^{2} \varrho(E', \mathbf{x})}{\partial \mathbf{x}^{2}} + \frac{1}{2\pi} \int \mathrm{d} \mathbb{E} \, g(E) \, \varrho(E, \mathbf{x}) \int V(\mathbf{x}, \mathbf{E}, t) \, e^{i(E'-E)t} \, dt. \end{split} \tag{14}$$

As a result, we obtain eq. (112) in the form

$$g(E')\to'\varphi(E',\mathbf{x}) = -g(E')\frac{\partial^2\varphi(E',\mathbf{x})}{\partial\mathbf{x}^2} + \frac{1}{2\pi}\int\_0^{E\_0} d\mathbf{E}\,g(\mathbf{E})\,\varphi(\mathbf{E},\mathbf{x})\int V(\mathbf{x},\mathbf{E},t)\,e^{i(\mathbf{E}'-\mathbf{E})t}dt.\tag{115}$$

#### **9.2 The case of the simple Albreht's potential**

Just as an example of a possible potential *V*(*x*, *t*), let us choose

$$V(\mathbf{x}, t) = V\_0(\mathbf{x}) + \gamma \, W\_A(\mathbf{x}) \,. \tag{116}$$

where *WA*(*x*) is the simple Albreht's dissipation term. Here, *γ* is a constant, *V*0(*x*) is the usual stationary component of *V*(*x*), and the dissipative component of *V*(*x*) has the form

$$W\_A(\mathbf{x}) = (\mathbf{x} - <\mathbf{x}>),\tag{117}

$$

where the averages are fulfilled by integrating over *x* by means of the functions Ψ∗(*x*, *t*) and Ψ(*x*, *t*). For the right part of eq. (117) one gets

$$\begin{split} \mathbf{x} < p > &= -i \int d\mathbf{x} \int d\mathbf{E}\_{1} \int d\mathbf{E}\_{2} \,\mathbf{g}(\mathbf{E}\_{1}) \,\mathbf{g}(\mathbf{E}\_{2}) \,\boldsymbol{\varrho}^{i}(\mathbf{E}\_{1}, \mathbf{x}) \,\boldsymbol{\varrho}^{\*}(\mathbf{E}\_{1}, \mathbf{x}) \,\frac{\partial \boldsymbol{\varrho}(\mathbf{E}\_{2}, \mathbf{x})}{\partial \mathbf{x}}, \\ \mathbf{x} < &= \int d\mathbf{x} \int d\mathbf{E}\_{3} \int d\mathbf{E}\_{4} \,\mathbf{g}(\mathbf{E}\_{3}) \,\mathbf{g}(\mathbf{E}\_{4}) \,\boldsymbol{\varrho}^{i(\mathbf{E}\_{3} - \mathbf{E}\_{4}) \mathbf{f}} \,\mathbf{x} \,\boldsymbol{\varrho}^{\*}(\mathbf{E}\_{3}, \mathbf{x}) \,\boldsymbol{\varrho}(\mathbf{E}\_{4}, \mathbf{x}); \end{split} \tag{118}$$

so that the total potential *V*(*x*, *t*) becomes

$$V(\mathbf{x},t) = V\_0(\mathbf{x}) - i\gamma \int d\mathbf{x}\_1 \int d\mathbf{x}\_2 \int d\mathbf{E}\_1 \int d\mathbf{E}\_2 \int d\mathbf{E}\_3 \int d\mathbf{E}\_4 \,\operatorname{g}(\mathbf{E}\_1) \,\operatorname{g}(\mathbf{E}\_2) \,\operatorname{g}(\mathbf{E}\_3) \,\operatorname{g}(\mathbf{E}\_4) \tag{119}$$

$$\times \epsilon^{i(E\_1 - E\_2 + E\_3 - E\_4)t} \left(\mathbf{x} - \mathbf{x}\_2\right) \,\boldsymbol{\varrho}^\*(\mathbf{E}\_1, \mathbf{x}\_1) \,\frac{\partial \boldsymbol{\varrho}(\mathbf{E}\_2, \mathbf{x}\_1)}{\partial \mathbf{x}\_1} \,\boldsymbol{\varrho}^\*(\mathbf{E}\_3, \mathbf{x}\_2) \,\boldsymbol{\varrho}(\mathbf{E}\_4, \mathbf{x}\_2).$$

Taking into account this, we find the second term, in the r.h.s. of eq. (115), to be:

1 2*π E*0 0 *dE g*(*E*) *ϕ*(*E*, *x*) *V*(*x*, *E*¯, *t*)*ei*(*E*� <sup>−</sup>*E*)*<sup>t</sup> dt* = = *E*0 0 *dE g*(*E*) *ϕ*(*E*, *x*) *V*0(*x*) *δ*(*E*� − *E*) − − *iγ E*0 0 *dE g*(*E*) *ϕ*(*E*, *x*) *dx*<sup>1</sup> *dx*<sup>2</sup> *E*0 0 *dE*<sup>1</sup> *E*0 0 *dE*<sup>2</sup> *E*0 0 *dE*<sup>3</sup> *E*0 0 *dE*<sup>4</sup> *g*(*E*1) *g*(*E*2) *g*(*E*3) *g*(*E*4) × *x* − *x*<sup>2</sup> *<sup>ϕ</sup>*∗(*E*1, *<sup>x</sup>*1) *∂ϕ*(*E*2, *<sup>x</sup>*1) *∂x*<sup>1</sup> *ϕ*∗(*E*3, *x*2) *ϕ*(*E*4, *x*2) *δ*(*E*� − *E* + *E*<sup>1</sup> − *E*<sup>2</sup> + *E*<sup>3</sup> − *E*4) = = *g*(*E*� ) *ϕ*(*E*� , *x*) *V*0(*x*) − − *iγ dx*<sup>1</sup> *dx*<sup>2</sup> *E*0 0 *dE*<sup>1</sup> *E*0 0 *dE*<sup>2</sup> *E*0 0 *dE*<sup>3</sup> *E*0 0 *dE*<sup>4</sup> *g*(*E*1) *g*(*E*2) *g*(*E*3) *g*(*E*4) *g*(*E*��) × *x* − *x*<sup>2</sup> *<sup>ϕ</sup>*∗(*E*1, *<sup>x</sup>*1) *∂ϕ*(*E*2, *<sup>x</sup>*1) *∂x*<sup>1</sup> *ϕ*∗(*E*3, *x*2) *ϕ*(*E*4, *x*2) *ϕ*(*E*��, *x*), (120)

where

34 Will-be-set-by-IN-TECH

*<sup>g</sup>*(*E*)*e*−*iEt <sup>∂</sup>*2*ϕ*(*E*, *<sup>x</sup>*)

<sup>−</sup>*iEt <sup>∂</sup>*2*ϕ*(*E*, *<sup>x</sup>*)

Let us now apply the inverse Fourier-transformation to this equation. Its left part becomes

2*π E*0

1 2*π* 

> 1 2*π E*0

0

*dE g*(*E*) *ϕ*(*E*, *x*)

*dt eiE*� *t E*0

0

*dE g*(*E*) *ϕ*(*E*, *x*)

*dE g*(*E*) *ϕ*(*E*, *x*)

*g*(*E*) *ϕ*(*E*, *x*)*δ*(*E*� − *E*) *EdE* = *g*(*E*�

*<sup>∂</sup>x*<sup>2</sup> *dE* <sup>+</sup>

1 2*π E*0

, *x*) *<sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

<sup>−</sup>*E*)*<sup>t</sup> dt* +

0

1 2*π E*0

0

where *WA*(*x*) is the simple Albreht's dissipation term. Here, *γ* is a constant, *V*0(*x*) is the usual

stationary component of *V*(*x*), and the dissipative component of *V*(*x*) has the form

0

*<sup>∂</sup>x*<sup>2</sup> *dE* <sup>+</sup>

*<sup>∂</sup>x*<sup>2</sup> *dE* <sup>+</sup>

 *E*0

*g*(*E*)*V*(*x*, *E*¯, *t*)*e*−*iEtϕ*(*E*, *x*) *dE*.

*g*(*E*) *V*(*x*, *E*¯, *t*) *e*−*iEt ϕ*(*E*, *x*) *dE*.

*g*(*E*) *V*(*x*, *E*¯, *t*)*e*−*iEt ϕ*(*E*, *x*) *dE* =

*V*(*x*, *E*¯, *t*)*ei*(*E*�

<sup>−</sup>*E*)*<sup>t</sup> dt*.

*V*(*x*, *E*¯, *t*) *ei*(*E*�

(111)

(112)

(113)

<sup>−</sup>*E*)−*<sup>t</sup> dt* =

<sup>−</sup>*E*)*<sup>t</sup> dt*. (115)

(114)

0

 *E*0

0

*dE E g*(*E*) *ϕ*(*E*, *x*)

) *E*� *ϕ*(*E*�

 *ei*(*E*� −*E*)*t dt* =

, *x*);

*V*(*x*, *E*¯, *t*) *ei*(*E*�

*V*(*x*, *t*) = *V*0(*x*) + *γ WA*(*x*). (116)

*WA*(*x*) =< *p* > (*x*− < *x* >), (117)

Afterwards, the r.h.s. of eq. (107) gets transformed into

 *E*0

0

Therefore, the whole equation (107) has been transformed into

 *E*0

*g*(*E*) *e*

0

*<sup>g</sup>*(*E*)*e*−*iEt <sup>ϕ</sup>*(*E*, *<sup>x</sup>*) *EdE* <sup>=</sup> <sup>1</sup>

Ψ(*x*, *t*) = −

 − *∂*2

 *E*0

0

1 2*π* 

− 1 2*π* 

<sup>=</sup> <sup>−</sup> <sup>1</sup> 2*π E*0

*g*(*E*�

*<sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *<sup>V</sup>*(*x*, *<sup>t</sup>*)

*<sup>g</sup>*(*E*)*e*−*iEt <sup>ϕ</sup>*(*E*, *<sup>x</sup>*) *EdE* <sup>=</sup> <sup>−</sup>

0

= *E*0

0

*∂*2*ϕ*(*E*, *x*) *∂x*<sup>2</sup>

> ) *∂*2*ϕ*(*E*�

As a result, we obtain eq. (112) in the form

, *x*) = −*g*(*E*�

**9.2 The case of the simple Albreht's potential**

) *∂*2*ϕ*(*E*�

Just as an example of a possible potential *V*(*x*, *t*), let us choose

*<sup>g</sup>*(*E*)*e*−*iEt <sup>∂</sup>*2*ϕ*(*E*, *<sup>x</sup>*)

 *e i*(*E*�

, *x*) *<sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

*dt eiE*� *t E*0

while its right part becomes

0

= −*g*(*E*�

*dE g*(*E*)

*dt eiE*� *t E*0

0

) *E*� *ϕ*(*E*�

$$E'' = E' + E\_1 - E\_2 + E\_3 - E\_4.\tag{121}$$

As a consequence, the whole eq. (115) gets transformed into

$$\begin{array}{c} \operatorname{g}(E')\operatorname{E}'\operatorname{\boldsymbol{\varrho}}(E',\mathbf{x}) = -\operatorname{g}(E')\frac{\partial^2\operatorname{\boldsymbol{\varrho}}(E',\mathbf{x})}{\partial\mathbf{x}^2} + \operatorname{g}(E')\operatorname{\boldsymbol{\varrho}}(E',\mathbf{x})\operatorname{V}\_{0}(\mathbf{x})\\ -\operatorname{i}\operatorname{\boldsymbol{\chi}}\int d\mathbf{x}\_1\int d\mathbf{x}\_2\int d\mathbf{E}\_1\int d\mathbf{E}\_2\int d\mathbf{E}\_3\int d\mathbf{E}\_4\operatorname{\boldsymbol{\varrho}}(\mathbf{E}\_1)\operatorname{\boldsymbol{\varrho}}(\mathbf{E}\_2)\operatorname{\boldsymbol{\varrho}}(\mathbf{E}\_3)\operatorname{\boldsymbol{\varrho}}(\mathbf{E}\_4)\operatorname{\boldsymbol{\varrho}}(E\_4)\operatorname{\boldsymbol{\varrho}}(E'')\\ \times \left(\operatorname{\mathbf{x}}-\operatorname{\mathbf{x}}\_2\right)\operatorname{\boldsymbol{\varrho}}^{\ast}(\operatorname{\mathbf{E}}\_{1\prime}\mathbf{x}\_1)\frac{\partial\operatorname{\boldsymbol{\varrho}}(\mathbf{E}\_2,\mathbf{x}\_1)}{\partial\mathbf{x}\_1}\operatorname{\boldsymbol{\varrho}}^{\ast}(\mathbf{E}\_3,\mathbf{x}\_2)\operatorname{\boldsymbol{\varrho}}(\mathbf{E}\_4,\mathbf{x}\_2)\operatorname{\boldsymbol{\varrho}}(E'',\mathbf{x}) \end{array}$$

where

**10. References**

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Albrecht, K. (1975). *Phys. Lett.* Vol. B56: 127.

Bonifacio, R. (1983). *Lett. N. Cim.* Vol. 37: 481.

Caldirola, P. (1941). *Nuovo Cimento* Vol. 18: 393. Caldirola, P. (1976). *Lett. N. Cim.* Vol. 16: 151. Caldirola, P. (1976). *Lett. N. Cim.* Vol. 17: 461. Caldirola, P. (1977). *Lett. N. Cim.* Vol. 18: 465.

Vol. 6: 1400–1433.

(No. 5): 441–457.

Caldirola, P. (1979). *Rivista N. Cim.* Vol. 2: issue no. 13.

Caldirola, P. & Montaldi, E. (1979). *Nuovo Cimento* Vol. B53: 291. Caldirola, P. & Lugiato, L. (1982). *Physica* Vol. A116: 248.

Fizmatgiz).

*E*�� = *E*<sup>0</sup> + *E*<sup>1</sup> − *E*<sup>2</sup> + *E*<sup>3</sup> − *E*4. (127)

The first equation holds when dissipation is absent. The second equation determines the unknown function *ϕ*<sup>1</sup> in terms of the given *ϕ*0: It results to be an ordinary differential equation

Time as Quantum Observable, Canonical Conjugated to Energy 53

Aharonov, Y., Oppenhem, J., Popescu, S., Reznik, B. & Unruh, W. (1998). *Phys. Rev.* Vol. A57:

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Akhiezer, A. I. & Berestezky, V. B. (1959). *Quantum Electrodynamics* [in Russian] (Moscow:

Anderson, A. (1989). Multiple scattering approach to one-dimensional potential problems,

Buck, B., Merchant, A. C. & Perez, S. M. (1993). Half-lives of favored alpha decays from nuclear

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or (with the change of variables *E*� → *E*)

$$
\begin{split}
\left(-\frac{\partial^2}{\partial \mathbf{x}^2} + V\_0(\mathbf{x}) - E\right)\boldsymbol{\varrho}(E, \mathbf{x}) &= \\
\mathbf{x} = i\gamma \int d\mathbf{x}\_1 \int d\mathbf{x}\_2 \int d\mathbf{E}\_1 \int d\mathbf{E}\_2 \int d\mathbf{E}\_3 \int d\mathbf{E}\_4 \frac{\boldsymbol{g}(\mathbf{E}\_1)\,\boldsymbol{g}(\mathbf{E}\_2)\,\boldsymbol{g}(\mathbf{E}\_3)\,\boldsymbol{g}(\mathbf{E}\_4)\,\boldsymbol{g}(E')}{\mathbf{g}(E)} \\
\times \left(\mathbf{x} - \mathbf{x}\_2\right) \boldsymbol{\varrho}^\*(E\_1, \mathbf{x}\_1) \frac{\partial \boldsymbol{g}(E\_2, \mathbf{x}\_1)}{\partial \mathbf{x}\_1} \boldsymbol{g}^\*(E\_3, \mathbf{x}\_2) \,\boldsymbol{g}(E\_4, \mathbf{x}\_2) \,\boldsymbol{g}(E'', \mathbf{x}) \,.
\end{split} \tag{122}
$$

We have thus obtained for this case the time-independent Schröedinger equation, by taking however into account dissipation via the parameter *γ*. Of course, when *γ* tends to zero, one goes back to the stationary Schrödinger equation.

#### **9.3 Method of the successive approximations**

Assuming the coefficient *γ* to be small, one can find the unknown function *ϕ*(*x*) in the simplified form

$$
\varphi(\mathbf{x}) = \varphi\_0(\mathbf{x}) + \gamma \,\,\varphi\_1(\mathbf{x}),
\tag{123}
$$

where as function *ϕ*0(*x*) it has been used the standard WF of the time-independent Schrödinger equation with potential *V*0(*x*) and energy *E*0:

$$\left(-\frac{\partial^2}{\partial x^2} + V\_0(\mathbf{x})\right)\varphi\_0(\mathbf{x}) = E\_0 \,\varphi\_0(\mathbf{x}).\tag{124}$$

Substituting solution (123) into eq. (124), we obtain a new equation containing all the powers *n* of *γ*, namely, the *γn*. Let us confine ourselves, however, to write down this equation with accuracy up to *γ*<sup>1</sup> only:

$$
\begin{split}
\left(-\frac{\partial^2}{\partial \mathbf{x}^2} + V\_0(\mathbf{x}) - E\right) \left(\boldsymbol{\varrho}\_0(\mathbf{E}, \mathbf{x}) + \boldsymbol{\gamma}\boldsymbol{\varrho}\_1(\mathbf{E}, \mathbf{x})\right) &= \\
\mathbf{x} = i\boldsymbol{\gamma} \int d\mathbf{x}\_1 \int d\mathbf{x}\_2 \int d\mathbf{E}\_1 \int d\mathbf{E}\_2 \int d\mathbf{E}\_3 \int d\mathbf{E}\_4 \frac{\mathbf{g}(\mathbf{E}\_1)\,\mathbf{g}(\mathbf{E}\_2)\,\mathbf{g}(\mathbf{E}\_3)\,\mathbf{g}(\mathbf{E}\_4)\,\mathbf{g}(\mathbf{E}^{\prime\prime})}{\mathbf{g}(E)} \\
\times \left(\mathbf{x} - \mathbf{x}\_2\right) \,\boldsymbol{\varrho}\_0^\*(\mathbf{E}\_1, \mathbf{x}\_1) \,\frac{\partial \boldsymbol{\varrho}\_0(\mathbf{E}\_2, \mathbf{x}\_1)}{\partial \mathbf{x}\_1} \,\boldsymbol{\varrho}\_0^\*(\mathbf{E}\_3, \mathbf{x}\_2) \,\boldsymbol{\varrho}\_0(\mathbf{E}\_4, \mathbf{x}\_2) \,\boldsymbol{\varrho}\_0(\mathbf{E}^{\prime\prime}, \mathbf{x})\,\end{split} \tag{125}
$$

where the unknown *ϕ*1(*x*) does not appear any longer, of course,into the r.h.s. of this equation. Taking *E* = *E*0, we can rewrite in eq. (125), separately, the various terms with different powers of *γ*. When limiting ourselves to *n* = 0, 1, we obtain

$$\begin{split} \gamma^{0}: & \qquad \left( -\frac{\partial^{2}}{\partial \mathbf{x}^{2}} + V\_{0}(\mathbf{x}) - E\_{0} \right) \boldsymbol{\varrho}\_{0}(\mathbf{E}\_{0}, \mathbf{x}) = \mathbf{0}, \\ \gamma^{1}: & \qquad \left( -\frac{\partial^{2}}{\partial \mathbf{x}^{2}} + V\_{0}(\mathbf{x}) - E\_{0} \right) \boldsymbol{\varrho}\_{1}(E\_{0}, \mathbf{x}) = \\ & \qquad \mathrel{\rm E\_{0}} \quad \int d\mathbf{x}\_{1} \int d\mathbf{x}\_{2} \int dE\_{1} \int dE\_{2} \int dE\_{3} \int dE\_{4} \frac{\boldsymbol{g}(E\_{1}) \cdot \boldsymbol{g}(E\_{2}) \cdot \boldsymbol{g}(E\_{3}) \cdot \boldsymbol{g}(E\_{4}) \cdot \boldsymbol{g}(E''\_{1})}{\operatorname{g}(E\_{0})} \\ & \qquad \times \left( \mathbf{x} - \mathbf{x}\_{2} \right) \boldsymbol{\varrho}\_{0}^{\*}(E\_{1}, \mathbf{x}\_{1}) \frac{\partial \boldsymbol{\varrho}\_{0}(E\_{2}, \mathbf{x}\_{1})}{\partial \mathbf{x}\_{1}} \, \boldsymbol{\varrho}\_{0}^{\*}(E\_{3}, \mathbf{x}\_{2}) \, \boldsymbol{\varrho}\_{0}(E\_{4}, \mathbf{x}\_{2}) \, \boldsymbol{\varrho}\_{0}(E''\_{1}, \mathbf{x}), \end{split} \tag{126}$$

where

36 Will-be-set-by-IN-TECH

*dE*<sup>4</sup>

*ϕ*(*E*, *x*) =

*g*(*E*1) *g*(*E*2) *g*(*E*3) *g*(*E*4) *g*(*E*��) *g*(*E*)

(122)

(125)

(126)

*ϕ*∗(*E*3, *x*2) *ϕ*(*E*4, *x*2) *ϕ*(*E*��, *x*).

*ϕ*(*x*) = *ϕ*0(*x*) + *γ ϕ*1(*x*), (123)

 =

*g*(*E*1) *g*(*E*2) *g*(*E*3) *g*(*E*4) *g*(*E*��) *g*(*E*)

*g*(*E*1) *g*(*E*2) *g*(*E*3) *g*(*E*4) *g*(*E*��) *g*(*E*0)

<sup>0</sup> (*E*3, *x*2) *ϕ*0(*E*4, *x*2) *ϕ*0(*E*��, *x*),

<sup>0</sup>(*E*3, *x*2) *ϕ*0(*E*4, *x*2) *ϕ*0(*E*��, *x*),

*ϕ*0(*E*0, *x*) = 0,

*ϕ*1(*E*0, *x*) =

*ϕ*0(*x*) = *E*<sup>0</sup> *ϕ*0(*x*). (124)

*<sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *<sup>V</sup>*0(*x*) <sup>−</sup> *<sup>E</sup>*

*dE*<sup>3</sup> *E*0

0

We have thus obtained for this case the time-independent Schröedinger equation, by taking however into account dissipation via the parameter *γ*. Of course, when *γ* tends to zero, one

Assuming the coefficient *γ* to be small, one can find the unknown function *ϕ*(*x*) in the

where as function *ϕ*0(*x*) it has been used the standard WF of the time-independent

Substituting solution (123) into eq. (124), we obtain a new equation containing all the powers *n* of *γ*, namely, the *γn*. Let us confine ourselves, however, to write down this equation with

*ϕ*0(*E*, *x*) + *γϕ*1(*E*, *x*)

or (with the change of variables *E*� → *E*)

= *iγ dx*<sup>1</sup> *dx*<sup>2</sup> *E*0

simplified form

accuracy up to *γ*<sup>1</sup> only:

= *iγ dx*<sup>1</sup> *dx*<sup>2</sup> *E*0

*γ*<sup>0</sup> :

*γ*<sup>1</sup> :

= *iγ dx*<sup>1</sup> *dx*<sup>2</sup> *E*0

> × *x* − *x*<sup>2</sup> *ϕ*∗

× *x* − *x*<sup>2</sup> *ϕ*∗

× *x* − *x*<sup>2</sup>   − *∂*2

*dE*<sup>1</sup> *E*0

Schrödinger equation with potential *V*0(*x*) and energy *E*0: − *∂*2

*<sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *<sup>V</sup>*0(*x*) <sup>−</sup> *<sup>E</sup>*

0

*dE*<sup>2</sup> *E*0

<sup>0</sup> (*E*1, *<sup>x</sup>*1) *∂ϕ*0(*E*2, *<sup>x</sup>*1)

0

*∂x*<sup>1</sup>

*<sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *<sup>V</sup>*0(*x*) <sup>−</sup> *<sup>E</sup>*<sup>0</sup>

*<sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *<sup>V</sup>*0(*x*) <sup>−</sup> *<sup>E</sup>*<sup>0</sup>

0

*∂x*<sup>1</sup>

*dE*<sup>3</sup> *E*0

0

*dE*<sup>1</sup> *E*0

0

*dE*<sup>2</sup> *E*0

*<sup>ϕ</sup>*∗(*E*1, *<sup>x</sup>*1) *∂ϕ*(*E*2, *<sup>x</sup>*1)

0

*<sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *<sup>V</sup>*0(*x*)

*dE*<sup>3</sup> *E*0

0

*ϕ*∗

where the unknown *ϕ*1(*x*) does not appear any longer, of course,into the r.h.s. of this equation. Taking *E* = *E*0, we can rewrite in eq. (125), separately, the various terms with different powers

*dE*<sup>4</sup>

*ϕ*∗

*dE*<sup>4</sup>

*∂x*<sup>1</sup>

0

goes back to the stationary Schrödinger equation.

**9.3 Method of the successive approximations**

 − *∂*2

0

of *γ*. When limiting ourselves to *n* = 0, 1, we obtain

 − *∂*2

> − *∂*2

> > *dE*<sup>1</sup> *E*0

> > > 0

*dE*<sup>2</sup> *E*0

<sup>0</sup> (*E*1, *<sup>x</sup>*1) *∂ϕ*0(*E*2, *<sup>x</sup>*1)

0

$$E'' = E\_0 + E\_1 - E\_2 + E\_3 - E\_4.\tag{127}$$

The first equation holds when dissipation is absent. The second equation determines the unknown function *ϕ*<sup>1</sup> in terms of the given *ϕ*0: It results to be an ordinary differential equation of the second order, that can be solved by the ordinary numerical methods.

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**0**

**3**

**Order of Time Derivatives in**

5B*, Pawi ´nskiego str., 02-106 Warsaw, Poland*

Jan Jerzy Sławianowski

**Quantum-Mechanical Equations**

*Institute of Fundamental Technological Research, Polish Academy of Sciences*

The problem is rather old, almost as quantum mechanics itself. The first, primary idea of Schrödinger was the relativistic one, with the d'Alembert operator on the left-hand side of quantum-mechanical equation, so, with the second-order time derivatives. Unfortunately, it turned out that the following results were in a rather clear contradiction with experimental data, although some kind of compatibility did exist. Schrödinger felt disappointed and at least temporarily he rejected his primary equation. Later on, basic on the idea of Lagrange-Hamilton optical-mechanical analogy and on certain de Broglie ideas, he in a sense derived his famous equation which seemed to remain in a beautiful agreement with spectroscopic data and was approximately compatible with the Bohr-Sommerfeld quantization rules. Nevertheless, it was of course drastically incompatible with the relativistic idea of Poincare symmetry. But, after the fall of the primary substantial interpretation by Schrödinger, it was compatible with the Born statistical interpretation of his formalism, and with the corresponding continuity equation for the probabilistic density (Veltman, 2003). Later on history was rather complicated. Dirac formulated his relativistic quantum theory of electrons based on first-order space-time derivatives of multicomponent wave functions. The multicomponent character of waves had to do obviously with the particle spin. It was also understood that the relativistic velocity-dependence of the electron mass and the spin phenomena act in opposite directions, and because of this non-relativistic Schrödinger equation seemed to be better than his second order equation, rediscovered later on by Klein and Gordon. The formalism of quantum field theory rehabilitated the Klein-Gordon equation, i.e., primaevally the relativistic Schrödinger equation, as one describing some physics. And, let us also mention that the field-theoretic approach based on the Pauli exclusion principle removed certain problems with the quantum-mechanical Dirac equation for a single electron. And certain inconsistencies on one-particle relativistic theory were resolved. The only, and fundamental inadequacy which remained, was one connected with the essential non-linearity of the quantum field-theoretic equations for the field operators and the resulting interpretation difficulties. Nevertheless, they were in a sense solvable on the basis of renormalization

But in spite of everything said, the problem is still alive. There are certain not completely clear facts within the framework of field theory based on the Dirac-Clifford paradigm of first-order differential equations of quantum mechanics with ¯*h*/2-spin. One can show that they become more clear when we assume that in a sense some second-order equations are primary and the first-order ones are some approximations valid for slowly-varying fields. There are also

**1. Introduction**

procedure.

	- Report NSF-ITP-02-62 (KIPT, UCSB; Santa Barbara, CA). URL: quant-ph/97060509v3.

### **Order of Time Derivatives in Quantum-Mechanical Equations**

### Jan Jerzy Sławianowski

*Institute of Fundamental Technological Research, Polish Academy of Sciences* 5B*, Pawi ´nskiego str., 02-106 Warsaw, Poland*

#### **1. Introduction**

40 Will-be-set-by-IN-TECH

56 Measurements in Quantum Mechanics

Olkhovsky, V. S., Recami, E. & Salesi, G. (2002). Tunneling through two successive barriers

Olkhovsky, V. S., Recami, E. & Zaichenko, A. K. (2005). Resonant & non-resonant

Olkhovsky, V. S., Dolinska, M. E. & Omelchenko, S. A. (2006). *Central European Journal of*

Olkhovsky, V. S. (2009). Time as a quantum observable, Canonically conjugated to energy,

Olkhovsky, V. S. (2011). On time as a quantum observable canonically conjugate to energy,

Recami, E. (1976). *An operator for the observable time* in *Recent Developments in Relativistic Q.F.T.*

Recami, E. (1977). *A time operator & the time-energy uncertainty relation* in *The Uncertainty*

Recami, E. & Farias, A. R. H. (2002). A simple quantum equation for decoherence and

Recami, E. (2004). Superluminal tunneling through successive barriers. Does QM predict

Santilli, R. M. (1983). *Foundations of Theoretical Mechanics, Vol.II: Birkhoffian Generalization of*

Schweber, S. (1961). *An Introduction to Relativistic Quantum Field Theory* chapter 5.3 (Row,

Sobiczewski, A. & Pomorski, K. (2007). Description of structure and properties of superheavy

Von Neumann, J. (1955). *Mathematical foundations of quantum mechanics* (Princeton Univ. Press,

*& Its Application (Proc. of the XIII Karpatz Winter School on Theor. Phys.)*, vol.II, ed. by

*Principle & Foundation of Quantum Mechanics*, ed. by C.Price & S. Chissik (London:

Report NSF-ITP-02-62 (KIPT, UCSB; Santa Barbara, CA). URL:

Olkhovsky, V. S., Recami, E. & Jakiel, J. (2004). *Phys. Rep.* Vol. 398: 133.

Olkhovsky, V. S. & Recami, E. (2007). *Intern. J. Mod. Phys* Vol. A22: 5063. Olkhovsky, V. S. & Recami, E. (2008). *Int. J. Mod. Phys.* Vol. B22: 1877.

Pauli, W. (1926). *Handbuch der Physik* vol. 5/1 (Berlin: Ed. by S. Fluegge): p. 60. Pauli, W. (1980). *General Principles of Quantum Theory* (Berlin: Springer).

W.Karwowski (Wroclaw Univ.Press; Wroclaw), pp.251-265.

Recami, E., Rodrigues, W. A. & Smrz, P. (1983). *Hadronic Journal* Vol. 6: 1773–1789.

infinite group-velocities? *Journal of Modern Optics* Vol. 51: 913–923.

quant-ph/0002022.

quant-th/0410128.

*Physics* Vol. 4 (No. 2): 1–18.

*Math. Phys.* Vol. 2009 83 p.

J. Wiley) Chap.4, pp.21–28.

quant-ph/97060509v3.

Peterson and Co).

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Princeton, N.J.).

Rosenbaum, D. M. (1969). *J. Math. Phys.* Vol. 10: 1127. Santilli, R. M. (1979). *Hadronic Journal* Vol. 2: 1460.

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*Hamiltonian Mechanics* (Springer; Berlin).

nuclei, *Prog. Part. Nucl. Phys.* Vol. 58: 292–349.

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dissipation,

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tunneling through a double barrier, *Europhysics Letters* Vol. 70: 712–718. URL:

and foundations of self-consistent Time Analysis of Quantum Processes, *Advances in*

The problem is rather old, almost as quantum mechanics itself. The first, primary idea of Schrödinger was the relativistic one, with the d'Alembert operator on the left-hand side of quantum-mechanical equation, so, with the second-order time derivatives. Unfortunately, it turned out that the following results were in a rather clear contradiction with experimental data, although some kind of compatibility did exist. Schrödinger felt disappointed and at least temporarily he rejected his primary equation. Later on, basic on the idea of Lagrange-Hamilton optical-mechanical analogy and on certain de Broglie ideas, he in a sense derived his famous equation which seemed to remain in a beautiful agreement with spectroscopic data and was approximately compatible with the Bohr-Sommerfeld quantization rules. Nevertheless, it was of course drastically incompatible with the relativistic idea of Poincare symmetry. But, after the fall of the primary substantial interpretation by Schrödinger, it was compatible with the Born statistical interpretation of his formalism, and with the corresponding continuity equation for the probabilistic density (Veltman, 2003).

Later on history was rather complicated. Dirac formulated his relativistic quantum theory of electrons based on first-order space-time derivatives of multicomponent wave functions. The multicomponent character of waves had to do obviously with the particle spin. It was also understood that the relativistic velocity-dependence of the electron mass and the spin phenomena act in opposite directions, and because of this non-relativistic Schrödinger equation seemed to be better than his second order equation, rediscovered later on by Klein and Gordon. The formalism of quantum field theory rehabilitated the Klein-Gordon equation, i.e., primaevally the relativistic Schrödinger equation, as one describing some physics. And, let us also mention that the field-theoretic approach based on the Pauli exclusion principle removed certain problems with the quantum-mechanical Dirac equation for a single electron. And certain inconsistencies on one-particle relativistic theory were resolved. The only, and fundamental inadequacy which remained, was one connected with the essential non-linearity of the quantum field-theoretic equations for the field operators and the resulting interpretation difficulties. Nevertheless, they were in a sense solvable on the basis of renormalization procedure.

But in spite of everything said, the problem is still alive. There are certain not completely clear facts within the framework of field theory based on the Dirac-Clifford paradigm of first-order differential equations of quantum mechanics with ¯*h*/2-spin. One can show that they become more clear when we assume that in a sense some second-order equations are primary and the first-order ones are some approximations valid for slowly-varying fields. There are also

We have formulated some arguments in favour of SU(2, 2) as a fundamental gauge group. Let us mention, incidentally, that this simply provokes the next question: Why the subgroup SU(2, 2) ⊂ GL(4, **C**) but not just the whole GL(4, **C**)? The latter group appears in a natural way as the structure group of the principal fibre bundle of the complexification of the usual bundle of frames over the four-dimensional space-time manifold. Obviously, it preserves the signature of sesquilinear forms, nevertheless changing them otherwise. Therefore, the bispinor sesquilinear Hermitian form *G* of signature (+, +, −, −) becomes an a priori free

Order of Time Derivatives in Quantum-Mechanical Equations 59

Hermitian form, the signature however being in a sense an integral of motion.

The generally-relativistic Lagrangian of the Dirac field is given by the expression:

<sup>Ψ</sup>*rDμ*Ψ*<sup>s</sup>* <sup>−</sup> *<sup>D</sup>μ*<sup>Ψ</sup>*<sup>r</sup>*Ψ*<sup>s</sup>*

<sup>Ψ</sup>*<sup>r</sup>* <sup>=</sup> <sup>Ψ</sup>*<sup>s</sup>*

where *G* denotes the Dirac-conjugation form of mass, i.e., sesquilinear Hermitian form of the natural signature (+, +, −, −). If the Finkelstein-Penrose-Weizsäcker-van der Waerden point of view on the two-component spinors is accepted, then *G* is intrinsic, because the **C**4-space is then expressed as the Cartesian product of two mutually antidual copies of **C**2. Without this

*<sup>A</sup>* are components of the tetrad field. Its dual cotetrad *e<sup>A</sup>*

*<sup>μ</sup>* in a quadratic way:

(e) The operation *D<sup>μ</sup>* symbolizes the covariant differentiation of bispinors. It is given by the

*γAωμγ<sup>B</sup>*

*<sup>β</sup>* + *e α Ae<sup>A</sup>*

*eA μe μ <sup>B</sup>* = *δ<sup>A</sup>*

*μe<sup>B</sup>*

*<sup>B</sup><sup>μ</sup>* <sup>=</sup> <sup>1</sup> 2 Tr

The Greek and capital Latin indices are shifted with the help of *gμν* and *ηAB*.

<sup>|</sup>*g*<sup>|</sup> <sup>−</sup> *<sup>m</sup>*<sup>Ψ</sup>*<sup>r</sup>*Ψ*<sup>r</sup>*

= *γAγ<sup>B</sup>* + *γBγ<sup>A</sup>* = 2*ηAB I*4, (3)

= diag (1, −1, −1, −1). (4)

*<sup>ν</sup>*, [*ηAB*] = diag (1, −1, −1, −1). (7)

*<sup>B</sup>*. (6)

*<sup>β</sup>*,*μ*, (8)

, (9)

Γ*Ars* = Γ*Asr* = Γ*Asr* = *GrzγAzs*. (5)

*Gsr*, (2)


*<sup>μ</sup>* is analytically

**2. On the track of the scalar Klein-Gordon-Dirac formalism**

(b) Dirac matrices *γ<sup>A</sup>* satisfy the following anticommutation rules:

 *γAB*

 *γA*, *γ<sup>B</sup>* 

*gμν* = *ηABe<sup>A</sup>*

Γ*α βμ* = *e α A*Γ*<sup>A</sup> Bμe<sup>B</sup>*

Γ*A*

*<sup>L</sup>* <sup>=</sup> *<sup>i</sup>* 2 *e μ <sup>A</sup>γArs* 

with the following meaning of symbols: (a) Ψ denote the Dirac-conjugation of Ψ,

point of view, analytically Ψ is **C**4-valued.

Besides, *γ<sup>A</sup>* are Hermitian with respect to Γ:

(c) The quantities *e<sup>μ</sup>*

given by the reciprocal expression

(d) The metric tensor *gμν* is built of *e<sup>A</sup>*

following sequence of expressions:

some arguments from geometrodynamics and gauge theories, one of fundamental methods in modern fields theory. It is known that there are certain disadvantages in geometrodynamical gauge models based on the Poincare group as a gauge group, in spite of certain correct results following from that approach. It seems that the main reason is the fact that Poincare group is not semi-simple. The best way out seems to take the simplest semisimple extension of this group, namely the conformal group of Minkowski space. It must be stressed that this group does not act in the space-time manifold, which in geometrodynamics is a general non-flat manifold of a dynamical structure. Instead, it acts as a purely internal group operating with internal degrees of freedom of our matter fields. To be more precise, instead its universal conformal group CO(1, 3), one should use its universal covering group SU(2, 2) of pseudo-unitary mappings with the signature (+, +, −, −), acting in the target spaces of matter fields and on the gauge connection components. The primary field equations are differential ones of the second-order in matter fields. After the careful rewriting in terms of the basic elements of Lie algebra SU(2, 2)� , the internal group rules both matter and geometry (gravitation). The use of conformal group is interesting in itself. It is the smallest semi-simple group containing Poincare group. It is also the largest group which in the geometrically Minkowskian formulation preserves the family of relativistic uniformly accelerated motions (described by the flat time-like hyperboles). It turns out that there are interesting aspects of this approach, having some correspondence with the usual gravitation theory and with generally-relativistic spinor fields. In the specially-relativistic limit, the theory seems to predict the existence of pairs of fundamental quarks and leptons, just as it is really in Nature. We mean here the quark pairs (*u*, *d*), (*c*,*s*), (*t*, *b*) and those of leptons (*νe*,*e*), (*νμ*, *μ*), (*ντ*, *τ*). It is interesting that in this limit the fermion fields are described by the Klein-Gordon-Dirac equation combining the Klein-Gordon and Dirac operators, and that in this limit the Dirac behaviour of fields seems to be more remarkable. There are certain interesting facts concerning the spin-statistics problem. It seems that on the very fundamental level some fermion-boson mixing may appear, or that the two possibilities will be unified by some quite new approach. This framework seems to be related, in a rather unexpected way, to another aspect of the problem of the order of time derivatives in quantum mechanics. Namely, certain quite interesting aspects of the quantum-mechanical and quantum field-theoretic problems appears, when one temporarily forgets about the quantum nature of equations, and considers them simply as some Hamiltonian systems of mathematical physics. Certain primary ideas concerning this problem were formulated in our papers a few years ago (Sławianowski & Kovalchuk, 2002; 2008; 2010; Sławianowski et al., 2004; 2005). There are some arguments which seem to show that there is some so-to-speak inadequacy in the first-order Schrödinger equation. In any case, the second-order corrections seem to be just admissible if not desirable. There are also certain indications for that from the theory of stochastic processes. This approach has certain common points with the former gauge-theoretic one. Namely, once using the language of Hamiltonian dynamics (perhaps infinite-dimensional one) we do not feel any longer the usual reluctance of quantum people to the idea of non linearity. In particular, it turns out that the dynamical scalar product, i.e., one non-constant, but satisfying a closed system of equations with the wave amplitudes, is a natural constituent of the approach. The theory becomes then essentially nonlinear. Essentially, i.e., in such a way that nonlinearity is not an accidental term imposed onto some basic linear background. Everything is then nonlinear in the zeroth-order approximation. Nonlinearity is an essential feature, similar to one used in non-Abelian gauge theories and may be perhaps responsible for the decoherence and measurement paradoxes.

2 Will-be-set-by-IN-TECH

some arguments from geometrodynamics and gauge theories, one of fundamental methods in modern fields theory. It is known that there are certain disadvantages in geometrodynamical gauge models based on the Poincare group as a gauge group, in spite of certain correct results following from that approach. It seems that the main reason is the fact that Poincare group is not semi-simple. The best way out seems to take the simplest semisimple extension of this group, namely the conformal group of Minkowski space. It must be stressed that this group does not act in the space-time manifold, which in geometrodynamics is a general non-flat manifold of a dynamical structure. Instead, it acts as a purely internal group operating with internal degrees of freedom of our matter fields. To be more precise, instead its universal conformal group CO(1, 3), one should use its universal covering group SU(2, 2) of pseudo-unitary mappings with the signature (+, +, −, −), acting in the target spaces of matter fields and on the gauge connection components. The primary field equations are differential ones of the second-order in matter fields. After the careful rewriting in terms of

(gravitation). The use of conformal group is interesting in itself. It is the smallest semi-simple group containing Poincare group. It is also the largest group which in the geometrically Minkowskian formulation preserves the family of relativistic uniformly accelerated motions (described by the flat time-like hyperboles). It turns out that there are interesting aspects of this approach, having some correspondence with the usual gravitation theory and with generally-relativistic spinor fields. In the specially-relativistic limit, the theory seems to predict the existence of pairs of fundamental quarks and leptons, just as it is really in Nature. We mean here the quark pairs (*u*, *d*), (*c*,*s*), (*t*, *b*) and those of leptons (*νe*,*e*), (*νμ*, *μ*), (*ντ*, *τ*). It is interesting that in this limit the fermion fields are described by the Klein-Gordon-Dirac equation combining the Klein-Gordon and Dirac operators, and that in this limit the Dirac behaviour of fields seems to be more remarkable. There are certain interesting facts concerning the spin-statistics problem. It seems that on the very fundamental level some fermion-boson mixing may appear, or that the two possibilities will be unified by some quite new approach. This framework seems to be related, in a rather unexpected way, to another aspect of the problem of the order of time derivatives in quantum mechanics. Namely, certain quite interesting aspects of the quantum-mechanical and quantum field-theoretic problems appears, when one temporarily forgets about the quantum nature of equations, and considers them simply as some Hamiltonian systems of mathematical physics. Certain primary ideas concerning this problem were formulated in our papers a few years ago (Sławianowski & Kovalchuk, 2002; 2008; 2010; Sławianowski et al., 2004; 2005). There are some arguments which seem to show that there is some so-to-speak inadequacy in the first-order Schrödinger equation. In any case, the second-order corrections seem to be just admissible if not desirable. There are also certain indications for that from the theory of stochastic processes. This approach has certain common points with the former gauge-theoretic one. Namely, once using the language of Hamiltonian dynamics (perhaps infinite-dimensional one) we do not feel any longer the usual reluctance of quantum people to the idea of non linearity. In particular, it turns out that the dynamical scalar product, i.e., one non-constant, but satisfying a closed system of equations with the wave amplitudes, is a natural constituent of the approach. The theory becomes then essentially nonlinear. Essentially, i.e., in such a way that nonlinearity is not an accidental term imposed onto some basic linear background. Everything is then nonlinear in the zeroth-order approximation. Nonlinearity is an essential feature, similar to one used in non-Abelian gauge theories and may be perhaps responsible for the decoherence

, the internal group rules both matter and geometry

the basic elements of Lie algebra SU(2, 2)�

and measurement paradoxes.

We have formulated some arguments in favour of SU(2, 2) as a fundamental gauge group. Let us mention, incidentally, that this simply provokes the next question: Why the subgroup SU(2, 2) ⊂ GL(4, **C**) but not just the whole GL(4, **C**)? The latter group appears in a natural way as the structure group of the principal fibre bundle of the complexification of the usual bundle of frames over the four-dimensional space-time manifold. Obviously, it preserves the signature of sesquilinear forms, nevertheless changing them otherwise. Therefore, the bispinor sesquilinear Hermitian form *G* of signature (+, +, −, −) becomes an a priori free Hermitian form, the signature however being in a sense an integral of motion.

#### **2. On the track of the scalar Klein-Gordon-Dirac formalism**

The generally-relativistic Lagrangian of the Dirac field is given by the expression:

$$L = \frac{i}{2} e^{\mu}{}\_A \gamma^{Ar}{}\_s \left( \tilde{\mathbf{Y}}\_I D\_\mu \mathbf{Y}^s - D\_\mu \tilde{\mathbf{Y}}\_r \mathbf{Y}^s \right) \sqrt{|\mathbf{g}|} - m \tilde{\mathbf{Y}}\_r \mathbf{Y}^r \sqrt{|\mathbf{g}|} \tag{1}$$

with the following meaning of symbols: (a) Ψ denote the Dirac-conjugation of Ψ,

$$
\tilde{\mathbf{Y}}\_{\mathbf{r}} = \overline{\mathbf{Y}}^{\overline{\mathbf{s}}} \mathbf{G}\_{\mathbf{\tilde{s}r}\prime} \tag{2}
$$

where *G* denotes the Dirac-conjugation form of mass, i.e., sesquilinear Hermitian form of the natural signature (+, +, −, −). If the Finkelstein-Penrose-Weizsäcker-van der Waerden point of view on the two-component spinors is accepted, then *G* is intrinsic, because the **C**4-space is then expressed as the Cartesian product of two mutually antidual copies of **C**2. Without this point of view, analytically Ψ is **C**4-valued.

(b) Dirac matrices *γ<sup>A</sup>* satisfy the following anticommutation rules:

$$\left\{\gamma^{A}\gamma^{B}\right\}=\gamma^{A}\gamma^{B}+\gamma^{B}\gamma^{A}=2\eta^{AB}I\_{4\prime}\tag{3}$$

$$\mathbb{E}\left[\gamma^{AB}\right] = \text{diag}\left(1, -1, -1, -1\right). \tag{4}$$

Besides, *γ<sup>A</sup>* are Hermitian with respect to Γ:

$$
\Gamma^A{}\_{\mathbb{F}s} = \overline{\Gamma^A{}\_{\mathbb{F}l}} = \overline{\Gamma^A{}\_{\mathbb{F}r}} = G\_{\mathbb{F}z} \gamma^{Az}{}\_s. \tag{5}
$$

(c) The quantities *e<sup>μ</sup> <sup>A</sup>* are components of the tetrad field. Its dual cotetrad *e<sup>A</sup> <sup>μ</sup>* is analytically given by the reciprocal expression

$$e^A{}\_{\mu}e^{\mu}{}\_{B} = \delta^A{}\_{B}.\tag{6}$$

(d) The metric tensor *gμν* is built of *e<sup>A</sup> <sup>μ</sup>* in a quadratic way:

$$g\_{\mu\nu} = \eta\_{AB} e^A{}\_{\mu} e^B{}\_{\nu} \quad [\eta\_{AB}] = \text{diag}\left(1, -1, -1, -1\right). \tag{7}$$

The Greek and capital Latin indices are shifted with the help of *gμν* and *ηAB*. (e) The operation *D<sup>μ</sup>* symbolizes the covariant differentiation of bispinors. It is given by the following sequence of expressions:

$$
\Gamma^{\mathfrak{a}}{}\_{\beta\mu} = e^{\mathfrak{a}}{}\_{A} \Gamma^{A}{}\_{B\mu} e^{B}{}\_{\beta} + e^{\mathfrak{a}}{}\_{A} e^{A}{}\_{\beta,\mu\nu} \tag{8}
$$

$$\Gamma^{A}{}\_{B\mu} = \frac{1}{2} \text{Tr}\left(\gamma^{A} \omega\_{\mu} \gamma\_{B}\right) , \tag{9}$$

and that the corresponding Lagrangian for Ψ should be just the complex-four-dimensional Lagrangian for the field Ψ, this time invariant under the total U(2, 2), no longer by SL(2, **C**). Nevertheless, some fundamental question remains, namely one concerning the relationship between Klein-Gordon equation of order two and first-order Dirac equation. It is though clear that the structural properties of differential equations are in very malicious way sensitive to

Order of Time Derivatives in Quantum-Mechanical Equations 61

It is interesting to begin the analysis from some rather academic example of the scalar complex field interacting in a minimal way with the gauge filed *eμ*, i.e., dynamically ruled by the unitary group U(1). Namely, let us assume the primeval globally invariant by U(1)

*<sup>g</sup>μν∂μ*Ψ*∂ν*<sup>Ψ</sup> <sup>−</sup> *<sup>c</sup>*

Now, as usually we introduce the covector gauge field *e<sup>μ</sup>* and the covariant derivative of Ψ,

Substituting *D<sup>μ</sup>* instead of *∂μ* to (16) we obtain as usual the locally-invariant expression

 <sup>|</sup>*g*<sup>|</sup> <sup>−</sup> *<sup>c</sup>* 2 ΨΨ

*<sup>g</sup>μ*κ*gνλ <sup>f</sup>μν <sup>f</sup>*κ*<sup>λ</sup>*

<sup>Ψ</sup>*∂ν*<sup>Ψ</sup> <sup>−</sup> *∂ν*ΨΨ

The first term, built of the first derivatives and of the algebraic expressions of fields, leads to first-order differential equations with respect to Ψ. The last, Klein-Gordon term leads through variational principle to the second-order equations in Ψ. The rigorous field equations have

> *<sup>μ</sup>e<sup>μ</sup>* <sup>−</sup> *iq* 2 *e μ* ;*μ* − 1 2

> > + *q*2*e*

Ψ*∂μ*Ψ − (*∂μ*Ψ)Ψ

 ΨΨ |*g*|


*gμνDμ*Ψ*Dν*Ψ

*Lg* <sup>=</sup> <sup>−</sup><sup>1</sup> 4

(the subscripts *m*, *g* refer respectively to the matter and gauge field).

*i* 2 

 *c* <sup>2</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup> 2 *e*

2 

<sup>2</sup> *<sup>g</sup>μνeμe<sup>ν</sup>*

2

ΨΨ

*Dμ*Ψ := *∂μ*Ψ − *iqeμ*Ψ. (17)

*fμν* = *∂μe<sup>ν</sup>* − *∂νe<sup>μ</sup>* (19)

*L* = *Lm* + *Lg* (21)

*gμν∂μ*Ψ*∂ν*Ψ





*gμν∂μ∂ν*Ψ = 0, (23)

*<sup>μ</sup>*ΨΨ. (24)

the removing highest-order derivative term.

*Lm* =

*Lm* <sup>=</sup> <sup>1</sup> 2

This is Lagrangian for Ψ. The corresponding term for

and the total Lagrangian for (Ψ, *f*) is given by the sum

Let us rewrite the matter term in the following form:

*Lm* = *qgμνe<sup>μ</sup>*

*qieμ∂μ*<sup>Ψ</sup> <sup>−</sup>

*∂ν <sup>f</sup> μν* <sup>=</sup> *qi*

− *c* <sup>2</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup> 1 2

Lagrangian for Ψ : *M* → **C**,

is as usually given by

the following form:

$$
\omega\_{\mu} = \frac{1}{2} \Gamma\_{L\mathbf{K}\mu} \Sigma^{L\mathbf{K}} = \frac{1}{2} \eta\_{LM} \Gamma^{M}{}\_{K\mu} \Sigma^{L\mathbf{K}}{}\_{\prime} \tag{10}
$$

$$
\Sigma^{LK} = \frac{1}{4} \left( \gamma^L \gamma^K - \gamma^K \gamma^L \right) . \tag{11}
$$

where, obviously, the following identities hold:

$$
\eta\_{A\mathbf{C}}\Gamma^{\mathbb{C}}{}\_{B\mu} + \eta\_{\mathbf{BC}}\Gamma^{\mathbb{C}}{}\_{A\mu} = 0,\qquad \nabla^{\Gamma}{}\_{\mu}g\_{a\beta} = 0.\tag{12}
$$

This means that the following is satisfied:

$$
\Gamma^{\alpha}{}\_{\beta\mu} = \left\{ ^{\alpha}\_{\beta\mu} \right\} + \mathcal{S}^{\alpha}{}\_{\beta\mu} + \mathcal{S}\_{\beta\mu}{}^{\alpha} - \mathcal{S}\_{\mu}{}^{\alpha}{}\_{\beta}.
$$

This means that *α βμ* are coefficients of the Levi-Civita connection built of the metric *g*, and *Sα βμ* = Γ*<sup>α</sup>* [*βμ*] is the torsion tensor of Γ*<sup>α</sup> βμ*.

In a sense this is a gauge theory. There are, however, certain objections against it, although from some point of view the theory works in a satisfactory way as a gauge frame. Let us quote mentioned objections:

(a) The use of tetrads seems to be essential here. This is important. In the standard gauge theories, e.g., in Salam-Weinberg model, or in quantum chromodynamics, the field of frames does not occur explicitly in the formalism. The idea of the local gauge Minkowski translations leads us far beyond the ground of the theory.

(b) Another non-pleasant and strange feature of the theory is the doubtful meaning of the gauge invariance under the group SU(2, 2), just as its quotient SL(2, **C**). Namely, for the massless particles it is so that field equations are SU(2, 2)/conformally invariant. But Lagrangian is not so, and because of this the Noether theorem is not either. And at the same time the mass form *Grs* just underlies the inertial properties of those Lagrangians. So, it is natural to expect that it should be essential for the invariance of Lagrangian. Therefore, perhaps Lagrangian should be transformed exactly up to the purely internal rule under SU(2, 2), just like it is done under the subgroup SL(2, **C**) in Einstein-Cartan theory. Namely, that for any *A* ∈ SU(2, 2) it should be modified as follows under *A* ∈ SL(2, **C**) in (1):

$$\left( (A\Psi)^{r} \left( \mathbf{x} \right) = A^{r}{}\_{s} \Psi^{s} \left( \mathbf{x} \right). \tag{13}$$

This is a purely internal rule, like those for SL(2, **C**) in (1), without any external correction. In this way, the number of geometric degrees of freedom is increased, but it should be so if SU(2, 2) are to be Hamiltonian symmetries.

(c) Finally, there is a strange feature of Lagrangian (1), namely, the one that it is essentially based on the covariant vector density

$$J^r{}\_{s\mu} := \left(D\_{\mu}\widetilde{\Psi}\_s\Psi^r - \widetilde{\Psi}\_s D\_{\mu}\Psi^r\right)\sqrt{|g|}\,. \tag{14}$$

or, to be more honest, on its contravariant upper-index version

$$J^r{}\_s{}^\mu := g^{\mu\nu} \left( D\_\nu \tilde{\mathbf{Y}}\_s \mathbf{Y}^r - \tilde{\mathbf{Y}}\_s D\_\nu \mathbf{Y}^r \right) \sqrt{|g|}. \tag{15}$$

The idea is that *J<sup>r</sup> s <sup>μ</sup>* looks as a typical bosonic current. What is the symmetry group responsible for it? The algebraic prescription for *J<sup>r</sup> s <sup>μ</sup>* does suggest that it is the group U(2, 2) and that the corresponding Lagrangian for Ψ should be just the complex-four-dimensional Lagrangian for the field Ψ, this time invariant under the total U(2, 2), no longer by SL(2, **C**). Nevertheless, some fundamental question remains, namely one concerning the relationship between Klein-Gordon equation of order two and first-order Dirac equation. It is though clear that the structural properties of differential equations are in very malicious way sensitive to the removing highest-order derivative term.

It is interesting to begin the analysis from some rather academic example of the scalar complex field interacting in a minimal way with the gauge filed *eμ*, i.e., dynamically ruled by the unitary group U(1). Namely, let us assume the primeval globally invariant by U(1) Lagrangian for Ψ : *M* → **C**,

$$L\_{\mathfrak{m}} = \left(\frac{1}{2}g^{\mu\nu}\overline{\partial\_{\mu}\Psi}\partial\_{\nu}\Psi - \frac{c}{2}\overline{\Psi}\Psi\right)\sqrt{|g|}.\tag{16}$$

Now, as usually we introduce the covector gauge field *e<sup>μ</sup>* and the covariant derivative of Ψ,

$$D\_{\mu}\Psi := \partial\_{\mu}\Psi - iqe\_{\mu}\Psi. \tag{17}$$

Substituting *D<sup>μ</sup>* instead of *∂μ* to (16) we obtain as usual the locally-invariant expression

$$L\_{\mathfrak{m}} = \frac{1}{2} g^{\mu \nu} \overline{D\_{\mu} \Psi} D\_{\nu} \Psi \sqrt{|g|} - \frac{c}{2} \overline{\Psi} \Psi \sqrt{|g|}. \tag{18}$$

This is Lagrangian for Ψ. The corresponding term for

$$f\_{\mu\nu} = \partial\_{\mu}\varepsilon\_{\nu} - \partial\_{\nu}\varepsilon\_{\mu} \tag{19}$$

is as usually given by

4 Will-be-set-by-IN-TECH

2

*<sup>A</sup><sup>μ</sup>* <sup>=</sup> 0, <sup>∇</sup><sup>Γ</sup>

*ηLM*Γ*<sup>M</sup>*

*βμ* <sup>+</sup> *<sup>S</sup>βμ<sup>α</sup>* <sup>−</sup> *<sup>S</sup><sup>μ</sup>*

are coefficients of the Levi-Civita connection built of the metric *g*, and

*α β*.

*<sup>K</sup>μ*Σ*LK*, (10)

*<sup>μ</sup>gαβ* = 0. (12)

, (11)

*<sup>s</sup>*Ψ*<sup>s</sup>* (*x*). (13)



*<sup>μ</sup>* does suggest that it is the group U(2, 2)

<sup>Γ</sup>*LKμ*Σ*LK* <sup>=</sup> <sup>1</sup>

*<sup>γ</sup>Lγ<sup>K</sup>* <sup>−</sup> *<sup>γ</sup>Kγ<sup>L</sup>*

*ωμ* <sup>=</sup> <sup>1</sup> 2

<sup>Σ</sup>*LK* <sup>=</sup> <sup>1</sup> 4 

*<sup>B</sup><sup>μ</sup>* + *ηBC*Γ*<sup>C</sup>*

*βμ*.

In a sense this is a gauge theory. There are, however, certain objections against it, although from some point of view the theory works in a satisfactory way as a gauge frame. Let us quote

(a) The use of tetrads seems to be essential here. This is important. In the standard gauge theories, e.g., in Salam-Weinberg model, or in quantum chromodynamics, the field of frames does not occur explicitly in the formalism. The idea of the local gauge Minkowski translations

(b) Another non-pleasant and strange feature of the theory is the doubtful meaning of the gauge invariance under the group SU(2, 2), just as its quotient SL(2, **C**). Namely, for the massless particles it is so that field equations are SU(2, 2)/conformally invariant. But Lagrangian is not so, and because of this the Noether theorem is not either. And at the same time the mass form *Grs* just underlies the inertial properties of those Lagrangians. So, it is natural to expect that it should be essential for the invariance of Lagrangian. Therefore, perhaps Lagrangian should be transformed exactly up to the purely internal rule under SU(2, 2), just like it is done under the subgroup SL(2, **C**) in Einstein-Cartan theory. Namely,

that for any *A* ∈ SU(2, 2) it should be modified as follows under *A* ∈ SL(2, **C**) in (1):

*<sup>r</sup>* (*x*) = *A<sup>r</sup>*

This is a purely internal rule, like those for SL(2, **C**) in (1), without any external correction. In this way, the number of geometric degrees of freedom is increased, but it should be so if

(c) Finally, there is a strange feature of Lagrangian (1), namely, the one that it is essentially

*<sup>D</sup>μ*Ψ*s*Ψ*<sup>r</sup>* <sup>−</sup> <sup>Ψ</sup>*sDμ*Ψ*<sup>r</sup>*

*<sup>D</sup>ν*Ψ*s*Ψ*<sup>r</sup>* <sup>−</sup> <sup>Ψ</sup>*sDν*Ψ*<sup>r</sup>*

*s*

*<sup>μ</sup>* looks as a typical bosonic current. What is the symmetry group

(*A*Ψ)

where, obviously, the following identities hold:

[*βμ*] is the torsion tensor of Γ*<sup>α</sup>*

leads us far beyond the ground of the theory.

SU(2, 2) are to be Hamiltonian symmetries.

*J r <sup>s</sup><sup>μ</sup>* := 

*J r s*

responsible for it? The algebraic prescription for *J<sup>r</sup>*

or, to be more honest, on its contravariant upper-index version

*<sup>μ</sup>* :<sup>=</sup> *<sup>g</sup>μν*

based on the covariant vector density

*s*

The idea is that *J<sup>r</sup>*

This means that the following is satisfied:

 *α βμ* 

This means that

mentioned objections:

*βμ* = Γ*<sup>α</sup>*

*Sα*

*ηAC*Γ*<sup>C</sup>*

Γ*α βμ* = *α βμ* + *S<sup>α</sup>*

$$L\_{\mathcal{S}} = -\frac{1}{4} g^{\mu \varkappa} g^{\nu \lambda} f\_{\mu \nu} f\_{\varkappa \lambda} \sqrt{|g|} \,\tag{20}$$

and the total Lagrangian for (Ψ, *f*) is given by the sum

$$L = L\_m + L\_\S \tag{21}$$

(the subscripts *m*, *g* refer respectively to the matter and gauge field). Let us rewrite the matter term in the following form:

$$\begin{split} L\_{\mathfrak{m}} &= q g^{\mu \nu} e\_{\mu} \frac{i}{2} \left( \overline{\Psi} \partial\_{\nu} \Psi - \overline{\partial\_{\nu} \Psi} \Psi \right) \sqrt{|g|} \\ &- \left( \frac{c}{2} - \frac{q^{2}}{2} g^{\mu \nu} e\_{\mu} e\_{\nu} \right) \overline{\Psi} \Psi \sqrt{|g|} + \frac{1}{2} g^{\mu \nu} \overline{\partial\_{\mu} \Psi} \partial\_{\nu} \Psi \sqrt{|g|}. \end{split} \tag{22}$$

The first term, built of the first derivatives and of the algebraic expressions of fields, leads to first-order differential equations with respect to Ψ. The last, Klein-Gordon term leads through variational principle to the second-order equations in Ψ. The rigorous field equations have the following form:

$$qie^{\mu}\partial\_{\mu}\Psi - \left(\frac{c}{2} - \frac{q^2}{2}e^{\mu}e\_{\mu} - \frac{iq}{2}e^{\mu}{}\_{;\mu}\right) - \frac{1}{2}g^{\mu\nu}\partial\_{\mu}\partial\_{\nu}\Psi = 0,\tag{23}$$

$$
\partial\_{\nu} f^{\mu \nu} = \frac{q i}{2} \left( \overline{\Psi} \partial\_{\mu} \Psi - (\partial\_{\mu} \overline{\Psi}) \Psi \right) + q^{2} e^{\mu} \overline{\Psi} \Psi. \tag{24}
$$

convenience. It is only its global signature that matters. As mentioned, in the Weyl, Penrose,

Order of Time Derivatives in Quantum-Mechanical Equations 63

 I2 0 0 −I2

Similarly, the Weyl, Penrose, Finkelstein and Weizsäcker procedure leads to the following

 =  0 *ηABσ<sup>B</sup> σ<sup>A</sup>* 0

 0 *σ<sup>R</sup>* <sup>−</sup>*σ<sup>R</sup>* <sup>0</sup>

*M* � *x* �→ *ϑ<sup>x</sup>* ∈ *L* (*TxM*, *u*(4, *G*)) (37)

(*Uϑ*)*<sup>x</sup>* <sup>=</sup> *<sup>U</sup>*(*x*)*ϑxU*(*x*)−<sup>1</sup> <sup>−</sup> *dUxU*(*x*)<sup>−</sup>1. (38)

 <sup>0</sup> *<sup>σ</sup><sup>A</sup> σ<sup>A</sup>* 0

, *γ<sup>R</sup>* =

Obviously, those are two particular choices, we quote them only as the two most important

The globally U(2, 2)-invariant second-order Klein-Gordon Lagrangian for the **C**4-valued

*∂ν*Ψ*<sup>s</sup> Grs*¯ <sup>|</sup>*g*<sup>|</sup> <sup>−</sup> *<sup>c</sup>* 2 *Grs*¯ <sup>Ψ</sup>*r*¯ Ψ*s* 

<sup>Ψ</sup>*<sup>r</sup>* :<sup>=</sup> <sup>Ψ</sup>*s*¯

*<sup>g</sup>μν∂μ*<sup>Ψ</sup>*∂ν*<sup>Ψ</sup>

First let us consider the problem of the local U(2, 2) � U(4, *G*)-invariance. To do that we must begin with introducing the connection form of the U(4, *G*)-connection. This is a u(4, *G*)-valued

transforming under the local U(4, *G*)-valued local transformations *U* : *M* → U(4, *G*) as

 <sup>|</sup>*g*<sup>|</sup> <sup>−</sup> *<sup>c</sup>* 2 ΨΨ

 0 I2 I2 0

. (29)

. (30)

. (31)

. (32)

, *R* = 1, 2, 3. (33)



*Gsr*¯ , (35)

[*Grs*¯ ] =

[*Grs*¯ ] =

1 √2 I2 I2 I2 −I2

Transition between these representations is described by the matrix

*γ<sup>A</sup>* = *ηABγ<sup>B</sup>* =

Finkelstein and Weizsäcker procedure the typical choice is

In the Dirac procedure one prefers the choice:

expressions for the Dirac matrices:

*γ*<sup>0</sup> = I2 0 0 −I2

scalar field on *M* is given by

differential form

follows:

ones.

In the Dirac representation we have that

Making use of the Dirac-conjugate field,

we can rewrite (34) in the following form:

 = *σ*<sup>0</sup> 0 <sup>0</sup> <sup>−</sup>*σ*<sup>0</sup>

*Lm*(Ψ; *<sup>g</sup>*) = *<sup>b</sup>*

2

*Lm*(Ψ; *<sup>g</sup>*) = *<sup>b</sup>*

*<sup>g</sup>μν∂μ*Ψ*r*¯

2

Obviously, the semicolon symbol in (23) denotes the *g*-metric Levi-Civita affine connection, or rather divergence. It is interesting that the vector field *e<sup>μ</sup>* plays a role similar to that of gravitational tetrad, in spite of all differences. And in general, the pair (Ψ,*eμ*) is formally analogous to the pair (Ψ*r*,*e<sup>μ</sup> <sup>A</sup>*), or equivalently Ψ*r*,*e<sup>A</sup> μ* , i.e., bispinor and tetrad/cotetrad. But there is no rigorous Clifford analogy. It is interesting that on the right hand side of (24) there is a combination of two terms: Dirac-like current and Schrödinger current.

It is interesting that the system of equations (23), (24) may be simplified by assuming that the system of first-order derivatives of Ψ is smaller than the system of quantities built of Ψ in an algebraic way, and similarly, the system of second derivatives of Ψ is smaller than the first- and zeroth-order derivatives of Ψ. But this means that the system (23), (24) may be approximated by the following one:

$$ie^{\mu}\partial\_{\mu}\Psi - \left(\frac{c}{2q} - \frac{q}{2}e^{\mu}e\_{\mu} - \frac{i}{2}e^{\mu}{}\_{;\mu}\right)\Psi = 0,\tag{25}$$

$$
\partial\_{\boldsymbol{\nu}} f^{\mu \boldsymbol{\nu}} = q^2 e^{\mu} \overline{\boldsymbol{\Psi}} \boldsymbol{\Psi}. \tag{26}
$$

It is interesting that this system, except the Clifford analogy, is structurally similar to the Dirac system of equations. It is difficult to state a priori if the essentially nonlinear system (25), (26) may have anything to do with reality. Nevertheless, the point is that it is both nonlinear, and as a system imposed on the pair Ψ,*e<sup>μ</sup>* it shows certain similarity to the Dirac-Maxwell system. And the bosonic current *i ∂μ*ΨΨ − Ψ*∂μ*Ψ |*g*<sup>|</sup> is an obvious counterpart of the SU(2, 2) current given by *i <sup>D</sup>μ*Ψ*s*Ψ*<sup>r</sup>* <sup>−</sup> <sup>Ψ</sup>*sDμ*Ψ*<sup>r</sup>* |*g*|. Obviously, the model (25), (26) is a bit non-physical and crazy, especially with its separation of terms. Nevertheless, it seems to follow from it that the above demands and objections concerning the U(2, 2)-invariance and the particular role of the bosonic currents and tetrads may be easily answered on the basis of the spinor counterpart of *Lm* (22) and its first-order limit (23), (24).

#### **3. Second order Klein-Gordon equation**

We need a few things, for instance, affine connection in space-time manifold, spinor connection, U(2, 2)-gauge field, metric tensor, and in certain approaches some field of frames, e.g., generalization of the tetrad field. The space-time manifold *M* is assumed structure-less and nothing but the differential-geometric structure is assumed in it. Unlike this, in the target space **C**4, we assume some internal geometry based on the use of some sesquilinear Hermitian *G* form of signature (+, +, −, −), as mentioned above. This form does belong to the internal structure of **C**4, and to be more rigorous, we can assume it to be a complex linear space of dimension four, endowed with the mentioned neutral signature. When this form is fixed, it distinguishes within the complex group GL(4, **C**), the pseudounitary group consisting of transformations preserving *G*, so that the following holds:

$$G\_{\rm ps} = G\_{\rm 27} \overline{\mathcal{U}}^2 \mathcal{U}^t{}\_s. \tag{27}$$

The Lie algebra of this group consists of linear mappings *u* which satisfy:

$$
\overline{G\_{\mathbb{P}2}\mu^2}\_{\mathbb{S}} + \overline{G\_{\mathbb{S}2}\mu^2}\_{\mathbb{S}} = 0. \tag{28}
$$

So, roughly speaking, U(2, 2)� consists of matrices (linear mapping of the target space) which are *G*-anti-Hermitian. Let us mention that any particular choice of Γ is only a matter of convenience. It is only its global signature that matters. As mentioned, in the Weyl, Penrose, Finkelstein and Weizsäcker procedure the typical choice is

$$[G\_{\rm l^{\ast}}] = \begin{bmatrix} 0 \ \mathbf{I}\_{2} \\ \mathbf{I}\_{2} \ \mathbf{0} \end{bmatrix}. \tag{29}$$

In the Dirac procedure one prefers the choice:

6 Will-be-set-by-IN-TECH

Obviously, the semicolon symbol in (23) denotes the *g*-metric Levi-Civita affine connection, or rather divergence. It is interesting that the vector field *e<sup>μ</sup>* plays a role similar to that of gravitational tetrad, in spite of all differences. And in general, the pair (Ψ,*eμ*) is formally

But there is no rigorous Clifford analogy. It is interesting that on the right hand side of (24)

It is interesting that the system of equations (23), (24) may be simplified by assuming that the system of first-order derivatives of Ψ is smaller than the system of quantities built of Ψ in an algebraic way, and similarly, the system of second derivatives of Ψ is smaller than the first- and zeroth-order derivatives of Ψ. But this means that the system (23), (24) may be approximated

It is interesting that this system, except the Clifford analogy, is structurally similar to the Dirac system of equations. It is difficult to state a priori if the essentially nonlinear system (25), (26) may have anything to do with reality. Nevertheless, the point is that it is both nonlinear,

*∂μ*ΨΨ − Ψ*∂μ*Ψ

bit non-physical and crazy, especially with its separation of terms. Nevertheless, it seems to follow from it that the above demands and objections concerning the U(2, 2)-invariance and the particular role of the bosonic currents and tetrads may be easily answered on the basis of

We need a few things, for instance, affine connection in space-time manifold, spinor connection, U(2, 2)-gauge field, metric tensor, and in certain approaches some field of frames, e.g., generalization of the tetrad field. The space-time manifold *M* is assumed structure-less and nothing but the differential-geometric structure is assumed in it. Unlike this, in the target space **C**4, we assume some internal geometry based on the use of some sesquilinear Hermitian *G* form of signature (+, +, −, −), as mentioned above. This form does belong to the internal structure of **C**4, and to be more rigorous, we can assume it to be a complex linear space of dimension four, endowed with the mentioned neutral signature. When this form is fixed, it distinguishes within the complex group GL(4, **C**), the pseudounitary group consisting of

*Grs*¯ <sup>=</sup> *Gzt*¯ *<sup>U</sup>z*¯

*<sup>s</sup>* + *Gsz*¯ *u<sup>z</sup>*

So, roughly speaking, U(2, 2)� consists of matrices (linear mapping of the target space) which are *G*-anti-Hermitian. Let us mention that any particular choice of Γ is only a matter of

The Lie algebra of this group consists of linear mappings *u* which satisfy:

*Grz*¯ *u<sup>z</sup>*

*rU<sup>t</sup>*

Ψ*r*,*e<sup>A</sup> μ* 

, i.e., bispinor and tetrad/cotetrad.

Ψ = 0, (25)

*<sup>μ</sup>*ΨΨ. (26)

it shows certain similarity to the Dirac-Maxwell


*<sup>s</sup>*. (27)

*<sup>r</sup>* = 0. (28)


*<sup>A</sup>*), or equivalently

there is a combination of two terms: Dirac-like current and Schrödinger current.

 *c* <sup>2</sup>*<sup>q</sup>* <sup>−</sup> *<sup>q</sup>* 2 *e <sup>μ</sup>e<sup>μ</sup>* <sup>−</sup> *<sup>i</sup>* 2 *e μ* ;*μ* 

Ψ,*e<sup>μ</sup>* 

*<sup>D</sup>μ*Ψ*s*Ψ*<sup>r</sup>* <sup>−</sup> <sup>Ψ</sup>*sDμ*Ψ*<sup>r</sup>*

*ieμ∂μ*<sup>Ψ</sup> <sup>−</sup>

transformations preserving *G*, so that the following holds:

*∂ν f μν* = *q*2*e*

the spinor counterpart of *Lm* (22) and its first-order limit (23), (24).

analogous to the pair (Ψ*r*,*e<sup>μ</sup>*

by the following one:

and as a system imposed on the pair

**3. Second order Klein-Gordon equation**

system. And the bosonic current *i*

SU(2, 2) current given by *i*

$$\begin{bmatrix} \mathbf{G}\_{\mathsf{P}s} \end{bmatrix} = \begin{bmatrix} \mathbf{I}\_2 & \mathbf{0} \\ \mathbf{0} & -\mathbf{I}\_2 \end{bmatrix}. \tag{30}$$

Transition between these representations is described by the matrix

$$\frac{1}{\sqrt{2}} \begin{bmatrix} \mathbf{I}\_2 & \mathbf{I}\_2 \\ \mathbf{I}\_2 & -\mathbf{I}\_2 \end{bmatrix}. \tag{31}$$

Similarly, the Weyl, Penrose, Finkelstein and Weizsäcker procedure leads to the following expressions for the Dirac matrices:

$$\gamma^{A} = \eta^{AB}\gamma\_{B} = \begin{bmatrix} 0 & \tilde{\sigma}^{A} \\ \sigma^{A} & 0 \end{bmatrix} = \begin{bmatrix} 0 & \eta^{AB}\sigma\_{B} \\ \sigma^{A} & 0 \end{bmatrix}. \tag{32}$$

In the Dirac representation we have that

$$\gamma^0 = \begin{bmatrix} \mathbf{I}\_2 & \mathbf{0} \\ \mathbf{0} & -\mathbf{I}\_2 \end{bmatrix} = \begin{bmatrix} \sigma^0 & \mathbf{0} \\ \mathbf{0} & -\sigma^0 \end{bmatrix}, \qquad \gamma^R = \begin{bmatrix} \mathbf{0} & \sigma^R \\ -\sigma^R & \mathbf{0} \end{bmatrix}, \qquad \mathbf{R} = \mathbf{1}, \mathbf{2}, \mathbf{3}. \tag{33}$$

Obviously, those are two particular choices, we quote them only as the two most important ones.

The globally U(2, 2)-invariant second-order Klein-Gordon Lagrangian for the **C**4-valued scalar field on *M* is given by

$$L\_{\mathfrak{m}}(\Psi; \mathfrak{g}) = \frac{b}{2} g^{\mu \nu} \partial\_{\mu} \overline{\Psi}^{\mathfrak{p}} \partial\_{\nu} \Psi^{s} G\_{\mathfrak{ps}} \sqrt{|g|} - \frac{c}{2} G\_{\mathfrak{rs}} \overline{\Psi}^{\mathfrak{p}} \Psi^{s} \sqrt{|g|}. \tag{34}$$

Making use of the Dirac-conjugate field,

$$
\widetilde{\mathbf{Y}}\_{\mathbf{r}} := \overline{\mathbf{Y}}^{\mathbf{\tilde{s}}} \mathbf{G}\_{\mathbf{\tilde{s}r}} \tag{35}
$$

we can rewrite (34) in the following form:

$$L\_{\mathfrak{m}}(\Psi; \mathbf{g}) = \frac{b}{2} \mathbf{g}^{\mu \nu} \partial\_{\mu} \tilde{\mathbf{Y}} \partial\_{\nu} \Psi \sqrt{|\mathbf{g}|} - \frac{c}{2} \tilde{\mathbf{Y}} \Psi \sqrt{|\mathbf{g}|}. \tag{36}$$

First let us consider the problem of the local U(2, 2) � U(4, *G*)-invariance. To do that we must begin with introducing the connection form of the U(4, *G*)-connection. This is a u(4, *G*)-valued differential form

$$M \ni \mathfrak{x} \mapsto \mathfrak{z}\_{\mathfrak{x}} \in L\left(T\_{\mathfrak{x}}M, \mu(\mathfrak{4}, \mathbb{G})\right) \tag{37}$$

transforming under the local U(4, *G*)-valued local transformations *U* : *M* → U(4, *G*) as follows:

$$\mathcal{U}(\mathcal{U}\theta)\_{\mathbf{x}} = \mathcal{U}(\mathbf{x})\theta\_{\mathbf{x}}\mathcal{U}(\mathbf{x})^{-1} - d\mathcal{U}\_{\mathbf{x}}\mathcal{U}(\mathbf{x})^{-1}.\tag{38}$$

Because of this the Yang-Mills Lagrangian (46) is invariant under the local U(2, 2) � U(*H*, *G*)

Order of Time Derivatives in Quantum-Mechanical Equations 65

This is the main, gauge constituent of the theory. Let us now mention only about the relationship of SU(2, 2)-matrices Φ to the representations SL(2, **C**) � *A* �→ *U*[*A*] ∈ U(2, 2) corresponding to the Weyl-Penrose-Finkelstein-Weizsäcker and to the Dirac representation of *Grs*¯ . In the first group of representation (W-P-F-W) we have the following realizations of *U*[*A*]:

for the group and algebra. The right "plus" superscript denotes obviously the Hermitian matrix conjugate. Obviously, quite independently of the choice of any representation the

where *P* : SL(2, **C**) → SO(1, 3)<sup>↑</sup> is the covering projection. The mappings *U* : SL(2, **C**) → U(2, 2) and *P* : SL(2, **C**) → SO(1, 3)<sup>↑</sup> generate the corresponding homomorphisms of Lie

[*u*[*a*],(*γ*)*K*] = *γ<sup>L</sup> p*[*a*]

*<sup>u</sup>*[*a*] = <sup>1</sup>

<sup>−</sup><sup>1</sup> = *γLP*[*A*]

*γLU*[*A*]*γKU*[*A*]

 <sup>=</sup> <sup>1</sup> 4 *γL*, *γ<sup>K</sup>* 

*γLu*[*a*]*γ<sup>K</sup>*

<sup>2</sup> *<sup>p</sup>*[*a*] *L <sup>K</sup>*Σ*<sup>L</sup>*

In the gauge Lagrangians above, the metric tensor in a sense played the parameter role. Our idea was to construct the U(2, 2) � U(*H*, *G*)-invariant theory of gravitation. The main constituents of the theory were the four-component complex Klein-Gordon field and the corresponding Maxwell-like gauge field. We will show that there are interesting and very important points for which this is important, perhaps even just exciting. But there is some weak point which was not yet completely explained. It is just the role and physical status of the metric tensor, which is present in the Klein-Gordon and gauge Lagrangian, however its geometric and physical sense is not yet full understood. It is clear that it must occur there if we are to be able to construct Lagrangians. But what is its meaning and how to identify properly

*<sup>γ</sup>Lγ<sup>K</sup>* <sup>−</sup> *<sup>γ</sup>Kγ<sup>L</sup>*

, *<sup>u</sup>*[*a*] =

, *<sup>u</sup>*[*a*] = <sup>1</sup>

*a* 0 <sup>0</sup> <sup>−</sup>*a*<sup>+</sup>

> 2

*L*

*L*

−1  *<sup>a</sup>* <sup>−</sup> *<sup>a</sup>*<sup>+</sup> *<sup>a</sup>* <sup>+</sup> *<sup>a</sup>*<sup>+</sup> *<sup>a</sup>* <sup>+</sup> *<sup>a</sup>*<sup>+</sup> *<sup>a</sup>* <sup>−</sup> *<sup>a</sup>*<sup>+</sup> *<sup>K</sup>*, (50)

. They are synchronized by

, (52)

. (55)

*<sup>K</sup>*. (51)

, (53)

*<sup>K</sup>*, (54)

(48)

(49)

transformations (38). And similarly, the matter Lagrangian (43) is invariant.

*A* 0 0 *A*−1<sup>+</sup>

*<sup>U</sup>*[*A*] =

algebras, *u* : SL(2, **C**)� → U(2, 2)� and *p* : SL(2, **C**)� → SO(1, 3)�

*P*[*A*] *L <sup>K</sup>* <sup>=</sup> <sup>1</sup> 4 Tr

*p*[*a*] *L <sup>K</sup>* <sup>=</sup> <sup>1</sup> 2 Tr

<sup>Σ</sup>*LK* <sup>=</sup> <sup>1</sup> 4 

It is also worth to note the following expressions:

where after the shift of indices we have that

**4. What about the metric tensor?**

*<sup>U</sup>*[*A*] = <sup>1</sup>

following holds:

and

2 

respectively for SL(2, **C**) and its Lie algebra. In Dirac representation

*<sup>A</sup>* <sup>+</sup> *<sup>A</sup>*−1<sup>+</sup> *<sup>A</sup>* <sup>−</sup> *<sup>A</sup>*−1<sup>+</sup> *<sup>A</sup>* <sup>−</sup> *<sup>A</sup>*−1<sup>+</sup> *<sup>A</sup>* <sup>+</sup> *<sup>A</sup>*−1<sup>+</sup>

*U*[*A*]*γKU*[*A*]

This connection form is controlled by the two real parameters corresponding to SU(2, 2) � SU(4, *G*) and to the one-parameter dilatation group. The corresponding covariant derivative of the four-component Klein-Gordon field has the following form:

$$\nabla\_{\mu}\Psi = \partial\_{\mu}\Psi + g\left(\theta\_{\mu} - \frac{1}{4}\text{Tr}\,\theta\_{\mu}\mathbf{I}\right)\Psi + \frac{g}{4}\text{Tr}\,\theta\_{\mu}\Psi = \partial\_{\mu}\Psi + g\theta\_{\mu}\Psi + \frac{g-g}{4}\text{Tr}\,\theta\_{\mu}\Psi. \tag{39}$$

Similarly for the Dirac-conjugate field we have the following dual formula:

$$\nabla\_{\mu}\tilde{\mathbf{V}} = \partial\_{\mu}\tilde{\mathbf{V}} - g\tilde{\mathbf{Y}}\left(\theta\_{\mu} - \frac{1}{4}\text{Tr}\,\theta\_{\mu}\mathbf{I}\right) - \frac{q}{4}\tilde{\mathbf{Y}}\text{Tr}\,\theta\_{\mu} = \partial\_{\mu}\tilde{\mathbf{Y}} - g\tilde{\mathbf{Y}}\theta\_{\mu} - \frac{q-g}{4}\tilde{\mathbf{Y}}\text{Tr}\,\theta\_{\mu}.\tag{40}$$

The curvature form Φ = *Dϑ* is then expressed as follows:

$$\Phi\_{\mu\nu} = d\theta\_{\mu\nu} + \lg\left[\theta\_{\mu\nu}\theta\_{\nu}\right] = \partial\_{\mu}\theta\_{\nu} - \partial\_{\nu}\theta\_{\mu} + \lg\left[\theta\_{\mu\nu}\theta\_{\nu}\right].\tag{41}$$

Let us observe that the conserved Noether current following from the Noether theorem applied to (34) is given by

$$j^r\_{\ s\mu} = \frac{b}{2} \left( \Psi^r \partial\_\mu \tilde{\Psi}\_s - \partial\_\mu \Psi^r \tilde{\Psi}\_s \right) \sqrt{|g|}. \tag{42}$$

One can show that the gauge invariant Lagrangian for the Ψ-matter has the following form:

$$L\_{\mathfrak{m}}(\Psi,\theta,\mathfrak{g}) = \frac{b}{2} \mathfrak{g}^{\mu\nu} \nabla\_{\mu} \tilde{\Psi} \nabla\_{\nu} \Psi \sqrt{|\mathfrak{g}|} - \frac{c}{2} \tilde{\Psi} \Psi \sqrt{|\mathfrak{g}|}. \tag{43}$$

The gauge-invariant current

$$J(\Psi, \theta, \mathbf{g})^r{}\_{s\mu} = \frac{b}{2} \left( \Psi^r \nabla\_{\mu} \tilde{\mathbf{V}}\_s - \nabla\_{\mu} \Psi^r \tilde{\mathbf{V}}\_s \right) \sqrt{|\mathbf{g}|} \tag{44}$$

may be obtained from the Lagrangian (43) by performing its differentiation with respect to the connection *ϑ*,

$$\frac{\partial L\_{m}(\Psi,\theta,\mathcal{g})}{\partial \theta^{r}{}\_{s\mu}} = gJ^{s}{}\_{r}{}^{\mu} + \frac{q-\mathcal{g}}{4}J^{z}{}\_{z\mu}\delta^{s}{}\_{r}.\tag{45}$$

This was about the matter Lagrangian. What concerns the gauge Lagrangian, the simplest possibility of the gauge-invariant model is the following one:

$$L\_{YM}(\theta,\mathcal{g}) = \frac{a}{4} \text{Tr}\left(\Phi\_{\mu\nu}\Phi\_{\varkappa\lambda}\right) \mathcal{g}^{\mu\varkappa}\mathcal{g}^{\nu\lambda}\sqrt{|\mathcal{g}|} + \frac{a'}{4} \text{Tr}\,\Phi\_{\mu\nu}\text{Tr}\,\Phi\_{\varkappa\lambda}\mathcal{g}^{\mu\varkappa}\mathcal{g}^{\nu\lambda}\sqrt{|\mathcal{g}|},\tag{46}$$

where *a*, *a*� are constants. The first term, controlled by the parameter *a* is the main, Maxwell-like expressions. The second term is additional one, built of the traces of field strengths. It is an auxiliary expression, nevertheless it is geometrically admissible and it may be some reasonable, helpful correction to the first one. In any case, it is a merely supplementary expression, although it may be convenient and physically justified.

Let us observe that in spite of the non-homogeneous transformation rule (38), the curvature two-form (41) transforms according to the tensorial homogeneous rule under (37), (38):

$$\Psi(\mathcal{U}\Phi)\_{\mathbf{x}} = \mathcal{U}(\mathbf{x})\Phi(\mathbf{x})\mathcal{U}(\mathbf{x})^{-1}.\tag{47}$$

Because of this the Yang-Mills Lagrangian (46) is invariant under the local U(2, 2) � U(*H*, *G*) transformations (38). And similarly, the matter Lagrangian (43) is invariant.

This is the main, gauge constituent of the theory. Let us now mention only about the relationship of SU(2, 2)-matrices Φ to the representations SL(2, **C**) � *A* �→ *U*[*A*] ∈ U(2, 2) corresponding to the Weyl-Penrose-Finkelstein-Weizsäcker and to the Dirac representation of *Grs*¯ . In the first group of representation (W-P-F-W) we have the following realizations of *U*[*A*]:

$$\mathcal{U}[A] = \begin{bmatrix} A & 0 \\ 0 & A^{-1+} \end{bmatrix}, \qquad \mathfrak{u}[a] = \begin{bmatrix} a & 0 \\ 0 & -a^+ \end{bmatrix} \tag{48}$$

respectively for SL(2, **C**) and its Lie algebra. In Dirac representation

$$u[A] = \frac{1}{2} \begin{bmatrix} A + A^{-1+} \ A - A^{-1+} \\ A - A^{-1+} \ A + A^{-1+} \end{bmatrix}, \qquad u[a] = \frac{1}{2} \begin{bmatrix} a - a^+ \ a + a^+ \\ a + a^+ \ a - a^+ \end{bmatrix} \tag{49}$$

for the group and algebra. The right "plus" superscript denotes obviously the Hermitian matrix conjugate. Obviously, quite independently of the choice of any representation the following holds:

$$
\mathcal{U}[A]\gamma\_K \mathcal{U}[A]^{-1} = \gamma\_L \mathcal{P}[A]^L \llcorner \tag{50}
$$

where *P* : SL(2, **C**) → SO(1, 3)<sup>↑</sup> is the covering projection. The mappings *U* : SL(2, **C**) → U(2, 2) and *P* : SL(2, **C**) → SO(1, 3)<sup>↑</sup> generate the corresponding homomorphisms of Lie algebras, *u* : SL(2, **C**)� → U(2, 2)� and *p* : SL(2, **C**)� → SO(1, 3)� . They are synchronized by

$$\mathbb{E}\left[u[a]\_\prime(\gamma)\_K\right] = \gamma\_L p[a]^L\_K. \tag{51}$$

It is also worth to note the following expressions:

$$P[A]^L{}\_K = \frac{1}{4} \text{Tr} \left( \gamma^L \mathcal{U}[A] \gamma\_K \mathcal{U}[A]^{-1} \right) , \tag{52}$$

$$p[a]^L{}\_K = \frac{1}{2} \text{Tr} \left( \gamma^L u[a] \gamma\_\mathcal{K} \right) \, , \tag{53}$$

and

8 Will-be-set-by-IN-TECH

This connection form is controlled by the two real parameters corresponding to SU(2, 2) � SU(4, *G*) and to the one-parameter dilatation group. The corresponding covariant derivative

Let us observe that the conserved Noether current following from the Noether theorem

*∂μ*<sup>Ψ</sup>*<sup>s</sup>* <sup>−</sup> *∂μ*Ψ*<sup>r</sup>*

One can show that the gauge invariant Lagrangian for the Ψ-matter has the following form:

*<sup>g</sup>μν*∇*μ*<sup>Ψ</sup><sup>∇</sup>*ν*<sup>Ψ</sup>

may be obtained from the Lagrangian (43) by performing its differentiation with respect to the

This was about the matter Lagrangian. What concerns the gauge Lagrangian, the simplest

 |*g*| + *a*�

where *a*, *a*� are constants. The first term, controlled by the parameter *a* is the main, Maxwell-like expressions. The second term is additional one, built of the traces of field strengths. It is an auxiliary expression, nevertheless it is geometrically admissible and it may be some reasonable, helpful correction to the first one. In any case, it is a merely

Let us observe that in spite of the non-homogeneous transformation rule (38), the curvature two-form (41) transforms according to the tensorial homogeneous rule under (37), (38):

= *g J<sup>s</sup> r*

Tr *ϑμ*<sup>Ψ</sup> <sup>=</sup> *∂μ*<sup>Ψ</sup> <sup>+</sup> *<sup>g</sup>ϑμ*<sup>Ψ</sup> <sup>+</sup> *<sup>q</sup>* <sup>−</sup> *<sup>g</sup>*

<sup>Ψ</sup>Tr *ϑμ* <sup>=</sup> *∂μ*<sup>Ψ</sup> <sup>−</sup> *<sup>g</sup>*<sup>Ψ</sup>*ϑμ* <sup>−</sup> *<sup>q</sup>* <sup>−</sup> *<sup>g</sup>*

 *ϑμ*, *ϑν* 

= *∂μϑν* − *∂νϑμ* + *g*

Ψ*s* 

 <sup>|</sup>*g*<sup>|</sup> <sup>−</sup> *<sup>c</sup>* 2 ΨΨ

<sup>∇</sup>*μ*<sup>Ψ</sup>*<sup>s</sup>* − ∇*μ*Ψ*<sup>r</sup>*

*<sup>μ</sup>* <sup>+</sup> *<sup>q</sup>* <sup>−</sup> *<sup>g</sup>* <sup>4</sup> *<sup>J</sup> z <sup>z</sup>μδ<sup>s</sup>*

Ψ*s* 

<sup>4</sup> Tr <sup>Φ</sup>*μν*Tr <sup>Φ</sup>κ*λgμ*κ*gνλ*

(*U*Φ)*<sup>x</sup>* = *U*(*x*)Φ(*x*)*U*(*x*)<sup>−</sup>1. (47)

<sup>4</sup> Tr *ϑμ*Ψ. (39)

<sup>4</sup> <sup>Ψ</sup>Tr *ϑμ*. (40)

. (41)




*<sup>r</sup>*. (45)


of the four-component Klein-Gordon field has the following form:

Similarly for the Dirac-conjugate field we have the following dual formula:

 *ϑμ*, *ϑν* 

2

*<sup>s</sup><sup>μ</sup>* <sup>=</sup> *<sup>b</sup>* 2 Ψ*r*

*∂Lm*(Ψ, *ϑ*, *g*) *∂ϑ<sup>r</sup> sμ*

 *ϑμ* <sup>−</sup> <sup>1</sup> 4 Tr *ϑμ*I <sup>Ψ</sup> <sup>+</sup> *<sup>q</sup>* 4

 *ϑμ* <sup>−</sup> <sup>1</sup> 4 Tr *ϑμ*I − *q* 4

The curvature form Φ = *Dϑ* is then expressed as follows:

Φ*μν* = *dϑμν* + *g*

*j r <sup>s</sup><sup>μ</sup>* <sup>=</sup> *<sup>b</sup>* 2 Ψ*r*

*Lm*(Ψ, *<sup>ϑ</sup>*, *<sup>g</sup>*) = *<sup>b</sup>*

*J*(Ψ, *ϑ*, *g*)*<sup>r</sup>*

possibility of the gauge-invariant model is the following one:

Φ*μν*Φκ*<sup>λ</sup>*

 *gμ*κ*gνλ*

supplementary expression, although it may be convenient and physically justified.

∇*μ*Ψ = *∂μ*Ψ + *g*

∇*μ*Ψ = *∂μ*Ψ − *g*Ψ

applied to (34) is given by

The gauge-invariant current

*LYM*(*ϑ*, *<sup>g</sup>*) = *<sup>a</sup>*

4 Tr

connection *ϑ*,

$$u[a] = \frac{1}{2} p[a]^L {}\_K \Sigma\_L {}^K \,, \tag{54}$$

where after the shift of indices we have that

$$
\Sigma^{LK} = \frac{1}{4} \left( \gamma^L \gamma^K - \gamma^K \gamma^L \right) = \frac{1}{4} \left[ \gamma^L, \gamma^K \right]. \tag{55}
$$

#### **4. What about the metric tensor?**

In the gauge Lagrangians above, the metric tensor in a sense played the parameter role. Our idea was to construct the U(2, 2) � U(*H*, *G*)-invariant theory of gravitation. The main constituents of the theory were the four-component complex Klein-Gordon field and the corresponding Maxwell-like gauge field. We will show that there are interesting and very important points for which this is important, perhaps even just exciting. But there is some weak point which was not yet completely explained. It is just the role and physical status of the metric tensor, which is present in the Klein-Gordon and gauge Lagrangian, however its geometric and physical sense is not yet full understood. It is clear that it must occur there if we are to be able to construct Lagrangians. But what is its meaning and how to identify properly

4. *Tμν* denotes the symmetric energy-momentum tensor of the fields *ϑ*, Ψ, so we have that

Order of Time Derivatives in Quantum-Mechanical Equations 67

*<sup>m</sup>* <sup>+</sup> *<sup>T</sup>μν*

 *∂Lm ∂gμν*,*<sup>α</sup>*

> *∂LYM ∂gμν*,*<sup>α</sup>*

,*α*

,*α*

*YM*, (64)

, (65)

. (66)

*<sup>s</sup>ν*, (67)

*<sup>μ</sup>*. (68)

. (69)


*<sup>x</sup> G*.

*<sup>μ</sup>*. And we assume that it is homogeneously

*<sup>s</sup>*(*x*)*W<sup>s</sup>*

*Tμν* = *Tμν*

 *∂Lm ∂gμν* −

 *∂LYM ∂gμν* −

As mentioned, the Hilbert-Einstein term of Lagrangian in (57) looks rather naive, although perhaps it may be reasonable. Equations resulting from the version with vanishing coefficients (or vanishing "*d*" at least) also seem to be not bad, and in any case not to be a priori rejected. And, as mentioned, the field equations following from the first two terms of (57) seem promisible. But, as said above, the Hilbert-Einstein term seems to spoil the whole taste of the gauge approach. In Einstein-Cartan theory it was the tetrad field who saved the situation, nevertheless, also for some price (as mentioned, in no other gauge theory one explicitly uses the field of frames as a dynamical variable). What may be done in our formalism to replace in a reasonable way the role of tetrad? We would like to answer this question before the further development of our theory. There are a few, at least three natural ways. Certainly there is no

*Tμν*

*Tμν*

We do not quote the explicit formulae.

elementary than the metric itself.

by

Analytically we describe this object as *W<sup>r</sup>*

transformable under the locally acting U(2, 2),

*<sup>m</sup>* <sup>=</sup> <sup>−</sup> <sup>2</sup>

*YM* <sup>=</sup> <sup>−</sup> <sup>2</sup>

possibility to build the metric tensor from the gauge field, in the sense:

*gμν* := *pϑ<sup>r</sup>*

*Wx* �→ *U*(*x*)*Wx*, i.e., �

This form gives rise to the metric tensor field on *M* as follows:

*<sup>g</sup>*(*W*)*μν* :<sup>=</sup> *Re*

*<sup>s</sup>μϑ<sup>s</sup>*

what apparently might seem natural. The point is, however that (67) is only globally, but not locally U(2, 2)-invariant. But one can do it in a local way, by introducing some fields more

1. We may assume that besides the connection form *ϑ*, the geometrodynamical sector involves some additional **C**4-valued (*H*-valued, let us say) differential one-form *W*:

*<sup>M</sup>* � *<sup>x</sup>* �→ *Wx* <sup>∈</sup> <sup>L</sup>(*TxM*, **<sup>C</sup>**4).

*<sup>W</sup>rμW<sup>r</sup> ν* 

Therefore, this expression is the symmetric, thus real part of the Hermitian tensor *W*∗

*<sup>L</sup>*(*W*, *<sup>ϑ</sup>*) = *<sup>a</sup>*∇*W μν*∇*W*κ*λgμ*κ*gνλ*

The quantity is locally U(2, 2)-invariant. The simplest gauge-invariant Lagrangian is given

*W<sup>r</sup>*

<sup>=</sup> *Re*

*<sup>μ</sup>* = *U<sup>r</sup>*

*W <sup>μ</sup>W<sup>ν</sup>* 


*<sup>r</sup><sup>ν</sup>* + *qϑ<sup>r</sup>*

*<sup>r</sup>μϑ<sup>s</sup>*

<sup>|</sup>*g*<sup>|</sup>

<sup>|</sup>*g*<sup>|</sup>

where, obviously,

its physical role? As one of potentials of gravitation, or as some secondary variable? And if the second possibility is to be chosen, what are the primary variables the byproduct of which is the metric tensor? Situation in this respect was clear only in the standard Einstein General Relativity. There it was just the only gravitational potential (or perhaps a superpotential if the connection coefficients were interpreted as proper potentials). But within any gauge framework the metric tensor is a merely one of a few potentials.In this paper we concentrate on the theory aspects not very sensitive to this problem. Instead, we shall present a few possibilities.

First of all, let us notice that quite naively, one can assume the Hilbert-Einstein term for the metric tensor *g*,

$$L\_{HE}(\mathbf{g}) = -d\mathcal{R}(\mathbf{g})\sqrt{|\mathbf{g}|} + l\sqrt{|\mathbf{g}|}.\tag{56}$$

where *d*, *l* are real constants. The special case *d* = 0 is not to be a priori rejected. Namely, if *d* = 0 and perhaps *l* = 0, then variation of the action functional with respect to *gμν* enables one to express *gμν* through the other variables. But of course, the choice (56) looks rather naive. In any case, the total Lagrangian of the form

$$L(\Psi, \theta, \mathbf{g}) := L\_{\mathfrak{m}}(\Psi, \theta, \mathbf{g}) + L\_{\text{YM}}(\theta, \mathbf{g}) + L\_{\text{HE}}(\mathbf{g}) \tag{57}$$

leads, after the variational procedure for the action, to the following system of equations:

$$g^{\mu\nu}\stackrel{\mathcal{S}}{\nabla}\_{\mu}\stackrel{\mathcal{S}}{\nabla}\_{\nu}\Psi + \frac{c}{b}\Psi = 0,\tag{58}$$

$$\chi^{\mu\nu}{}\_{;\nu} + \lg \left[ \mathfrak{G}\_{\nu} \chi^{\mu\nu} \right] = \lg^{\mu} + \frac{q - \lg}{4} \text{Tr} \, J\_{\mu} \text{I}\_{\prime} \tag{59}$$

$$d\left(\mathcal{R}(g)^{\mu\nu} - \frac{1}{2}\mathcal{R}(g)g^{\mu\nu}\right) = \frac{1}{2}g^{\mu\nu} + \frac{1}{2}T^{\mu\nu},\tag{60}$$

with the meaning of symbols as below:


$$\stackrel{\mathcal{S}}{\nabla}\_{\mu}Y^{r}\_{\ \nu} + \mathcal{g}\theta^{r}\_{s\mu}Y^{s}\_{\ \nu} + \frac{q-\mathcal{g}}{4}\theta^{z}\_{\ \ z\mu}Y^{r}\_{\ \ \nu} - \stackrel{\mathcal{S}}{\cdot}\left\{\begin{array}{c}\lambda\\\mu\nu\end{array}\right\}Y^{r}\_{\ \ \lambda\nu} \tag{61}$$

and similarly, i.e., dually, or in the Leibniz-multiplication sense, for other quantities.

3. *χ* is the field momentum conjugate to *ϑ*, so

$$\chi^{r}\_{\ s}{}^{\mu\nu} = \frac{\partial L\_{YM}}{\partial \theta^{s}{}\_{r\mu,\nu}} = -a \Phi^{r}\_{\ s\alpha\beta} \mathcal{g}^{a\mu} \mathcal{g}^{\beta\nu} \sqrt{|g|} \tag{62}$$

$$-a' \delta^{r}{}\_{s} \Phi^{z}{}\_{za\beta} \mathcal{g}^{a\mu} \mathcal{g}^{\beta\nu} \sqrt{|g|} \,\tag{63}$$

or in the shortened form with the *g*-shifting of indices,

$$\chi^{\mu\nu} = -a\Phi^{\mu\nu}\sqrt{|g|} - a'\text{Tr}\,\Phi^{\mu\nu}\sqrt{|g|}.\tag{63}$$

4. *Tμν* denotes the symmetric energy-momentum tensor of the fields *ϑ*, Ψ, so we have that

$$T^{\mu\nu} = T\_m^{\mu\nu} + T\_{YM'}^{\mu\nu} \tag{64}$$

where, obviously,

10 Will-be-set-by-IN-TECH

its physical role? As one of potentials of gravitation, or as some secondary variable? And if the second possibility is to be chosen, what are the primary variables the byproduct of which is the metric tensor? Situation in this respect was clear only in the standard Einstein General Relativity. There it was just the only gravitational potential (or perhaps a superpotential if the connection coefficients were interpreted as proper potentials). But within any gauge framework the metric tensor is a merely one of a few potentials.In this paper we concentrate on the theory aspects not very sensitive to this problem. Instead, we shall present a few

First of all, let us notice that quite naively, one can assume the Hilbert-Einstein term for the

where *d*, *l* are real constants. The special case *d* = 0 is not to be a priori rejected. Namely, if *d* = 0 and perhaps *l* = 0, then variation of the action functional with respect to *gμν* enables one to express *gμν* through the other variables. But of course, the choice (56) looks rather naive.

leads, after the variational procedure for the action, to the following system of equations:

*c b*

*<sup>χ</sup>μν*;*<sup>ν</sup>* <sup>+</sup> *<sup>g</sup>* [*ϑν*, *<sup>χ</sup>μν*] <sup>=</sup> *g J<sup>μ</sup>* <sup>+</sup> *<sup>q</sup>* <sup>−</sup> *<sup>g</sup>*

 = *l* 2 *gμν* + 1 2

differentiation of the space-time indices is joined there with the U(2, 2) � U(*H*, *G*)

*<sup>ν</sup>* <sup>+</sup> *<sup>q</sup>* <sup>−</sup> *<sup>g</sup>* <sup>4</sup> *<sup>ϑ</sup><sup>z</sup>*

and similarly, i.e., dually, or in the Leibniz-multiplication sense, for other quantities.

<sup>=</sup> <sup>−</sup>*a*Φ*<sup>r</sup>*

−*a*� *δr <sup>s</sup>*Φ*<sup>z</sup>*


∇*<sup>μ</sup>* denotes the complete covariant differentiation. The Levi-Civita covariant

*<sup>z</sup>μY<sup>r</sup>*

*sαβgαμgβν*

I Tr Φ*μν*

*<sup>ν</sup>* <sup>−</sup> *<sup>g</sup>*

 *λ μν Yr*

 |*g*|,

*zαβgαμgβν*

 |*g*| + *l* 

*L*(Ψ, *ϑ*, *g*) := *Lm*(Ψ, *ϑ*, *g*) + *LYM*(*ϑ*, *g*) + *LHE*(*g*) (57)


<sup>4</sup> Tr *<sup>J</sup>μ*I, (59)

*Tμν*, (60)

*<sup>λ</sup>*, (61)



Ψ = 0, (58)

*LHE*(*g*) = −*dR*(*g*)

possibilities.

metric tensor *g*,

2. The symbol

In any case, the total Lagrangian of the form

*d* 

*g* <sup>∇</sup>*μY<sup>r</sup>*

3. *χ* is the field momentum conjugate to *ϑ*, so

*χr s*

or in the shortened form with the *g*-shifting of indices,

with the meaning of symbols as below:

*g*

*gμν g* ∇*μ g* ∇*ν*Ψ +

*<sup>R</sup>*(*g*)*μν* <sup>−</sup> <sup>1</sup>

1. The semicolon ";" is the *g*-Levi-Civita covariant differentiation.

*<sup>ν</sup>* + *gϑ<sup>r</sup>*

2

covariant differentials of internal indices. Let us quote a typical example:

*<sup>s</sup>μY<sup>s</sup>*

*μν* <sup>=</sup> *<sup>∂</sup>LYM ∂ϑ<sup>s</sup> rμ*,*ν*

*<sup>χ</sup>μν* <sup>=</sup> <sup>−</sup>*a*Φ*μν*

*R*(*g*)*gμν*

$$T\_{\mathfrak{m}}^{\mu\nu} = -\frac{2}{\sqrt{|g|}} \left( \frac{\partial L\_{\mathfrak{m}}}{\partial \mathbf{g}\_{\mu\nu}} - \left( \frac{\partial L\_{\mathfrak{m}}}{\partial \mathbf{g}\_{\mu\nu,\mathfrak{a}}} \right)\_{,\mathfrak{a}} \right), \tag{65}$$

$$T\_{YM}^{\mu\nu} = -\frac{2}{\sqrt{|g|}} \left( \frac{\partial L\_{YM}}{\partial g\_{\mu\nu}} - \left( \frac{\partial L\_{YM}}{\partial g\_{\mu\nu,a}} \right)\_{,a} \right). \tag{66}$$

We do not quote the explicit formulae.

As mentioned, the Hilbert-Einstein term of Lagrangian in (57) looks rather naive, although perhaps it may be reasonable. Equations resulting from the version with vanishing coefficients (or vanishing "*d*" at least) also seem to be not bad, and in any case not to be a priori rejected. And, as mentioned, the field equations following from the first two terms of (57) seem promisible. But, as said above, the Hilbert-Einstein term seems to spoil the whole taste of the gauge approach. In Einstein-Cartan theory it was the tetrad field who saved the situation, nevertheless, also for some price (as mentioned, in no other gauge theory one explicitly uses the field of frames as a dynamical variable). What may be done in our formalism to replace in a reasonable way the role of tetrad? We would like to answer this question before the further development of our theory. There are a few, at least three natural ways. Certainly there is no possibility to build the metric tensor from the gauge field, in the sense:

$$\mathcal{g}\_{\mu\nu} := p\theta^r{}\_{s\mu}\theta^s{}\_{r\nu} + q\theta^r{}\_{r\mu}\theta^s{}\_{s\nu} \tag{67}$$

what apparently might seem natural. The point is, however that (67) is only globally, but not locally U(2, 2)-invariant. But one can do it in a local way, by introducing some fields more elementary than the metric itself.

1. We may assume that besides the connection form *ϑ*, the geometrodynamical sector involves some additional **C**4-valued (*H*-valued, let us say) differential one-form *W*:

$$M \ni \mathfrak{x} \mapsto \mathcal{W}\_{\mathfrak{x}} \in \operatorname{L}(T\_{\mathfrak{x}}M, \mathbb{C}^{4}).$$

Analytically we describe this object as *W<sup>r</sup> <sup>μ</sup>*. And we assume that it is homogeneously transformable under the locally acting U(2, 2),

$$\mathcal{W}\_{\mathbf{x}} \mapsto \mathcal{U}(\mathbf{x}) \mathcal{W}\_{\mathbf{x}} \qquad \text{i.e.,} \qquad \ulcorner \mathcal{W}\_{\boldsymbol{\mu}} = \mathcal{U}\_{\boldsymbol{s}} (\mathbf{x}) \mathcal{W}\_{\boldsymbol{\mu}}^{\boldsymbol{s}}.\tag{68}$$

This form gives rise to the metric tensor field on *M* as follows:

$$\log(\mathcal{W})\_{\mu\nu} := \text{Re}\left(\tilde{\mathcal{W}}\_{r\mu}\mathcal{W}^r{}\_{\nu}\right) = \text{Re}\left(\tilde{\mathcal{W}}\_{\mu}\mathcal{W}\_{\nu}\right). \tag{69}$$

Therefore, this expression is the symmetric, thus real part of the Hermitian tensor *W*∗ *<sup>x</sup> G*. The quantity is locally U(2, 2)-invariant. The simplest gauge-invariant Lagrangian is given by

$$L(\mathcal{W}, \theta) = a \nabla \tilde{W}\_{\mu \nu} \nabla W\_{\varkappa \lambda} g^{\mu \varkappa} g^{\nu \lambda} \sqrt{|g|} + b \sqrt{|g|}. \tag{70}$$

This is the most natural Born-Infeld scheme.

*<sup>L</sup>*(Ψ, *<sup>ϑ</sup>*) = *<sup>a</sup>*

4 Tr

In both expressions (77), (78) the space-time metric is given by (76), and *a*, *a*�

+ *a*�

**5. The main ideas of the Klein-Gordon** U(2, 2)**-ruled theory**

the first two terms of (57), or even by the total (57).

The basis elements are built algebraically of *γA*-matrices.

Let us introduce the following matrices built algebraically of *γA*-s:

It is clear that *<sup>A</sup>γ*-s satisfy the opposite-sign anticommutation rules

*iγA*, *i*

<sup>Σ</sup>*AB* <sup>=</sup> <sup>1</sup> 4 

*Aγ*, *<sup>B</sup><sup>γ</sup>*

*<sup>A</sup><sup>γ</sup>* <sup>=</sup> *<sup>B</sup>γηBA* <sup>=</sup> <sup>−</sup> *<sup>i</sup>*

The Lie algebra U(2, 2)� = U(*H*, *G*)� may be spanned in the **R**-sense on the matrices

6

Removing from this system the imaginary matrix *iI*4, we obtain the basis of SU(2, 2)� =

<sup>2</sup> (*γ<sup>A</sup>* <sup>+</sup> *<sup>A</sup>γ*), *<sup>χ</sup><sup>A</sup>* :<sup>=</sup> <sup>1</sup>

[*τA*, *<sup>τ</sup>B*] <sup>=</sup> 0,

e.g.,

constants.

One can show that

SU(*H*, *G*)�

where the convention *ε*<sup>0123</sup> = 1 is used.

. The matrices *iγA*, *i*

and difference are bases of Abelian Lie subalgebras:

*<sup>τ</sup><sup>A</sup>* :<sup>=</sup> <sup>1</sup>

Let us mention, there are also similar, but modified expressions with some "potential" terms,

Order of Time Derivatives in Quantum-Mechanical Equations 69

<sup>4</sup> Tr *<sup>F</sup>μν*Tr *<sup>F</sup>*κ*λgμ*κ*gνλ*

Without judging the three presented models of the metric field, or rather the four of them if the Hilbert-Einstein possibility is admitted, we have nevertheless formulated them. And from the point of view of aesthetic criteria, they look quite reasonable. But now, let us discuss the main results of the very U(2, 2)-invariant Klein-Gordon gauge model of gravity as ruled by

The field equations (58), (59), (60) become qualitatively readable when some special basis is chosen in the Lie algebra of U(2, 2). This basis is somehow related to the twistor geometry, although literally it is something else than the conformal geometry in Minkowskian space.

*<sup>γ</sup>Aγ<sup>B</sup>* <sup>−</sup> *<sup>γ</sup>Bγ<sup>A</sup>*

*gμ*κ*gνλ*



*<sup>γ</sup>*<sup>5</sup> <sup>=</sup> <sup>−</sup>*γ*<sup>5</sup> <sup>=</sup> <sup>−</sup>*γ*0*γ*1*γ*2*γ*3, (79) *<sup>A</sup><sup>γ</sup>* <sup>=</sup> *<sup>i</sup>γAγ*<sup>5</sup> <sup>=</sup> <sup>−</sup>*iγ*5*γA*, (80)

. (81)

<sup>=</sup> <sup>−</sup>2*ηAB <sup>I</sup>*. (82)

*εABCDγBγCγD*, (83)

*<sup>A</sup>γ*, Σ*AB*, *iγ*5, *iI*4. (84)

*<sup>A</sup>γ* do not **R**-span Lie algebras. But it is clear that their sum

*<sup>γ</sup><sup>A</sup>* <sup>−</sup> *<sup>A</sup><sup>γ</sup>*

, (85)

= 0. (86)

2 

*χA*, *χ<sup>B</sup>*  |*g*|. (78)

, *b* are some real

*FμνF*κ*<sup>λ</sup>* 

In this expression *<sup>a</sup>*, *<sup>b</sup>* are some real constants, *<sup>g</sup>μαgαν* <sup>=</sup> *<sup>δ</sup>μν*, and <sup>∇</sup>*<sup>W</sup>* denotes the exterior covariant differential of *W*, so that

$$\nabla \mathcal{W}\_{\mu\nu} = d\mathcal{W}\_{\mu\nu} + \mathcal{g} \left( \theta\_{\mu} \mathcal{W}\_{\nu} - \theta\_{\nu} \mathcal{W}\_{\mu} \right) + \frac{\mathcal{'}q - \mathcal{g}}{4} \left( \text{Tr} \,\theta\_{\mu} \mathcal{W}\_{\nu} - \text{Tr} \,\theta\_{\nu} \mathcal{W}\_{\mu} \right) \,. \tag{71}$$

Here � *q*, the coupling constant, is the kind of electric charge of *W*. Let us stress, the Lagrangian (70) is locally invariant under U(2, 2). It is interesting that after the SL(2, **C**)-reduction *W* is a 3/2-spin particle. This resembles the super-symmetric idea of gravitino.

2. Let us suppose that besides of *ϑ*, the geometric sector contains also another U(2, 2)� -valued differential form *W*, *M* � *x* �→ *Wx* ∈ L(*TxM*, U(2, 2)� ). Analytically it is represented by the system of quantities *W<sup>r</sup> <sup>s</sup>μ*. But unlike the connection form *ϑ*, just like in the previous idea, it suffers a homogeneous transformation rule under U(2, 2),

$$\mathcal{W}\_{\mathbf{x}} \mapsto \mathcal{U}(\mathbf{x}) \mathcal{W}\_{\mathbf{x}} \mathcal{U}(\mathbf{x})^{-1}, \qquad \mathcal{W}\_{s\mu}^{r} = \mathcal{U}\_{z}^{r} \mathcal{W}\_{t\mu}^{z} \mathcal{U}^{-1t} \,\_{s}. \tag{72}$$

The corresponding metric field *g*(*W*) on *M* is given by

$$\log(\mathcal{W})\_{\mu\nu} = a \text{Tr}\left(\mathcal{W}\_{\mu}\mathcal{W}\_{\nu}\right) + b \text{Tr}\,\mathcal{W}\_{\mu}\text{Tr}\,\mathcal{W}\_{\nu} \tag{73}$$

where *a*, *b* are constants and obviously *a* �= 0; the *a*-term is dominant, whereas the *b*-term is a merely correction.

The exterior covariant differential of *W* is given by

$$
\nabla \mathcal{W}\_{\mu \nu} = d\mathcal{W}\_{\mu \nu} + \mathcal{g} \left[ \mathcal{\theta}\_{\mu \prime} \mathcal{W}\_{\nu} \right] - \mathcal{g} \left[ \mathcal{W}\_{\nu \prime} \mathcal{\theta}\_{\mu} \right]. \tag{74}$$

The corresponding Maxwell Lagrangian for the form *W* is given by

$$\begin{split} L(\mathcal{W}, \mathfrak{d}) &= a \text{Tr} \left( \nabla \mathcal{W}\_{\mu \nu} \nabla \mathcal{W}\_{\varkappa \lambda} \right) \mathcal{g}^{\mu \varkappa} \mathcal{g}^{\nu \lambda} \sqrt{|\mathcal{g}|} \\ &+ b \text{Tr} \left( \nabla \mathcal{W}\_{\mu \nu} \right) \text{Tr} \left( \nabla \mathcal{W}\_{\varkappa \lambda} \right) \mathcal{g}^{\mu \varkappa} \mathcal{g}^{\nu \lambda} \sqrt{|\mathcal{g}|} + c \sqrt{|\mathcal{g}|} \end{split} \tag{75}$$

with constant coefficients *a*, *b*, *c*.

3. There is also a different model, maximally economic in the sense that its only dynamical variables are *ϑ* and Ψ. There is neither *g* nor any other geometric quantity used as Lagrangian argument. Instead, we use the metric-like tensor built in a locally-invariant gauge way from the basic field quantities:

$$\log(\Psi\_{\prime}\theta)\_{\mu\nu} = a\text{Re}\left(\nabla\_{\mu}\widetilde{\Psi}\nabla\_{\nu}\Psi\right) = a\text{Re}\left(G\_{\mathbb{P}s}\nabla\_{\mu}\widetilde{\Psi}^{\prime}\nabla\_{\nu}\Psi^{s}\right). \tag{76}$$

Obviously, it would be meaningless to substitute this metric to the usual Klein-Gordon Lagrangian, because the result would be trivial. However, there are modified Born-Infeld type schemes, in a sense very interesting ones, as usual Born-Infeld schemes are. The typical Born-Infeld scheme for (Ψ, *ϑ*) is the following one:

$$L(\Psi, \theta) = \sqrt{\left| \det \left[ \frac{b}{2} g\_{\mu \nu} + \frac{a}{4} \text{Tr} \left( F\_{\mu \varkappa} F\_{\nu \lambda} \right) g^{\varkappa \lambda} + \frac{a'}{4} \text{Tr} \, F\_{\mu \varkappa} \text{Tr} \, F\_{\nu \lambda} g^{\varkappa \lambda} \right] \right|} . \tag{77}$$

This is the most natural Born-Infeld scheme.

12 Will-be-set-by-IN-TECH

covariant differential of *W*, so that

Here �

gravitino.

system of quantities *W<sup>r</sup>*

is a merely correction.

with constant coefficients *a*, *b*, *c*.

gauge way from the basic field quantities:

Born-Infeld scheme for (Ψ, *ϑ*) is the following one:

 det *b* 2 *gμν* + *a* 4 Tr

*L*(Ψ, *ϑ*) =

*g*(Ψ, *ϑ*)*μν* = *aRe*

∇*Wμν* = *dWμν* + *g*

differential form *W*, *M* � *x* �→ *Wx* ∈ L(*TxM*, U(2, 2)�

The corresponding metric field *g*(*W*) on *M* is given by

The exterior covariant differential of *W* is given by

*L*(*W*, *ϑ*) = *a*Tr

+ *b*Tr

it suffers a homogeneous transformation rule under U(2, 2),

*Wx* �→ *<sup>U</sup>*(*x*)*WxU*(*x*)<sup>−</sup>1, �

*g*(*W*)*μν* = *a*Tr

∇*Wμν* = *dWμν* + *g*

The corresponding Maxwell Lagrangian for the form *W* is given by

∇*Wμν*∇*W*κ*<sup>λ</sup>*

∇*Wμν* 

∇*μ*Ψ∇*ν*Ψ

Obviously, it would be meaningless to substitute this metric to the usual Klein-Gordon Lagrangian, because the result would be trivial. However, there are modified Born-Infeld type schemes, in a sense very interesting ones, as usual Born-Infeld schemes are. The typical

> *Fμ*κ*Fνλ g*κ*<sup>λ</sup>* +

*ϑμW<sup>ν</sup>* − *ϑνW<sup>μ</sup>*

2. Let us suppose that besides of *ϑ*, the geometric sector contains also another U(2, 2)�

In this expression *<sup>a</sup>*, *<sup>b</sup>* are some real constants, *<sup>g</sup>μαgαν* <sup>=</sup> *<sup>δ</sup>μν*, and <sup>∇</sup>*<sup>W</sup>* denotes the exterior

 + � *q* − *g* 4 

*q*, the coupling constant, is the kind of electric charge of *W*. Let us stress, the Lagrangian (70) is locally invariant under U(2, 2). It is interesting that after the SL(2, **C**)-reduction *W* is a 3/2-spin particle. This resembles the super-symmetric idea of

*W<sup>r</sup>*

*WμW<sup>ν</sup>* 

where *a*, *b* are constants and obviously *a* �= 0; the *a*-term is dominant, whereas the *b*-term

 *ϑμ*, *W<sup>ν</sup>* − *g Wν*, *ϑμ* 

3. There is also a different model, maximally economic in the sense that its only dynamical variables are *ϑ* and Ψ. There is neither *g* nor any other geometric quantity used as Lagrangian argument. Instead, we use the metric-like tensor built in a locally-invariant

> = *aRe*

 *gμ*κ*gνλ*

Tr(∇*W*κ*λ*) *<sup>g</sup>μ*κ*gνλ*

*<sup>s</sup>μ*. But unlike the connection form *ϑ*, just like in the previous idea,

*zW<sup>z</sup>*

*<sup>t</sup>μU*−1*<sup>t</sup>*

+ *b*Tr *Wμ*Tr *Wν*, (73)

*<sup>s</sup><sup>μ</sup>* = *U<sup>r</sup>*

 |*g*|

> |*g*| + *c*

*Grs*¯ <sup>∇</sup>*μ*Ψ*r*¯

*a*�

<sup>∇</sup>*ν*Ψ*<sup>s</sup>* 

<sup>4</sup> Tr *<sup>F</sup>μ*κTr *<sup>F</sup>νλg*κ*<sup>λ</sup>*

Tr *ϑμW<sup>ν</sup>* − Tr *ϑνW<sup>μ</sup>*

). Analytically it is represented by the

. (71)


*<sup>s</sup>*. (72)

. (74)


. (76)

. (77)

  Let us mention, there are also similar, but modified expressions with some "potential" terms, e.g.,

$$\begin{split} L(\boldsymbol{\Psi}, \boldsymbol{\theta}) &= \frac{a}{4} \text{Tr} \left( F\_{\mu\nu} F\_{\varkappa\lambda} \right) g^{\mu\varkappa} g^{\nu\lambda} \sqrt{|\boldsymbol{g}|} \\ &+ \frac{a'}{4} \text{Tr} \, F\_{\mu\nu} \text{Tr} \, F\_{\varkappa\lambda} g^{\mu\varkappa} g^{\nu\lambda} \sqrt{|\boldsymbol{g}|} + b \sqrt{|\boldsymbol{g}|}. \end{split} \tag{78}$$

In both expressions (77), (78) the space-time metric is given by (76), and *a*, *a*� , *b* are some real constants.

#### **5. The main ideas of the Klein-Gordon** U(2, 2)**-ruled theory**

Without judging the three presented models of the metric field, or rather the four of them if the Hilbert-Einstein possibility is admitted, we have nevertheless formulated them. And from the point of view of aesthetic criteria, they look quite reasonable. But now, let us discuss the main results of the very U(2, 2)-invariant Klein-Gordon gauge model of gravity as ruled by the first two terms of (57), or even by the total (57).

The field equations (58), (59), (60) become qualitatively readable when some special basis is chosen in the Lie algebra of U(2, 2). This basis is somehow related to the twistor geometry, although literally it is something else than the conformal geometry in Minkowskian space. The basis elements are built algebraically of *γA*-matrices.

Let us introduce the following matrices built algebraically of *γA*-s:

$$
\gamma^5 = -\gamma\_5 = -\gamma^0 \gamma^1 \gamma^2 \gamma^3.\tag{79}
$$

$$\prescript{A}{}{\gamma} = i\gamma^A \prescript{5}{}{\gamma} = -i\gamma^5 \gamma^A,\tag{80}$$

$$
\Sigma^{AB} = \frac{1}{4} \left( \gamma^A \gamma^B - \gamma^B \gamma^A \right). \tag{81}
$$

It is clear that *<sup>A</sup>γ*-s satisfy the opposite-sign anticommutation rules

$$\left\{{}^{A}\gamma\,{}^{B}\gamma\right\} = -2\eta {}^{AB}I.\tag{82}$$

One can show that

$$
\sigma\_A \gamma = {}^B \gamma \eta\_{BA} = -\frac{i}{6} \varepsilon\_{ABCD} \gamma {}^B \gamma {}^C \gamma {}^D,\tag{83}
$$

where the convention *ε*<sup>0123</sup> = 1 is used. The Lie algebra U(2, 2)� = U(*H*, *G*)� may be spanned in the **R**-sense on the matrices

$$\dot{\imath}\gamma^{A}\iota \qquad \dot{\imath}^{A}\gamma\_{\prime} \qquad \Sigma^{AB}\iota \qquad \dot{\imath}\gamma^{5} \qquad \text{i}I\_{4}.\tag{84}$$

Removing from this system the imaginary matrix *iI*4, we obtain the basis of SU(2, 2)� = SU(*H*, *G*)� . The matrices *iγA*, *i <sup>A</sup>γ* do not **R**-span Lie algebras. But it is clear that their sum and difference are bases of Abelian Lie subalgebras:

$$\pi\_A := \frac{1}{2} \left( \gamma\_A + {}\_A \gamma \right) , \quad \chi^A := \frac{1}{2} \left( \gamma^A - {}^A \gamma \right) , \tag{85}$$

$$\left[\tau\_{A\prime}\,\tau\_{B}\right] = 0, \qquad \left[\chi^{A}\,,\chi^{B}\right] = 0. \tag{86}$$

necessity to use an additional version of the cotetrad, transforming under dilatations in the

Order of Time Derivatives in Quantum-Mechanical Equations 71

Let us stress that in certain formulae it is still more convenient to use the *γA*, *<sup>A</sup>γ*-expansion

<sup>=</sup> *<sup>η</sup>ABEB*, *<sup>F</sup><sup>A</sup>* <sup>=</sup> <sup>1</sup>

*<sup>B</sup>* <sup>∧</sup> *<sup>e</sup><sup>B</sup>* <sup>=</sup> *de<sup>A</sup>* <sup>+</sup> <sup>Γ</sup>*<sup>A</sup>*

*Cδ<sup>A</sup>*

*g* 1 8 *R*(Γ)*<sup>A</sup>*

*<sup>B</sup>*, *R*(Ω)*<sup>A</sup>*

Let us remember that the torsion of a linear connection may be interpreted as a contribution to affine connection. The corresponding space-time objects, i.e., connections, torsions and

> [*ij*] <sup>=</sup> <sup>−</sup><sup>1</sup> 2 *e k AT*(*e*)*<sup>A</sup>*

[*ij*] <sup>=</sup> <sup>−</sup><sup>1</sup>

1 2

*<sup>A</sup>* <sup>=</sup> *d fA* <sup>+</sup> *fB* <sup>∧</sup> <sup>Γ</sup>*<sup>B</sup>*

*g R*(Γ)*<sup>A</sup>*

∇*λgμν* = −*Qλgμν*, (96)

*<sup>e</sup><sup>A</sup>* <sup>−</sup> *<sup>η</sup>AB fB*

*i*

*<sup>A</sup>γ*, (97)

= *ηABFB*. (98)

*γ*<sup>5</sup> + *FiI*, (99)

*<sup>B</sup>* <sup>∧</sup> *<sup>e</sup>B*, (100)

*Cδ<sup>A</sup> B*

*<sup>C</sup>* <sup>∧</sup> <sup>Ω</sup>*<sup>C</sup>*

*<sup>i</sup>*,*j*, (106)

*ij*, (108)

*Bj f kB* + *f kA fAi*,*j*, (107)

<sup>2</sup> *<sup>f</sup> kAT*(*f*)*Aij*, (109)

*Bij*, (110)

*Bij*. (111)

*<sup>A</sup>*, (101)

, (102)

, (103)

*<sup>B</sup>*. (105)

*<sup>γ</sup>*<sup>5</sup> + *<sup>A</sup>μiI* + *EAμiγ<sup>A</sup>* + *FAμ<sup>i</sup>*

2 

*<sup>R</sup>AB*Σ*AB* <sup>+</sup> *<sup>G</sup>* <sup>1</sup>

*<sup>B</sup>* <sup>−</sup> <sup>2</sup>*ge<sup>A</sup>* <sup>∧</sup> *fB*

*F* = *dA*, (104)

*<sup>B</sup>* = *d*Ω*<sup>A</sup>*

*<sup>B</sup>* <sup>−</sup> <sup>1</sup> 4 *R*(Γ)*<sup>C</sup>*

*<sup>A</sup>* <sup>−</sup> *<sup>g</sup>*2*e<sup>A</sup>* <sup>∧</sup> *fa*

*<sup>B</sup>* + *g*Ω*<sup>A</sup>*

inverse way. If we used the world metric *gμν*, then *Q<sup>μ</sup>* was the Killing covector, i.e.,

1 *i*

Let us now take the following expansion of the curvature vector-valued two-form:

in the sense of the GL(2, **C**)-part of the U(2, 2)-connection.

*e<sup>A</sup>* + *ηAB fB*

where the following partially clear symbols are used: *T*(*e*)*<sup>A</sup>* = *de<sup>A</sup>* + *g*Ω*<sup>A</sup>*

*<sup>T</sup>*(*f*)*<sup>A</sup>* <sup>=</sup> *d fA* <sup>+</sup> *g fB* <sup>∧</sup> <sup>Ω</sup>*<sup>B</sup>*

*<sup>B</sup>* = *R*(Ω)*<sup>A</sup>*

*<sup>G</sup>* <sup>=</sup> <sup>1</sup> 4*g*

where *R*(Γ), *R*(Ω) denote the curvature two-form:

*<sup>B</sup>* + Γ*<sup>A</sup>*

Γ(*e*)*<sup>k</sup>*

Γ(*f*)*<sup>k</sup>*

*S*(*e*)*<sup>k</sup>*

*S*(*f*)*<sup>k</sup>*

*R*(*e*)*mkij* = *e*

*<sup>B</sup>* = *d*Γ*<sup>A</sup>*

*<sup>μ</sup>*Σ*AB* + *B<sup>μ</sup>*

Φ = *T*(*e*)*Aiτ<sup>A</sup>* + *T*(*f*)*Aiχ<sup>A</sup>* +

*<sup>B</sup>* <sup>−</sup> <sup>1</sup> 4 *R*(Ω)*<sup>C</sup>*

<sup>+</sup>2*gηACηBDe<sup>D</sup>* <sup>∧</sup> *fC* <sup>=</sup> <sup>1</sup>

*dQ* <sup>−</sup> *ge<sup>A</sup>* <sup>∧</sup> *fA* <sup>=</sup> <sup>1</sup>

*<sup>C</sup>* <sup>∧</sup> <sup>Γ</sup>*<sup>C</sup>*

*ij* = *e k A*Γ*<sup>A</sup> Bje<sup>B</sup> <sup>i</sup>* + *e k Ae<sup>A</sup>*

*ij* <sup>=</sup> <sup>−</sup>*fAi*Γ*<sup>A</sup>*

*ij* = Γ(*e*)*<sup>k</sup>*

*ij* = Γ(*f*)*<sup>k</sup>*

*<sup>R</sup>*(*f*)*mkij* <sup>=</sup> <sup>−</sup>*fAk <sup>f</sup> mBRA*

*mAeB kR<sup>A</sup>*

<sup>−</sup>2*g*2*e<sup>A</sup>* <sup>∧</sup> *fB* <sup>+</sup> <sup>2</sup>*g*2*ηACηBDe<sup>D</sup>* <sup>∧</sup> *fC*

than those based on *τA*, *χA*, namely,

where the following notation is used:

*RA*

*R*(Γ)*<sup>A</sup>*

curvatures are given by

*<sup>E</sup><sup>A</sup>* <sup>=</sup> <sup>1</sup> 2 

*ϑμ* <sup>=</sup> <sup>1</sup> 2 Ω*AB*

In the twistor language the quantities *τ<sup>A</sup>* generate Minkowskian translations, while *χ<sup>A</sup>* are generators of the group of proper conformal mappings. Obviously, this interpretation is true only within the framework of Minkowskian-conformal geometry. The literal meaning of this interpretation is lost within the internal interpretation of those mappings. Nevertheless, the commutation rules of the conformal group are still valid. It becomes internal group just like the Lorentz group in Einstein-Cartan theory.

The connection from *ϑ* may be expanded as follows:

$$\mathfrak{G}\_{\mu} = \frac{1}{2} \check{\Omega}^{AB}{}\_{\mu} \Sigma\_{AB} + B\_{\mu} \frac{1}{\dot{\mathfrak{i}}} \gamma\_{5} + A\_{\mu} i I + e^{A}{}\_{\mu} i \tau\_{A} + f\_{A\mu} i \chi^{A} \,. \tag{87}$$

where, obviously, Ω˘ *AB <sup>μ</sup>* <sup>=</sup> <sup>−</sup>Ω˘ *BA μ*. It may be convenient to introduce the object

$$
\Omega^{A}{}\_{B\mu} := \check{\Omega}^{A}{}\_{B\mu} + 2B\_{\mu}\delta^{A}{}\_{B\prime} \tag{88}
$$

where the following holds:

$$B\_{\mu} = \frac{1}{8} \Omega^{A}{}\_{A\mu} \qquad \acute{\Omega}{}^{A}{}\_{B\mu} = \Omega^{A}{}\_{B\mu} - \frac{1}{4} \Omega^{C}{}\_{C\mu} \delta^{A}{}\_{B} . \tag{89}$$

Therefore, 8*B<sup>μ</sup>* may be identified with the trace, and Ω˘ *<sup>A</sup> <sup>B</sup><sup>μ</sup>* — with the trace-less part of the object Ω*<sup>A</sup> <sup>B</sup>μ*. So, it may be natural to write *ϑμ* as follows:

$$\mathcal{O}\_{\mu} = \frac{1}{2} \Omega^{AB}{}\_{\mu} \left( \Sigma\_{AB} + \frac{1}{4} n\_{AB} \frac{1}{i} \gamma^{5} \right) + e^{A}{}\_{\mu} i \tau\_{A} + f\_{A\mu} i \chi^{A} + A\_{\mu} i I\_{\nu} \tag{90}$$

or alternatively

$$\theta\_{\mu} = \frac{1}{2g} \dot{\Gamma}^{AB}{}\_{\mu} \Sigma\_{AB} + \frac{1}{4g} Q\_{\mu} \frac{1}{\dot{\imath}} \gamma^{5} + \frac{1}{g} \varepsilon^{A}{}\_{\mu} i \tau\_{A} + \frac{1}{g} \varphi\_{A\mu} i \chi^{A} + A\_{\mu} i I,\tag{91}$$

or just as

$$\boldsymbol{\theta}\_{\mu} = \frac{1}{2g} \boldsymbol{\Gamma}^{AB}{}\_{\mu} \left( \Sigma\_{AB} + \frac{1}{4} n\_{AB} \frac{1}{i} \boldsymbol{\gamma}^{5} \right) + \frac{1}{g} \boldsymbol{\varepsilon}^{A}{}\_{\mu} i \boldsymbol{\tau}\_{A} + \frac{1}{g} \boldsymbol{\rho}\_{A\mu} i \boldsymbol{\chi}^{A} + A\_{\mu} i \boldsymbol{I}\_{\prime} \tag{92}$$

where the following auxiliary gauge symbols are used:

$$
\Gamma^{A}{}\_{B\mu} = \text{g}\,\Omega^{A}{}\_{B\mu} \qquad \mathring{\Gamma}^{A}{}\_{B\mu} = \Gamma^{A}{}\_{B\mu} - \frac{1}{4}\Gamma^{C}{}\_{C\mu}\delta^{A}{}\_{B\prime} \tag{93}
$$

$$Q\_{\mu} = 4gB\_{\mu} = \frac{g}{2} \Omega^{A}{}\_{A\mu} = \frac{1}{2} \Gamma^{A}{}\_{A\mu} \tag{94}$$

$$
\varepsilon^{A}{}\_{\mu} = \mathcal{g} \varepsilon^{A}{}\_{\mu} \qquad \varphi\_{A\mu} = \mathcal{g} f\_{A\mu}.\tag{95}
$$

The systems of differential forms Ω*<sup>A</sup> <sup>B</sup>* = (1/*g*)Γ*<sup>A</sup> B* , *e<sup>A</sup>* , [ *fA*] are parts of the connection form *ϑ*, and because of this, the action of *x*-dependent matrices *U* on them is inhomogeneous. But when we restrict ourselves to the *U*-injected group SL(2, **C**), the transformation rule for *e<sup>A</sup>* , [ *fA*] becomes homogeneous. It is just the correspondence rule with the situation of Einstein-Cartan theory where *e<sup>A</sup>* was a gravitational cotetrad. Let us remind that in GL(2, **C**)-invariant spinor theory, *Q<sup>μ</sup>* = (1/2)Γ*<sup>A</sup> <sup>A</sup><sup>μ</sup>* was the Weyl covector, and there was necessity to use an additional version of the cotetrad, transforming under dilatations in the inverse way. If we used the world metric *gμν*, then *Q<sup>μ</sup>* was the Killing covector, i.e.,

$$
\nabla\_{\lambda} \mathcal{g}\_{\mu \nu} = -Q\_{\lambda} \mathcal{g}\_{\mu \nu \prime} \tag{96}
$$

in the sense of the GL(2, **C**)-part of the U(2, 2)-connection.

Let us stress that in certain formulae it is still more convenient to use the *γA*, *<sup>A</sup>γ*-expansion than those based on *τA*, *χA*, namely,

$$\boldsymbol{\theta}\_{\mu} = \frac{1}{2} \boldsymbol{\Omega}^{AB}{}\_{\mu} \boldsymbol{\Sigma}\_{AB} + \boldsymbol{B}\_{\mu} \frac{1}{i} \gamma^{5} + A\_{\mu} i \boldsymbol{I} + E\_{A\mu} i \gamma^{A} + F\_{A\mu} i^{A} \gamma\_{\prime} \tag{97}$$

where the following notation is used:

14 Will-be-set-by-IN-TECH

In the twistor language the quantities *τ<sup>A</sup>* generate Minkowskian translations, while *χ<sup>A</sup>* are generators of the group of proper conformal mappings. Obviously, this interpretation is true only within the framework of Minkowskian-conformal geometry. The literal meaning of this interpretation is lost within the internal interpretation of those mappings. Nevertheless, the commutation rules of the conformal group are still valid. It becomes internal group just like

the Lorentz group in Einstein-Cartan theory.

*ϑμ* <sup>=</sup> <sup>1</sup> 2 Ω˘ *AB*

It may be convenient to introduce the object

*ϑμ* <sup>=</sup> <sup>1</sup> 2 Ω*AB μ* Σ*AB* +

*ϑμ* <sup>=</sup> <sup>1</sup> 2*g* Γ˘ *AB*

*ϑμ* <sup>=</sup> <sup>1</sup> 2*g* Γ*AB μ* Σ*AB* +

The systems of differential forms

of Einstein-Cartan theory where

where, obviously, Ω˘ *AB*

where the following holds:

object Ω*<sup>A</sup>*

or just as

for *e<sup>A</sup>*

or alternatively

The connection from *ϑ* may be expanded as follows:

*<sup>μ</sup>* <sup>=</sup> <sup>−</sup>Ω˘ *BA*

*<sup>B</sup><sup>μ</sup>* <sup>=</sup> <sup>1</sup> 8 Ω*<sup>A</sup>*

Therefore, 8*B<sup>μ</sup>* may be identified with the trace, and Ω˘ *<sup>A</sup>*

*<sup>B</sup>μ*. So, it may be natural to write *ϑμ* as follows:

*<sup>μ</sup>*Σ*AB* +

where the following auxiliary gauge symbols are used:

*<sup>B</sup><sup>μ</sup>* = *g*Ω*<sup>A</sup>*

*<sup>Q</sup><sup>μ</sup>* <sup>=</sup> <sup>4</sup>*gB<sup>μ</sup>* <sup>=</sup> *<sup>g</sup>*

*<sup>μ</sup>* = *ge<sup>A</sup>*

Γ*A*

*εA*

GL(2, **C**)-invariant spinor theory, *Q<sup>μ</sup>* = (1/2)Γ*<sup>A</sup>*

*<sup>μ</sup>*Σ*AB* + *B<sup>μ</sup>*

*μ*.

Ω*<sup>A</sup>*

1 *i*

*<sup>B</sup><sup>μ</sup>* := Ω˘ *<sup>A</sup>*

*<sup>A</sup>μ*, <sup>Ω</sup>˘ *<sup>A</sup>*

1 4 *nAB* 1 *i γ*5 + *e<sup>A</sup>*

1 4*g Q<sup>μ</sup>* 1 *i γ*<sup>5</sup> + 1 *g εA <sup>μ</sup>iτ<sup>A</sup>* +

1 4 *nAB* 1 *i γ*5 + 1 *g εA <sup>μ</sup>iτ<sup>A</sup>* +

*<sup>B</sup>μ*, Γ˘ *<sup>A</sup>*

2 Ω*<sup>A</sup>*

Ω*<sup>A</sup>*

*<sup>B</sup><sup>μ</sup>* = Γ*<sup>A</sup>*

*B* , *e<sup>A</sup>*

, [ *fA*] becomes homogeneous. It is just the correspondence rule with the situation

*<sup>A</sup><sup>μ</sup>* <sup>=</sup> <sup>1</sup> 2 Γ*A*

*<sup>B</sup>* = (1/*g*)Γ*<sup>A</sup>*

form *ϑ*, and because of this, the action of *x*-dependent matrices *U* on them is inhomogeneous. But when we restrict ourselves to the *U*-injected group SL(2, **C**), the transformation rule

*γ*<sup>5</sup> + *AμiI* + *e<sup>A</sup>*

*<sup>B</sup><sup>μ</sup>* + 2*Bμδ<sup>A</sup>*

*<sup>B</sup><sup>μ</sup>* <sup>−</sup> <sup>1</sup> 4 Ω*<sup>C</sup> Cμδ<sup>A</sup>*

1

1

*<sup>μ</sup>*, *ϕA<sup>μ</sup>* = *g fAμ*. (95)

*e<sup>A</sup>* was a gravitational cotetrad. Let us remind that in

*<sup>B</sup><sup>μ</sup>* <sup>−</sup> <sup>1</sup> 4 Γ*C Cμδ<sup>A</sup>*

*<sup>B</sup><sup>μ</sup>* = Ω*<sup>A</sup>*

*<sup>μ</sup>iτ<sup>A</sup>* + *fAμiχA*, (87)

*<sup>B</sup>*, (88)

*<sup>B</sup><sup>μ</sup>* — with the trace-less part of the

*<sup>μ</sup>iτ<sup>A</sup>* + *fAμiχ<sup>A</sup>* + *<sup>A</sup>μiI*, (90)

*<sup>g</sup> <sup>ϕ</sup>Aμiχ<sup>A</sup>* <sup>+</sup> *<sup>A</sup>μiI*, (91)

*<sup>g</sup> <sup>ϕ</sup>Aμiχ<sup>A</sup>* <sup>+</sup> *<sup>A</sup>μiI*, (92)

*<sup>A</sup>μ*, (94)

*<sup>A</sup><sup>μ</sup>* was the Weyl covector, and there was

, [ *fA*] are parts of the connection

*<sup>B</sup>*, (93)

*<sup>B</sup>*. (89)

$$E^A = \frac{1}{2} \left( e^A + \eta^{AB} f\_B \right) = \eta^{AB} E\_{B'} \quad \text{ } \quad \mathbf{F}^A = \frac{1}{2} \left( e^A - \eta^{AB} f\_B \right) = \eta^{AB} \mathbf{F}\_B. \tag{98}$$

Let us now take the following expansion of the curvature vector-valued two-form:

$$\Phi = T(e)^A i \tau\_A + T(f)^A i \chi\_A + \frac{1}{2} \tilde{\mathcal{R}}^{AB} \Sigma\_{AB} + \mathcal{G} \frac{1}{i} \gamma^5 + \text{Fi}I,\tag{99}$$

where the following partially clear symbols are used:

$$T(e)^A = de^A + g\Omega^A{}\_B \wedge e^B = de^A + \Gamma^A{}\_B \wedge e^B{}\_\dots \tag{100}$$

$$T(f)\_A = df\_A + \operatorname{gf}\_B \wedge \Omega^B\_{~A} = df\_A + f\_B \wedge \Gamma^B\_{~A\nu} \tag{101}$$

$$\begin{split} \tilde{\mathcal{R}}^A{}\_B &= R(\Omega)^A{}\_B - \frac{1}{4} R(\Omega)^C{}\_C \delta^A{}\_B - 2g e^A \wedge f\_B \\ &+ 2g \eta^{AC} \eta\_{BD} e^D \wedge f\_\mathbb{C} = \frac{1}{g} \left( R(\Gamma)^A{}\_B - \frac{1}{4} R(\Gamma)^C{}\_C \delta^A{}\_B \right. \\ &- 2g^2 e^A \wedge f\_B + 2g^2 \eta^{AC} \eta\_{BD} e^D \wedge f\_\mathbb{C} \Big), \end{split} \tag{102}$$

$$G = \frac{1}{4g}dQ - ge^A \wedge f\_A = \frac{1}{g} \left(\frac{1}{8}R(\Gamma)^A{}\_A - g^2e^A \wedge f\_a\right),\tag{103}$$

$$F = \stackrel{\circ}{dA} \tag{104}$$

where *R*(Γ), *R*(Ω) denote the curvature two-form:

$$R(\Gamma)^A{}\_B = d\Gamma^A{}\_B + \Gamma^A{}\_C \wedge \Gamma^C{}\_B \quad R(\Omega)^A{}\_B = d\Omega^A{}\_B + \g \Omega^A{}\_C \wedge \Omega^C{}\_B. \tag{105}$$

Let us remember that the torsion of a linear connection may be interpreted as a contribution to affine connection. The corresponding space-time objects, i.e., connections, torsions and curvatures are given by

$$
\Gamma(e)^k{}\_{i\dot{j}} = \varepsilon^k{}\_A \Gamma^A{}\_{B\dot{j}} e^B{}\_{\dot{i}} + \varepsilon^k{}\_A e^A{}\_{i\dot{j}\prime} \tag{106}
$$

$$
\Gamma(f)^k\_{\,\,ij} = -f\_{A\dot{\imath}}\Gamma^A{}\_{B\dot{\jmath}}f^{kB} + f^{kA}f\_{Ai\_{\jmath}\nu} \tag{107}
$$

$$\left(S(e)^{k}\right)\_{\vec{i}\vec{j}} = \Gamma(e)^{k}\_{[\vec{i}\vec{j}]} = -\frac{1}{2}e^{k}{}\_{A}T(e)^{A}{}\_{\vec{i}\vec{j}}\tag{108}$$

$$S(f)^k\_{\,\,ij} = \Gamma(f)^k\_{\,\,[ij]} = -\frac{1}{2} f^{kA} T(f)\_{A\,\,ij\,\,} \tag{109}$$

$$\mathcal{R}(\boldsymbol{e})^{m}\_{k\text{ij}} = \boldsymbol{e}^{m}{}\_{A}\boldsymbol{e}^{B}{}\_{k}\mathcal{R}^{A}{}\_{B\text{ij}},\tag{110}$$

$$R(f)^{\mathfrak{m}}\_{\text{kij}} = -f\_{\text{Ak}} f^{\mathfrak{m}\mathbf{B}} R^A{}\_{\text{Bij}}.\tag{111}$$

with the first-order Dirac equation. However, the situation is not very bad, on the contrary, it may seem promising and desirable. If we substitute to the matter equation the Dirac-Einstein assumption, e.g., with *p* = 1, *k* = 1, then we obtain the following Dirac-Klein-Gordon

Order of Time Derivatives in Quantum-Mechanical Equations 73

<sup>Ψ</sup> <sup>−</sup> <sup>4</sup>*bg*<sup>2</sup> <sup>−</sup> *<sup>c</sup>*

the *g*-Levi-Civita differentiation. It is interesting that the first two terms of (119) correspond exactly with the Dirac theory in Einstein-Cartan space-time. However, there is quite a natural question if the third, second-order d'Alembert term does not destroy completely this

The simplest way to answer this question is to consider the specially-relativistic situation,

1 2*g*

> *c bg*<sup>2</sup> <sup>−</sup> <sup>3</sup>

<sup>2</sup>*bg* <sup>Ψ</sup> <sup>+</sup>

*<sup>b</sup>* <sup>−</sup> <sup>2</sup>*g*<sup>2</sup>

It is obvious that the general solution of this equation is a combination of two Dirac waves

 1 ±

means that there is a range of "Dirac" behaviour, *c*/*b* > 3*g*2. The primeval mass parameter must be sufficiently large for that. Then the solution of (120) will be a superposition of two Dirac fields. If *c*/*b* = 3*g*2, then one obtains the exactly Dirac behaviour. This means that there is no splitting of mass and that *m* = |*g*|. Below this threshold we are dealing with tachyonic or decay phenomena. It is also very interesting that if *c*/*b* = 4*g*2, then one of the partner states is massless, namely, *m*<sup>−</sup> = 0, *m*<sup>+</sup> = 2|*g*|. Obviously, the doubling of mass within the Dirac behaviour needs some explanation in terms of experimental data. Let us quote three

• If the energy gap *m*<sup>+</sup> − *m*<sup>−</sup> is very small (i.e., |*g*| is small enough), then perhaps it is below

• If the energy gap *m*<sup>+</sup> − *m*<sup>−</sup> is so large that perhaps it is too difficult to create/excite the

• And finally, the most important and promising explanation. Perhaps the mass-state-doubling does exist and is just observed. This would be just the explanation of the mysterious relationship between fundamental quarks and fermions in the standard model of weak interactions. We mean their occurance in pairs (*u*, *d*), (*c*,*s*), (*t*, *b*) (quarks) and (*νe*,*e*), (*νμ*, *μ*), (*ντ*, *τ*) (leptons) (Veltman, 2003). For example, the situation *c*/*b* = 4*g*<sup>2</sup> might be a naive explanation of the pairing between heavy leptons and their nuetrinos.

<sup>2</sup>*bg* <sup>Ψ</sup> <sup>+</sup>

1 2*g g* ∇*μ g*

*<sup>B</sup>*-differentiation of objects with the capital Lorentz indices and with

*<sup>μ</sup>*. It is clear that under this substitution one obtains the

*ipμx<sup>μ</sup>*

*ημν∂μ∂ν*Ψ = 0. (120)

. (121)

are substituted to (120). This

∇*ν*Ψ = 0, (119)

*g*

∇*<sup>μ</sup>* joints that

equation:

similarity.

*e μ Aiγ<sup>A</sup>*

differentiation with Γ*<sup>A</sup>*

when *gμν* = *ημν*, Γ*<sup>A</sup>*

<sup>∇</sup>*<sup>μ</sup>* <sup>+</sup> *<sup>S</sup><sup>ν</sup>*

*<sup>B</sup><sup>μ</sup>* = 0, *e<sup>A</sup>*

Dirac-Klein-Gordon differential equation

with two possible masses, namely, *m*±, where

possibilities (Sławianowski & Kovalchuk, 2008):

state of higher mass.

the present accuracy of our experimental abilities.

*νμ I*<sup>4</sup> 

where ∇*<sup>μ</sup>* is the SL(2, **C**)-part of the U(2, 2)-covariant derivative and

*<sup>μ</sup>* = *δ<sup>A</sup>*

*<sup>i</sup>γμ∂μ*<sup>Ψ</sup> <sup>−</sup> <sup>4</sup>*bg*<sup>2</sup> <sup>−</sup> *<sup>c</sup>*

*m*2 <sup>±</sup> <sup>=</sup> *<sup>c</sup>*

This result is obtained when the amplitudes U(*p*) exp

We do not quote the total explicit form of field equations. Being generally-covariant, they are over-determined, therefore, as usual suspected to be inconsistent. Nevertheless, substituting to the field equations the non-excited matter Ψ = 0, and the above equations (106)–(111), one can show that there are certain non-trivial solutions. Namely, let us take the following Einstein-Dirac metrics:

$$h(e\_\prime \eta)\_{\mu \nu} = \eta\_{AB} e^A{}\_\mu e^B{}\_{\nu \nu}$$

$$h(f\_\prime \eta)\_{\mu \nu} = \eta^{AB} f\_{A\mu} f\_{B\nu}.\tag{112}$$

And now, let us substitute the following conditions to the field equations (58), (59), (60):

$$\Psi = 0, \qquad f\_{A\mu} = k\eta\_{AB}e^B\_{\ \mu\prime} \qquad \text{g}\_{\mu\nu} = \text{ph}(e,\eta)\_{\mu\nu} \tag{113}$$

$$Q\_{\mu} = 0, \qquad A\_{\mu} = 0, \qquad S(e)^{\lambda}{}\_{\mu\nu} = S(f)^{\lambda}{}\_{\mu\nu} = 0. \tag{114}$$

It is simply marvellous that the very complicated system of equations following from (58), (59), (60) after substituting (106)–(114) is solvable, moreover, it is reducible to something very simple. Namely, the very complicated system of equations for the geometric fields reduces step by step to

$$\mathcal{R}^{\mu\nu} - \frac{1}{2} \mathcal{R} \mathcal{g}^{\mu\nu} = -12 \frac{\mathcal{g}^2 k}{p} \mathcal{g}^{\mu\nu},\tag{115}$$

where *Rμν* denotes the twice contravariant Ricci tensor built of *gαβ*, and *R* is the curvature scalar. Substituting there *k* = 1, *p* = 1, one obtains simply the following equation

$$\mathcal{R}^{\mu\nu} - \frac{1}{2}\mathcal{R}\mathcal{g}^{\mu\nu} = -12\mathcal{g}^2\mathcal{g}^{\mu\nu}.\tag{116}$$

In any case one deals here with the Einstein equation with the cosmological constant. It is remarkable that this coupling constant is proportional to the gauge coupling constant. Such a coupling between microphysical model and macroscopic or even just cosmic scale physics is marvellous and philosophically fascinating. When the Einstein-Hilbert dynamics of *gμν* is used, we obtain also the condition

$$T^{\mu\nu} = 0.\tag{117}$$

But there is no contradiction between (117) and (115)/(116). There exist some common solutions, namely, ones corresponding to the constant curvature spaces:

$$R\_{a\\$\mu\nu} = \frac{4g^2k}{p} \left(\mathbf{g}\_{a\mu}\mathbf{g}\_{\beta\nu} - \mathbf{g}\_{a\nu}\mathbf{g}\_{\beta\mu}\right). \tag{118}$$

It is again the fascinating idea that the conformal flatness of space-time, expressed by (118) is so nicely compatible with the assumption that the theory is invariant under the covering group of the conformal group.

The field sector of the model corresponds smoothly to the gauge Poincare gravitation. When the field Lagrangian is expressed in terms of the above quantities *e*, *f* , *S*(*e*), *S*(*f*), *R*(*e*), *R*(*f*), one obtains expression quadratic in field variables just like in Poincare gauge models. However, the use of the semisimple SU(2, 2) results in well-defined, rigorous ratio of constant coefficients, not accidental one like in Poincare model.

What concerns material sector, the use of the second-order Klein-Gordon equation and the presence of second-order derivatives of the matter field, is a drastic difference in comparison 16 Will-be-set-by-IN-TECH

We do not quote the total explicit form of field equations. Being generally-covariant, they are over-determined, therefore, as usual suspected to be inconsistent. Nevertheless, substituting to the field equations the non-excited matter Ψ = 0, and the above equations (106)–(111), one can show that there are certain non-trivial solutions. Namely, let us take the following

*h*(*e*, *η*)*μν* = *ηABe<sup>A</sup>*

And now, let us substitute the following conditions to the field equations (58), (59), (60):

It is simply marvellous that the very complicated system of equations following from (58), (59), (60) after substituting (106)–(114) is solvable, moreover, it is reducible to something very simple. Namely, the very complicated system of equations for the geometric fields reduces

*Rgμν* <sup>=</sup> <sup>−</sup><sup>12</sup> *<sup>g</sup>*2*<sup>k</sup>*

where *Rμν* denotes the twice contravariant Ricci tensor built of *gαβ*, and *R* is the curvature

In any case one deals here with the Einstein equation with the cosmological constant. It is remarkable that this coupling constant is proportional to the gauge coupling constant. Such a coupling between microphysical model and macroscopic or even just cosmic scale physics is marvellous and philosophically fascinating. When the Einstein-Hilbert dynamics of *gμν* is

But there is no contradiction between (117) and (115)/(116). There exist some common

It is again the fascinating idea that the conformal flatness of space-time, expressed by (118) is so nicely compatible with the assumption that the theory is invariant under the covering

The field sector of the model corresponds smoothly to the gauge Poincare gravitation. When the field Lagrangian is expressed in terms of the above quantities *e*, *f* , *S*(*e*), *S*(*f*), *R*(*e*), *R*(*f*), one obtains expression quadratic in field variables just like in Poincare gauge models. However, the use of the semisimple SU(2, 2) results in well-defined, rigorous ratio of constant

What concerns material sector, the use of the second-order Klein-Gordon equation and the presence of second-order derivatives of the matter field, is a drastic difference in comparison

*gαμgβν* − *gαν gβμ*

<sup>Ψ</sup> = 0, *fA<sup>μ</sup>* = *<sup>k</sup>ηABe<sup>B</sup>*

*Q<sup>μ</sup>* = 0, *A<sup>μ</sup>* = 0, *S*(*e*)*<sup>λ</sup>*

*<sup>R</sup>μν* <sup>−</sup> <sup>1</sup> 2

> *<sup>R</sup>μν* <sup>−</sup> <sup>1</sup> 2

solutions, namely, ones corresponding to the constant curvature spaces:

*<sup>R</sup>αβμν* <sup>=</sup> <sup>4</sup>*g*2*<sup>k</sup>*

*p* 

scalar. Substituting there *k* = 1, *p* = 1, one obtains simply the following equation

*μe<sup>B</sup> ν*,

*<sup>h</sup>*(*<sup>f</sup>* , *<sup>η</sup>*)*μν* = *<sup>η</sup>AB fA<sup>μ</sup> fBν*. (112)

*μν* = *S*(*f*)*<sup>λ</sup>*

*<sup>μ</sup>*, *gμν* = *ph*(*e*, *η*)*μν*, (113)

*<sup>p</sup> <sup>g</sup>μν*, (115)

*Rgμν* <sup>=</sup> <sup>−</sup>12*g*2*gμν*. (116)

*Tμν* = 0. (117)

. (118)

*μν* = 0. (114)

Einstein-Dirac metrics:

step by step to

used, we obtain also the condition

group of the conformal group.

coefficients, not accidental one like in Poincare model.

with the first-order Dirac equation. However, the situation is not very bad, on the contrary, it may seem promising and desirable. If we substitute to the matter equation the Dirac-Einstein assumption, e.g., with *p* = 1, *k* = 1, then we obtain the following Dirac-Klein-Gordon equation:

$$\varepsilon^{\mu}{}\_{A}i\gamma^{A}\left(\nabla\_{\mu} + S^{\nu}{}\_{\nu\mu}I\_{4}\right)\Psi - \frac{4bg^{2}-c}{2bg}\Psi + \frac{1}{2g}\stackrel{\mathcal{S}}{\nabla}\_{\mu}\stackrel{\mathcal{S}}{\nabla}\_{\nu}\Psi = 0,\tag{119}$$

where ∇*<sup>μ</sup>* is the SL(2, **C**)-part of the U(2, 2)-covariant derivative and *g* ∇*<sup>μ</sup>* joints that differentiation with Γ*<sup>A</sup> <sup>B</sup>*-differentiation of objects with the capital Lorentz indices and with the *g*-Levi-Civita differentiation. It is interesting that the first two terms of (119) correspond exactly with the Dirac theory in Einstein-Cartan space-time. However, there is quite a natural question if the third, second-order d'Alembert term does not destroy completely this similarity.

The simplest way to answer this question is to consider the specially-relativistic situation, when *gμν* = *ημν*, Γ*<sup>A</sup> <sup>B</sup><sup>μ</sup>* = 0, *e<sup>A</sup> <sup>μ</sup>* = *δ<sup>A</sup> <sup>μ</sup>*. It is clear that under this substitution one obtains the Dirac-Klein-Gordon differential equation

$$i\gamma^{\mu}\partial\_{\mu}\Psi - \frac{4bg^2 - c}{2bg}\Psi + \frac{1}{2g}\eta^{\mu\nu}\partial\_{\mu}\partial\_{\nu}\Psi = 0.\tag{120}$$

It is obvious that the general solution of this equation is a combination of two Dirac waves with two possible masses, namely, *m*±, where

$$m\_{\pm}^2 = \frac{c}{b} - 2g^2 \left(1 \pm \sqrt{\frac{c}{bg^2} - 3}\right). \tag{121}$$

This result is obtained when the amplitudes U(*p*) exp *ipμx<sup>μ</sup>* are substituted to (120). This means that there is a range of "Dirac" behaviour, *c*/*b* > 3*g*2. The primeval mass parameter must be sufficiently large for that. Then the solution of (120) will be a superposition of two Dirac fields. If *c*/*b* = 3*g*2, then one obtains the exactly Dirac behaviour. This means that there is no splitting of mass and that *m* = |*g*|. Below this threshold we are dealing with tachyonic or decay phenomena. It is also very interesting that if *c*/*b* = 4*g*2, then one of the partner states is massless, namely, *m*<sup>−</sup> = 0, *m*<sup>+</sup> = 2|*g*|. Obviously, the doubling of mass within the Dirac behaviour needs some explanation in terms of experimental data. Let us quote three possibilities (Sławianowski & Kovalchuk, 2008):


**0**

**4**

*Japan*

Yutaka Shikano

**Theory of "Weak Value" and Quantum**

Quantum mechanics provides us many perspectives and insights on Nature and our daily life. However, its mathematical axiom initiated by von Neumann (121) is not satisfied to describe nature phenomena. For example, it is impossible not to explain a non self-adjoint operator, i.e., the momentum operator on a half line (See, e.g., Ref. (154).), as the physical observable. On considering foundations of quantum mechanics, the simple and specific expression is needed. One of the candidates is the *weak value* initiated by Yakir Aharonov and his colleagues (4). It is remarked that the idea of their seminal work is written in ref. (3). Furthermore, this quantity has a potentiality to explain the counter-factual phenomena, in which there is the contradiction under the classical logic, e.g., the Hardy paradox (64). If so, it may be possible to quantitatively explain quantum mechanics in the particle picture. In this review based on the author thesis (152), we consider the theory of the weak value and construct a measurement

model to extract the weak value. See the other reviews in Refs. (12; 14; 15; 20).

*f*�*A*�*<sup>w</sup>*

*<sup>i</sup>* :<sup>=</sup> �*<sup>f</sup>* <sup>|</sup>*A*|*i*�

where |*i*� and | *f*� are called a pre- and post-selected state, respectively. As the naming of the "weak value", this quantity is experimentally accessible by the weak measurement as explained below. As seen in Fig. 1, the weak value can be measured as the shift of a meter of the probe after the weak interaction between the target and the probe with the specific post-selection of the target. Due to the weak interaction, the quantum state of the target is only slightly changed but the information of the desired observable *A* is encoded in the probe by the post-selection. While the previous studies of the weak value since the seminal paper (4), which will be reviewed in Sec. 3, are based on the measurement scheme, there are few works that the weak value is focused on and is independent of the measurement scheme. Furthermore, in these 20 years, we have not yet understood the mathematical properties of the weak value. In this chapter, we review the historical backgrounds of the weak value and the weak measurement and recent development on the measurement model to extract the weak

The time evolution for the quantum state and the operation for the measurement are called a quantum operation. In this section, we review a general description of the quantum operation. Therefore, the quantum operation can describe the time evolution for the quantum state, the

Let the weak value for an observable *A* be defined as

**1. Introduction**

value.

**2. Review of quantum operation**

**Mechanical Measurements**

�*<sup>f</sup>* <sup>|</sup>*i*� , (1)

*Department of Physics, Tokyo Institute of Technology, Tokyo*

#### **6. Some additional interpretation problems**

In one of our earlier papers we have discussed the idea of higher-order derivatives just from the point of view of the order of time derivatives (Sławianowski & Kovalchuk, 2002; 2010; Sławianowski et al., 2004; 2005), because the other continuous variables were absent. The Schrödinger equation was then interpreted as a Hamiltonian system of mathematical physics. And it was just then where the second-order time derivatives seemed not only admissible, but just necessary. In the field problems, this concerns, of course, the occurrence of all second-order space-time derivatives of the field quantities/wave functions. And, as shown above, those second-order space-time derivatives just seem desirable, not only admissible. The next important question is: why just U(2, 2) ⊂ GL(4, **C**), not the total GL(4, **C**)? But it is seen that it is only signature (+, +, −, −) of Hermitian *G*-forms, not the group U(2, 2) itself that matters. Namely, the mass form *Grs* must by present in Lagrangian, however not as a fixed constant Hermitian form but as a dynamical, *x*-dependent form of signature (+, +, −, −). The corresponding Lagrangian term would be proportional to

$$\mathcal{G}^{s\overline{z}}\mathcal{G}^{t\overline{r}}\frac{\partial \mathcal{G}\_{\overline{rs}}}{\partial \mathcal{x}^{\mu}}\frac{\partial \mathcal{G}\_{\overline{z}t}}{\partial \mathcal{x}^{\nu}}\mathcal{g}^{\mu\nu},\tag{122}$$

or perhaps to the Born-Infeld term

$$\sqrt{\det\left[G^{s\overline{\pi}}G^{t\overline{\pi}}G\_{\overline{\pi}s,\mu}G\_{\overline{\pi}t,\nu}\right]}\tag{123}$$

independent on the mentioned choice of the space-time metric *gμν*. This term is evidently GL(4, **C**)-invariant.

#### **7. References**


## **Theory of "Weak Value" and Quantum Mechanical Measurements**

Yutaka Shikano

*Department of Physics, Tokyo Institute of Technology, Tokyo Japan*

#### **1. Introduction**

18 Will-be-set-by-IN-TECH

74 Measurements in Quantum Mechanics

In one of our earlier papers we have discussed the idea of higher-order derivatives just from the point of view of the order of time derivatives (Sławianowski & Kovalchuk, 2002; 2010; Sławianowski et al., 2004; 2005), because the other continuous variables were absent. The Schrödinger equation was then interpreted as a Hamiltonian system of mathematical physics. And it was just then where the second-order time derivatives seemed not only admissible, but just necessary. In the field problems, this concerns, of course, the occurrence of all second-order space-time derivatives of the field quantities/wave functions. And, as shown above, those second-order space-time derivatives just seem desirable, not only admissible. The next important question is: why just U(2, 2) ⊂ GL(4, **C**), not the total GL(4, **C**)? But it is seen that it is only signature (+, +, −, −) of Hermitian *G*-forms, not the group U(2, 2) itself that matters. Namely, the mass form *Grs* must by present in Lagrangian, however not as a fixed constant Hermitian form but as a dynamical, *x*-dependent form of signature

(+, +, −, −). The corresponding Lagrangian term would be proportional to

 det

M. Mladenov, (Ed.), 66–132, SOFTEX, Sofia.

*<sup>G</sup>szGtr <sup>∂</sup>Grs ∂x<sup>μ</sup>*

*∂Gzt*

*GszGtrGrs*,*μGzt*,*<sup>ν</sup>*

independent on the mentioned choice of the space-time metric *gμν*. This term is evidently

Sławianowski, J.J. & Kovalchuk, V. (2002). Klein-Gordon-Dirac equation: physical justification

Sławianowski, J.J. & Kovalchuk, V. (2008). Search for the geometrodynamical gauge group.

Sławianowski, J.J. & Kovalchuk, V. (2010). Schrödinger and related equations as Hamiltonian

quantum mechanics. *Reports on Mathematical Physics*, Vol. 65, No. 1, 29–76. Sławianowski, J.J.; Kovalchuk, V.; Sławianowska A.; Gołubowska B.; Martens A.; Rozko ˙

Sławianowski, J.J.; Kovalchuk, V.; Sławianowska A.; Gołubowska B.; Martens A.; Rozko E.E. ˙

Veltman M. (2003). *Facts and mysteries in elementary particle physics*, World Scientific, New

and quantization attempts. *Reports on Mathematical Physics*, Vol. 49, No. 2/3, 249–257.

Hypotheses and some results, In: *Geometry, Integrability and Quantization IX*, Ivailo

systems, manifolds of second-order tensors and new ideas of nonlinearity in

E.E. & Zawistowski Z.J. (2004). Affine symmetry in mechanics of collective and internal modes. Part I. Classical models. *Reports on Mathematical Physics*, Vol. 54, No.

& Zawistowski Z.J. (2005). Affine symmetry in mechanics of collective and internal modes. Part II. Quantum models. *Reports on Mathematical Physics*, Vol. 55, No. 1, 1–45.

*<sup>∂</sup>x<sup>ν</sup> <sup>g</sup>μν*, (122)

(123)

**6. Some additional interpretation problems**

or perhaps to the Born-Infeld term

GL(4, **C**)-invariant.

3, 373–427.

Jersey.

**7. References**

Quantum mechanics provides us many perspectives and insights on Nature and our daily life. However, its mathematical axiom initiated by von Neumann (121) is not satisfied to describe nature phenomena. For example, it is impossible not to explain a non self-adjoint operator, i.e., the momentum operator on a half line (See, e.g., Ref. (154).), as the physical observable. On considering foundations of quantum mechanics, the simple and specific expression is needed. One of the candidates is the *weak value* initiated by Yakir Aharonov and his colleagues (4). It is remarked that the idea of their seminal work is written in ref. (3). Furthermore, this quantity has a potentiality to explain the counter-factual phenomena, in which there is the contradiction under the classical logic, e.g., the Hardy paradox (64). If so, it may be possible to quantitatively explain quantum mechanics in the particle picture. In this review based on the author thesis (152), we consider the theory of the weak value and construct a measurement model to extract the weak value. See the other reviews in Refs. (12; 14; 15; 20). Let the weak value for an observable *A* be defined as

$$\,\_{f}\langle A\rangle\_{i}^{w} := \frac{\langle f|A|i\rangle}{\langle f|i\rangle},\tag{1}$$

where |*i*� and | *f*� are called a pre- and post-selected state, respectively. As the naming of the "weak value", this quantity is experimentally accessible by the weak measurement as explained below. As seen in Fig. 1, the weak value can be measured as the shift of a meter of the probe after the weak interaction between the target and the probe with the specific post-selection of the target. Due to the weak interaction, the quantum state of the target is only slightly changed but the information of the desired observable *A* is encoded in the probe by the post-selection. While the previous studies of the weak value since the seminal paper (4), which will be reviewed in Sec. 3, are based on the measurement scheme, there are few works that the weak value is focused on and is independent of the measurement scheme. Furthermore, in these 20 years, we have not yet understood the mathematical properties of the weak value. In this chapter, we review the historical backgrounds of the weak value and the weak measurement and recent development on the measurement model to extract the weak value.

#### **2. Review of quantum operation**

The time evolution for the quantum state and the operation for the measurement are called a quantum operation. In this section, we review a general description of the quantum operation. Therefore, the quantum operation can describe the time evolution for the quantum state, the

the quantum dynamics in the compound system should be also described as the positive map since the compound system should be subject to quantum mechanics. Given the positive map, the positive map is called a CP map if and only if the positive map is also in the compound system coupled to any auxiliary system. One of the important aspects of the CP map is that all physically realizable quantum operations can be described only by operators defined in the target system. Furthermore, the auxiliary system can be environmental system, the probe system, and the controlled system. Regardless to the role of the auxiliary system, the CP map gives the same description for the target system. On the other hand, both quantum

Theory of "Weak Value" and Quantum Mechanical Measurements 77

Let E be a positive map from L(H*s*), a set of linear operations on the Hilbert space H*s*, to L(H*s*). If E is completely positive, its trivial extension K from L(H*s*) to L(H*<sup>s</sup>* ⊗ H*e*) is also

for an arbitrary state |*α*�∈H*<sup>s</sup>* ⊗ H*p*, where **1** is the identity operator. We assume without loss of generality dimH*<sup>s</sup>* = dimH*<sup>e</sup>* < ∞. Throughout this chapter, we concentrate on the case that the target state is pure though the generalization to mixed states is straightforward. From the

**Theorem 2.1.** *Let* E *be a CP map from* H*<sup>s</sup> to* H*s. For any quantum state* |*ψ*�*<sup>s</sup>* ∈ H*s, there exist a*

*<sup>ψ</sup>k*|*k*�*s*, <sup>|</sup>*ψ*˜�*<sup>e</sup>* <sup>=</sup> ∑

*k ψ*∗


complete positivity, we obtain the following theorem for quantum state changes.


> ∑ *m*,*n*

From the complete positivity, K(|*α*�) > 0 for all |*α*�∈H*<sup>s</sup>* ⊗ H*e*, we can express *σ*(|*α*�) as

*sm*|*s*ˆ*m*��*s*ˆ*m*| = ∑

*m*


K(|*α*�)=(E ⊗ **1**)

K(|*α*�) = ∑

*m*

= ∑ *m*,*n*

K(|*α*�) := (E ⊗ **1**)(|*α*��*α*|) > 0, (4)

<sup>E</sup>(|*ψ*�*s*�*ψ*|) = *<sup>e</sup>*�*ψ*˜|K(|*α*�)|*ψ*˜�*e*, (5)

*<sup>k</sup>* |*k*�*e*, (6)




*<sup>e</sup>*�*m*|K(|*α*�)|*n*�*<sup>e</sup>* = E(|*m*�*s*�*n*|). (9)

measurement and decoherence give the same role for the target system.

positive such that

*where*

to obtain

*map σ and a pure state* |*α*�∈H*<sup>s</sup>* ⊗ H*<sup>e</sup> such that*

*Proof.* We can write in the Schmidt form as

We rewrite the right hand sides of Eq. (5) as

*which represents the state change for the density operator.*

By linearity, the desired equation (5) can be derived.

Fig. 1. Schematic figure of the weak measurement.

control of the quantum state, the quantum measurement, and the noisy quantum system in the same formulation.

#### **2.1 Historical remarks**

Within the mathematical postulates of quantum mechanics (121), the state change is subject to the Schrödinger equation. However, the state change on the measurement is not subject to this but is subject to another axiom, conventionally, von Neumann-Lüders projection postulate (105). See more details on quantum measurement theory in the books (31; 40; 194). Let us consider a state change from the initial state <sup>|</sup>*ψ*� on the *projective measurement* <sup>1</sup> for the operator *A* = ∑*<sup>j</sup> aj*|*aj*��*aj*|. From the Born rule, the probability to obtain the measurement outcome, that is, the eigenvalue of the observable *A*, is given by

$$\Pr[A = a\_{\mathfrak{m}}] = |\langle a\_{\mathfrak{m}} | \psi \rangle|^2 = \text{Tr}\left[ |\psi\rangle\langle\psi| \cdot |a\_{\mathfrak{m}}\rangle\langle a\_{\mathfrak{m}}| \right] = \text{Tr}\,\rho P\_{\mathfrak{a}\_{\mathfrak{m}}} \tag{2}$$

where *ρ* := |*ψ*��*ψ*| and *Pam* = |*am*��*am*|. After the measurement with the measurement outcome *am*, the quantum state change is given by

$$|\psi\rangle \to |a\_m\rangle,\tag{3}$$

which is often called the "collapse of wavefunction" or "state reduction". This implies that it is necessary to consider the **non-unitary process** even in the isolated system. To understand the measuring process as quantum dynamics, we need consider the general theory of quantum operations.

#### **2.2 Operator-sum representation**

Let us recapitulate the general theory of quantum operations of a finite dimensional quantum system (122). All physically realizable quantum operations can be generally described by a completely positive (CP) map (127; 128), since the isolated system of a target system and an auxiliary system always undergoes the unitary evolution according to the axiom of quantum mechanics (121). Physically speaking, the operation of the target system should be described as a positive map, that is, the map from the positive operator to the positive operator, since the density operator is positive. Furthermore, if any auxiliary system is coupled to the target one,

<sup>1</sup> This measurement is often called the *von Neumann measurement* or the *strong measurement*.

the quantum dynamics in the compound system should be also described as the positive map since the compound system should be subject to quantum mechanics. Given the positive map, the positive map is called a CP map if and only if the positive map is also in the compound system coupled to any auxiliary system. One of the important aspects of the CP map is that all physically realizable quantum operations can be described only by operators defined in the target system. Furthermore, the auxiliary system can be environmental system, the probe system, and the controlled system. Regardless to the role of the auxiliary system, the CP map gives the same description for the target system. On the other hand, both quantum measurement and decoherence give the same role for the target system.

Let E be a positive map from L(H*s*), a set of linear operations on the Hilbert space H*s*, to L(H*s*). If E is completely positive, its trivial extension K from L(H*s*) to L(H*<sup>s</sup>* ⊗ H*e*) is also positive such that

$$\mathcal{K}(|\mathfrak{a}\rangle) := (\mathcal{E} \otimes \mathbf{1})(|\mathfrak{a}\rangle\langle\mathfrak{a}|) > 0,\tag{4}$$

for an arbitrary state |*α*�∈H*<sup>s</sup>* ⊗ H*p*, where **1** is the identity operator. We assume without loss of generality dimH*<sup>s</sup>* = dimH*<sup>e</sup>* < ∞. Throughout this chapter, we concentrate on the case that the target state is pure though the generalization to mixed states is straightforward. From the complete positivity, we obtain the following theorem for quantum state changes.

**Theorem 2.1.** *Let* E *be a CP map from* H*<sup>s</sup> to* H*s. For any quantum state* |*ψ*�*<sup>s</sup>* ∈ H*s, there exist a map σ and a pure state* |*α*�∈H*<sup>s</sup>* ⊗ H*<sup>e</sup> such that*

$$\mathcal{E}(|\psi\rangle\_s \langle \psi|) = \,\_\varepsilon \langle \tilde{\psi} | \mathcal{K}(|\alpha\rangle) |\tilde{\psi}\rangle\_{\varepsilon} \tag{5}$$

*where*

2 Will-be-set-by-IN-TECH

control of the quantum state, the quantum measurement, and the noisy quantum system in

Within the mathematical postulates of quantum mechanics (121), the state change is subject to the Schrödinger equation. However, the state change on the measurement is not subject to this but is subject to another axiom, conventionally, von Neumann-Lüders projection postulate (105). See more details on quantum measurement theory in the books (31; 40; 194). Let us consider a state change from the initial state <sup>|</sup>*ψ*� on the *projective measurement* <sup>1</sup> for the operator *A* = ∑*<sup>j</sup> aj*|*aj*��*aj*|. From the Born rule, the probability to obtain the measurement

where *ρ* := |*ψ*��*ψ*| and *Pam* = |*am*��*am*|. After the measurement with the measurement outcome

which is often called the "collapse of wavefunction" or "state reduction". This implies that it is necessary to consider the **non-unitary process** even in the isolated system. To understand the measuring process as quantum dynamics, we need consider the general theory of quantum

Let us recapitulate the general theory of quantum operations of a finite dimensional quantum system (122). All physically realizable quantum operations can be generally described by a completely positive (CP) map (127; 128), since the isolated system of a target system and an auxiliary system always undergoes the unitary evolution according to the axiom of quantum mechanics (121). Physically speaking, the operation of the target system should be described as a positive map, that is, the map from the positive operator to the positive operator, since the density operator is positive. Furthermore, if any auxiliary system is coupled to the target one,

<sup>1</sup> This measurement is often called the *von Neumann measurement* or the *strong measurement*.

Pr[*<sup>A</sup>* <sup>=</sup> *am*] = |�*am*|*ψ*�|<sup>2</sup> <sup>=</sup> Tr[|*ψ*��*ψ*|·|*am*��*am*|] <sup>=</sup> Tr *<sup>ρ</sup>Pam* , (2)


Fig. 1. Schematic figure of the weak measurement.

*am*, the quantum state change is given by

**2.2 Operator-sum representation**

outcome, that is, the eigenvalue of the observable *A*, is given by

the same formulation.

**2.1 Historical remarks**

operations.

$$|\psi\rangle\_{\mathbf{s}} = \sum\_{k} \psi\_{k} |k\rangle\_{\mathbf{s}\prime} \quad |\tilde{\psi}\rangle\_{\mathbf{c}} = \sum\_{k} \psi\_{k}^{\*} |k\rangle\_{\mathbf{c}\prime} \tag{6}$$

*which represents the state change for the density operator.*

*Proof.* We can write in the Schmidt form as

$$|\mathfrak{a}\rangle = \sum\_{m} |m\rangle\_{\mathfrak{s}} |m\rangle\_{\mathfrak{e}}.\tag{7}$$

We rewrite the right hand sides of Eq. (5) as

$$\begin{split} \mathcal{K}(|n\rangle) &= (\mathcal{E}\otimes \mathbf{1}) \left( \sum\_{m,n} |m\rangle\_{s} |m\rangle\_{\mathcal{E}\,s} \langle n|\_{\mathcal{E}} \langle n| \right) \\ &= \sum\_{m,n} |m\rangle\_{\mathcal{E}} \langle n| \mathcal{E}(|m\rangle\_{s} \langle n|) \end{split} \tag{8}$$

to obtain

$$
\mathcal{L}\_{\varepsilon} \langle m | \mathcal{K}(|n\rangle) | n \rangle\_{\varepsilon} = \mathcal{E} \left( |m\rangle\_{s} \langle n| \right). \tag{9}
$$

By linearity, the desired equation (5) can be derived.

From the complete positivity, K(|*α*�) > 0 for all |*α*�∈H*<sup>s</sup>* ⊗ H*e*, we can express *σ*(|*α*�) as

$$\mathcal{K}(|\alpha\rangle) = \sum\_{m} s\_{m} |\mathfrak{s}\_{m}\rangle \langle \mathfrak{s}\_{m}| = \sum\_{m} |\mathfrak{s}\_{m}\rangle \langle \mathfrak{s}\_{m}| \,\tag{10}$$

**3. Review of weak value**

For an observable *A*, the *weak value* �*A*�*<sup>w</sup>* is defined as

depends on the observable (151, Sec. 4.1) 3.

Ex[*A*] = �*ψ*|*A*|*ψ*� =

where *hA*[|*φ*�] = *<sup>φ</sup>*�*A*�*<sup>w</sup>*

= 

<sup>2</sup> This concept is shared in Refs. (1; 49; 51; 78; 81; 82; 117; 130).

weak value.

*f*�*A*�*<sup>w</sup>*

In Secs. 2.1 and 2.3, the direct and indirect quantum measurement schemes, we only get the probability distribution. However, the probability distribution is not the only thing that is experimentally accessible in quantum mechanics. In quantum mechanics, the phase is also an essential ingredient and in particular the geometric phase is a notable example of an experimentally accessible quantity (150). The general experimentally accessible quantity which contains complete information of the probability and the phase seems to be the *weak value* advocated by Aharonov and his collaborators (4; 14). They proposed a model of weakly coupled system and probe, see Sec. 4.3, to obtain information to a physical quantity as a "weak value" only slightly disturbing the state. Here, we briefly review the formal aspects of the

Theory of "Weak Value" and Quantum Mechanical Measurements 79

�*A*�*<sup>w</sup>* :<sup>=</sup> �*<sup>f</sup>* <sup>|</sup>*U*(*tf* , *<sup>t</sup>*)*AU*(*t*, *ti*)|*i*�

where |*i*� and �*f* | are normalized pre-selected ket and post-selected bra state vectors, respectively (4). Here, *U*(*t*2, *t*1) is an evolution operator from the time *t*<sup>1</sup> to *t*2. The weak value �*A*�*<sup>w</sup>* actually depends on the pre- and post-selected states |*i*� and �*f* | but we omit them for notational simplicity in the case that we fix them. Otherwise, we write them explicitly as

Let us calculate the expectation value in quantum mechanics for the quantum state |*ψ*� as

*<sup>ψ</sup>* is complex random variable and *dP* :<sup>=</sup> |�*φ*|*ψ*�|2*d<sup>φ</sup>* is the probability

*dφ* �*ψ*|*φ*�·�*φ*|*ψ*�

�*φ*|*A*|*ψ*� �*φ*|*ψ*� ,

*<sup>ψ</sup>*, (19)

*dφ* �*ψ*|*φ*��*φ*|*A*|*ψ*� =

*φ*�*A*�*<sup>w</sup>*

measure and is independent of the observable *A*. Therefore, the event space Ω = {|*φ*�} is taken as the set of the post-selected state. This formula means that the extended probability theory corresponds to the Born rule. From the conventional definition of the variance in

<sup>3</sup> Due to this, the probability in quantum mechanics cannot be applied to the standard probability theory.

As another approach to resolve this, there is the quantum probability theory (138).

*<sup>d</sup><sup>φ</sup>* |�*ψ*|*φ*�|<sup>2</sup>

*<sup>i</sup>* instead for �*A*�*w*. The denominator is assumed to be non-vanishing. This quantity is, in general, in the complex number **C**. Historically, the terminology "weak value" comes from the *weak measurement*, where the coupling between the target system and the probe is weak, explained in the following section. Apart from their original concept of the weak value and the weak measurement, we emphasize that the concept of the weak value is independent of the weak measurement 2. To take the weak value as *a priori* given quantity in quantum mechanics, we will construct the observable-independent probability space. In the conventional quantum measurement theory, the probability space, more precisely speaking, the probability measure,

�*<sup>f</sup>* <sup>|</sup>*U*(*tf* , *ti*)|*i*� <sup>∈</sup> **<sup>C</sup>**, (18)

where *sm*'s are positive and {|*s*ˆ*m*�} is a complete orthonormal set with <sup>|</sup>*sm*� :<sup>=</sup> <sup>√</sup>*sm*|*s*ˆ*m*�. We define the *Kraus operator Em* (95) as

$$E\_{\mathfrak{m}}|\psi\rangle\_{\mathfrak{s}} := \, \_{\mathfrak{e}}\langle \tilde{\psi}|s\_{\mathfrak{m}}\rangle. \tag{11}$$

Then, the quantum state change becomes the operator-sum representation,

$$\sum\_{m} E\_{m} |\psi\rangle\_{\mathfrak{s}} \langle \psi | E\_{m}^{\dagger} = \sum\_{m} \, \_{\mathfrak{e}} \langle \tilde{\psi} | s\_{m} \rangle \langle s\_{m} | \tilde{\psi} \rangle\_{\mathfrak{e}} = \, \_{\mathfrak{e}} \langle \tilde{\psi} | \mathcal{K}(|\mathfrak{a}\rangle) | \tilde{\psi} \rangle\_{\mathfrak{e}} = \mathcal{E}(|\psi\rangle\_{\mathfrak{s}} \langle \psi|).$$

It is emphasized that the quantum state change is described solely in terms of the quantities of the target system.

#### **2.3 Indirect quantum measurement**

In the following, the operator-sum representation of the quantum state change is related to the indirect measurement model. Consider the observable *As* and *Bp* for the target and probe systems given by

$$A\_s = \sum\_j a\_j |a\_j\rangle\_s \langle a\_j|\_\prime \quad B\_p = \sum\_j b\_j |b\_j\rangle\_p \langle b\_j|\_\prime \tag{12}$$

respectively. We assume that the interaction Hamiltonian is given by

$$H\_{\rm int}(t) = \lg(A\_{\rm s} \otimes B\_p) \,\delta(t - t\_0), \tag{13}$$

where *t*<sup>0</sup> is measurement time. Here, without loss of generality, the interaction is impulsive and the coupling constant *g* is scalar. The quantum dynamics for the compound system is given by

$$|s\_m\rangle\langle s\_m| = \mathcal{U}(|\psi\rangle\_s\langle\psi| \otimes |\phi\rangle\_p\langle\phi|) \mathcal{U}^\dagger,\tag{14}$$

where |*ψ*�*<sup>s</sup>* and |*φ*�*<sup>p</sup>* are the initial quantum state on the target and probe systems, respectively. For the probe system, we perform the projective measurement for the observable *Bp*. The probability to obtain the measurement outcome *bm* is given by

$$\begin{split} \Pr[B\_p = b\_m] &= \text{Tr}\_s \langle b\_m | \mathcal{U}(|\psi\rangle\_s \langle \psi| \otimes |\phi\rangle\_p \langle \phi|) \mathcal{U}^\dagger | b\_m \rangle\_s \\ &= \text{Tr}\_s E\_m |\psi\rangle\_s \langle \psi| E\_m^\dagger = \text{Tr}\_s |\psi\rangle\_s \langle \psi| M\_{m\_s} \end{split} \tag{15}$$

where the Kraus operator *Em* is defined as

$$E\_m := \,\_p\langle \mathfrak{b}\_m | \mathcal{U} | \mathfrak{\phi} \rangle\_{p\prime} \tag{16}$$

and *Mm* := *E*† *mEm* is called a *positive operator valued measure* (POVM) (45). The POVM has the same role of the spectrum of the operator *As* in the case of the projective measurement. To derive the projective measurement from the indirect measurement, we set the spectrum of the operator *As* as the POVM, that is, *Mm* = |*am*�*s*�*am*|. Since the sum of the probability distribution over the measurement outcome equals to one, we obtain

$$\sum\_{m} \Pr[B\_p = b\_m] = 1 \iff \sum\_m \text{Tr}\left|\psi\right\rangle\_s \langle \psi | M\_m = \text{Tr}\left|\psi\right\rangle\_s \langle \psi | \sum\_m M\_m = 1$$

$$\to \sum\_m M\_m = \mathbf{1}.\tag{17}$$

Here, the last line uses the property of the density operator, Tr|*ψ*�*s*�*ψ*| = 1 for any |*ψ*�.

#### **3. Review of weak value**

4 Will-be-set-by-IN-TECH

where *sm*'s are positive and {|*s*ˆ*m*�} is a complete orthonormal set with <sup>|</sup>*sm*� :<sup>=</sup> <sup>√</sup>*sm*|*s*ˆ*m*�. We

It is emphasized that the quantum state change is described solely in terms of the quantities

In the following, the operator-sum representation of the quantum state change is related to the indirect measurement model. Consider the observable *As* and *Bp* for the target and probe

*aj*|*aj*�*s*�*aj*|, *Bp* = ∑

where *t*<sup>0</sup> is measurement time. Here, without loss of generality, the interaction is impulsive and the coupling constant *g* is scalar. The quantum dynamics for the compound system is

where |*ψ*�*<sup>s</sup>* and |*φ*�*<sup>p</sup>* are the initial quantum state on the target and probe systems, respectively. For the probe system, we perform the projective measurement for the observable *Bp*. The

Pr[*Bp* <sup>=</sup> *bm*] = Tr*s*�*bm*|*U*(|*ψ*�*s*�*ψ*|⊗|*φ*�*p*�*φ*|)*U*†|*bm*�,

the same role of the spectrum of the operator *As* in the case of the projective measurement. To derive the projective measurement from the indirect measurement, we set the spectrum of the operator *As* as the POVM, that is, *Mm* = |*am*�*s*�*am*|. Since the sum of the probability

*mEm* is called a *positive operator valued measure* (POVM) (45). The POVM has

Tr|*ψ*�*s*�*ψ*|*Mm* = Tr|*ψ*�*s*�*ψ*|∑

<sup>=</sup> Tr*<sup>s</sup> Em*|*ψ*�*s*�*ψ*|*E*†

Then, the quantum state change becomes the operator-sum representation,

*<sup>m</sup>* = ∑ *m*

*As* = ∑ *j*

probability to obtain the measurement outcome *bm* is given by

distribution over the measurement outcome equals to one, we obtain

→ ∑ *m*

*m*

Here, the last line uses the property of the density operator, Tr|*ψ*�*s*�*ψ*| = 1 for any |*ψ*�.

Pr[*Bp* = *bm*] = <sup>1</sup> ⇐⇒ ∑

where the Kraus operator *Em* is defined as

respectively. We assume that the interaction Hamiltonian is given by

*Em*|*ψ*�*<sup>s</sup>* :<sup>=</sup> *<sup>e</sup>*�*ψ*˜|*sm*�. (11)

*bj*|*bj*�*p*�*bj*|, (12)

*<sup>m</sup>* = Tr*<sup>s</sup>* |*ψ*�*s*�*ψ*|*Mm*, (15)

*m*

*Mm* = **1**. (17)

*Mm* = 1

*Em* := *<sup>p</sup>*�*bm*|*U*|*φ*�*p*, (16)

*<sup>e</sup>*�*ψ*˜|*sm*��*sm*|*ψ*˜�*<sup>e</sup>* <sup>=</sup> *<sup>e</sup>*�*ψ*˜|K(|*α*�)|*ψ*˜�*<sup>e</sup>* <sup>=</sup> <sup>E</sup>(|*ψ*�*s*�*ψ*|).

*j*

*Hint*(*t*) = *g*(*As* ⊗ *Bp*) *δ*(*t* − *t*0), (13)

<sup>|</sup>*sm*��*sm*<sup>|</sup> <sup>=</sup> *<sup>U</sup>*(|*ψ*�*s*�*ψ*|⊗|*φ*�*p*�*φ*|)*U*†, (14)

define the *Kraus operator Em* (95) as

*Em*|*ψ*�*s*�*ψ*|*E*†

**2.3 Indirect quantum measurement**

∑ *m*

of the target system.

systems given by

given by

and *Mm* := *E*†

∑ *m*

In Secs. 2.1 and 2.3, the direct and indirect quantum measurement schemes, we only get the probability distribution. However, the probability distribution is not the only thing that is experimentally accessible in quantum mechanics. In quantum mechanics, the phase is also an essential ingredient and in particular the geometric phase is a notable example of an experimentally accessible quantity (150). The general experimentally accessible quantity which contains complete information of the probability and the phase seems to be the *weak value* advocated by Aharonov and his collaborators (4; 14). They proposed a model of weakly coupled system and probe, see Sec. 4.3, to obtain information to a physical quantity as a "weak value" only slightly disturbing the state. Here, we briefly review the formal aspects of the weak value.

For an observable *A*, the *weak value* �*A*�*<sup>w</sup>* is defined as

$$\langle A \rangle\_{\mathcal{W}} := \frac{\langle f | \mathcal{U}(t\_f, t) A \mathcal{U}(t, t\_i) | i \rangle}{\langle f | \mathcal{U}(t\_f, t\_i) | i \rangle} \in \mathbb{C},\tag{18}$$

where |*i*� and �*f* | are normalized pre-selected ket and post-selected bra state vectors, respectively (4). Here, *U*(*t*2, *t*1) is an evolution operator from the time *t*<sup>1</sup> to *t*2. The weak value �*A*�*<sup>w</sup>* actually depends on the pre- and post-selected states |*i*� and �*f* | but we omit them for notational simplicity in the case that we fix them. Otherwise, we write them explicitly as *f*�*A*�*<sup>w</sup> <sup>i</sup>* instead for �*A*�*w*. The denominator is assumed to be non-vanishing. This quantity is, in general, in the complex number **C**. Historically, the terminology "weak value" comes from the *weak measurement*, where the coupling between the target system and the probe is weak, explained in the following section. Apart from their original concept of the weak value and the weak measurement, we emphasize that the concept of the weak value is independent of the weak measurement 2. To take the weak value as *a priori* given quantity in quantum mechanics, we will construct the observable-independent probability space. In the conventional quantum measurement theory, the probability space, more precisely speaking, the probability measure, depends on the observable (151, Sec. 4.1) 3.

Let us calculate the expectation value in quantum mechanics for the quantum state |*ψ*� as

$$\operatorname{Ex}[A] = \langle \psi | A | \psi \rangle = \int d\phi \,\langle \psi | \phi \rangle \langle \phi | A | \psi \rangle = \int d\phi \, \langle \psi | \phi \rangle \cdot \langle \phi | \psi \rangle \frac{\langle \phi | A | \psi \rangle}{\langle \phi | \psi \rangle},$$

$$= \int d\phi \, \left| \langle \psi | \phi \rangle \right|^2 \,\_\phi \langle A \rangle\_{\Psi'}^w \tag{19}$$

where *hA*[|*φ*�] = *<sup>φ</sup>*�*A*�*<sup>w</sup> <sup>ψ</sup>* is complex random variable and *dP* :<sup>=</sup> |�*φ*|*ψ*�|2*d<sup>φ</sup>* is the probability measure and is independent of the observable *A*. Therefore, the event space Ω = {|*φ*�} is taken as the set of the post-selected state. This formula means that the extended probability theory corresponds to the Born rule. From the conventional definition of the variance in

<sup>2</sup> This concept is shared in Refs. (1; 49; 51; 78; 81; 82; 117; 130).

<sup>3</sup> Due to this, the probability in quantum mechanics cannot be applied to the standard probability theory. As another approach to resolve this, there is the quantum probability theory (138).

OPERA experiment (125). The framework of weak values has been theoretically applied to foundations of quantum physics, e.g., the derivation of the Born rule from the alternative assumption for *a priori* measured value (74), the relationship to the uncertainty relationship (72), the quantum stochastic process (190), the tunneling traverse time (135; 170; 171), arrival time and time operator (21; 39; 146; 147), the decay law (46; 187), the non-locality (32; 180; 181), especially, quantum non-locality, which is characterized by the modular variable, consistent history (87; 188), Bohmian quantum mechanics (98), semi-classical weak values on the tunneling (175), the quantum trajectory (192), and classical stochastic theory (177). Also, in quantum information science, the weak value was analyzed on quantum computation (35; 126), quantum communications (29; 36), quantum estimation, e.g., state tomography (67–69; 111; 158) and the parameter estimation (70; 73; 157), the entanglement concentration (113), the quasi-probability distribution (24; 61; 148; 183) and the cloning of the unknown quantum state with hint (161). Furthermore, this was applied to the cosmological situations in quantum-mechanical region, e.g., the causality (22), the inflation theory (42), backaction of the Hawking radiation from the black hole (34; 54; 55), and the new interpretation of the universe (9; 53; 62). However, the most important fact is that the weak value is experimentally accessible so that the intuitive argument based on the weak values can be either verified or falsified by experiments. There are many experimental proposals to obtain the weak value in the optical (2; 44; 88; 101; 112; 159; 197) and the solid-state (83; 84; 94; 115; 143; 144; 191; 200) systems. Recently, the unified viewpoint was

Theory of "Weak Value" and Quantum Mechanical Measurements 81

On the realized experiments on the weak value, we can classify the three concepts: (i) testing the quantum theory, (ii) the amplification of the tiny effect in quantum mechanics, and (iii) the

(i) Testing the quantum theory. The weak value can solve many quantum paradoxes seen in the book (14). The Hardy paradox (64), which there occurs in two Mach-Zehnder interferometers of the electron and the position, was resolved by the weak value (8) and was analyzed deeper (75). This paradoxical situation was experimentally demonstrated in the optical setup (107; 198). By the interference by the polarization (131) and shifting the optical axis (141), the spin beyond the eigenvalue is verified. By the latter technique, the three-box paradox (16; 188) was realized (139). Thereafter, the theoretical progresses are the contextuality on quantum mechanics (178), the generalized N-box paradox (99), and the relationship to the Kirkpatrick game (137). The weak value is used to show the violation of the Leggett-Garg inequality (110; 191). This experimental realizations were demonstrated in the system of the superconducting qubit (97), the optical systems (50; 134). Furthermore, since the weak value for the position observable |*x*��*x*| with the pre-selected state |*ψ*� and

�|*x*��*x*|�*<sup>w</sup>* <sup>=</sup> �*p*|*x*��*x*|*ψ*�

the Bohmian trajectory (91; 149) on the base of the theoretical analysis (193).

we obtain the wavefunction *ψ*(*x*) := �*x*|*ψ*� as the weak value with the multiplication factor 1/*φ*(0) with *φ*(*p*) := �*p*|*ψ*� in the case of *p* = 0. Using the photon transverse wavefunction, there are experimentally demonstrated by replacing the weak measurement for the position as the polarization measurement (109). This paper was theoretically criticized to compare the standard quantum state tomography for the phase space in Ref. (63) and was generalized to a conventionally unobservable (108). As other examples, there are the detection of the superluminal signal (37), the quantum non-locality (165), and

�*p*|*ψ*� <sup>=</sup> *<sup>e</sup>ixpψ*(*x*)

*<sup>φ</sup>*(*p*) , (23)

found in the weak measurement (92).

the post-selection |*p*� is given by

quantum phase.

quantum mechanics, we obtain the variance as

$$\begin{split} \text{Var}[A] &= \int |h\_A[|\phi\rangle]|^2 dP - \left(\int h\_A[|\phi\rangle] dP\right)^2 \\ &= \int \left|\frac{\langle\phi|A|\psi\rangle}{\langle\phi|\psi\rangle}\right|^2 |\langle\phi|\psi\rangle|^2 d\phi - \left(\int \frac{\langle\phi|A|\psi\rangle}{\langle\phi|\psi\rangle} |\langle\phi|\psi\rangle|^2 d\phi\right)^2 \\ &= \int |\langle\phi|A|\psi\rangle|^2 d\phi - \left(\int \langle\psi|\phi\rangle \langle\phi|A|\psi\rangle d\phi\right)^2 \\ &= \int \langle\psi|A|\phi\rangle \langle\phi|A|\psi\rangle d\phi - \left(\langle\psi|A|\psi\rangle\right)^2 \\ &= \langle\psi|A^2|\psi\rangle - \left(\langle\psi|A|\psi\rangle\right)^2. \end{split} \tag{20}$$

This means that the observable-independent probability space can be characterized by the weak value (155). From another viewpoint of the weak value, the statistical average of the weak value coincides with the expectation value in quantum mechanics (7). This can be interpreted as the probability while this allows the "negative probability" 4. On this idea, the uncertainty relationship was analyzed on the Robertson inequality (58; 163) and on the Ozawa inequality (106), which the uncertainty relationships are reviewed in Ref. (151, Appendix A). Also, the joint probability for the compound system was analyzed in Refs. (27; 30). Furthermore, if the operator *A* is a projection operator *A* = |*a*��*a*|, the above identity becomes an analog of the Bayesian formula,

$$\left| \langle a|\psi\rangle \right|^2 = \int \,\_{\phi} \langle |a\rangle \langle a| \rangle\_{\Psi}^{w} |\langle \phi|\psi\rangle|^2 d\phi. \tag{21}$$

The left hand side is the probability to obtain the state |*a*� given the initial state |*ψ*�. From this, one may get some intuition by interpreting the weak value *<sup>φ</sup>*�|*a*��*a*|�*<sup>w</sup> <sup>ψ</sup>* as the complex conditional probability of obtaining the result |*a*� under an initial condition |*i*� and a final condition <sup>|</sup> *<sup>f</sup>*� in the process <sup>|</sup>*i*�→|*a*�→| *<sup>f</sup>*� (170; 171) 5. Of course, we should not take the strange weak values too literally but the remarkable consistency of the framework of the weak values due to Eq. (21) and a consequence of the completeness relation,

$$\sum\_{a} \langle |a\rangle \langle a| \rangle\_{\text{tr}} = 1,\tag{22}$$

may give a useful concept to further push theoretical consideration by intuition. This interpretation of the weak values gives many possible examples of strange phenomena like a negative kinetic energy (11), a spin 100¯*h* for an electron (4; 23; 52; 60) and a superluminal propagation of light (142; 162) and neutrino (28; 176) motivated by the

<sup>4</sup> The concept of negative probability is not new, e.g., see Refs. (47; 57; 65; 66; 71). The weak value defined by Eq. (18) is normally called the transition amplitude from the state |*ψ*� to �*φ*| via the intermediate state |*a*� for *A* = |*a*��*a*|, the absolute value squared of which is the probability for the process. But the three references quoted above seem to suggest that they might be interpreted as probabilities in the case that the process is counter-factual, i.e., the case that the intermediate state |*a*� is not projectively measured. The description of intermediate state |*a*� in the present work is counter-factual or virtual in the sense that the intermediate state would not be observed by projective measurements. Feynman's example is the counter-factual "probability" for an electron to have its spin up in the *x*-direction and also spin down in the *z*-direction (57).

<sup>5</sup> The interpretation of the weak value as a complex probability is suggested in the literature (118).

6 Will-be-set-by-IN-TECH

*hA*[|*φ*�]*dP*

<sup>2</sup>

�*φ*|*A*|*ψ*�

�*ψ*|*φ*��*φ*|*A*|*ψ*�*dφ*

<sup>=</sup> �*ψ*|*A*2|*ψ*� − (�*ψ*|*A*|*ψ*�)2. (20)

�*φ*|*ψ*� |�*φ*|*ψ*�|2*d<sup>φ</sup>*

<sup>2</sup>

<sup>2</sup>

*<sup>ψ</sup>* |�*φ*|*ψ*�|2*dφ*. (21)

�|*a*��*a*|�*<sup>w</sup>* = 1, (22)

*<sup>ψ</sup>* as the complex


This means that the observable-independent probability space can be characterized by the weak value (155). From another viewpoint of the weak value, the statistical average of the weak value coincides with the expectation value in quantum mechanics (7). This can be interpreted as the probability while this allows the "negative probability" 4. On this idea, the uncertainty relationship was analyzed on the Robertson inequality (58; 163) and on the Ozawa inequality (106), which the uncertainty relationships are reviewed in Ref. (151, Appendix A). Also, the joint probability for the compound system was analyzed in Refs. (27; 30). Furthermore, if the operator *A* is a projection operator *A* = |*a*��*a*|, the above identity becomes

*φ*�|*a*��*a*|�*<sup>w</sup>*

The left hand side is the probability to obtain the state |*a*� given the initial state |*ψ*�. From

conditional probability of obtaining the result |*a*� under an initial condition |*i*� and a final condition <sup>|</sup> *<sup>f</sup>*� in the process <sup>|</sup>*i*�→|*a*�→| *<sup>f</sup>*� (170; 171) 5. Of course, we should not take the strange weak values too literally but the remarkable consistency of the framework of the weak

This interpretation of the weak values gives many possible examples of strange phenomena like a negative kinetic energy (11), a spin 100¯*h* for an electron (4; 23; 52; 60) and a superluminal propagation of light (142; 162) and neutrino (28; 176) motivated by the

<sup>4</sup> The concept of negative probability is not new, e.g., see Refs. (47; 57; 65; 66; 71). The weak value defined by Eq. (18) is normally called the transition amplitude from the state |*ψ*� to �*φ*| via the intermediate state |*a*� for *A* = |*a*��*a*|, the absolute value squared of which is the probability for the process. But the three references quoted above seem to suggest that they might be interpreted as probabilities in the case that the process is counter-factual, i.e., the case that the intermediate state |*a*� is not projectively measured. The description of intermediate state |*a*� in the present work is counter-factual or virtual in the sense that the intermediate state would not be observed by projective measurements. Feynman's example is the counter-factual "probability" for an electron to have its spin up in the *x*-direction and

<sup>5</sup> The interpretation of the weak value as a complex probability is suggested in the literature (118).

�*ψ*|*A*|*φ*��*φ*|*A*|*ψ*�*d<sup>φ</sup>* <sup>−</sup> (�*ψ*|*A*|*ψ*�)<sup>2</sup>

quantum mechanics, we obtain the variance as

= 

= 

= 

an analog of the Bayesian formula,

also spin down in the *z*-direction (57).


�*φ*|*A*|*ψ*� �*φ*|*ψ*�



values due to Eq. (21) and a consequence of the completeness relation,

this, one may get some intuition by interpreting the weak value *<sup>φ</sup>*�|*a*��*a*|�*<sup>w</sup>*

∑*a*

may give a useful concept to further push theoretical consideration by intuition.

<sup>2</sup>*dP* <sup>−</sup>

<sup>2</sup> *<sup>d</sup><sup>φ</sup>* <sup>−</sup>

 2

Var[*A*] =

OPERA experiment (125). The framework of weak values has been theoretically applied to foundations of quantum physics, e.g., the derivation of the Born rule from the alternative assumption for *a priori* measured value (74), the relationship to the uncertainty relationship (72), the quantum stochastic process (190), the tunneling traverse time (135; 170; 171), arrival time and time operator (21; 39; 146; 147), the decay law (46; 187), the non-locality (32; 180; 181), especially, quantum non-locality, which is characterized by the modular variable, consistent history (87; 188), Bohmian quantum mechanics (98), semi-classical weak values on the tunneling (175), the quantum trajectory (192), and classical stochastic theory (177). Also, in quantum information science, the weak value was analyzed on quantum computation (35; 126), quantum communications (29; 36), quantum estimation, e.g., state tomography (67–69; 111; 158) and the parameter estimation (70; 73; 157), the entanglement concentration (113), the quasi-probability distribution (24; 61; 148; 183) and the cloning of the unknown quantum state with hint (161). Furthermore, this was applied to the cosmological situations in quantum-mechanical region, e.g., the causality (22), the inflation theory (42), backaction of the Hawking radiation from the black hole (34; 54; 55), and the new interpretation of the universe (9; 53; 62). However, the most important fact is that the weak value is experimentally accessible so that the intuitive argument based on the weak values can be either verified or falsified by experiments. There are many experimental proposals to obtain the weak value in the optical (2; 44; 88; 101; 112; 159; 197) and the solid-state (83; 84; 94; 115; 143; 144; 191; 200) systems. Recently, the unified viewpoint was found in the weak measurement (92).

On the realized experiments on the weak value, we can classify the three concepts: (i) testing the quantum theory, (ii) the amplification of the tiny effect in quantum mechanics, and (iii) the quantum phase.

(i) Testing the quantum theory. The weak value can solve many quantum paradoxes seen in the book (14). The Hardy paradox (64), which there occurs in two Mach-Zehnder interferometers of the electron and the position, was resolved by the weak value (8) and was analyzed deeper (75). This paradoxical situation was experimentally demonstrated in the optical setup (107; 198). By the interference by the polarization (131) and shifting the optical axis (141), the spin beyond the eigenvalue is verified. By the latter technique, the three-box paradox (16; 188) was realized (139). Thereafter, the theoretical progresses are the contextuality on quantum mechanics (178), the generalized N-box paradox (99), and the relationship to the Kirkpatrick game (137). The weak value is used to show the violation of the Leggett-Garg inequality (110; 191). This experimental realizations were demonstrated in the system of the superconducting qubit (97), the optical systems (50; 134). Furthermore, since the weak value for the position observable |*x*��*x*| with the pre-selected state |*ψ*� and the post-selection |*p*� is given by

$$
\langle\langle|\mathbf{x}\rangle\langle\mathbf{x}|\rangle\_{\rm w} = \frac{\langle p|\mathbf{x}\rangle\langle\mathbf{x}|\boldsymbol{\psi}\rangle}{\langle p|\boldsymbol{\psi}\rangle} = \frac{e^{i\mathbf{x}\cdot\boldsymbol{p}}\psi(\mathbf{x})}{\phi(p)},\tag{23}
$$

we obtain the wavefunction *ψ*(*x*) := �*x*|*ψ*� as the weak value with the multiplication factor 1/*φ*(0) with *φ*(*p*) := �*p*|*ψ*� in the case of *p* = 0. Using the photon transverse wavefunction, there are experimentally demonstrated by replacing the weak measurement for the position as the polarization measurement (109). This paper was theoretically criticized to compare the standard quantum state tomography for the phase space in Ref. (63) and was generalized to a conventionally unobservable (108). As other examples, there are the detection of the superluminal signal (37), the quantum non-locality (165), and the Bohmian trajectory (91; 149) on the base of the theoretical analysis (193).

**4.1 Time symmetric quantum measurement**

becomes

formalism".

**4.2 Protective measurement**

second law of thermodynamics.

motivation to construct the time symmetric quantum measurement.

the probability to obtain the measurement outcome *aj* can be rewritten as

measurement at time *t*0. Therefore, the quantum state at time *ti* is given by

probability to obtain the measurement outcome *aj* is given by

While the fundamental equations of the microscopic physics are time symmetric, for example, the Newton equation, the Maxwell equation, and the Schrödinger equation 8, the quantum measurement is not time symmetric. This is because the quantum state after quantum measurement depends on the measurement outcome seen in Sec. 2. The fundamental equations of the microscopic physics can be solved to give the initial boundary condition. To construct the time symmetric quantum measurement, the two boundary conditions, which is called pre- and post-selected states, are needed. The concept of the pre- and post-selected states is called the two-state vector formalism (6). In the following, we review the original

Theory of "Weak Value" and Quantum Mechanical Measurements 83

Let us consider the projective measurement for the observable *A* = ∑*<sup>i</sup> ai*|*ai*��*ai*| with the initial boundary condition denoted as |*i*� at time *ti*. To take quantum measurement at time *t*0, the

with the time evolution *U* := *U*(*t*0, *ti*). After the projective measurement, the quantum state becomes |*aj*�. Thereafter, the quantum state at *tf* is given by |*ϕj*� := *V*|*aj*� with *V* = *U*(*tf* , *t*0).

Pr[*<sup>A</sup>* <sup>=</sup> *aj*] = � �*ϕj*|*V*|*aj*� �2� �*aj*|*U*|*i*� �<sup>2</sup>

It is noted that � �*ϕj*|*V*|*aj*� �2<sup>=</sup> 1. Here, we consider the backward time evolution from the quantum state |*ϕj*� at time *tf* . We always obtain the quantum state |*aj*� after the projective

In general, <sup>|</sup>˜*i*� is different from <sup>|</sup>*i*�. Therefore, projective measurement is time asymmetric. To construct the time-symmetric quantum measurement, we add the boundary condition at time *tf* . Substituting the quantum state |*ϕj*� to the *specific* one denoted as | *f*�, which is called the post-selected state, the probability to obtain the measurement outcome *aj*, Eq. (26),

Pr[*<sup>A</sup>* <sup>=</sup> *aj*] = � �*<sup>f</sup>* <sup>|</sup>*V*|*aj*� �2� �*aj*|*U*|*i*� �<sup>2</sup>

This is called the Aharonov-Bergmann-Lebowitz (ABL) formula (6). From the analogous discussion to the above, this measurement is time symmetric. Therefore, describing quantum mechanics by the pre- and post-selected states, |*i*� and �*f* |, is called the "two-state vector

In this subsection, we will see the noninvasive quantum measurement for the specific quantum state on the target system. Consider a system of consisting of a target and a probe defined in the Hilbert space H*<sup>s</sup>* ⊗ H*p*. The interaction between the target and the probe is

<sup>8</sup> It is, of course, noted that thermodynamics does not have the time symmetric properties from the

Pr[*<sup>A</sup>* <sup>=</sup> *aj*] =� �*aj*|*U*|*i*� �2, (25)

<sup>∑</sup>*<sup>j</sup>* � �*ϕj*|*V*|*aj*� �2� �*aj*|*U*|*i*� �<sup>2</sup> . (26)

<sup>∑</sup>*<sup>j</sup>* � �*<sup>f</sup>* <sup>|</sup>*V*|*aj*� �2� �*aj*|*U*|*i*� �<sup>2</sup> . (28)

<sup>|</sup>˜*i*� :<sup>=</sup> *<sup>U</sup>*†|*aj*��*aj*|*V*†|*ϕj*� <sup>=</sup> *<sup>U</sup>*†|*aj*�. (27)


$$\begin{split} \gamma &:= \arg \langle \psi\_1 | \psi\_2 \rangle \langle \psi\_2 | \psi\_3 \rangle \langle \psi\_3 | \psi\_1 \rangle \\ &= \arg \frac{\langle \psi\_1 | \psi\_2 \rangle \langle \psi\_2 | \psi\_3 \rangle \langle \psi\_3 | \psi\_1 \rangle}{|\langle \psi\_3 | \psi\_1 \rangle|^2} = \arg \frac{\langle \psi\_1 | \psi\_2 \rangle \langle \psi\_2 | \psi\_3 \rangle}{\langle \psi\_1 | \psi\_3 \rangle} \\ &= \arg \, \_\psi \langle | \psi\_2 \rangle \langle \psi\_2 | \rangle\_{\psi\_3}^w. \end{split} \tag{24}$$

where the quantum states, |*ψ*1�, |*ψ*2�, and |*ψ*3�, are the pure states (160). Here, the quantum states, |*ψ*1� and |*ψ*3�, are the post- and pre-selected states, respectively. Therefore, we can evaluate the weak value from the phase shift (174). Of course, vice versa (38). Tamate *et al.* proposal was demonstrated on the relationship to quantum eraser (90) and by the a three-pinhole interferometer (89). The phase shift from the zero mode to *π* mode was observed by using the interferometer with a Cs vapor (41) and the phase shift in the which-way path experiment was demonstrated (116). Furthermore, by the photonic crystal, phase singularity was demonstrated (164).

(iv)Miscellaneous. The backaction of the weak measurement is experimentally realized in the optical system (79). Also, the parameter estimation using the weak value is demonstrated (73). (iv) Miscellaneous.

#### **4. Historical background – two-state vector formalism**

In this section, we review the original concept of the two-state vector formalism. This theory is seen in the reviewed papers (15; 20).

<sup>6</sup> Unfortunately, the signal to noise ratio is not drastically changed under the assumption that the probe wavefunction is Gaussian on a one-dimensional parameter space.

<sup>7</sup> Unfortunately, the experimental data are mismatched to the theoretical prediction. While the authors claimed that this differences results from the stray of light, the full-order calculation even is not mismatched (93). However, this difference remains the open problem.

#### **4.1 Time symmetric quantum measurement**

8 Will-be-set-by-IN-TECH

(ii) Amplification of the tiny effect in quantum mechanics. Since the weak value has the denominator, the weak value is very large when the pre- and post-selected states are almost orthogonal6. This is practical advantage to use the weak value. While the spin Hall effect of light (124) is too tiny effect to observe its shift in the conventional scheme, by almost orthogonal polarizations for the input and output, this effect was experimentally verified (76) to be theoretically analyzed from the viewpoint of the spin moments (96). Also, some interferometers were applied. The beam deflection on the Sagnac interferometer (48) was shown to be supported by the classical and quantum theoretical analyses (77) 7. Thereafter, optimizing the signal-to-noise ratio (166; 184), the phase amplification (168; 169), and the precise frequency measurement (167) were demonstrated. As another example, there is shaping the laser pulse beyond the diffraction limit (136). According to Steinberg (172), in his group, the amplification on the single-photon nonlinearity has been progressed to be based on the theoretical proposal (56). While the charge sensing amplification was proposed in the solid-state system (200), there is no experimental demonstration on the amplification for the solid-state system. Furthermore, the upper bound of the amplification has not yet solved. Practically, this open problem is so important to understand the relationship to the weak

(iii)Quantum phase. The argument of the weak value for the projection operator is the

where the quantum states, |*ψ*1�, |*ψ*2�, and |*ψ*3�, are the pure states (160). Here, the quantum states, |*ψ*1� and |*ψ*3�, are the post- and pre-selected states, respectively. Therefore, we can evaluate the weak value from the phase shift (174). Of course, vice versa (38). Tamate *et al.* proposal was demonstrated on the relationship to quantum eraser (90) and by the a three-pinhole interferometer (89). The phase shift from the zero mode to *π* mode was observed by using the interferometer with a Cs vapor (41) and the phase shift in the which-way path experiment was demonstrated (116). Furthermore, by the photonic

(iv)Miscellaneous. The backaction of the weak measurement is experimentally realized in the optical system (79). Also, the parameter estimation using the weak value is

In this section, we review the original concept of the two-state vector formalism. This theory

<sup>6</sup> Unfortunately, the signal to noise ratio is not drastically changed under the assumption that the probe

<sup>7</sup> Unfortunately, the experimental data are mismatched to the theoretical prediction. While the authors claimed that this differences results from the stray of light, the full-order calculation even is not


�*ψ*1|*ψ*3�

*<sup>ψ</sup>*<sup>3</sup> . (24)

*γ* := arg�*ψ*1|*ψ*2��*ψ*2|*ψ*3��*ψ*3|*ψ*1� <sup>=</sup> arg �*ψ*1|*ψ*2��*ψ*2|*ψ*3��*ψ*3|*ψ*1�

<sup>=</sup> arg *<sup>ψ</sup>*<sup>1</sup> �|*ψ*2��*ψ*2|�*<sup>w</sup>*

crystal, phase singularity was demonstrated (164).

**4. Historical background – two-state vector formalism**

wavefunction is Gaussian on a one-dimensional parameter space.

mismatched (93). However, this difference remains the open problem.

measurement regime.

geometric phase as

(iii) Quantum phase.

demonstrated (73).

(iv) Miscellaneous.

is seen in the reviewed papers (15; 20).

While the fundamental equations of the microscopic physics are time symmetric, for example, the Newton equation, the Maxwell equation, and the Schrödinger equation 8, the quantum measurement is not time symmetric. This is because the quantum state after quantum measurement depends on the measurement outcome seen in Sec. 2. The fundamental equations of the microscopic physics can be solved to give the initial boundary condition. To construct the time symmetric quantum measurement, the two boundary conditions, which is called pre- and post-selected states, are needed. The concept of the pre- and post-selected states is called the two-state vector formalism (6). In the following, we review the original motivation to construct the time symmetric quantum measurement.

Let us consider the projective measurement for the observable *A* = ∑*<sup>i</sup> ai*|*ai*��*ai*| with the initial boundary condition denoted as |*i*� at time *ti*. To take quantum measurement at time *t*0, the probability to obtain the measurement outcome *aj* is given by

Pr[*<sup>A</sup>* <sup>=</sup> *aj*] =� �*aj*|*U*|*i*� �2, (25)

with the time evolution *U* := *U*(*t*0, *ti*). After the projective measurement, the quantum state becomes |*aj*�. Thereafter, the quantum state at *tf* is given by |*ϕj*� := *V*|*aj*� with *V* = *U*(*tf* , *t*0). the probability to obtain the measurement outcome *aj* can be rewritten as

$$\Pr[A = a\_j] = \frac{||\,\langle \varphi\_j | V | a\_j \rangle||^2 ||\,\langle a\_j | U | i \rangle||^2}{\sum\_j ||\,\langle \varphi\_j | V | a\_j \rangle||^2 ||\,\langle a\_j | U | i \rangle||^2}. \tag{26}$$

It is noted that � �*ϕj*|*V*|*aj*� �2<sup>=</sup> 1. Here, we consider the backward time evolution from the quantum state |*ϕj*� at time *tf* . We always obtain the quantum state |*aj*� after the projective measurement at time *t*0. Therefore, the quantum state at time *ti* is given by

$$\langle \tilde{l} \rangle := \mathcal{U}^{\dagger} |a\_{\hat{l}}\rangle \langle a\_{\hat{l}}|V^{\dagger}|\varphi\_{\hat{l}}\rangle = \mathcal{U}^{\dagger} |a\_{\hat{l}}\rangle. \tag{27}$$

In general, <sup>|</sup>˜*i*� is different from <sup>|</sup>*i*�. Therefore, projective measurement is time asymmetric. To construct the time-symmetric quantum measurement, we add the boundary condition at time *tf* . Substituting the quantum state |*ϕj*� to the *specific* one denoted as | *f*�, which is called the post-selected state, the probability to obtain the measurement outcome *aj*, Eq. (26), becomes

$$\Pr[A = a\_j] = \frac{\|\mid \langle f|V|a\_j\rangle \parallel^2 \|\mid \langle a\_j|U|i\rangle \parallel^2}{\sum\_j \|\, \langle f|V|a\_j\rangle \parallel^2 \|\mid \langle a\_j|U|i\rangle \parallel^2}. \tag{28}$$

This is called the Aharonov-Bergmann-Lebowitz (ABL) formula (6). From the analogous discussion to the above, this measurement is time symmetric. Therefore, describing quantum mechanics by the pre- and post-selected states, |*i*� and �*f* |, is called the "two-state vector formalism".

#### **4.2 Protective measurement**

In this subsection, we will see the noninvasive quantum measurement for the specific quantum state on the target system. Consider a system of consisting of a target and a probe defined in the Hilbert space H*<sup>s</sup>* ⊗ H*p*. The interaction between the target and the probe is

<sup>8</sup> It is, of course, noted that thermodynamics does not have the time symmetric properties from the second law of thermodynamics.

In the limit of *N* → ∞, by quadrature by parts, we obtain

× <sup>1</sup> <sup>−</sup> *<sup>T</sup> N*

post-selected states and in Ref. (10) by the meta-stable state.

<sup>=</sup> <sup>|</sup>*Ei*(*T*)� exp

−*i T* 0

−*i T* 0

 *T* 0

<sup>Δ</sup>[*Q*] = *<sup>T</sup>*

0

*<sup>g</sup>*(*t*) Ex[*A*(*t*)]*dt*

Theory of "Weak Value" and Quantum Mechanical Measurements 85

*<sup>g</sup>*(*t*) Ex[*A*(*t*)]*dt*

Therefore, the shift of the expectation value for the position operator on the probe system is

It is emphasized that the quantum state on the target system remains to be the energy eigenstate of *Hs*. Therefore, this is called the *protective measurement* (5; 18). It is remarked that the generalized version of the protective measurement in Ref. (19) by the pre- and

From the above discussions, is it possible to combine the above two concepts, i.e., the time-symmetric quantum measurement without destroying the quantum state (189)? This answer is the *weak measurement* (4). Consider a target system and a probe defined in the Hilbert space H*<sup>s</sup>* ⊗ H*p*. The interaction of the target system and the probe is assumed to be weak and

where an observable *<sup>A</sup>* is defined in <sup>H</sup>*s*, while *<sup>P</sup>*<sup>ˆ</sup> is the momentum operator of the probe. The time evolution operator becomes *<sup>e</sup>*−*ig*(*A*⊗*P*ˆ). Suppose the probe initial state is <sup>|</sup>*ξ*�. For the transition from the pre-selected state |*i*� to the post-selected state | *f*�, the probe wave function

*<sup>U</sup>*|*i*�|*ξ*� <sup>=</sup> �*<sup>f</sup>* <sup>|</sup>*V*[**<sup>1</sup>** <sup>−</sup> *ig*(*<sup>A</sup>* <sup>⊗</sup> *<sup>P</sup>*ˆ)]*U*|*i*�|*ξ*� <sup>+</sup> *<sup>O</sup>*(*g*2)

where �*f* |*VAU*|*i*�/�*f* |*VU*|*i*� = �*A*�*w*. Here, the last equation uses the approximation that *<sup>g</sup>*�*A*�*<sup>w</sup>* � <sup>1</sup> 10. We obtain the shifts of the expectation values for the position and momentum

**Theorem 4.1** (Jozsa (85))**.** *We obtain the shifts of the expectation values for the position and*

<sup>10</sup> It is remarked that Wu and Li showed the second-order correction of the weak measurement (196). A

*momentum operators on the probe after the weak measurement with the post-selection as*

<sup>Δ</sup>[*Q*ˆ] = *<sup>g</sup>*Re�*A*�*<sup>w</sup>* <sup>+</sup> *mg*Im�*A*�*<sup>w</sup>*

� <sup>=</sup> �*<sup>f</sup>* <sup>|</sup>*Ve*−*ig*(*A*⊗*P*ˆ)*U*|*i*�|*ξ*�, which is in the weak coupling case,

<sup>=</sup> �*<sup>f</sup>* <sup>|</sup>*VU*|*i*� − *ig*�*<sup>f</sup>* <sup>|</sup>*VAU*|*i*� ⊗ *<sup>P</sup>*ˆ|*ξ*� <sup>+</sup> *<sup>O</sup>*(*g*2) = �*<sup>f</sup>* <sup>|</sup>*VU*|*i*�

operators on the probe as the following theorem:

further analysis was shown in Refs. (129; 132).

*<sup>g</sup>*2(*t*) Var[*A*(*t*)]*dt*

*P*ˆ 

*P*ˆ  *P*ˆ2 

*Hint*(*t*) = *<sup>g</sup>*(*<sup>A</sup>* <sup>⊗</sup> *<sup>P</sup>*ˆ)*δ*(*<sup>t</sup>* <sup>−</sup> *<sup>t</sup>*0), (38)

*d* Var[*Q*ˆ] *dt*

<sup>Δ</sup>[*P*ˆ] = <sup>2</sup>*g*Im�*A*�*<sup>w</sup>* Var[*P*ˆ], (41)

 *t*=*t*<sup>0</sup>

<sup>1</sup> <sup>−</sup> *ig*�*A*�*wP*<sup>ˆ</sup>

<sup>|</sup>*ξ*� <sup>+</sup> *<sup>O</sup>*(*g*2) (39)

, (40)


*g*(*t*) Ex[*A*(*t*)]*dt*. (37)

 1 *N* 


<sup>|</sup>Φ(*T*)�∼|*Ei*(*T*)� exp

given by

**4.3 Weak measurement**

instantaneous,

becomes |*ξ*�

� <sup>=</sup> �*<sup>f</sup>* <sup>|</sup>*Ve*−*ig*(*A*⊗*P*ˆ)


given by

$$H\_{\rm int}(t) = \mathcal{g}(t)(A \otimes \vec{P}),\tag{29}$$

where

$$\int\_{0}^{T} \mathbf{g}(t)dt = \colon \mathbf{g}\_{0}. \tag{30}$$

The total Hamiltonian is given by

$$H\_{\rm tot}(t) = H\_{\rm s}(t) + H\_p(t) + H\_{\rm int}(t). \tag{31}$$

Here, we suppose that *Hs*(*t*) has discrete and non-degenerate eigenvalues denoted as *Ei*(*t*). Its corresponding eigenstate is denoted as |*Ei*(*t*)� for any time *t*. Furthermore, we consider the discretized time from the time interval [0, *T*];

$$t\_n = \frac{n}{N} T\left(n = 0, 1, 2, \dots, N\right),\tag{32}$$

where *N* is a sufficiently large number. We assume that the initial target state is the energy eigenvalue <sup>|</sup>*Ei*(*t*)� <sup>9</sup> the initial probe state is denoted as <sup>|</sup>*ξ*(0)�. Under the adiabatic condition, the compound state for the target and probe systems at time *T* is given by

$$|\Phi(T)\rangle := |E\_i(t\_N)\rangle\langle E\_i(t\_N)|e^{-i\frac{\mathsf{i}}{N}H\_{\mathrm{lol}}(t\_N)}|E\_i(t\_{N-1})\rangle\langle E\_i(t\_{N-1})|e^{-i\frac{\mathsf{i}}{N}H\_{\mathrm{lol}}(t\_{N-1})}\dots$$

$$\times|E\_i(t\_2)\rangle\langle E\_i(t\_2)|e^{-i\frac{\mathsf{i}}{N}H\_{\mathrm{lol}}(t\_2)}|E\_i(t\_1)\rangle\langle E\_i(t\_1)|e^{-i\frac{\mathsf{i}}{N}H\_{\mathrm{lol}}(t\_1)}|E\_i(0)\rangle\otimes|\xi(0)\rangle.\tag{33}$$

Applying the Trotter-Suzuki theorem (173; 182), one has

$$|\Phi(T)\rangle := |E\_i(t\_N)\rangle\langle E\_i(t\_N)|e^{-i\frac{\mathsf{I}}{N}H\_{\mathrm{int}}(t\_N)}|E\_i(t\_N)\rangle\langle E\_i(t\_{N-1})|e^{-i\frac{\mathsf{I}}{N}H\_{\mathrm{int}}(t\_{N-1})}\dots$$

$$\times |E\_i(t\_3)\rangle\langle E\_i(t\_2)|e^{-i\frac{\mathsf{I}}{N}H\_{\mathrm{int}}(t\_2)}|E\_i(t\_2)\rangle\langle E\_i(t\_1)|e^{-i\frac{\mathsf{I}}{N}H\_{\mathrm{int}}(t\_1)}|E\_i(1)\rangle\otimes|\xi(T)\rangle. \tag{34}$$

By the Taylor expansion with the respect to *N*, the expectation value is

$$\begin{split} \langle E\_{i}(t\_{n}) \vert e^{-i\frac{T}{N}g(t\_{n})A\otimes\hat{P}} \vert E\_{i}(t\_{n}) \rangle &= 1 - i\frac{T}{N}g(t\_{n})\operatorname{Ex}[A(t\_{n})]\hat{P} - \frac{1}{2}\frac{T^{2}}{N^{2}}g^{2}(t\_{n})(\operatorname{Ex}[A(t\_{n})])^{2}\hat{P}^{2} \\ & \quad - \frac{1}{2}\frac{T^{2}}{N^{2}}g^{2}(t\_{n})\operatorname{Var}[A(t\_{n})]\hat{P}^{2} + O\left(\frac{1}{N^{3}}\right) \\ & \sim e^{-i\frac{T}{N}g(t\_{n})\operatorname{Ex}[A(t\_{n})]\hat{P}} \left(1 - \frac{1}{2}\frac{T^{2}}{N^{2}}g^{2}(t\_{n})\operatorname{Var}[A(t\_{n})]\hat{P}^{2}\right). \end{split} \tag{35}$$

<sup>9</sup> Due to this assumption, it is impossible to apply this to the arbitrary quantum state. Furthermore, while we seemingly need the projective measurement, that is, destructive measurement, for the target system to confirm whether the initial quantum state is in the eigenstates (145; 186), they did not apply this to the arbitrary state. For example, if the system is cooled down, we can pickup the ground state of the target Hamiltonian *Hs*(0).

In the limit of *N* → ∞, by quadrature by parts, we obtain

$$\begin{split} |\Phi(T)\rangle &\sim |E\_i(T)\rangle \exp\left[-i\left(\int\_0^T \mathbf{g}(t) \operatorname{Ex}[A(t)]dt\right)\hat{P}\right] \\ &\times \left[1 - \frac{T}{N} \left(\int\_0^T \mathbf{g}^2(t) \operatorname{Var}[A(t)]dt\right)\hat{P}^2\right]|\xi(T)\rangle + O\left(\frac{1}{N}\right) \\ &= |E\_i(T)\rangle \exp\left[-i\left(\int\_0^T \mathbf{g}(t) \operatorname{Ex}[A(t)]dt\right)\hat{P}\right]|\xi(T)\rangle. \end{split} \tag{36}$$

Therefore, the shift of the expectation value for the position operator on the probe system is given by

$$
\Delta[Q] = \int\_0^T g(t) \operatorname{Ex}[A(t)] dt. \tag{37}
$$

It is emphasized that the quantum state on the target system remains to be the energy eigenstate of *Hs*. Therefore, this is called the *protective measurement* (5; 18). It is remarked that the generalized version of the protective measurement in Ref. (19) by the pre- and post-selected states and in Ref. (10) by the meta-stable state.

#### **4.3 Weak measurement**

10 Will-be-set-by-IN-TECH

Here, we suppose that *Hs*(*t*) has discrete and non-degenerate eigenvalues denoted as *Ei*(*t*). Its corresponding eigenstate is denoted as |*Ei*(*t*)� for any time *t*. Furthermore, we consider the

where *N* is a sufficiently large number. We assume that the initial target state is the energy eigenvalue <sup>|</sup>*Ei*(*t*)� <sup>9</sup> the initial probe state is denoted as <sup>|</sup>*ξ*(0)�. Under the adiabatic condition,





*<sup>N</sup> <sup>g</sup>*(*tn*) Ex[*A*(*tn*)]*P*<sup>ˆ</sup> <sup>−</sup> <sup>1</sup>

 <sup>1</sup> <sup>−</sup> <sup>1</sup> 2 *T*2

− 1 2 *T*2

*<sup>N</sup> <sup>g</sup>*(*tn*) Ex[*A*(*tn*)]*P*<sup>ˆ</sup>

<sup>9</sup> Due to this assumption, it is impossible to apply this to the arbitrary quantum state. Furthermore, while we seemingly need the projective measurement, that is, destructive measurement, for the target system to confirm whether the initial quantum state is in the eigenstates (145; 186), they did not apply this to the arbitrary state. For example, if the system is cooled down, we can pickup the ground state of the

 *T* 0

*tn* <sup>=</sup> *<sup>n</sup>*

the compound state for the target and probe systems at time *T* is given by

<sup>−</sup>*<sup>i</sup> <sup>T</sup> <sup>N</sup> Htot*(*tN* )

<sup>−</sup>*<sup>i</sup> <sup>T</sup> <sup>N</sup> Htot*(*t*2)

<sup>−</sup>*<sup>i</sup> <sup>T</sup> <sup>N</sup> Hint*(*tN* )

<sup>−</sup>*<sup>i</sup> <sup>T</sup> <sup>N</sup> Hint*(*t*2)

By the Taylor expansion with the respect to *N*, the expectation value is

*T*


∼ *e* <sup>−</sup>*<sup>i</sup> <sup>T</sup>*

*Hint*(*t*) = *<sup>g</sup>*(*t*)(*<sup>A</sup>* <sup>⊗</sup> *<sup>P</sup>*ˆ), (29)

*Htot*(*t*) = *Hs*(*t*) + *Hp*(*t*) + *Hint*(*t*). (31)

*<sup>N</sup> <sup>T</sup>* (*<sup>n</sup>* <sup>=</sup> 0, 1, 2, . . . , *<sup>N</sup>*), (32)

<sup>−</sup>*<sup>i</sup> <sup>T</sup>*

<sup>−</sup>*<sup>i</sup> <sup>T</sup> <sup>N</sup> Htot*(*t*1)

<sup>−</sup>*<sup>i</sup> <sup>T</sup> <sup>N</sup> Hint*(*t*1)

> 2 *T*2

*<sup>N</sup>*<sup>2</sup> *<sup>g</sup>*2(*tn*) Var[*A*(*tn*)]*P*ˆ2 <sup>+</sup> *<sup>O</sup>*

<sup>−</sup>*<sup>i</sup> <sup>T</sup>*

*<sup>N</sup> Htot*(*tN*−<sup>1</sup>) ···

*<sup>N</sup> Hint*(*tN*−<sup>1</sup>) ···



 1 *N*<sup>3</sup> 

. (35)

*<sup>N</sup>*<sup>2</sup> *<sup>g</sup>*2(*tn*)(Ex[*A*(*tn*)])2*P*ˆ2

*<sup>N</sup>*<sup>2</sup> *<sup>g</sup>*2(*tn*) Var[*A*(*tn*)]*P*ˆ2

*g*(*t*)*dt* =: *g*0. (30)

given by

where

The total Hamiltonian is given by

discretized time from the time interval [0, *T*];



*<sup>N</sup> <sup>g</sup>*(*tn*)*A*⊗*P*<sup>ˆ</sup>

�*Ei*(*tn*)|*e*

<sup>−</sup>*<sup>i</sup> <sup>T</sup>*

target Hamiltonian *Hs*(0).

× |*Ei*(*t*2)��*Ei*(*t*2)|*e*

× |*Ei*(*t*3)��*Ei*(*t*2)|*e*

Applying the Trotter-Suzuki theorem (173; 182), one has

From the above discussions, is it possible to combine the above two concepts, i.e., the time-symmetric quantum measurement without destroying the quantum state (189)? This answer is the *weak measurement* (4). Consider a target system and a probe defined in the Hilbert space H*<sup>s</sup>* ⊗ H*p*. The interaction of the target system and the probe is assumed to be weak and instantaneous,

$$H\_{\rm int}(t) = \lg(A \otimes \hat{P})\delta(t - t\_0),\tag{38}$$

where an observable *<sup>A</sup>* is defined in <sup>H</sup>*s*, while *<sup>P</sup>*<sup>ˆ</sup> is the momentum operator of the probe. The time evolution operator becomes *<sup>e</sup>*−*ig*(*A*⊗*P*ˆ). Suppose the probe initial state is <sup>|</sup>*ξ*�. For the transition from the pre-selected state |*i*� to the post-selected state | *f*�, the probe wave function becomes |*ξ*� � <sup>=</sup> �*<sup>f</sup>* <sup>|</sup>*Ve*−*ig*(*A*⊗*P*ˆ)*U*|*i*�|*ξ*�, which is in the weak coupling case,

$$\begin{split} \left| \mathfrak{f}' \right\rangle &= \langle f \vert V e^{-i\mathfrak{g}\left(A \otimes \hat{P}\right)} \mathcal{U} \vert i \rangle \vert \mathfrak{f} \rangle = \langle f \vert V [\mathbf{1} - i\mathfrak{g}(A \otimes \hat{P})] \mathcal{U} \vert i \rangle \vert \mathfrak{f} \rangle + \mathcal{O}(\mathfrak{g}^2) \\ &= \langle f \vert V \mathcal{U} \vert i \rangle - i\mathfrak{g} \langle f \vert V \mathcal{U} \mathcal{U} \vert i \rangle \otimes \hat{P} \vert \mathfrak{f} \rangle + \mathcal{O}(\mathfrak{g}^2) = \langle f \vert V \mathcal{U} \vert i \rangle \left(1 - i\mathfrak{g} \langle A \rangle\_{\mathfrak{w}} \hat{P}\right) \vert \mathfrak{f} \rangle + \mathcal{O}(\mathfrak{g}^2) \tag{39} \end{split}$$

where �*f* |*VAU*|*i*�/�*f* |*VU*|*i*� = �*A*�*w*. Here, the last equation uses the approximation that *<sup>g</sup>*�*A*�*<sup>w</sup>* � <sup>1</sup> 10. We obtain the shifts of the expectation values for the position and momentum operators on the probe as the following theorem:

**Theorem 4.1** (Jozsa (85))**.** *We obtain the shifts of the expectation values for the position and momentum operators on the probe after the weak measurement with the post-selection as*

$$\Delta[\hat{Q}] = g \text{Re}\langle A \rangle\_{\mathcal{W}} + mg \text{Im}\langle A \rangle\_{\mathcal{W}} \left. \frac{d \,\text{Var}[\hat{Q}]}{dt} \right|\_{t=t\_0} \text{ \,\text{s}}\tag{40}$$

$$
\Delta[\hat{P}] = 2g \text{Im} \langle A \rangle\_w \text{Var}[\hat{P}]\_\prime \tag{41}
$$

<sup>10</sup> It is remarked that Wu and Li showed the second-order correction of the weak measurement (196). A further analysis was shown in Refs. (129; 132).

Substituting them into Eq. (49), we derive

*can be reduced to*

*∂*

*<sup>∂</sup><sup>Q</sup> <sup>S</sup>*(*Q*) = 0 to obtain *<sup>∂</sup>*

<sup>Δ</sup>[*Q*ˆ] = *<sup>g</sup>*Re�*A*�*<sup>w</sup>* <sup>+</sup> *mg*Im�*A*�*<sup>w</sup>*

Putting together, we can measure the weak value �*A*�*<sup>w</sup>* by observing the shift of the expectation value of the probe both in the coordinate and momentum representations. The shift of the probe position contains the future information up to the post-selected state.

Theory of "Weak Value" and Quantum Mechanical Measurements 87

**Corollary 4.2.** *When the probe wavefunction is real-valued in the coordinate representation, Eq. (40)*

since the interaction to the target system is taken at time *t* = *t*0.

*Proof.* From the Schrödinger equation in the coordinate representation;

*<sup>ξ</sup>*(*Q*) = <sup>1</sup>

*<sup>R</sup>*(*Q*) + *<sup>∂</sup>*

*<sup>∂</sup><sup>t</sup> R* = 0. Therefore, we obtain

2*m*

*∂Q*

*∂*2

where *<sup>ξ</sup>*(*Q*) ≡ �*Q*|*ξ*�, putting *<sup>ξ</sup>*(*Q*) = *<sup>R</sup>*(*Q*)*eiS*(*Q*), we obtain the equation for the real part as

*R*(*Q*) *<sup>∂</sup>*

Therefore, if the probe wavefunction is real-valued in the coordinate representation, one has

*d* Var[*Q*ˆ]

for any time *t*. Vice versa. From this statement, we obtain the desired result from Eq. (40).

It is noted that there are many analyses on the weak measurement, e.g., on the phase space (102), on the finite sample (179), on the counting statics (26; 104), on the non-local

Summing up this section, the two-state vector formalism is called if the pre- and post-selected states are prepared and the weak or strong measurement is taken in the von-Neumann type Hamiltonian, *<sup>H</sup>* <sup>=</sup> *gAP*ˆ*δ*(*<sup>t</sup>* <sup>−</sup> *<sup>t</sup>*0) between the pre- and post-selected states. In the case of the strong measurement, we obtain the expectation value Ex(*A*) in the probe. On the other hand,

In this subsection, we consider the weak measurement in the case that the probe system is a

where *v* is a unit vector. Expanding the interaction Hamiltonian for the pre- and post-selected states, |*ψ*� and |*φ*�, respectively up to the first order for *g*, we obtain the shift of the expectation

in the case of the weak measurement, we obtain the weak value �*A*�*<sup>w</sup>* in the probe.

*<sup>∂</sup><sup>Q</sup> S*(*Q*) *m*

*i ∂ ∂t*

> *∂ ∂t*

observable (32; 33), and on the complementary observable (197).

qubit system (195). In general, the interaction Hamiltonian is given by

**5. Weak-value measurement for a qubit system**

*d* Var[*Q*ˆ] *dt*

 *t*=*t*<sup>0</sup>

<sup>Δ</sup>[*Q*ˆ] = *<sup>g</sup>*Re�*A*�*w*. (53)

*<sup>∂</sup>Q*<sup>2</sup> *<sup>ξ</sup>*(*Q*) + *<sup>V</sup>*(*Q*)*ξ*(*Q*), (54)

*dt* <sup>=</sup> <sup>0</sup> (56)

*Hint* = *g*[*A* ⊗ (*v* ·*σ*)]*δ*(*t* − *t*0), (57)

= 0. (55)

(52)

*where*

$$
\Delta[\hat{Q}] := \frac{\langle \mathfrak{f}' | \hat{Q} | \mathfrak{f}' \rangle}{\langle \mathfrak{f}' | \mathfrak{f}' \rangle} - \langle \mathfrak{f} | \hat{Q} | \mathfrak{f} \rangle. \tag{42}
$$

$$
\Delta[\hat{P}] := \frac{\langle \mathfrak{F}' | \hat{P} | \mathfrak{F}' \rangle}{\langle \mathfrak{F}' | \mathfrak{F}' \rangle} - \langle \mathfrak{F} | \hat{P} | \mathfrak{F} \rangle,\tag{43}
$$

$$\text{Var}[\hat{Q}] := \langle \mathfrak{f} | \hat{Q}^2 | \mathfrak{f} \rangle - \left( \langle \mathfrak{f} | \hat{Q} | \mathfrak{f} \rangle \right)^2 \rangle \tag{44}$$

$$\text{Var}[\mathcal{P}] := \langle \mathfrak{f} | \hat{P}^2 | \mathfrak{f} \rangle - (\langle \mathfrak{f} | \mathcal{P} | \mathfrak{f} \rangle)^2. \tag{45}$$

*Here, the probe Hamiltonian is assumed as*

$$
\hat{H} = \frac{\hat{P}^2}{2m} + V(Q)\_\prime \tag{46}
$$

*where V*(*Q*) *is the potential on the coordinate space.*

*Proof.* For the probe observable *M*ˆ , we obtain

$$\begin{split} \frac{\langle\xi'|\hat{M}|\xi'\rangle}{\langle\xi'|\xi'\rangle} &= \frac{\langle\xi|\hat{M}|\xi\rangle - ig\langle A\rangle\_{w} \langle\xi|\hat{M}\hat{P}|\xi\rangle + ig\langle A\rangle\_{w} \langle\xi|\hat{P}\hat{M}|\xi\rangle}{\langle\xi|\xi\rangle - ig\langle A\rangle\_{w} \langle\xi|\hat{P}|\xi\rangle + ig\langle\overline{A\rangle\_{w}} \langle\xi|\hat{P}|\xi\rangle} \\ &= \frac{\langle\xi|\hat{M}|\xi\rangle + ig\text{Re}\langle A\rangle\_{w} \langle\xi|[\hat{P},\hat{M}]|\xi\rangle + g\text{Im}\langle A\rangle\_{w} \langle\xi|\left\langle\hat{P},\hat{M}\right\rangle|\xi\rangle}{\langle\xi|\xi\rangle + 2g\text{Im}\langle A\rangle\_{w} \langle\xi|\hat{P}|\xi\rangle} \\ &= \left(\langle\xi|\hat{M}|\xi\rangle + ig\text{Re}\langle A\rangle\_{w}\langle\xi|[\hat{P},\hat{M}]|\xi\rangle + g\text{Im}\langle A\rangle\_{w} \langle\xi|\left\langle\hat{P},\hat{M}\right\rangle|\xi\rangle\right) \\ &\qquad \times \left(1 - 2g\text{Im}\langle A\rangle\_{w}\langle\xi|\hat{P}|\xi\rangle\right) + O(g^{2}) \\ &= \langle\xi|\hat{M}|\xi\rangle + ig\text{Re}\langle A\rangle\_{w}\langle\xi|[\hat{P},\hat{M}]|\xi\rangle \\ &\qquad + g\text{Im}\langle A\rangle\_{w}\left(\langle\xi|\{\hat{P},\hat{M}\}|\xi\rangle - 2\langle\xi|\hat{M}|\xi\rangle\langle\xi|\hat{P}|\xi\rangle\right) + O(g^{2}). \end{split} \tag{47}$$

If we set *M*ˆ = *P*ˆ, one has

$$
\Delta[\hat{P}] = 2g \text{Im}\langle A \rangle\_w \text{Var}[\hat{P}].\tag{48}
$$

If instead we set *M*ˆ = *Q*ˆ, one has

$$\Delta[\hat{\mathbb{Q}}] = \mathrm{g} \mathrm{Re}\langle A \rangle\_{\mathcal{W}} + \mathrm{g} \mathrm{Im}\langle A \rangle\_{\mathcal{W}} \left( \langle \dddot{\xi} | \{ \hat{P}, \hat{\mathbb{Q}} \} | \xi \rangle - 2 \mathfrak{g} \langle \tilde{\xi} | \hat{\mathbb{Q}} | \xi \rangle \langle \tilde{\xi} | \hat{P} | \xi \rangle \right) \tag{49}$$

since [*P*ˆ, *<sup>Q</sup>*ˆ] = <sup>−</sup>*i*. From the Heisenberg equation with the probe Hamiltonian (46), we obtain the Ehrenfest theorem;

$$\mathrm{i}\frac{d}{dt}\langle\mathfrak{F}|\hat{\mathcal{Q}}|\mathfrak{F}\rangle = \langle\mathfrak{F}|[\hat{\mathcal{Q}},\hat{H}]|\mathfrak{F}\rangle = \mathrm{i}\frac{\langle\mathfrak{F}|\hat{P}|\mathfrak{F}\rangle}{m} \tag{50}$$

$$i\frac{d}{dt}\langle\tilde{\xi}|\hat{Q}^2|\xi\rangle = \langle\tilde{\xi}|[\hat{Q}^2,\hat{H}]|\xi\rangle = i\frac{\langle\tilde{\xi}|\{\hat{P},\hat{Q}\}|\xi\rangle}{m}.\tag{51}$$

Substituting them into Eq. (49), we derive

12 Will-be-set-by-IN-TECH

<sup>|</sup>*Q*ˆ|*ξ*� �

<sup>|</sup>*P*ˆ|*ξ*� �

�*ξ*�|*ξ*�� − �*ξ*|*Q*ˆ|*ξ*�, (42)

�*ξ*�|*ξ*�� − �*ξ*|*P*ˆ|*ξ*�, (43)

+ *V*(*Q*), (46)

+ *O*(*g*2)

*<sup>m</sup>* (50)

*<sup>m</sup>* . (51)

+ *O*(*g*2). (47)

(49)

Var[*Q*ˆ] :<sup>=</sup> �*ξ*|*Q*ˆ2|*ξ*� − (�*ξ*|*Q*ˆ|*ξ*�)2, (44)

Var[*P*ˆ] :<sup>=</sup> �*ξ*<sup>|</sup> <sup>ˆ</sup>*P*2|*ξ*� − (�*ξ*|*P*ˆ|*ξ*�)2. (45)

<sup>Δ</sup>[*Q*ˆ] :<sup>=</sup> �*ξ*�

<sup>Δ</sup>[*P*ˆ] :<sup>=</sup> �*ξ*�

*<sup>H</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>P</sup>*ˆ2 2*m*

�*ξ*�|*ξ*�� <sup>=</sup> �*ξ*|*M*<sup>ˆ</sup> <sup>|</sup>*ξ*� − *ig*�*A*�*w*�*ξ*|*M*<sup>ˆ</sup> *<sup>P</sup>*ˆ|*ξ*� <sup>+</sup> *ig*�*A*�*w*�*ξ*|*P*ˆ*M*<sup>ˆ</sup> <sup>|</sup>*ξ*�

<sup>=</sup> �*ξ*|*M*<sup>ˆ</sup> <sup>|</sup>*ξ*� <sup>+</sup> *ig*Re�*A*�*w*�*ξ*|[*P*ˆ, *<sup>M</sup>*<sup>ˆ</sup> ]|*ξ*�

+ *g*Im�*A*�*<sup>w</sup>*

<sup>Δ</sup>[*Q*ˆ] = *<sup>g</sup>*Re�*A*�*<sup>w</sup>* <sup>+</sup> *<sup>g</sup>*Im�*A*�*<sup>w</sup>*

*i d*

*i d* �*ξ*|*ξ*� − *ig*�*A*�*w*�*ξ*|*P*ˆ|*ξ*� <sup>+</sup> *ig*�*A*�*w*�*ξ*|*P*ˆ|*ξ*�

×

*dt*�*ξ*|*Q*ˆ|*ξ*� <sup>=</sup> �*ξ*|[*Q*ˆ, *<sup>H</sup>*<sup>ˆ</sup> ]|*ξ*� <sup>=</sup> *<sup>i</sup>*

*dt*�*ξ*|*Q*<sup>ˆ</sup> <sup>2</sup>|*ξ*� <sup>=</sup> �*ξ*|[*Q*<sup>ˆ</sup> 2, *<sup>H</sup>*<sup>ˆ</sup> ]|*ξ*� <sup>=</sup> *<sup>i</sup>*

since [*P*ˆ, *<sup>Q</sup>*ˆ] = <sup>−</sup>*i*. From the Heisenberg equation with the probe Hamiltonian (46), we obtain

<sup>=</sup> �*ξ*|*M*<sup>ˆ</sup> <sup>|</sup>*ξ*� <sup>+</sup> *ig*Re�*A*�*w*�*ξ*|[*P*ˆ, *<sup>M</sup>*<sup>ˆ</sup> ]|*ξ*� <sup>+</sup> *<sup>g</sup>*Im�*A*�*w*�*ξ*|{*P*ˆ, *<sup>M</sup>*<sup>ˆ</sup> }|*ξ*� �*ξ*|*ξ*� <sup>+</sup> <sup>2</sup>*g*Im�*A*�*w*�*ξ*|*P*ˆ|*ξ*�

�*ξ*|*M*<sup>ˆ</sup> <sup>|</sup>*ξ*� <sup>+</sup> *ig*Re�*A*�*w*�*ξ*|[*P*ˆ, *<sup>M</sup>*<sup>ˆ</sup> ]|*ξ*� <sup>+</sup> *<sup>g</sup>*Im�*A*�*w*�*ξ*|{*P*ˆ, *<sup>M</sup>*<sup>ˆ</sup> }|*ξ*�

�*ξ*|{*P*ˆ, *<sup>M</sup>*<sup>ˆ</sup> }|*ξ*� − <sup>2</sup>�*ξ*|*M*<sup>ˆ</sup> <sup>|</sup>*ξ*��*ξ*|*P*ˆ|*ξ*�

<sup>1</sup> <sup>−</sup> <sup>2</sup>*g*Im�*A*�*w*�*ξ*|*P*ˆ|*ξ*�

<sup>Δ</sup>[*P*ˆ] = <sup>2</sup>*g*Im�*A*�*<sup>w</sup>* Var[*P*ˆ]. (48)

�*ξ*|{*P*ˆ, *<sup>Q</sup>*<sup>ˆ</sup> }|*ξ*� − <sup>2</sup>*g*�*ξ*|*Q*ˆ|*ξ*��*ξ*|*P*ˆ|*ξ*�

�*ξ*|{*P*ˆ, *<sup>Q</sup>*<sup>ˆ</sup> }|*ξ*�

�*ξ*|*P*ˆ|*ξ*�

*Here, the probe Hamiltonian is assumed as*

�*ξ*� <sup>|</sup>*M*<sup>ˆ</sup> <sup>|</sup>*ξ*� �

If we set *M*ˆ = *P*ˆ, one has

the Ehrenfest theorem;

If instead we set *M*ˆ = *Q*ˆ, one has

*where V*(*Q*) *is the potential on the coordinate space. Proof.* For the probe observable *M*ˆ , we obtain

=

*where*

$$\Delta[\hat{\mathcal{Q}}] = g \text{Re}\langle A \rangle\_w + mg \text{Im}\langle A \rangle\_w \left. \frac{d \operatorname{Var}[\hat{\mathcal{Q}}]}{dt} \right|\_{t=t\_0} \tag{52}$$

since the interaction to the target system is taken at time *t* = *t*0.

Putting together, we can measure the weak value �*A*�*<sup>w</sup>* by observing the shift of the expectation value of the probe both in the coordinate and momentum representations. The shift of the probe position contains the future information up to the post-selected state.

**Corollary 4.2.** *When the probe wavefunction is real-valued in the coordinate representation, Eq. (40) can be reduced to*

$$
\Delta[\hat{Q}] = g \text{Re}\langle A \rangle\_w. \tag{53}
$$

*Proof.* From the Schrödinger equation in the coordinate representation;

$$i\frac{\partial}{\partial t}\mathfrak{f}(Q) = \frac{1}{2m}\frac{\partial^2}{\partial Q^2}\mathfrak{f}(Q) + V(Q)\mathfrak{f}(Q),\tag{54}$$

where *<sup>ξ</sup>*(*Q*) ≡ �*Q*|*ξ*�, putting *<sup>ξ</sup>*(*Q*) = *<sup>R</sup>*(*Q*)*eiS*(*Q*), we obtain the equation for the real part as

$$\frac{\partial}{\partial t}R(Q) + \frac{\partial}{\partial Q}\left(\frac{R(Q)\frac{\partial}{\partial Q}S(Q)}{m}\right) = 0. \tag{55}$$

Therefore, if the probe wavefunction is real-valued in the coordinate representation, one has *∂ <sup>∂</sup><sup>Q</sup> <sup>S</sup>*(*Q*) = 0 to obtain *<sup>∂</sup> <sup>∂</sup><sup>t</sup> R* = 0. Therefore, we obtain

$$\frac{d\operatorname{Var}[\hat{\mathbb{Q}}]}{dt} = 0\tag{56}$$

for any time *t*. Vice versa. From this statement, we obtain the desired result from Eq. (40).

It is noted that there are many analyses on the weak measurement, e.g., on the phase space (102), on the finite sample (179), on the counting statics (26; 104), on the non-local observable (32; 33), and on the complementary observable (197).

Summing up this section, the two-state vector formalism is called if the pre- and post-selected states are prepared and the weak or strong measurement is taken in the von-Neumann type Hamiltonian, *<sup>H</sup>* <sup>=</sup> *gAP*ˆ*δ*(*<sup>t</sup>* <sup>−</sup> *<sup>t</sup>*0) between the pre- and post-selected states. In the case of the strong measurement, we obtain the expectation value Ex(*A*) in the probe. On the other hand, in the case of the weak measurement, we obtain the weak value �*A*�*<sup>w</sup>* in the probe.

#### **5. Weak-value measurement for a qubit system**

In this subsection, we consider the weak measurement in the case that the probe system is a qubit system (195). In general, the interaction Hamiltonian is given by

$$H\_{\rm int} = \lg[A \otimes (\vec{v} \cdot \vec{\sigma})] \delta(t - t\_0) \,\tag{57}$$

where *v* is a unit vector. Expanding the interaction Hamiltonian for the pre- and post-selected states, |*ψ*� and |*φ*�, respectively up to the first order for *g*, we obtain the shift of the expectation

and similarly, in the case of *η* ∼ 1, one has

*<sup>s</sup>*�*φ*| ⊗ *<sup>p</sup>*�*k*|

<sup>=</sup> <sup>|</sup>(*<sup>γ</sup>* <sup>−</sup> *<sup>η</sup>*) *<sup>φ</sup>*�|*k*�*s*�*k*|�*<sup>w</sup>*

Putting together Eqs. (67) and (69), we obtain

<sup>2</sup>*η*(*<sup>γ</sup>* <sup>−</sup> *<sup>η</sup>*)Re *<sup>φ</sup>*�|*k*�*s*�*k*|�*<sup>w</sup>*

 <sup>|</sup>Ψ*c*� �<sup>2</sup>

 |Ψ*c*� �<sup>2</sup>

<sup>=</sup> <sup>|</sup>(*<sup>γ</sup>* <sup>−</sup> *<sup>η</sup>*) *<sup>s</sup>*�*φ*|*k*�*s*�*k*|*ψ*�*<sup>s</sup>* <sup>+</sup> *<sup>η</sup> <sup>s</sup>*�*φ*|*ψ*�*s*<sup>|</sup>

<sup>∑</sup>*m*∈{0,1} �

*<sup>ψ</sup>* + *η*| 2

*<sup>s</sup>*�*φ*| ⊗ *<sup>p</sup>*�*m*|

<sup>1</sup> <sup>−</sup> (*<sup>γ</sup>* <sup>−</sup> *<sup>η</sup>*)2(<sup>1</sup> <sup>−</sup> <sup>∑</sup>*m*∈{0,1} <sup>|</sup> *<sup>φ</sup>*�|*m*�*s*�*m*|�*<sup>w</sup>*

post-selection, the POVM to obtain the measurement outcome *k* is

*γ* =

*<sup>ψ</sup>* + *�* 

2 <sup>1</sup> − *�*<sup>2</sup> 

*<sup>ψ</sup>* <sup>−</sup> *�* 2  1

Re *<sup>φ</sup>*�|*k*�*s*�*k*|�*<sup>w</sup>*

<sup>|</sup> *<sup>φ</sup>*�|*k*�*s*�*k*|�*<sup>w</sup>*

Pr[*k*] :<sup>=</sup> �

= 

<sup>∑</sup>*m*∈{0,1} �

measurement outcome *k* as

*R*[*k*] =

one has

Setting the parameters;

<sup>=</sup> <sup>1</sup> 2

*<sup>R</sup>*[*k*] = (<sup>1</sup> <sup>−</sup> *�*)Re *<sup>φ</sup>*�|*k*�*s*�*k*|�*<sup>w</sup>*

Re *<sup>φ</sup>*�|*k*�*s*�*k*|�*<sup>w</sup>*

*α*|0�*s*|1�*<sup>p</sup>* + *β*|1�*s*|0�*p*. (66)

 2

Those cases can be taken as the standard von Neumann projective measurement. For the post-selected state |*φ*�, the probability to obtain the measurement outcome *k* on the probe is

Theory of "Weak Value" and Quantum Mechanical Measurements 89

( *<sup>s</sup>*�*φ*|0�*s*�0|*ψ*�*sγ* + *<sup>s</sup>*�*φ*|1�*s*�1|*ψ*�*sη*) *δk*,0 + ( *<sup>s</sup>*�*φ*|0�*s*�0|*ψ*�*sη* + *<sup>s</sup>*�*φ*|1�*s*�1|*ψ*�*sγ*) *δk*,1

*<sup>ψ</sup>* |2)


Here, in the last line, the parameters *γ* and *η* are assumed to be real. Without the

Here, the coefficient of the first term means that the strength of measurement and the second term is always added. Therefore, we define the quantity to distinguish the probability for the

*<sup>R</sup>*[*k*] :<sup>=</sup> Pr[*k*] <sup>−</sup> *<sup>η</sup>*<sup>2</sup>

*<sup>ψ</sup>* + (*<sup>γ</sup>* <sup>−</sup> *<sup>η</sup>*)2[<sup>|</sup> *<sup>φ</sup>*�|*k*�*s*�*k*|�*<sup>w</sup>*

(*γ*<sup>2</sup> <sup>−</sup> *<sup>η</sup>*2)[<sup>1</sup> <sup>−</sup> (*<sup>γ</sup>* <sup>−</sup> *<sup>η</sup>*)2(<sup>1</sup> <sup>−</sup> <sup>∑</sup>*m*∈{0,1} <sup>|</sup> *<sup>φ</sup>*�|*m*�*s*�*m*|�*<sup>w</sup>*

<sup>2</sup> <sup>+</sup> *�*, *<sup>η</sup>* <sup>=</sup>

*ψ* | <sup>2</sup> + <sup>1</sup> <sup>2</sup> − *�* 

*<sup>ψ</sup>* <sup>−</sup> <sup>1</sup> 2

From Eq. (72), it is possible to obtain the real part of the weak value from the first term and its imaginary part from the second term. Since the first order of the parameter *�* is the gradient on

<sup>1</sup> <sup>−</sup> <sup>∑</sup>*m*∈{0,1} <sup>|</sup> *<sup>φ</sup>*�|*m*�*s*�*m*|�*<sup>w</sup>*

(*γ*<sup>2</sup> − *<sup>η</sup>*2)

*ψ* |

1

<sup>|</sup> *<sup>φ</sup>*�|*k*�*s*�*k*|�*<sup>w</sup>*

*<sup>s</sup>*�*φ*| ⊗ *<sup>p</sup>*�*m*|

 |Ψ*c*� �<sup>2</sup>

2

*Ek* = (*γ*<sup>2</sup> <sup>−</sup> *<sup>η</sup>*2)|*k*�*s*�*k*<sup>|</sup> <sup>+</sup> *<sup>η</sup>*2. (68)

. (67)

. (69)

<sup>2</sup> <sup>−</sup> *�*, (71)

*ψ* | 2) 

<sup>+</sup> *<sup>O</sup>*(*�*2),

+ *O*(*�*2). (72)

*ψ* | 2)]

*<sup>ψ</sup>* <sup>|</sup>2)] . (70)

<sup>2</sup> <sup>+</sup> *<sup>η</sup>*2(<sup>1</sup> − | *<sup>φ</sup>*�|*k*�*s*�*k*|�*<sup>w</sup>*

(<sup>1</sup> − | *<sup>φ</sup>*�|*k*�*s*�*k*|�*<sup>w</sup>*

*ψ* |2

*ψ* | 2 

value for *q* ·*σ* as

$$\begin{split} \Delta[\vec{q}\cdot\vec{v}] &= \frac{\langle\xi'|[\vec{q}\cdot\vec{v}]|\xi'\rangle}{\langle\xi'|\xi'\rangle} - \langle\xi|[\vec{q}\cdot\vec{v}]|\xi\rangle = g\langle\xi|i[\vec{v}\cdot\vec{v},\vec{q}\cdot\vec{v}]|\xi\rangle \text{Re}\langle A\rangle\_{w} \\ &+ g\left(\langle\xi|\left\{\vec{v}\cdot\vec{v},\vec{q}\cdot\vec{v}\right\}|\xi\rangle - 2\langle\xi|\vec{v}\cdot\vec{v}|\xi\rangle\langle\xi|\vec{q}\cdot\vec{v}|\xi\rangle\right) \text{Im}\langle A\rangle\_{w} + O(g^{2}) \\ &= 2g\{(\vec{q}\cdot\vec{v})\cdot\vec{m}\} \text{Re}\langle A\rangle\_{w} + 2g\{\vec{v}\cdot\vec{q} - (\vec{v}\cdot\vec{m})(\vec{q}\cdot\vec{m})\} \text{Im}\langle A\rangle\_{w} + O(g^{2}), \end{split} \tag{58}$$

where

$$|\mathfrak{F}'\rangle = \langle \phi | e^{-i\mathfrak{g}[A \otimes (\vec{v} \cdot \vec{\sigma})]} |\psi\rangle |\xi\rangle. \tag{59}$$

$$|\xi\rangle\langle\xi| = :\frac{1}{2}(\mathbf{1} + \vec{m} \cdot \vec{\sigma}).\tag{60}$$

Furthermore, the pre- and post-selected states are assumed to be

$$\begin{aligned} |\psi\rangle\langle\psi| &= : \frac{1}{2}(\mathbf{1} + \vec{r}\_i \cdot \vec{\sigma}), \\ |\phi\rangle\langle\phi| &= : \frac{1}{2}(\mathbf{1} + \vec{r}\_f \cdot \vec{\sigma}). \end{aligned} \tag{61}$$

Since the weak value of the observable *n* ·*σ* is

$$
\langle \vec{n} \cdot \vec{\sigma} \rangle\_{\vec{w}} = \frac{\langle \phi | \vec{n} \cdot \vec{\sigma} | \psi \rangle \langle \psi | \phi \rangle}{|\langle \phi | \psi \rangle|^{2}} = \vec{n} \cdot \frac{\vec{r}\_{i} + \vec{r}\_{f} + i(\vec{r}\_{i} \times \vec{r}\_{f})}{1 + \vec{r}\_{i} \cdot \vec{r}\_{f}},\tag{62}
$$

we obtain

$$\Delta[\vec{q}\cdot\vec{\sigma}] = 2g\{(\vec{q}\times\vec{v})\cdot\vec{m}\}\frac{\vec{n}\cdot(\vec{r}\_i+\vec{r}\_f)}{1+\vec{r}\_i\cdot\vec{r}\_f} + 2g\{\vec{v}\cdot\vec{q}-(\vec{v}\cdot\vec{m})(\vec{q}\cdot\vec{m})\}\frac{\vec{n}\cdot(\vec{r}\_i\times\vec{r}\_f)}{1+\vec{r}\_i\cdot\vec{r}\_f} + O(g^2). \tag{63}$$

From Eq. (63), we can evaluate the real and imaginary parts of the weak value changing the parameter of the measurement direction *q*. This calculation is used in the context of the Hamiltonian estimation (157).

Next, as mentioned before, we emphasize that the weak measurement is only one of the methods to obtain the weak value. There are many other approaches to obtain the weak value, e.g., on changing the probe state (59; 80; 103; 119), and on the entangled probe state (114). Here, we show another method to obtain the weak value in the case that the target and the probe systems are both qubit systems (133).

Let |*ψ*�*<sup>s</sup>* := *α*|0�*<sup>s</sup>* + *β*|1�*<sup>s</sup>* be the pre-selected state for the target system. The initial probe state can described as |*ξ*�*<sup>p</sup>* := *γ*|0�*<sup>p</sup>* + *η*|1�*p*. It is emphasized that the initial probe state is controllable. Here, the initial states are normalized, that is, |*α*| <sup>2</sup> <sup>+</sup> <sup>|</sup>*β*<sup>|</sup> <sup>2</sup> <sup>=</sup> 1 and <sup>|</sup>*γ*<sup>|</sup> <sup>2</sup> <sup>+</sup> <sup>|</sup>*η*<sup>|</sup> <sup>2</sup> = 1. Applying the Controlled-NOT (C-NOT) gate, we make a transform of the quantum state for the compound system to

$$<\langle \psi \rangle\_s \otimes |\xi\rangle\_p \xrightarrow{\mathbb{C}-\text{NOT}} |\Psi\_c\rangle := (a\gamma|0\rangle\_s + \beta\eta|1\rangle\_s)|0\rangle\_p + (a\eta|0\rangle\_s + \beta\gamma|1\rangle\_s)|1\rangle\_p. \tag{64}$$

In the case of *γ* ∼ 1, we obtain the compound state as

$$
\alpha|0\rangle\_{\rm s}|0\rangle\_{\rm p} + \beta|1\rangle\_{\rm s}|1\rangle\_{\rm p} \tag{65}
$$

and similarly, in the case of *η* ∼ 1, one has

14 Will-be-set-by-IN-TECH

�*ξ*�|*ξ*�� − �*ξ*|[*<sup>q</sup>* ·*<sup>σ</sup>*]|*ξ*� <sup>=</sup> *<sup>g</sup>*�*ξ*|*i*[*<sup>v</sup>* ·*<sup>σ</sup>*,*<sup>q</sup>* ·*<sup>σ</sup>*]|*ξ*�Re�*A*�*<sup>w</sup>* <sup>+</sup> *<sup>g</sup>* (�*ξ*<sup>|</sup> {*<sup>v</sup>* ·*<sup>σ</sup>*,*<sup>q</sup>* ·*<sup>σ</sup>*} <sup>|</sup>*ξ*� − <sup>2</sup>�*ξ*<sup>|</sup>*<sup>v</sup>* ·*<sup>σ</sup>*|*ξ*��*ξ*<sup>|</sup>*<sup>q</sup>* ·*<sup>σ</sup>*|*ξ*�)Im�*A*�*<sup>w</sup>* <sup>+</sup> *<sup>O</sup>*(*g*2)

<sup>=</sup> <sup>2</sup>*g*{(*<sup>q</sup>* <sup>×</sup>*<sup>v</sup>*) · *<sup>m</sup>* }Re�*A*�*<sup>w</sup>* <sup>+</sup> <sup>2</sup>*g*{*<sup>v</sup>* ·*<sup>q</sup>* <sup>−</sup> (*<sup>v</sup>* · *<sup>m</sup>* )(*<sup>q</sup>* · *<sup>m</sup>* )}Im�*A*�*<sup>w</sup>* <sup>+</sup> *<sup>O</sup>*(*g*2), (58)

<sup>−</sup>*ig*[*A*⊗(*<sup>v</sup>*·*<sup>σ</sup>*)]|*ψ*�|*ξ*�, (59)

(**1** + *m* ·*σ*). (60)

*ri* +*rf* + *i*(*ri* ×*rf*) 1 +*ri* ·*rf*

<sup>2</sup> <sup>+</sup> <sup>|</sup>*β*<sup>|</sup>

*α*|0�*s*|0�*<sup>p</sup>* + *β*|1�*s*|1�*p*, (65)

<sup>C</sup>−NOT −−−−−→ |Ψ*c*� := (*αγ*|0�*<sup>s</sup>* <sup>+</sup> *βη*|1�*s*)|0�*<sup>p</sup>* + (*αη*|0�*<sup>s</sup>* <sup>+</sup> *βγ*|1�*s*)|1�*p*. (64)

*n* · (*ri* ×*rf*) 1 +*ri* ·*rf*

<sup>2</sup> <sup>=</sup> 1 and <sup>|</sup>*γ*<sup>|</sup>

(61)

, (62)

+ *O*(*g*2). (63)

<sup>2</sup> <sup>+</sup> <sup>|</sup>*η*<sup>|</sup>

<sup>2</sup> = 1.

value for *q* ·*σ* as

where

we obtain

Δ[*q* ·*σ*] = 2*g*{(*q* ×*v*) · *m* }

Hamiltonian estimation (157).

the compound system to


systems are both qubit systems (133).

<sup>Δ</sup>[*<sup>q</sup>* ·*<sup>σ</sup>*] = �*ξ*�


Since the weak value of the observable *n* ·*σ* is

�


Furthermore, the pre- and post-selected states are assumed to be

�*<sup>n</sup>* ·*<sup>σ</sup>*�*<sup>w</sup>* <sup>=</sup> �*φ*<sup>|</sup>*<sup>n</sup>* ·*<sup>σ</sup>*|*ψ*��*ψ*|*φ*�

*n* · (*ri* +*rf*) 1 +*ri* ·*rf*

controllable. Here, the initial states are normalized, that is, |*α*|

In the case of *γ* ∼ 1, we obtain the compound state as


� = �*φ*|*e*



1 2

1 2


From Eq. (63), we can evaluate the real and imaginary parts of the weak value changing the parameter of the measurement direction *q*. This calculation is used in the context of the

Next, as mentioned before, we emphasize that the weak measurement is only one of the methods to obtain the weak value. There are many other approaches to obtain the weak value, e.g., on changing the probe state (59; 80; 103; 119), and on the entangled probe state (114). Here, we show another method to obtain the weak value in the case that the target and the probe

Let |*ψ*�*<sup>s</sup>* := *α*|0�*<sup>s</sup>* + *β*|1�*<sup>s</sup>* be the pre-selected state for the target system. The initial probe state can described as |*ξ*�*<sup>p</sup>* := *γ*|0�*<sup>p</sup>* + *η*|1�*p*. It is emphasized that the initial probe state is

Applying the Controlled-NOT (C-NOT) gate, we make a transform of the quantum state for

(**1** +*ri* ·*σ*),

(**1** +*rf* ·*σ*).

+ 2*g*{*v* ·*q* − (*v* · *m* )(*q* · *m* )}

1 2

$$
\alpha|0\rangle\_s|1\rangle\_p + \beta|1\rangle\_s|0\rangle\_p. \tag{66}
$$

Those cases can be taken as the standard von Neumann projective measurement. For the post-selected state |*φ*�, the probability to obtain the measurement outcome *k* on the probe is

$$\begin{split} \Pr[k] := \frac{\|\left(\,\_{s}\langle\phi\rangle\otimes\,\_{p}\langle k\rangle\right)\|\,\_{\varphi}^{2}\|}{\sum\_{m\in\{0,1\}}\|\left(\,\_{s}\langle\phi\rangle\otimes\,\_{p}\langle m\rangle\right)\|\,\_{\varphi}^{2}\|} \\ &= \frac{\left|\left(\,\_{s}\langle\phi\vert\-0\right)\delta\_{s}\langle\theta\vert\-\delta\_{s}\langle\phi\vert\-1\rangle\right|\delta\_{s}\langle\phi\vert\-1\vert\right|}{\sum\_{m\in\{0,1\}}\|\left(\,\_{s}\langle\phi\vert\-p\vert\-m\right)\|\,\_{p}\langle\phi\vert\-p\vert}\,\_{\varphi}\langle\phi\vert\-1\rangle\_{s}\langle\phi\vert\-1\rangle\_{s}\langle\phi\vert\-1\rangle\_{s}\langle\phi\vert\-p\vert \\ &= \frac{|\langle\gamma-\eta\rangle\,\_{s}\langle\phi\vert\-\delta\vert\_{s}\langle\phi\vert\-p\vert\-m\rangle\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle}{|\langle\gamma-\eta\rangle\,\_{s}\langle\phi\vert\-\delta\vert\_{s}\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle} \\ &= \frac{|\langle(\gamma-\eta\rangle\,\_{s}\langle\phi\vert\-\delta\vert\_{s}\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle\langle\phi\vert\-p\rangle}}{1-(\gamma-\eta\rfloor\,\_{s}\langle\phi\vert\-\delta\vert\_{s}\langle\phi\vert\-m\vert\_{\forall}\langle\phi\vert\-\frac{\vert$$

Here, in the last line, the parameters *γ* and *η* are assumed to be real. Without the post-selection, the POVM to obtain the measurement outcome *k* is

$$E\_k = (\gamma^2 - \eta^2)|k\rangle\_s \langle k| + \eta^2. \tag{68}$$

Here, the coefficient of the first term means that the strength of measurement and the second term is always added. Therefore, we define the quantity to distinguish the probability for the measurement outcome *k* as

$$R[k] := \frac{\Pr[k] - \eta^2}{(\gamma^2 - \eta^2)}.\tag{69}$$

Putting together Eqs. (67) and (69), we obtain

$$R[k] = \frac{2\eta(\gamma - \eta)\text{Re}\_{\phi}\langle |k\rangle\_{\text{s}}\langle k|\rangle\_{\text{\#}}^{\text{w}} + (\gamma - \eta)^{2}[|\,\_{\phi}\langle |k\rangle\_{\text{s}}\langle k|\rangle\_{\text{\#}}^{\text{w}}]^{2} + \eta^{2}(1 - |\,\_{\phi}\langle |k\rangle\_{\text{s}}\langle k|\rangle\_{\text{\#}}^{\text{w}})^{2}]}{(\gamma^{2} - \eta^{2})[1 - (\gamma - \eta)^{2}(1 - \sum\_{m \in \{0,1\}} |\,\_{\phi}\langle |m\rangle\_{\text{s}}\langle m|\rangle\_{\text{\#}}^{\text{w}}|^{2})]}.\tag{70}$$

Setting the parameters;

$$
\gamma = \sqrt{\frac{1}{2} + \epsilon}, \quad \eta = \sqrt{\frac{1}{2} - \epsilon}. \tag{71}
$$

one has

$$R[k] = \frac{(1 - \varepsilon) \text{Re}\_{\phi} \langle |k\rangle\_{s} \langle k| \rangle\_{\Psi}^{w} + \varepsilon \left[ |\,\_{\phi} \langle |k\rangle\_{s} \langle k| \rangle\_{\Psi}^{w} |^{2} + \left( \frac{1}{2} - \varepsilon \right) \left( 1 - |\,\_{\phi} \langle |k\rangle\_{s} \langle k| \rangle\_{\Psi}^{w} |^{2} \right) \right]}{2 \left[ 1 - \varepsilon^{2} \left( 1 - \sum\_{m \in \{0, 1\}} |\,\_{\phi} \langle |m\rangle\_{s} \langle m| \rangle\_{\Psi}^{w} |^{2} \right) \right]} + O(\varepsilon^{2}),$$

$$= \frac{1}{2} \text{Re}\_{\phi} \langle |k\rangle\_{s} \langle k| \rangle\_{\Psi}^{w} - \frac{\varepsilon}{2} \left( \text{Re}\_{\phi} \langle |k\rangle\_{s} \langle k| \rangle\_{\Psi}^{w} - \frac{1}{2} |\,\_{\phi} \langle |k\rangle\_{s} \langle k| \rangle\_{\Psi}^{w} |^{2} \right) + O(\varepsilon^{2}). \tag{72}$$

From Eq. (72), it is possible to obtain the real part of the weak value from the first term and its imaginary part from the second term. Since the first order of the parameter *�* is the gradient on

By construction, the two states |*ψ*(*t*)� and �*φ*(*t*)| satisfy the Schrödinger equations with the same Hamiltonian with the initial and final conditions |*ψ*(*ti*)� = |*i*� and �*φ*(*tf*)| = �*f* |. In a sense, |*ψ*(*t*)� evolves forward in time while �*φ*(*t*)| evolves backward in time. The time reverse of the W operator (76) is *<sup>W</sup>*† <sup>=</sup> <sup>|</sup>*φ*(*t*)��*ψ*(*t*)|. Thus, we can say the W operator is based on the two-state vector formalism formally described in Refs. (16; 17). Even an apparently similar quantity to the W operator (76) was introduced by Reznik and Aharonov (140) in the name of "two-state" with the conceptually different meaning. This is because the W operator acts on

Theory of "Weak Value" and Quantum Mechanical Measurements 91

while the generalized two-state, which is called a multiple-time state, was introduced (13), this is essentially reduced to the two-state vector formalism. The W operator gives the weak

�*A*�*<sup>W</sup>* <sup>=</sup> Tr(*WA*)

Ex[*A*] = Tr(*ρA*)

from Born's rule. Furthermore, the W operator (74) can be regarded as a special case of a standard purification of the density operator (185). In our opinion, the W operator should be considered on the same footing of the density operator. For a closed system, both satisfy the Schrödinger equation. In a sense, the W operator *W* is the square root of the density operator

which describes a state evolving forward in time for a given initial state |*ψ*(*ti*)��*ψ*(*ti*)| = |*i*��*i*|,

which describes a state evolving backward in time for a given final state |*φ*(*tf*)��*φ*(*tf*)| = | *f*��*f* |. The W operator describes the entire history of the state from the past (*ti*) to the future (*tf* ) and measurement performed at the time *t*<sup>0</sup> as we shall see in Appendix 4.3. This description is conceptually different from the conventional one by the time evolution of the density operator. From the viewpoint of geometry, the W operator can be taken as the

When the dimension of the Hilbert space is *N*: dimH = *N*, the structure group of this bundle is *U*(*N*) (25, Sec. 9.3). Therefore, the W operator has richer information than the density operator formalism as we shall see a typical example of a geometric phase (155). Furthermore, we can express the probability to get the measurement outcome *an* ∈ *A* due to the ABL

Pr[*<sup>A</sup>* <sup>=</sup> *an*] = <sup>|</sup> Tr *WPan* <sup>|</sup>

<sup>12</sup> While the original notation of the weak values is �*A*�*<sup>w</sup>* indicating the "w"eak value of an observable *A*, our notation is motivated by one of which the pre- and post-selected states are explicitly shown as

*<sup>W</sup>*(*t*)*W*†(*t*) = <sup>|</sup>*ψ*(*t*)��*ψ*(*t*)<sup>|</sup> <sup>=</sup> *<sup>U</sup>*(*t*, *ti*)|*i*��*i*|*U*†(*t*, *ti*), (79)

*<sup>W</sup>*†(*t*)*W*(*t*) = <sup>|</sup>*φ*(*t*)��*φ*(*t*)<sup>|</sup> <sup>=</sup> *<sup>U</sup>*(*tf* , *<sup>t</sup>*)<sup>|</sup> *<sup>f</sup>*��*<sup>f</sup>* <sup>|</sup>*U*†(*tf* , *<sup>t</sup>*), (80)

<sup>Π</sup> : *<sup>W</sup>*(*t*) <sup>→</sup> *<sup>ρ</sup>i*(*t*) :<sup>=</sup> *<sup>W</sup>*(*t*)*W*†(*t*). (81)

2

<sup>∑</sup>*<sup>n</sup>* <sup>|</sup> Tr *WPan* <sup>|</sup><sup>2</sup> , (82)

←−H2. Furthermore,

Tr *<sup>W</sup>* , (77)

Tr *<sup>ρ</sup>* (78)

a Hilbert space <sup>H</sup> but the two-state vector acts on the Hilbert space −→H<sup>1</sup> <sup>⊗</sup>

in parallel with the expectation value of the observable *A* by

Hilbert-Schmidt bundle. The bundle projection is given by

formula (28) using the W operator *W* as

value of the observable *A* <sup>12</sup> as

since

while

*f*�*A*�*<sup>w</sup> i* .

changing the initial probe state from |*ξ*�*<sup>p</sup>* = <sup>√</sup> 1 2 (|0�*<sup>p</sup>* + |1�*p*), realistically, we can evaluate the imaginary part of the weak value from the gradient of the readout. This method is also used in Ref. (198) on the joint weak value. It is emphasized that the weak value can be experimentally accessible by changing the initial probe state while the interaction is **not weak** 11.

#### **6. Weak values for arbitrary coupling quantum measurement**

We just calculate an arbitrary coupling between the target and the probe systems (93; 120; 199). Throughout this section, we assume that the desired observable is the projection operator to be denoted as *A*<sup>2</sup> = *A* (153). In the case of the von-Neumann interaction motivated by the original work (4), when the pre- and post-selected states are |*i*� and | *f*�, respectively, and the probe state is |*ξ*�, the probe state |*ξ*� � after the interaction given by *Hint* <sup>=</sup> *gAP*<sup>ˆ</sup> becomes

$$\begin{split} \langle |\xi'\rangle &= \langle f| \varepsilon^{-igA\hat{P}} |i\rangle |\xi\rangle = \langle f| \left( 1 + \sum\_{k=1}^{\infty} \frac{1}{k!} (-igA\hat{P})^{k} \right) |i\rangle |\xi\rangle = \langle f| \left( 1 + A \sum\_{k=1}^{\infty} \frac{1}{k!} (-ig\hat{P})^{k} \right) |i\rangle |\xi\rangle \\ &= \langle f| \left( 1 - A + A \sum\_{k=0}^{\infty} \frac{1}{k!} (-ig\hat{P})^{k} \right) |i\rangle |\xi\rangle = \langle f| \left( 1 - A + Ae^{-i\xi\hat{P}} \right) |i\rangle |\xi\rangle \\ &= \langle f|i\rangle \left( 1 - \langle A \rangle\_{w} + \langle A \rangle\_{w} e^{-i\xi\hat{P}} \right) |\xi\rangle. \end{split} \tag{73}$$

It is remarked that the desired observable *B*, which satisfies *B*<sup>2</sup> = 1 (93; 120), corresponds to *B* = 2*A* − 1. Analogous to Theorem 4.1, we can derive the expectation values of the position and the momentum after the weak measurement. These quantities depends on the weak value �*A*�*<sup>w</sup>* and the generating function for the position and the momentum of the initial probe state |*ξ*�.

#### **7. Weak value with decoherence**

The decoherence results from the coupled system to the environment and leads to the transition from the quantum to classical systems. The general framework of the decoherence was discussed in Sec. 2. In this section, we discuss the analytical expressions for the weak value.

While we directly discuss the weak value with decoherence, the weak value is defined as a complex number. To analogously discuss the density operator formalism, we need the operator associated with the weak value. Therefore, we define a *W operator W*(*t*) as

$$\mathcal{W}(t) := \mathcal{U}(t, t\_i) |i\rangle\langle f| \mathcal{U}(t\_{f'}t). \tag{74}$$

To facilitate the formal development of the weak value, we introduce the ket state |*ψ*(*t*)� and the bra state �*φ*(*t*)| as

$$
\langle \psi(t) \rangle = \mathcal{U}(t, t\_i) |i\rangle , \langle \phi(t) | = \langle f | \mathcal{U}(t\_f, t) \rangle \tag{75}
$$

so that the expression for the W operator simplifies to

$$\mathcal{W}(t) = |\psi(t)\rangle\langle\phi(t)|.\tag{76}$$

<sup>11</sup> This point seems to be misunderstood. According to Ref. (134), the violation of the Leggett-Garg inequality (100) was shown, but the macroscopic realism cannot be denied since the noninvasive measurability is not realized.

16 Will-be-set-by-IN-TECH

1 2

accessible by changing the initial probe state while the interaction is **not weak** 11.

1 *k*!

**6. Weak values for arbitrary coupling quantum measurement**

imaginary part of the weak value from the gradient of the readout. This method is also used in Ref. (198) on the joint weak value. It is emphasized that the weak value can be experimentally

We just calculate an arbitrary coupling between the target and the probe systems (93; 120; 199). Throughout this section, we assume that the desired observable is the projection operator to be denoted as *A*<sup>2</sup> = *A* (153). In the case of the von-Neumann interaction motivated by the original work (4), when the pre- and post-selected states are |*i*� and | *f*�, respectively, and the

(−*igAP*ˆ)*<sup>k</sup>*


It is remarked that the desired observable *B*, which satisfies *B*<sup>2</sup> = 1 (93; 120), corresponds to *B* = 2*A* − 1. Analogous to Theorem 4.1, we can derive the expectation values of the position and the momentum after the weak measurement. These quantities depends on the weak value �*A*�*<sup>w</sup>* and the generating function for the position and the momentum of the initial probe state

The decoherence results from the coupled system to the environment and leads to the transition from the quantum to classical systems. The general framework of the decoherence was discussed in Sec. 2. In this section, we discuss the analytical expressions for the weak

While we directly discuss the weak value with decoherence, the weak value is defined as a complex number. To analogously discuss the density operator formalism, we need the

To facilitate the formal development of the weak value, we introduce the ket state |*ψ*(*t*)� and

<sup>11</sup> This point seems to be misunderstood. According to Ref. (134), the violation of the Leggett-Garg inequality (100) was shown, but the macroscopic realism cannot be denied since the noninvasive

operator associated with the weak value. Therefore, we define a *W operator W*(*t*) as

(|0�*<sup>p</sup>* + |1�*p*), realistically, we can evaluate the

� after the interaction given by *Hint* <sup>=</sup> *gAP*<sup>ˆ</sup> becomes

 1 + *A*


*W*(*t*) := *U*(*t*, *ti*)|*i*��*f* |*U*(*tf* , *t*). (74)

*W*(*t*) = |*ψ*(*t*)��*φ*(*t*)|. (76)


 |*i*�|*ξ*�

∞ ∑ *k*=1

1 *k*!

(−*igP*ˆ)*<sup>k</sup>*



<sup>1</sup> <sup>−</sup> *<sup>A</sup>* <sup>+</sup> *Ae*−*igP*<sup>ˆ</sup>

changing the initial probe state from |*ξ*�*<sup>p</sup>* = <sup>√</sup>

probe state is |*ξ*�, the probe state |*ξ*�

1 − *A* + *A*

**7. Weak value with decoherence**


1 − �*A*�*<sup>w</sup>* + �*A*�*we*

∞ ∑ *k*=0

1 *k*!

so that the expression for the W operator simplifies to

(−*igP*ˆ)*<sup>k</sup>*

<sup>−</sup>*igP*<sup>ˆ</sup> 

 1 + ∞ ∑ *k*=1

<sup>−</sup>*igAP*<sup>ˆ</sup>



value.

the bra state �*φ*(*t*)| as

measurability is not realized.

� = �*f* |*e*

= �*f* |

= �*f* |*i*�

By construction, the two states |*ψ*(*t*)� and �*φ*(*t*)| satisfy the Schrödinger equations with the same Hamiltonian with the initial and final conditions |*ψ*(*ti*)� = |*i*� and �*φ*(*tf*)| = �*f* |. In a sense, |*ψ*(*t*)� evolves forward in time while �*φ*(*t*)| evolves backward in time. The time reverse of the W operator (76) is *<sup>W</sup>*† <sup>=</sup> <sup>|</sup>*φ*(*t*)��*ψ*(*t*)|. Thus, we can say the W operator is based on the two-state vector formalism formally described in Refs. (16; 17). Even an apparently similar quantity to the W operator (76) was introduced by Reznik and Aharonov (140) in the name of "two-state" with the conceptually different meaning. This is because the W operator acts on a Hilbert space <sup>H</sup> but the two-state vector acts on the Hilbert space −→H<sup>1</sup> <sup>⊗</sup> ←−H2. Furthermore, while the generalized two-state, which is called a multiple-time state, was introduced (13), this is essentially reduced to the two-state vector formalism. The W operator gives the weak value of the observable *A* <sup>12</sup> as

$$
\langle A \rangle\_W = \frac{\text{Tr}(WA)}{\text{Tr}\, W},
\tag{77}
$$

in parallel with the expectation value of the observable *A* by

$$\operatorname{Ex}[A] = \frac{\operatorname{Tr}(\rho A)}{\operatorname{Tr}\rho} \tag{78}$$

from Born's rule. Furthermore, the W operator (74) can be regarded as a special case of a standard purification of the density operator (185). In our opinion, the W operator should be considered on the same footing of the density operator. For a closed system, both satisfy the Schrödinger equation. In a sense, the W operator *W* is the square root of the density operator since

$$W(t)W^\dagger(t) = |\psi(t)\rangle\langle\psi(t)| = \mathcal{U}(t\_\prime t\_\mathrm{i})|i\rangle\langle i|\mathcal{U}^\dagger(t\_\prime t\_\mathrm{i}),\tag{79}$$

which describes a state evolving forward in time for a given initial state |*ψ*(*ti*)��*ψ*(*ti*)| = |*i*��*i*|, while

$$\mathcal{W}^{\dagger}(t)\mathcal{W}(t) = |\phi(t)\rangle\langle\phi(t)| = \mathcal{U}(t\_f, t)|f\rangle\langle f|\mathcal{U}^{\dagger}(t\_f, t),\tag{80}$$

which describes a state evolving backward in time for a given final state |*φ*(*tf*)��*φ*(*tf*)| = | *f*��*f* |. The W operator describes the entire history of the state from the past (*ti*) to the future (*tf* ) and measurement performed at the time *t*<sup>0</sup> as we shall see in Appendix 4.3. This description is conceptually different from the conventional one by the time evolution of the density operator. From the viewpoint of geometry, the W operator can be taken as the Hilbert-Schmidt bundle. The bundle projection is given by

$$\Pi: \mathcal{W}(t) \to \rho\_l(t) := \mathcal{W}(t)\mathcal{W}^\dagger(t). \tag{81}$$

When the dimension of the Hilbert space is *N*: dimH = *N*, the structure group of this bundle is *U*(*N*) (25, Sec. 9.3). Therefore, the W operator has richer information than the density operator formalism as we shall see a typical example of a geometric phase (155). Furthermore, we can express the probability to get the measurement outcome *an* ∈ *A* due to the ABL formula (28) using the W operator *W* as

$$\Pr[A = a\_{\mathcal{U}}] = \frac{|\operatorname{Tr} WP\_{a\_{\mathcal{U}}}|^2}{\sum\_{n} |\operatorname{Tr} WP\_{a\_{n}}|^2} \tag{82}$$

<sup>12</sup> While the original notation of the weak values is �*A*�*<sup>w</sup>* indicating the "w"eak value of an observable *A*, our notation is motivated by one of which the pre- and post-selected states are explicitly shown as *f*�*A*�*<sup>w</sup> i* .

Similarly to the Kraus operator (16), we define the two operators, *Em* and *F*†

*<sup>m</sup>* = ∑ *m*

using Theorem 7.1 in the last line. By linearity, we obtain the desired result.

*mEm* = 1. As discussed in Eq. (17), the proof goes through as

∑ *m*

<sup>1</sup> <sup>=</sup> Tr(E(*ρ*)) = Tr

∑ *m F*†

the Hawking radiation (55), respectively.

**8. Weak measurement with environment**

decomposition of unity because

*mEm* = ∑ *m*

operator as

∑*<sup>m</sup> E*†

where

∑ *m*

*Em*|*ψ*(*t*)�*s*�*φ*(*t*)|*F*†

*Em*|*ψ*(*t*)�*<sup>s</sup>* :<sup>=</sup> *<sup>e</sup>*�*ψ*˜(*t*)|*sm*�, *<sup>s</sup>*�*φ*(*t*)|*F*†

where <sup>|</sup>*ψ*˜(*t*)�*<sup>e</sup>* and <sup>|</sup>*φ*˜(*t*)�*<sup>e</sup>* are defined in Eq. (86). Therefore, we obtain the change of the W

Theory of "Weak Value" and Quantum Mechanical Measurements 93

Summing up, we have introduced the W operator (74) and obtained the general form of the quantum operation of the W operator (88) in an analogous way to the quantum operation of the density operator assuming the complete positivity of the physical operation. This can be also described from information-theoretical approach (43) to solve the open problem listed in Ref. (13, Sec. XII). However, this geometrical meaning has still been an open problem.

It is well established that the trace preservation, Tr(E(*ρ*)) = Tr *ρ* = 1 for all *ρ*, implies that

because this is the density operator in the time reversed world in the two-state vector

are the evolution operators, which act on H*<sup>s</sup>* ⊗ H*e*. |*ei*� and |*ef*� are some basis vectors and

The above equality (98) may be interpreted as a decomposition of the history in analogy to the

is the S-matrix element. On this idea, Ojima and Englert have developed the formulation on the S-matrix in the context of the algebraic quantum field theory (123) and the backaction of

Let us consider a target system coupled with an environment and a general weak measurement for the compound of the target system and the environment. We assume that

<sup>=</sup> Tr

∑ *m E*† *mEmρ*

*<sup>m</sup>* = *<sup>e</sup>*�*ef* |*V*|*em*�*e*, (96)

*U* = *U*(*t*, *ti*), *V* = *U*(*tf* , *t*), (97)

*<sup>e</sup>*�*ef* |*V*|*em*�*e*�*em*|*U*|*ei*�*<sup>e</sup>* = *<sup>e</sup>*�*ef* |*VU*|*ei*�*e*. (98)

*<sup>e</sup>*�*ef* |*VU*|*ei*�*<sup>e</sup>* = *<sup>e</sup>*�*ef* |*S*|*ei*�*<sup>e</sup>* = *Sf i* (99)

(∀*ρ*). (95)

*mFm* = 1

*EmρE*† *m* 

This argument for the density operator *ρ* = *WW*† applies also for *W*†*W* to obtain ∑*<sup>m</sup> F*†

formulation as reviewed in Sec. 4. Therefore, we can express the Kraus operators,

*Em* <sup>=</sup> *<sup>e</sup>*�*em*|*U*|*ei*�*e*, *<sup>F</sup>*†


*<sup>e</sup>*�*ψ*˜(*t*)|*sm*��*tm*|*φ*˜(*t*)�*<sup>e</sup>* <sup>=</sup> *<sup>e</sup>*�*ψ*˜(*t*)|X |*φ*˜(*t*)�*<sup>e</sup>*

= E (|*ψ*(*t*)�*s*�*φ*(*t*)|), (94)

*<sup>m</sup>*, as

*<sup>m</sup>* :<sup>=</sup> �*tm*|*φ*˜(*t*)�*e*, (93)

where *A* = ∑*<sup>n</sup> an*|*an*��*an*| =: ∑*<sup>n</sup> anPan* . This shows the usefulness of the W operator. Let us discuss a state change in terms of the W operator and define a map X as

$$\mathcal{X}(|\alpha\rangle, |\beta\rangle) := (\mathcal{E} \otimes \mathbf{1}) \left( |\alpha\rangle \langle \beta| \right) , \tag{83}$$

for an arbitrary |*α*�, |*β*�∈H*<sup>s</sup>* ⊗ H*e*. Then, we obtain the following theorem on the change of the W operator such as Theorem 2.1.

**Theorem 7.1.** *For any W operator W* = |*ψ*(*t*)�*s*�*φ*(*t*)|*, we expand*

$$
\langle \psi(t) \rangle\_{\rm s} = \sum\_{m} \psi\_{m} |a\_{m}\rangle\_{\rm s} \, |\phi(t)\rangle\_{\rm s} = \sum\_{m} \phi\_{m} |\beta\_{m}\rangle\_{\rm s} \,. \tag{84}
$$

*with fixed complete orthonormal sets* {|*αm*�*s*} *and* {|*βm*�*s*}*. Then, a change of the W operator can be written as*

$$\mathcal{E}\left(|\psi(t)\rangle\_{s}\langle\phi(t)|\right) = {}\_{\varepsilon}\langle\tilde{\psi}(t)|\mathcal{X}(|a\rangle,|\beta\rangle)|\tilde{\phi}(t)\rangle\_{\varepsilon\prime} \tag{85}$$

*where*

$$
\langle |\tilde{\psi}(t)\rangle\_{\varepsilon} = \sum\_{k} \psi\_{k}^{\*} |a\_{k}\rangle\_{\varepsilon\prime} \,|\tilde{\phi}(t)\rangle\_{\varepsilon} = \sum\_{k} \phi\_{k}^{\*} |\beta\_{k}\rangle\_{\varepsilon\prime} \tag{86}
$$

*and* |*α*� *and* |*β*� *are maximally entangled states defined by*

$$\langle \alpha \rangle := \sum\_{m} |\alpha\_{m}\rangle\_{\mathbf{s}} |\alpha\_{m}\rangle\_{\mathcal{E}'} \ |\mathcal{S}\rangle := \sum\_{m} |\mathcal{S}\_{m}\rangle\_{\mathbf{s}} |\mathcal{S}\_{m}\rangle\_{\mathcal{E}}.\tag{87}$$

*Here,* {|*αm*�*e*} *and* {|*βm*�*e*} *are complete orthonormal sets corresponding to* {|*αm*�*s*} *and* {|*βm*�*s*}*, respectively.*

The proof is completely parallel to that of Theorem 2.1.

**Theorem 7.2.** *For any W operator W* = |*ψ*(*t*)�*s*�*φ*(*t*)|*, given the CP map* E*, the operator-sum representation is written as*

$$\mathcal{E}(\mathcal{W}) = \sum\_{m} E\_{m} W F\_{m\prime}^{\dagger} \tag{88}$$

*where Em and Fm are the Kraus operators.*

It is noted that, in general, <sup>E</sup>(*W*)E(*W*†) �<sup>=</sup> <sup>E</sup>(*ρ*) although *<sup>ρ</sup>* <sup>=</sup> *WW*†.

*Proof.* We take the polar decomposition of the map *X* to obtain

$$\mathcal{X} = \mathcal{K}u,\tag{89}$$

noting that

$$
\mathcal{K}\mathcal{X}^{\dagger} = \mathcal{K}u\iota^{\dagger}\mathcal{K} = \mathcal{K}^2. \tag{90}
$$

The unitary operator *u* is well-defined on H*<sup>s</sup>* ⊗ H*<sup>e</sup>* because K defined in Eq. (4) is positive. This is a crucial point to obtain this result (88), which is the operator-sum representation for the quantum operation of the W operator. From Eq. (10), we can rewrite X as

$$\mathcal{X} = \sum\_{m} |s\_{m}\rangle \langle s\_{m}|u = \sum\_{m} |s\_{m}\rangle \langle t\_{m}| \,\tag{91}$$

where

$$
\langle t\_m \vert = \langle \mathbf{s}\_m \vert \mathfrak{u}.\tag{92}
$$

Similarly to the Kraus operator (16), we define the two operators, *Em* and *F*† *<sup>m</sup>*, as

$$E\_m|\psi(t)\rangle\_s := \,\_\varepsilon\langle \tilde{\psi}(t)|s\_m\rangle,\quad \_s\langle \phi(t)|F\_m^\dagger := \langle t\_m|\tilde{\phi}(t)\rangle\_{\varepsilon\prime} \tag{93}$$

where <sup>|</sup>*ψ*˜(*t*)�*<sup>e</sup>* and <sup>|</sup>*φ*˜(*t*)�*<sup>e</sup>* are defined in Eq. (86). Therefore, we obtain the change of the W operator as

$$\sum\_{m} E\_{m} |\psi(t)\rangle\_{s} \langle \phi(t) | F\_{m}^{\dagger} = \sum\_{m} \, \_{\varepsilon} \langle \tilde{\psi}(t) | s\_{m} \rangle \langle t\_{m} | \tilde{\phi}(t) \rangle\_{\varepsilon} = \, \_{\varepsilon} \langle \tilde{\psi}(t) | \mathcal{X} | \tilde{\phi}(t) \rangle\_{\varepsilon}$$

$$= \mathcal{E} \left( |\psi(t)\rangle\_{s} \langle \phi(t) | \right) \,, \tag{94}$$

using Theorem 7.1 in the last line. By linearity, we obtain the desired result.

Summing up, we have introduced the W operator (74) and obtained the general form of the quantum operation of the W operator (88) in an analogous way to the quantum operation of the density operator assuming the complete positivity of the physical operation. This can be also described from information-theoretical approach (43) to solve the open problem listed in Ref. (13, Sec. XII). However, this geometrical meaning has still been an open problem.

It is well established that the trace preservation, Tr(E(*ρ*)) = Tr *ρ* = 1 for all *ρ*, implies that ∑*<sup>m</sup> E*† *mEm* = 1. As discussed in Eq. (17), the proof goes through as

$$1 = \text{Tr}(\mathcal{E}(\rho)) = \text{Tr}\left(\sum\_{m} E\_{m}\rho E\_{m}^{\dagger}\right) = \text{Tr}\left(\sum\_{m} E\_{m}^{\dagger}E\_{m}\rho\right) \tag{95}$$

This argument for the density operator *ρ* = *WW*† applies also for *W*†*W* to obtain ∑*<sup>m</sup> F*† *mFm* = 1 because this is the density operator in the time reversed world in the two-state vector formulation as reviewed in Sec. 4. Therefore, we can express the Kraus operators,

$$E\_m = \,\_\varepsilon \langle \varepsilon\_m | \mathcal{U} | \varepsilon\_{\dot{\iota}} \rangle\_{\mathcal{E}\prime} \, F\_m^\dagger = \,\_\varepsilon \langle \varepsilon\_{\dot{f}} | V | \varepsilon\_m \rangle\_{\mathcal{E}\prime} \tag{96}$$

where

18 Will-be-set-by-IN-TECH

for an arbitrary |*α*�, |*β*�∈H*<sup>s</sup>* ⊗ H*e*. Then, we obtain the following theorem on the change of

*<sup>ψ</sup>m*|*αm*�*s*, |*φ*(*t*)�*<sup>s</sup>* = ∑

*<sup>k</sup>* <sup>|</sup>*αk*�*e*, <sup>|</sup>*φ*˜(*t*)�*<sup>e</sup>* <sup>=</sup> ∑

*with fixed complete orthonormal sets* {|*αm*�*s*} *and* {|*βm*�*s*}*. Then, a change of the W operator can be*


*Here,* {|*αm*�*e*} *and* {|*βm*�*e*} *are complete orthonormal sets corresponding to* {|*αm*�*s*} *and* {|*βm*�*s*}*,*

**Theorem 7.2.** *For any W operator W* = |*ψ*(*t*)�*s*�*φ*(*t*)|*, given the CP map* E*, the operator-sum*

*m*

The unitary operator *u* is well-defined on H*<sup>s</sup>* ⊗ H*<sup>e</sup>* because K defined in Eq. (4) is positive. This is a crucial point to obtain this result (88), which is the operator-sum representation for


*m*

the quantum operation of the W operator. From Eq. (10), we can rewrite X as

X = ∑ *m*

*EmWF*†

E(*W*) = ∑

It is noted that, in general, <sup>E</sup>(*W*)E(*W*†) �<sup>=</sup> <sup>E</sup>(*ρ*) although *<sup>ρ</sup>* <sup>=</sup> *WW*†.

*Proof.* We take the polar decomposition of the map *X* to obtain

X (|*α*�, |*β*�) := (E ⊗ **1**)(|*α*��*β*|), (83)

*φm*|*βm*�*s*, (84)

*<sup>k</sup>* |*βk*�*e*, (86)


*<sup>m</sup>*, (88)


X = K*u*, (89)

X X † <sup>=</sup> <sup>K</sup>*uu*†<sup>K</sup> <sup>=</sup> <sup>K</sup>2. (90)

�*tm*| = �*sm*|*u*. (92)

*m*

<sup>E</sup> (|*ψ*(*t*)�*s*�*φ*(*t*)|) <sup>=</sup> *<sup>e</sup>*�*ψ*˜(*t*)|X (|*α*�, <sup>|</sup>*β*�)|*φ*˜(*t*)�*e*, (85)

*k φ*∗

*m*

where *A* = ∑*<sup>n</sup> an*|*an*��*an*| =: ∑*<sup>n</sup> anPan* . This shows the usefulness of the W operator. Let us discuss a state change in terms of the W operator and define a map X as

the W operator such as Theorem 2.1.

*written as*

*respectively.*

noting that

where

*representation is written as*

*where Em and Fm are the Kraus operators.*

*where*

**Theorem 7.1.** *For any W operator W* = |*ψ*(*t*)�*s*�*φ*(*t*)|*, we expand*


<sup>|</sup>*ψ*˜(*t*)�*<sup>e</sup>* <sup>=</sup> ∑


*and* |*α*� *and* |*β*� *are maximally entangled states defined by*

The proof is completely parallel to that of Theorem 2.1.

*m*

*k ψ*∗

$$\mathcal{U} = \mathcal{U}(t, t\_i), \; V = \mathcal{U}(t\_f, t), \tag{97}$$

are the evolution operators, which act on H*<sup>s</sup>* ⊗ H*e*. |*ei*� and |*ef*� are some basis vectors and |*em*� is a complete set of basis vectors with ∑*<sup>m</sup>* |*em*��*em*| = 1. We can compute

$$
\sum\_{m} F\_{m}^{\dagger} E\_{m} = \sum\_{m} \left< e\_{f} \middle| V \middle| e\_{m} \right> \left< e\_{m} \middle| \mathcal{U} \middle| e\_{i} \right>\_{\varepsilon} = \left. \left< e\_{f} \middle| V \mathcal{U} \middle| e\_{i} \right>\_{\varepsilon}. \tag{98}
$$

The above equality (98) may be interpreted as a decomposition of the history in analogy to the decomposition of unity because

$$\iota\_{\varepsilon} \langle e\_f | V \mathcal{U} | e\_i \rangle\_{\varepsilon} = \iota\_{\varepsilon} \langle e\_f | S | e\_i \rangle\_{\varepsilon} = \mathcal{S}\_{fi} \tag{99}$$

is the S-matrix element. On this idea, Ojima and Englert have developed the formulation on the S-matrix in the context of the algebraic quantum field theory (123) and the backaction of the Hawking radiation (55), respectively.

#### **8. Weak measurement with environment**

Let us consider a target system coupled with an environment and a general weak measurement for the compound of the target system and the environment. We assume that

Analogous to Theorem 4.1, the shift of the expectation value of the position operator on the

Theory of "Weak Value" and Quantum Mechanical Measurements 95

From an analogous discussion, we obtain the shift of the expectation value of the momentum

Thus, we have shown that the probe shift in the weak measurement is exactly given by the weak value defined by the quantum operation of the W operator due to the environment.

We have reviewed that the weak value is defined independent of the weak measurement in the original idea (4) and have explained its properties. Furthermore, to extract the weak value, we have constructed some measurement model to extract the weak value. I hope that the weak value becomes the fundamental quantity to describe quantum mechanics and quantum field

The author acknowledges useful collaborations and discussion with Akio Hosoya, Yuki Susa, and Shu Tanaka. The author thanks Yakir Aharonov, Richard Jozsa, Sandu Popescu, Aephraim Steinberg, and Jeff Tollaksen for useful discussion. The author would like to thank the use of the utilities of Tokyo Institute of Technology and Massachusetts Institute of Technology and many technical and secretary supports. The author is grateful to the financial supports from JSPS Research Fellowships for Young Scientists (No. 21008624), JSPS Excellent Young Researcher Overseas Visit Program, Global Center of Excellence Program "Nanoscience and

[3] Y. Aharonov, D. Z. Albert, A. Casher, and L. Vaidman, Phys. Lett. A 124, 199 (1987).

[8] Y. Aharonov, A. Botero, S. Popescu, B. Reznik, and J. Tollaksen, Phys. Lett. A 301, 130

[10] Y. Aharonov, S. Massar, S. Popescu, J. Tollaksen, and L. Vaidman, Phys. Rev. Lett. 77,

[11] Y. Aharonov, S. Popescu, D. Rohrlich, and L. Vaidman, Phys. Rev. A 48, 4084 (1993).

[13] Y. Aharonov, S. Popescu, J. Tollaksen, and L. Vaidman, Phys. Rev. A 79, 052110 (2009). [14] Y. Aharonov and D. Rohrlich, *Quantum Paradoxes* (Wiley-VCH, Weibheim, 2005).

[12] Y. Aharonov, S. Popescu, and J. Tollaksen, Physics Today 63, 27 (2010).

[4] Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). [5] Y. Aharonov, J. Anandan, and L. Vaidman, Phys. Rev. A 47, 4616 (1993). [6] Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, Phys. Rev. 134, B1410 (1964).

theory and has practical advantage in the quantum-mechanical world.

Quantum Physics" at Tokyo Institute of Technology during his Ph.D study.

[1] J. Åberg and G. Mitchison, J. Math. Phys. 50, 042103 (2009). [2] G. S. Agarwal and P. K. Pathak, Phys. Rev. A 75, 032108 (2007).

[7] Y. Aharonov and A. Botero, Phys. Rev. A 72, 052111 (2005).

[9] Y. Aharonov and E. Y. Gruss, arXiv:quant-ph/0507269.

*dt*

<sup>Δ</sup>[*P*] = <sup>2</sup>*<sup>g</sup>* · Var[*P*] · Im[�*A*�E(*W*)]. (110)

 *t*=*t*<sup>0</sup>

. (109)

<sup>Δ</sup>[*Q*] = *<sup>g</sup>* · Re[�*A*�E(*W*)] + *mg* · Im[�*A*�E(*W*)] *<sup>d</sup>* Var[*Q*]

probe is

**9. Summary**

operator on the probe as

**10. Acknowledgment**

**11. References**

(2002).

983 (1996).

there is no interaction between the probe and the environment and the same interaction between the target and probe systems (38). The Hamiltonian for the target system and the environment is given by

$$H = H\_0 \otimes \mathbf{1}\_{\mathcal{E}} + H\_{1\prime} \tag{100}$$

where *H*<sup>0</sup> acts on the target system H*<sup>s</sup>* and the identity operator **1***<sup>e</sup>* is for the environment H*e*, while *H*<sup>1</sup> acts on H*<sup>s</sup>* ⊗ H*e*. The evolution operators *U* := *U*(*t*, *ti*) and *V* := *U*(*tf* , *t*) as defined in Eq. (97) can be expressed by

$$\mathcal{U} = \mathcal{U}\_0 \mathcal{K}(t\_0, t\_i), \; V = \mathcal{K}(t\_f, t\_0) V\_{0\prime} \tag{101}$$

where *U*<sup>0</sup> and *V*<sup>0</sup> are the evolution operators forward in time and backward in time, respectively, by the target Hamiltonian *H*0. *K*'s are the evolution operators in the interaction picture,

$$\mathcal{K}(t\_0, t\_i) = \mathcal{T}e^{-i\int\_{t\_i}^{t\_0} dt \mathcal{U}\_0^\dagger H\_1 \mathcal{U}\_0},\\ \mathcal{K}(t\_f, t\_0) = \overline{\mathcal{T}}e^{-i\int\_{t\_0}^{t\_f} dt V\_0 H\_1 V\_0^\dagger},\tag{102}$$

where T and T stand for the time-ordering and anti time-ordering products. Let the initial and final environmental states be |*ei*� and |*ef*�, respectively. The probe state now becomes

$$|\xi'\rangle = \langle f|\langle e\_f|V\mathcal{U}I|e\_i\rangle|i\rangle \left(\mathbf{1} - g\frac{\langle f|\langle e\_f|V\mathcal{U}I|e\_i\rangle|i\rangle}{\langle f|\langle e\_f|V\mathcal{U}I|e\_i\rangle|i\rangle}\hat{P} + \mathcal{O}(g^2)\right)|\xi\rangle. \tag{103}$$

Plugging the expressions for *U* and *V* into the above, we obtain the probe state as

$$|\xi'\rangle = N\xi \left(\mathbf{1} - g \frac{\langle f|\langle e\_f|\mathbf{K}(t\_f, t\_0) V\_0 A \mathbf{J} I\_0 \mathbf{K}(t\_0, t\_i)|e\_i\rangle |i\rangle}{N} \hat{P}\right) |\xi\rangle + O(g^2),\tag{104}$$

where *N* = �*f* |�*ef* |*K*(*tf* , *t*0)*V*0*U*0*K*(*t*0, *ti*)|*ei*�|*i*� is the normalization factor. We define the dual quantum operation as

$$\mathcal{E}^\*(A) := \langle e\_f | K(t\_f, t\_0) V\_0 A L I\_0 K(t\_0, t\_i) | e\_i \rangle = \sum\_m V\_0 F\_m^\dagger A E\_m L I\_0. \tag{105}$$

where

$$F\_m^\dagger := V\_0^\dagger \langle e\_f | K(t\_f, t\_0) | e\_m \rangle V\_{0\prime} \tag{106}$$

$$E\_m := \mathcal{U}l\_0 \langle e\_m | K(t\_0, t\_i) | e\_i \rangle \mathcal{U}\_0^\dagger \tag{107}$$

are the Kraus operators. Here, we have inserted the completeness relation ∑*<sup>m</sup>* |*em*��*em*| = 1 with |*em*� being not necessarily orthogonal. The basis |*ei*� and |*ef*� are the initial and final environmental states, respectively. Thus, we obtain the wave function of the probe as

$$\begin{split} |\xi'\rangle &= N\left(\mathbf{1} - g\frac{\langle f|\mathcal{E}^\*(A)|i\rangle}{N}\hat{P}\right)|\xi\rangle + O(g^2) = N\left(\mathbf{1} - g\frac{\sum\_m \langle f|V\_0\mathcal{F}\_m^\dagger A E\_m \mathcal{U}\_0|i\rangle}{\sum\_m \langle f|V\_0\mathcal{F}\_m^\dagger E\_m \mathcal{U}\_0|i\rangle}\hat{P}\right)|\xi\rangle + O(g^2) \\ &= N\left(\mathbf{1} - g\frac{\text{Tr}\left[A\sum\_m E\_m \mathcal{U}\_0|i\rangle\langle f|V\_0\mathcal{F}\_m^\dagger\right]}{\text{Tr}\left[\sum\_m E\_m \mathcal{U}\_0|i\rangle\langle f|V\_0\mathcal{F}\_m^\dagger\right]}\hat{P}\right)|\xi\rangle + O(g^2) \\ &= N\left(\mathbf{1} - g\frac{\text{Tr}[\mathcal{E}(W)A]}{\text{Tr}[\mathcal{E}(W)]}\hat{P}\right)|\xi\rangle + O(g^2) = N(1 - g\langle A\rangle\_{\mathcal{E}(W)}\hat{P})|\xi\rangle + O(g^2), \end{split} \tag{108}$$

Analogous to Theorem 4.1, the shift of the expectation value of the position operator on the probe is

$$\Delta[\mathcal{Q}] = \mathcal{g} \cdot \text{Re}[\langle A \rangle\_{\mathcal{E}(W)}] + m \mathbf{g} \cdot \text{Im}[\langle A \rangle\_{\mathcal{E}(W)}] \left. \frac{d \operatorname{Var}[\mathcal{Q}]}{dt} \right|\_{t=t\_0} . \tag{109}$$

From an analogous discussion, we obtain the shift of the expectation value of the momentum operator on the probe as

$$
\Delta[P] = 2\text{g} \cdot \text{Var}[P] \cdot \text{Im}[\langle A \rangle\_{\mathcal{E}(W)}].\tag{110}
$$

Thus, we have shown that the probe shift in the weak measurement is exactly given by the weak value defined by the quantum operation of the W operator due to the environment.

#### **9. Summary**

20 Will-be-set-by-IN-TECH

there is no interaction between the probe and the environment and the same interaction between the target and probe systems (38). The Hamiltonian for the target system and the

where *H*<sup>0</sup> acts on the target system H*<sup>s</sup>* and the identity operator **1***<sup>e</sup>* is for the environment H*e*, while *H*<sup>1</sup> acts on H*<sup>s</sup>* ⊗ H*e*. The evolution operators *U* := *U*(*t*, *ti*) and *V* := *U*(*tf* , *t*) as defined

where *U*<sup>0</sup> and *V*<sup>0</sup> are the evolution operators forward in time and backward in time, respectively, by the target Hamiltonian *H*0. *K*'s are the evolution operators in the interaction

Let the initial and final environmental states be |*ei*� and |*ef*�, respectively. The probe state now

�*f* |�*ef* |*K*(*tf* , *t*0)*V*0*AU*0*K*(*t*0, *ti*)|*ei*�|*i*�

where *N* = �*f* |�*ef* |*K*(*tf* , *t*0)*V*0*U*0*K*(*t*0, *ti*)|*ei*�|*i*� is the normalization factor. We define the dual

*Em* :<sup>=</sup> *<sup>U</sup>*0�*em*|*K*(*t*0, *ti*)|*ei*�*U*†

are the Kraus operators. Here, we have inserted the completeness relation ∑*<sup>m</sup>* |*em*��*em*| = 1 with |*em*� being not necessarily orthogonal. The basis |*ei*� and |*ef*� are the initial and final environmental states, respectively. Thus, we obtain the wave function of the probe as

<sup>|</sup>*ξ*� <sup>+</sup> *<sup>O</sup>*(*g*2)

, *K*(*tf* , *t*0) = T *e*

�*f* |�*ef* |*VAU*|*ei*�|*i*�

�*<sup>f</sup>* |�*ef* <sup>|</sup>*VU*|*ei*�|*i*� *<sup>P</sup>*<sup>ˆ</sup> <sup>+</sup> *<sup>O</sup>*(*g*2)

*m V*0*F*†

**<sup>1</sup>** <sup>−</sup> *<sup>g</sup>* <sup>∑</sup>*m*�*<sup>f</sup>* <sup>|</sup>*V*0*F*†

<sup>∑</sup>*m*�*<sup>f</sup>* <sup>|</sup>*V*0*F*†

<sup>|</sup>*ξ*� <sup>+</sup> *<sup>O</sup>*(*g*2) = *<sup>N</sup>*(**<sup>1</sup>** <sup>−</sup> *<sup>g</sup>*�*A*�E(*W*)*P*ˆ)|*ξ*� <sup>+</sup> *<sup>O</sup>*(*g*2), (108)

*<sup>N</sup> <sup>P</sup>*<sup>ˆ</sup>

*H* = *H*<sup>0</sup> ⊗ **1***<sup>e</sup>* + *H*1, (100)

*U* = *U*0*K*(*t*0, *ti*), *V* = *K*(*tf* , *t*0)*V*0, (101)

−*i tf <sup>t</sup>*<sup>0</sup> *dtV*0*H*1*V*†

<sup>0</sup> , (102)


<sup>|</sup>*ξ*� <sup>+</sup> *<sup>O</sup>*(*g*2), (104)

*<sup>m</sup> AEmU*0, (105)

<sup>|</sup>*ξ*� <sup>+</sup> *<sup>O</sup>*(*g*2)

<sup>0</sup> (107)

*<sup>m</sup> AEmU*0|*i*�

*mEmU*0|*i*� *<sup>P</sup>*<sup>ˆ</sup>

<sup>0</sup> �*ef* |*K*(*tf* , *t*0)|*em*�*V*0, (106)

environment is given by

picture,

becomes

where


in Eq. (97) can be expressed by


� = *Nξ*

 **1** − *g*

�*f* |E∗(*A*)|*i*� *<sup>N</sup> <sup>P</sup>*<sup>ˆ</sup>

Tr[E(*W*)*A*] Tr[E(*W*)] *<sup>P</sup>*<sup>ˆ</sup>

Tr

Tr


quantum operation as

 **1** − *g*

= *N* **1** − *g*

= *N* **1** − *g* *K*(*t*0, *ti*) = T *e*

� = �*f* |�*ef* |*VU*|*ei*�|*i*�

−*i <sup>t</sup>*<sup>0</sup> *t <sup>i</sup> dtU*† <sup>0</sup> *H*1*U*<sup>0</sup>

where T and T stand for the time-ordering and anti time-ordering products.

 **1** − *g*

Plugging the expressions for *U* and *V* into the above, we obtain the probe state as

E∗(*A*) := �*ef* |*K*(*tf* , *<sup>t</sup>*0)*V*0*AU*0*K*(*t*0, *ti*)|*ei*� = ∑

<sup>|</sup>*ξ*� <sup>+</sup> *<sup>O</sup>*(*g*2) = *<sup>N</sup>*

*m* 

*m P*ˆ 

*F*† *<sup>m</sup>* := *<sup>V</sup>*†

*<sup>A</sup>* <sup>∑</sup>*<sup>m</sup> EmU*0|*i*��*<sup>f</sup>* <sup>|</sup>*V*0*F*†

<sup>∑</sup>*<sup>m</sup> EmU*0|*i*��*<sup>f</sup>* <sup>|</sup>*V*0*F*†

We have reviewed that the weak value is defined independent of the weak measurement in the original idea (4) and have explained its properties. Furthermore, to extract the weak value, we have constructed some measurement model to extract the weak value. I hope that the weak value becomes the fundamental quantity to describe quantum mechanics and quantum field theory and has practical advantage in the quantum-mechanical world.

#### **10. Acknowledgment**

The author acknowledges useful collaborations and discussion with Akio Hosoya, Yuki Susa, and Shu Tanaka. The author thanks Yakir Aharonov, Richard Jozsa, Sandu Popescu, Aephraim Steinberg, and Jeff Tollaksen for useful discussion. The author would like to thank the use of the utilities of Tokyo Institute of Technology and Massachusetts Institute of Technology and many technical and secretary supports. The author is grateful to the financial supports from JSPS Research Fellowships for Young Scientists (No. 21008624), JSPS Excellent Young Researcher Overseas Visit Program, Global Center of Excellence Program "Nanoscience and Quantum Physics" at Tokyo Institute of Technology during his Ph.D study.

#### **11. References**


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**0**

**5**

*USA*

**Quantum Mechanics**

and Donald J. Kouri *University of Houston*

**Generalized Non-Relativistic Supersymmetric**

Symmetry has long been recognized as a powerful formal and computational tool in quantum mechanics, beginning with the seminal work of Wigner Wigner & Fano (1960) and Weyl Weyl (1950). Indeed, it is well understood that so-called "accidental degeneracies" were, in fact, not accidents at all but rather the result of "hidden symmetry" (*e.g.* the 2*l*+1 degeneracy of the hydrogen atom energy states ). Because of this fundamental role in quantum mechanics, the discovery of new symmetries (and their possible "breaking" by interactions) is of enormous interest Griffiths (1987). In the latter half of the 20*th* century, a new hidden symmetry was discovered that led to much speculation in relativistic quantum field theory as applied to elementary particles. The essence of this so-called "supersymmetry" (SUSY) is that for every boson, there is also a fermion of the same mass (energy) and vice versa. Of course, this has not been observed in nature, leading to speculation that there exists some interaction in nature that "breaks" the symmetry. On the other hand, there is also substantial opinion held by many

This chapter is not aimed at addressing such issues. Rather, it was observed by many Junker (1996) that one did not have to deal with quantum field theory to encounter SUSY. Indeed, SUSY is also an intrinsic feature of ordinary, non-relativistic quantum mechanics (SUSY-QM). In this case, attention has not been focused on whether the symmetry exists in nature. Instead, it has been used primarily as a pedagogical tool. The reason for this lies in the intimate connection of SUSY-QM and the ladder operator approach to the harmonic oscillator, angular

The essence of SUSY-QM is the factorization of the Hamiltonian for a one dimensional system in analogy with the harmonic oscillator. For most bound-state quantum systems, it is possible to define operators analogous to the lowering (*a*ˆ) and raising (*a*ˆ†) operators that factor the harmonic oscillator Hamiltonian. However, in the general case, these operators do not possess all the properties of the harmonic oscillator *a*ˆ and *a*ˆ†, but rather they behave as so-called "charge operators". As such, the SUSY charge operators not only allow factorization of the one dimensional Hamiltonian, they form a Lie algebraic structure. This structure results in the generation of isospectral "sector Hamiltonians". Unfortunately, almost all previous research concentrated on exactly soluble, one dimensional model systems. We became interested in the possibility of taking computational advantage of SUSY. Our idea was that symmetry in QM has long been known to lead to significant computational simplifications and advantages. We

physicists that SUSY has no connection to physical reality.

momentum, and the hydrogen atom Dirac (1958).

**1. Introduction**

Thomas L. Markovich, Mason T. Biamonte, Eric R. Bittner


## **Generalized Non-Relativistic Supersymmetric Quantum Mechanics**

Thomas L. Markovich, Mason T. Biamonte, Eric R. Bittner and Donald J. Kouri *University of Houston USA*

#### **1. Introduction**

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> Symmetry has long been recognized as a powerful formal and computational tool in quantum mechanics, beginning with the seminal work of Wigner Wigner & Fano (1960) and Weyl Weyl (1950). Indeed, it is well understood that so-called "accidental degeneracies" were, in fact, not accidents at all but rather the result of "hidden symmetry" (*e.g.* the 2*l*+1 degeneracy of the hydrogen atom energy states ). Because of this fundamental role in quantum mechanics, the discovery of new symmetries (and their possible "breaking" by interactions) is of enormous interest Griffiths (1987). In the latter half of the 20*th* century, a new hidden symmetry was discovered that led to much speculation in relativistic quantum field theory as applied to elementary particles. The essence of this so-called "supersymmetry" (SUSY) is that for every boson, there is also a fermion of the same mass (energy) and vice versa. Of course, this has not been observed in nature, leading to speculation that there exists some interaction in nature that "breaks" the symmetry. On the other hand, there is also substantial opinion held by many physicists that SUSY has no connection to physical reality.

> This chapter is not aimed at addressing such issues. Rather, it was observed by many Junker (1996) that one did not have to deal with quantum field theory to encounter SUSY. Indeed, SUSY is also an intrinsic feature of ordinary, non-relativistic quantum mechanics (SUSY-QM). In this case, attention has not been focused on whether the symmetry exists in nature. Instead, it has been used primarily as a pedagogical tool. The reason for this lies in the intimate connection of SUSY-QM and the ladder operator approach to the harmonic oscillator, angular momentum, and the hydrogen atom Dirac (1958).

> The essence of SUSY-QM is the factorization of the Hamiltonian for a one dimensional system in analogy with the harmonic oscillator. For most bound-state quantum systems, it is possible to define operators analogous to the lowering (*a*ˆ) and raising (*a*ˆ†) operators that factor the harmonic oscillator Hamiltonian. However, in the general case, these operators do not possess all the properties of the harmonic oscillator *a*ˆ and *a*ˆ†, but rather they behave as so-called "charge operators". As such, the SUSY charge operators not only allow factorization of the one dimensional Hamiltonian, they form a Lie algebraic structure. This structure results in the generation of isospectral "sector Hamiltonians". Unfortunately, almost all previous research concentrated on exactly soluble, one dimensional model systems. We became interested in the possibility of taking computational advantage of SUSY. Our idea was that symmetry in QM has long been known to lead to significant computational simplifications and advantages. We

The Hamiltonian operator now can be factored in the form

<sup>1</sup> (*x*) <sup>−</sup> *dW*<sup>1</sup>

*<sup>Q</sup>*<sup>1</sup> <sup>=</sup> *<sup>d</sup>*

Then the "first sector" Hamiltonian is defined as

We then apply *Q*<sup>1</sup> to the equation, to obtain

Application of *Q*†, then implies that *Q*†

which shows that the quantity *Q*1*ψ*<sup>1</sup>

Because of the uniqueness of the *E*<sup>1</sup>

must be modified. Consider

<sup>0</sup> ≡ 0. So

and we conclude that

sector. Indeed, Equation (2.13) indicates that such a *ψ*<sup>2</sup>

*Q*† 1*Q*1(*Q*† 1*ψ*<sup>1</sup>

Thus, *Q*1*ψ*<sup>1</sup>

we recall that

Then

since *Qψ*<sup>1</sup>

the eigenstates of H2:

ground state, since the *E*<sup>1</sup>

Then it follows that for *n* > 0, (since for *n* = 0, *E*<sup>0</sup> = 0),

*dx* <sup>=</sup>

 − *d*

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 103

We define the "charge" operator and its adjoint by (assuming *W*<sup>1</sup> is hermitian; *i.e.*, *ψ*<sup>1</sup>

*dx* <sup>+</sup> *<sup>W</sup>*1, *<sup>Q</sup>*†

*Q*† 1*Q*1*ψ*<sup>1</sup>

1(*Q*1*ψ*<sup>1</sup>

*<sup>n</sup>* = *<sup>Q</sup>*1*Q*†

1*ψ*<sup>2</sup>

*Q*1*ψ*<sup>1</sup>

*Q*1*Q*† 1*ψ*<sup>2</sup> *<sup>n</sup>* = *<sup>E</sup>*<sup>2</sup>

> *E*2 *<sup>n</sup>* = *<sup>E</sup>*<sup>1</sup>

*<sup>n</sup>*+1) = *<sup>E</sup>*<sup>1</sup>

It follows that the Hamiltonians H<sup>1</sup> and H<sup>2</sup> have identical spectra with the exception of the

1*ψ*<sup>2</sup>

1*Q*1)(*Q*†

*Q*1*Q*†

*<sup>n</sup>* is an eigenstate of <sup>H</sup><sup>2</sup> with the same energy, *<sup>E</sup>*<sup>1</sup>

<sup>H</sup>2*ψ*<sup>2</sup>

(*Q*†

<sup>H</sup><sup>1</sup> <sup>=</sup> *<sup>Q</sup>*†

*<sup>n</sup>* = *<sup>E</sup>*<sup>1</sup>

*<sup>n</sup>*) = *<sup>E</sup>*<sup>1</sup>

1*ψ*<sup>2</sup> *<sup>n</sup>* = *<sup>E</sup>*<sup>2</sup>

*<sup>n</sup>*) = *<sup>E</sup>*<sup>2</sup>

*<sup>n</sup>* is an eigenstate of H1:

*nψ*<sup>2</sup>

*n*+1(*Q*† 1*ψ*<sup>1</sup>

*nQ*† 1*ψ*<sup>2</sup>

*dx* <sup>+</sup> *<sup>W</sup>*1(*x*)

<sup>1</sup> <sup>=</sup> <sup>−</sup> *<sup>d</sup>*

*nψ*<sup>1</sup>

*nQ*1*ψ*<sup>1</sup>

*nψ*<sup>2</sup>

<sup>0</sup> <sup>=</sup> 0 wave function is unique. In the case of the ground state *<sup>ψ</sup>*<sup>1</sup>

*<sup>n</sup>*, as the state *ψ*<sup>1</sup>

*d*

*dx* <sup>+</sup> *<sup>W</sup>*1(*x*)

*dx* <sup>+</sup> *<sup>W</sup>*<sup>1</sup> (2.7)

<sup>1</sup>*Q*<sup>1</sup> (2.8)

*<sup>n</sup>* (2.9)

*<sup>n</sup>* (2.10)

*<sup>n</sup>*. (2.11)

*<sup>n</sup>* (2.12)

<sup>0</sup> = 0 (2.13)

*<sup>n</sup>* (2.14)

*<sup>n</sup>*+<sup>1</sup> (2.16)

*<sup>n</sup>*+1) (2.15)

<sup>0</sup> cannot be used to generate the ground state of the second

<sup>0</sup> = 0 state, the indexing of the first and second sector levels

<sup>0</sup> would vanish identically.

*<sup>n</sup>*. Similarly, consider

0,

(2.6)

<sup>0</sup> is real)

− *d*2 *dx*<sup>2</sup> <sup>+</sup> *<sup>W</sup>*<sup>2</sup>

therefore asked whether this could be the case for SUSY. It turns out that SUSY does, in fact, lead to significant computational advantages Kouri et al. (2010a); Kouri, Markovich, Maxwell & Bittner (2009); Kouri, Markovich, Maxwell & Bodman (2009). In particular, the structure of the degeneracies between sector Hamiltonians makes it possible to achieve significant progress in more accurate calculations of excited state energies and wave functions. Below we outline how the theory can be used as a new computational tool, first for one dimension and later for higher dimensional systems. In addition, we also introduce an entirely new class of system dependent coherent states.

There have been a number of suggested generalizations of SUSY-QM to treat more than one dimensional systems Andrianov et al. (1985); Andrianov, Borisov & Ioffe (1984a;b;c); Andrianov et al. (1986); Andrianov, Borisov, Ioffe & Eides (1984); Andrianov & Ioffe (1988); Andrianov et al. (2002); Cannata et al. (2002); Das & Pernice (1996). For the most part, these have involved the introduction of new "spin-like" variables. One early study instead introduced tensorial operators Stedman (1985), but at the cost of seriously affecting the nature of the energy level degeneracies. In addition, in the tensorial operator approach, he did not consider the application to any system in detail other than writing down the equations for the hydrogen atom without exploring their solutions. In the following sections, we present our generalization of SUSY-QM to allow the treatment of multi-particle, multi-dimensional systems. These include clusters of distinguishable particles and the electronic structure of atoms.

#### **2. Introduction to supersymmetric quantum mechanics in one dimension**

The general starting point is to define the so-called "superpotential", usually denoted as *W*. In the theory, *W* is related to the ground state wave function through the well-known Riccati substitution Jafarpour & Afshar (2002):

$$\psi\_0^1(\mathbf{x}) = \text{Ne}^{-\int\_0^\mathbf{x} \mathcal{W}(\mathbf{x}') d\mathbf{x}'}.\tag{2.1}$$

The relationship between the superpotential *W* and the physical interaction *V*(*x*) results from assuming that Equation (2.1) solves the standard Schrödinger equation with energy zero. This does not impose any restriction since the energy can be changed by adding any constant to the Hamiltonian. Thus,

$$-\frac{\hbar^2}{2m}\frac{d^2\psi\_0^1}{dx^2} + V\_1\psi\_0^1 \tag{2.2}$$

If we solve for *W*<sup>1</sup> in Equation (2.1), we find that

$$\mathcal{W}\_1 = -\frac{\frac{d\psi\_0^1}{dx}}{\psi\_0^1} = -\frac{d}{dx}\ln\psi\_0^1 \tag{2.3}$$

and, if *W*<sup>1</sup> is known, *V*<sup>1</sup> is given by

$$V\_1(\mathbf{x}) = \frac{\hbar^2}{2m} \left( W\_1^2(\mathbf{x}) - \frac{dW\_1}{d\mathbf{x}} \right) \tag{2.4}$$

It is then evident that

$$-\frac{d^2\psi\_0^1}{dx^2} + \left(W\_1^2(x) - \frac{dW\_1}{dx}\right)\psi\_0^1 = 0\tag{2.5}$$

The Hamiltonian operator now can be factored in the form

$$-\frac{d^2}{dx^2} + \mathcal{W}\_1^2(\mathbf{x}) - \frac{d\mathcal{W}\_1}{d\mathbf{x}} = \left[ -\frac{d}{d\mathbf{x}} + \mathcal{W}\_1(\mathbf{x}) \right] \left[ \frac{d}{d\mathbf{x}} + \mathcal{W}\_1(\mathbf{x}) \right] \tag{2.6}$$

We define the "charge" operator and its adjoint by (assuming *W*<sup>1</sup> is hermitian; *i.e.*, *ψ*<sup>1</sup> <sup>0</sup> is real)

$$Q\_1 = \frac{d}{d\mathbf{x}} + W\_{1\prime} \ Q\_1^{\dagger} = -\frac{d}{d\mathbf{x}} + W\_1 \tag{2.7}$$

Then the "first sector" Hamiltonian is defined as

$$\mathcal{H}\_1 = Q\_1^\dagger Q\_1 \tag{2.8}$$

Then it follows that for *n* > 0, (since for *n* = 0, *E*<sup>0</sup> = 0),

$$Q\_1^\dagger Q\_1 \psi\_n^1 = \, \_nE\_n^1 \psi\_n^1 \tag{2.9}$$

We then apply *Q*<sup>1</sup> to the equation, to obtain

$$\left(Q\_1 Q\_1^\dagger (Q\_1 \psi\_n^1)\right) = \left. E\_n^1 Q\_1 \psi\_n^1 \right. \tag{2.10}$$

Thus, *Q*1*ψ*<sup>1</sup> *<sup>n</sup>* is an eigenstate of <sup>H</sup><sup>2</sup> with the same energy, *<sup>E</sup>*<sup>1</sup> *<sup>n</sup>*, as the state *ψ*<sup>1</sup> *<sup>n</sup>*. Similarly, consider the eigenstates of H2:

$$
\mathcal{H}\_2 \psi\_n^2 = Q\_1 Q\_1^\dagger \psi\_n^2 = E\_n^2 \psi\_n^2. \tag{2.11}
$$

Application of *Q*†, then implies that *Q*† 1*ψ*<sup>2</sup> *<sup>n</sup>* is an eigenstate of H1:

$$(Q\_1^\dagger Q\_1)(Q\_1^\dagger \psi\_n^2) \quad = E\_n^2 Q\_1^\dagger \psi\_n^2 \tag{2.12}$$

It follows that the Hamiltonians H<sup>1</sup> and H<sup>2</sup> have identical spectra with the exception of the ground state, since the *E*<sup>1</sup> <sup>0</sup> <sup>=</sup> 0 wave function is unique. In the case of the ground state *<sup>ψ</sup>*<sup>1</sup> 0, we recall that

$$Q\_1 \psi\_0^1 = \begin{array}{c} 0 \end{array} \tag{2.13}$$

which shows that the quantity *Q*1*ψ*<sup>1</sup> <sup>0</sup> cannot be used to generate the ground state of the second sector. Indeed, Equation (2.13) indicates that such a *ψ*<sup>2</sup> <sup>0</sup> would vanish identically.

Because of the uniqueness of the *E*<sup>1</sup> <sup>0</sup> = 0 state, the indexing of the first and second sector levels must be modified. Consider

$$Q\_1 Q\_1^\dagger \psi\_n^2 = E\_n^2 \psi\_n^2 \tag{2.14}$$

Then

2

atoms.

of system dependent coherent states.

substitution Jafarpour & Afshar (2002):

If we solve for *W*<sup>1</sup> in Equation (2.1), we find that

and, if *W*<sup>1</sup> is known, *V*<sup>1</sup> is given by

the Hamiltonian. Thus,

It is then evident that

therefore asked whether this could be the case for SUSY. It turns out that SUSY does, in fact, lead to significant computational advantages Kouri et al. (2010a); Kouri, Markovich, Maxwell & Bittner (2009); Kouri, Markovich, Maxwell & Bodman (2009). In particular, the structure of the degeneracies between sector Hamiltonians makes it possible to achieve significant progress in more accurate calculations of excited state energies and wave functions. Below we outline how the theory can be used as a new computational tool, first for one dimension and later for higher dimensional systems. In addition, we also introduce an entirely new class

There have been a number of suggested generalizations of SUSY-QM to treat more than one dimensional systems Andrianov et al. (1985); Andrianov, Borisov & Ioffe (1984a;b;c); Andrianov et al. (1986); Andrianov, Borisov, Ioffe & Eides (1984); Andrianov & Ioffe (1988); Andrianov et al. (2002); Cannata et al. (2002); Das & Pernice (1996). For the most part, these have involved the introduction of new "spin-like" variables. One early study instead introduced tensorial operators Stedman (1985), but at the cost of seriously affecting the nature of the energy level degeneracies. In addition, in the tensorial operator approach, he did not consider the application to any system in detail other than writing down the equations for the hydrogen atom without exploring their solutions. In the following sections, we present our generalization of SUSY-QM to allow the treatment of multi-particle, multi-dimensional systems. These include clusters of distinguishable particles and the electronic structure of

**2. Introduction to supersymmetric quantum mechanics in one dimension**

<sup>0</sup>(*x*) = *Ne*<sup>−</sup> *<sup>x</sup>*

*d*2*ψ*<sup>1</sup> 0 *dx*<sup>2</sup> <sup>+</sup> *<sup>V</sup>*1*ψ*<sup>1</sup>

> *dψ*<sup>1</sup> 0 *dx ψ*1 0

2*m W*<sup>2</sup>

 *W*<sup>2</sup> <sup>=</sup> <sup>−</sup> *<sup>d</sup>*

<sup>1</sup> (*x*) <sup>−</sup> *dW*<sup>1</sup> *dx*

*dx* ln *<sup>ψ</sup>*<sup>1</sup>

<sup>1</sup> (*x*) <sup>−</sup> *dW*<sup>1</sup> *dx*

> *ψ*1

*ψ*1

− *h*¯ 2 2*m*

*W*<sup>1</sup> = −

*<sup>V</sup>*1(*x*) = *<sup>h</sup>*¯ <sup>2</sup>

<sup>−</sup> *<sup>d</sup>*2*ψ*<sup>1</sup> 0 *dx*<sup>2</sup> <sup>+</sup>

The general starting point is to define the so-called "superpotential", usually denoted as *W*. In the theory, *W* is related to the ground state wave function through the well-known Riccati

The relationship between the superpotential *W* and the physical interaction *V*(*x*) results from assuming that Equation (2.1) solves the standard Schrödinger equation with energy zero. This does not impose any restriction since the energy can be changed by adding any constant to

<sup>0</sup> *W*(*x*� ) *dx*�

. (2.1)

<sup>0</sup> = 0 (2.2)

<sup>0</sup> (2.3)

<sup>0</sup> = 0 (2.5)

(2.4)

$$(Q\_1^\dagger Q\_1 (Q\_1^\dagger \psi\_{n+1}^1) \ = E\_{n+1}^1 (Q\_1^\dagger \psi\_{n+1}^1) \tag{2.15}$$

since *Qψ*<sup>1</sup> <sup>0</sup> ≡ 0. So

$$E\_n^2 = E\_{n+1}^1 \tag{2.16}$$

and we conclude that

Fig. 1. Above, we display graphically how this hierarchical degenarcy is realized.

*W*(*x*) =

*djdj*� *<sup>x</sup>*2(*j*+*<sup>j</sup>*

*dx* <sup>+</sup> *<sup>W</sup>*<sup>1</sup> and *<sup>Q</sup>*†

<sup>0</sup> (*x*) = *Ne*<sup>−</sup> <sup>∑</sup>*<sup>J</sup>*

*Q*†*ψ*<sup>2</sup>

We stress that contrary to the periodic case Kouri, Markovich, Maxwell & Bodmann (2009),

is not allowed because it is not normalizable. Thus, the ground state for the second sector

Then the corresponding sector one potential, *V*1(*x*), is

*J* ∑ *j*=0

*<sup>Q</sup>*<sup>1</sup> <sup>=</sup> *<sup>d</sup>*

*J* ∑ *j*�=0

Then the first sector ground state for a general member of this family is

*ψ*+

*V*1(*x*) =

The "charge" operators are given by

the solution of the sector two equation,

satisfies

sector Hamiltonian, H*j*, *j* > 1.

functions for H<sup>1</sup> involves solving for the ground state energies and wave functions for each

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 105

For a more concrete example, we present a general family of anharmonic oscillator. The ubiquity of this example in chemistry and physics is best exemplified by the fact that all nuclear vibrations in molecules are anharmonic, with the effect increasing as the vibrational energy gets closer to the dissociation limit Wilson et al. (1955). Additionally, anharmonicity is also present when studying the effects of rotation, through the centrifugal potential. For the one dimensional case, we consider an oscillator on the domain −∞ < *x* < ∞. In order to have potentials that are guaranteed to possess bound states, we shall postulate a superpotential

> *J* ∑ *j*=0

> > � <sup>+</sup>1) <sup>−</sup>

*J* ∑ *j* = 0

<sup>1</sup> <sup>=</sup> <sup>−</sup> *<sup>d</sup>*

*j*=0 *dj <sup>x</sup>*2*j*+<sup>2</sup>

*djx*<sup>2</sup>*j*+<sup>1</sup> (2.30)

*dJ*(2*j* + 1)*x*2*<sup>j</sup>* (2.31)

*dx* <sup>+</sup> *<sup>W</sup>*1. (2.32)

(2*j*+2) (2.33)

<sup>0</sup>(*x*) = 0 (2.34)

$$
\psi\_n^2 = \frac{Q\_1 \psi\_{n+1}^1}{\sqrt{E\_{n+1}^1}} \text{ and } \psi\_{n+1}^1 = \frac{Q\_1^\dagger \psi\_n^2}{\sqrt{E\_n^2}} \tag{2.17}
$$

The next step in building a hierarchy of isospectral Hamiltonians is to define a second superpotential, *W*2, according to

$$W\_2 = -\frac{d}{d\chi} \ln \psi\_0^2\tag{2.18}$$

in much the same way as we did before. It is then clear that we can define an alternate form for H2, given by

$$\mathcal{H}\_2 = Q\_2^\dagger Q\_2 + E\_{0\prime}^2 \tag{2.19}$$

where

$$Q\_2 = \frac{d}{d\mathbf{x}} + W\_2 \tag{2.20}$$

We observe that *ψ*<sup>2</sup> <sup>0</sup> is automatically an eigenstate of this form for H<sup>2</sup>

$$\mathcal{H}\_2 \psi\_0^2 = E\_0^2 \psi\_0^2 \tag{2.21}$$

Next consider the first excited state eigenvalue equation for the second sector:

$$\mathcal{H}\_2 \psi\_1^2 = E\_1^2 \psi\_1^2 \tag{2.22}$$

We apply *Q*<sup>2</sup> to Equation (2.22) to find

$$\left(Q\_2\mathbf{Q}\_2^\dagger + E\_0^2\right)Q\_2\boldsymbol{\psi}\_1^2 = E\_n^2Q\_2\boldsymbol{\psi}\_1^2\tag{2.23}$$

Then, by similar reasoning, we deduce that

$$Q\_2 \psi\_1^2 = \sqrt{E\_1^2 - E\_0^2} \psi\_0^3. \tag{2.24}$$

Using the new charge operators *Q*<sup>2</sup> and *Q*† <sup>2</sup>, we then define the third sector Hamiltonian,

$$\mathcal{H}\_3 = Q\_2 Q\_2^\dagger + E\_{0'}^2 \tag{2.25}$$

with ground state equation

$$\mathcal{H}\_3 \psi\_0^3 = E\_0^3 \psi\_0^3 \tag{2.26}$$

It follows that

$$Q\_2^\dagger \psi\_0^3 = \sqrt{E\_1^2 - E\_0^2} \psi\_1^2 \tag{2.27}$$

and,

$$Q\_2 Q\_2^\dagger \psi\_0^3 = \left(E\_1^2 - E\_0^2\right) \psi\_0^3 = E\_0^3 \psi\_0^3\tag{2.28}$$

Thus, we conclude that

$$E\_0^3 = E\_1^2 - E\_0^2 \tag{2.29}$$

It is clear that this procedure can be continued until one exhausts the number of bound excited states supported by H1. We also see that determining the excited state energies and wave 4

*ψ*2

superpotential, *W*2, according to

We apply *Q*<sup>2</sup> to Equation (2.22) to find

Then, by similar reasoning, we deduce that

Using the new charge operators *Q*<sup>2</sup> and *Q*†

with ground state equation

Thus, we conclude that

It follows that

and,

for H2, given by

We observe that *ψ*<sup>2</sup>

where

*<sup>n</sup>* <sup>=</sup> *<sup>Q</sup>*1*ψ*<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> *E*1 *n*+1

and *ψ*<sup>1</sup>

*dx* ln *<sup>ψ</sup>*<sup>2</sup>

<sup>2</sup>*Q*<sup>2</sup> <sup>+</sup> *<sup>E</sup>*<sup>2</sup>

The next step in building a hierarchy of isospectral Hamiltonians is to define a second

in much the same way as we did before. It is then clear that we can define an alternate form

*<sup>W</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup> *<sup>d</sup>*

<sup>H</sup><sup>2</sup> <sup>=</sup> *<sup>Q</sup>*†

*<sup>Q</sup>*<sup>2</sup> <sup>=</sup> *<sup>d</sup>*

<sup>0</sup> is automatically an eigenstate of this form for H<sup>2</sup>

<sup>0</sup> <sup>=</sup> *<sup>E</sup>*<sup>2</sup> 0*ψ*<sup>2</sup>

<sup>1</sup> <sup>=</sup> *<sup>E</sup>*<sup>2</sup> 1*ψ*<sup>2</sup>

> *E*2 <sup>1</sup> <sup>−</sup> *<sup>E</sup>*<sup>2</sup> 0*ψ*3

<sup>H</sup><sup>3</sup> <sup>=</sup> *<sup>Q</sup>*2*Q*†

<sup>0</sup> <sup>=</sup> *<sup>E</sup>*<sup>3</sup> 0*ψ*<sup>3</sup>

> *E*2 <sup>1</sup> <sup>−</sup> *<sup>E</sup>*<sup>2</sup> 0*ψ*2

> > <sup>1</sup> <sup>−</sup> *<sup>E</sup>*<sup>2</sup>

It is clear that this procedure can be continued until one exhausts the number of bound excited states supported by H1. We also see that determining the excited state energies and wave

<sup>H</sup>3*ψ*<sup>3</sup>

*E*3 <sup>0</sup> <sup>=</sup> *<sup>E</sup>*<sup>2</sup> <sup>1</sup> <sup>=</sup> *<sup>E</sup>*<sup>2</sup>

<sup>2</sup> <sup>+</sup> *<sup>E</sup>*<sup>2</sup>

*nQ*2*ψ*<sup>2</sup>

<sup>2</sup>, we then define the third sector Hamiltonian,

<sup>H</sup>2*ψ*<sup>2</sup>

<sup>H</sup>2*ψ*<sup>2</sup>

<sup>2</sup> <sup>+</sup> *<sup>E</sup>*<sup>2</sup> 0 *Q*2*ψ*<sup>2</sup>

*Q*2*ψ*<sup>2</sup> <sup>1</sup> =

*Q*† 2*ψ*<sup>3</sup> <sup>0</sup> =

*Q*2*Q*† 2*ψ*<sup>3</sup> <sup>0</sup> = *E*2 <sup>1</sup> <sup>−</sup> *<sup>E</sup>*<sup>2</sup> 0 *ψ*3 <sup>0</sup> <sup>=</sup> *<sup>E</sup>*<sup>3</sup> 0*ψ*<sup>3</sup>

Next consider the first excited state eigenvalue equation for the second sector:

 *Q*2*Q*† *<sup>n</sup>*+<sup>1</sup> <sup>=</sup> *<sup>Q</sup>*†

1*ψ*<sup>2</sup> *<sup>n</sup> E*2 *n*

<sup>0</sup> (2.18)

<sup>0</sup>, (2.19)

<sup>0</sup> (2.21)

<sup>1</sup> (2.22)

<sup>1</sup> (2.23)

<sup>0</sup>. (2.24)

<sup>0</sup>, (2.25)

<sup>0</sup> (2.26)

<sup>1</sup> (2.27)

<sup>0</sup> (2.29)

<sup>0</sup> (2.28)

*dx* <sup>+</sup> *<sup>W</sup>*<sup>2</sup> (2.20)

(2.17)

Fig. 1. Above, we display graphically how this hierarchical degenarcy is realized.

functions for H<sup>1</sup> involves solving for the ground state energies and wave functions for each sector Hamiltonian, H*j*, *j* > 1.

For a more concrete example, we present a general family of anharmonic oscillator. The ubiquity of this example in chemistry and physics is best exemplified by the fact that all nuclear vibrations in molecules are anharmonic, with the effect increasing as the vibrational energy gets closer to the dissociation limit Wilson et al. (1955). Additionally, anharmonicity is also present when studying the effects of rotation, through the centrifugal potential. For the one dimensional case, we consider an oscillator on the domain −∞ < *x* < ∞. In order to have potentials that are guaranteed to possess bound states, we shall postulate a superpotential

$$\mathcal{W}(\mathbf{x}) \;= \sum\_{j=0}^{J} d\_{j} \mathbf{x}^{2j+1} \tag{2.30}$$

Then the corresponding sector one potential, *V*1(*x*), is

$$W\_1(\mathbf{x}) = \sum\_{j=0}^{I} \sum\_{\vec{j}'=0}^{I} d\_{\vec{j}} d\_{\vec{j}'} \mathbf{x}^{2(j+\vec{j}'+1)} - \sum\_{j=0}^{I} d\_{\vec{j}} (2j+1) \mathbf{x}^{2j} \tag{2.31}$$

The "charge" operators are given by

$$Q\_1 = \frac{d}{d\mathbf{x}} + W\_\mathbf{l} \text{ and } Q\_\mathbf{l}^\dagger = -\frac{d}{d\mathbf{x}} + W\_\mathbf{l}.\tag{2.32}$$

Then the first sector ground state for a general member of this family is

$$\psi\_0^+ (\mathbf{x}) = N e^{-\sum\_{j=0}^l \frac{d\_j x^{2j+2}}{(2j+2)}} \tag{2.33}$$

We stress that contrary to the periodic case Kouri, Markovich, Maxwell & Bodmann (2009), the solution of the sector two equation,

$$Q^\dagger \psi\_0^2(\mathfrak{x}) = \begin{array}{c} \ \ \ \ \ \end{array} \tag{2.34}$$

is not allowed because it is not normalizable. Thus, the ground state for the second sector satisfies

so

Possessing *<sup>ψ</sup>*(2)

where

with

problem.

and we must solve the equation

by expressing H<sup>2</sup> in the following form

H<sup>2</sup> =

 − *d* H2*ψ*(2)

*dx* <sup>+</sup> *<sup>W</sup>*2(*x*)

*<sup>ψ</sup>*(3)

*<sup>W</sup>*2(*x*) = <sup>−</sup> *<sup>d</sup>*

<sup>0</sup> <sup>=</sup> *<sup>Q</sup>*2*ψ*(2) 

*E*(2) <sup>1</sup> <sup>−</sup> *<sup>E</sup>*(2) 0

From this point, one can obviously generate as many Hamiltonians as needed. It should also be noted that the excited state wave functions can be obtained by using the charge operators we have previously defined. We now turn to the proof of principle for this approach as a computational scheme to obtain improved excited state energies and wave functions in the Rayleigh-Ritz variational method. We should note that these results can be generalized to any system where a hierarchy of hamiltonians can be generated because of the nature of the Rayleigh-Ritz scheme. In the standard approach one calculates the energies and wave functions variationally, relying on the Hylleraas-Undheim theorem for convergence Hylleraas & Undheim (1930). This, however, is unattractive for higher energy states because they require a much larger basis to converge to the same error. We stress that this is true regardless of the specific basis set used. Of course, some bases will be more efficient than others but it is generally true that for a given basis, the Rayleigh-Ritz result is less accurate for excited states. We address this situation by always solving for ground states in the variational part of the

To demonstrate our computational scheme, we investigate the first example system from the previous section. For this potential Equation (3.2), exact solutions are known for all states of H1. We use the exact results to assess the accuracy of the variational calculations. For our first

<sup>√</sup>*<sup>π</sup>*

Using the hierarchy of hamiltonians, we present the converged eigenvalues in Table 1. In Table 1, all energies were obtained for each of the Hamiltonians, H<sup>1</sup> and H2, by standard variational calculations using basis set sizes to achieve an accuracy of 10−6. It is easily seen that the ground state of H<sup>2</sup> is degenerate with the first excited state of H1. More interesting is the behavior of the excited state wave functions. Using the Cauchy criterion to measure

H*n*(*x*)*e*

− *x*2

�*φn*�|H*i*|*φn*� (3.13)

variational calculations, we use the harmonic oscillator basis functions where:

*<sup>φ</sup>n*(*x*) = <sup>1</sup>

with each matrix element determined using

2*nn*!

<sup>0</sup> <sup>=</sup> *<sup>E</sup>*(2)

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 107

<sup>0</sup> *<sup>ψ</sup>*(2)

<sup>0</sup> , we may develop the next Hamiltonian in the hierarchy. To do so, we begin

*dx* ln *<sup>ψ</sup>*(2)

1

*dx* <sup>+</sup> *<sup>W</sup>*2(*x*)

*d*

*V*<sup>2</sup> = *x*<sup>6</sup> + 4*x*<sup>4</sup> + 7*x*<sup>2</sup> + 2 (3.7)

 + *E*(2)

<sup>0</sup> . (3.8)

<sup>0</sup> . (3.10)

. (3.11)

<sup>2</sup> , (3.12)

<sup>0</sup> (3.9)

$$Q\_1 Q\_1^\dagger \psi\_0^2(\mathbf{x}) \ = E\_0^2 \psi\_0^2(\mathbf{x}) \ = E\_1^1 \psi\_0^2(\mathbf{x}) \tag{2.35}$$

where

$$E\_1^1 \neq 0\tag{2.36}$$

However, once *ψ*<sup>2</sup> <sup>0</sup>(*x*) is known, one can generate the first excited state *<sup>ψ</sup>*<sup>1</sup> <sup>1</sup>(*x*) according to

$$Q\_1^\dagger \psi\_0^2(\mathbf{x}) = \sqrt{E\_0^2} \psi\_0^1(\mathbf{x}) \tag{2.37}$$

The energy, *E*<sup>1</sup> <sup>1</sup>, of *<sup>ψ</sup>*<sup>1</sup> <sup>1</sup>(*x*) is, of course, equal to *<sup>E</sup>*<sup>2</sup> 0.

We remark that the ground state, *ψ*<sup>1</sup> <sup>0</sup>(*x*) is equal to the product of the ground states for each separate term in W. This is to say that,

$$\psi\_0^1(\mathbf{x}) = N \prod\_{j=0}^{J} e^{\frac{4j^{2j+2}}{(2j+2)}},\tag{2.38}$$

where N is the normalization constant. This is true even though *V*1(*x*) contains cross terms of the form

$$d\_j d\_{j'} x^{2(j+j'+1)}, \ \ j \neq j' \tag{2.39}$$

In fact, even more general anharmonic oscillators can be dealt with. Thus, any function, *g*(*x*) can be added to *<sup>W</sup>*(*x*) in Equation (2.30), provided only that *<sup>e</sup>*<sup>−</sup> *<sup>x</sup>* <sup>0</sup> *g*(*x*� ) *dx*� is L2. Thus, not only polynomic anharmonic potentials can be treated but many others.

#### **3. Computational examples**

In the following section we will explore the computational aspects of our SUSY-QM approach using two example anharmonic oscillator systems. To illustrate this approach to polynomic anharmonic oscillation we define *W*(*x*) to be

$$\mathcal{W}\_1(\mathbf{x}) = \mathbf{x}^3 + \mathbf{2}\mathbf{x},\tag{3.1}$$

which obviously yields a potential for the first sector of

$$V\_1(\mathbf{x}) = \mathbf{x}^6 + 4\mathbf{x}^4 + \mathbf{x}^2 - 2. \tag{3.2}$$

Where *x* is defined on the domain −∞ < *x* < ∞. We can thus define H<sup>1</sup> as:

$$\mathcal{H}\_1 = \left[ -\frac{d}{d\mathbf{x}} + \mathcal{W}\_1(\mathbf{x}) \right] \left[ \frac{d}{d\mathbf{x}} + \mathcal{W}\_1(\mathbf{x}) \right] \tag{3.3}$$

which satisfies the equation

$$\mathcal{H}\_1 \psi(\mathfrak{x})\_0^{(1)} = \mathbf{0},\tag{3.4}$$

and possesses an analytic ground state wave function of

$$
\psi\_0^{(1)} = N e^{-\left(\frac{\mathbf{x}^4}{4} + \mathbf{x}^2\right)}.\tag{3.5}
$$

To get the second Hamiltonian in the hierarchy we next define H<sup>2</sup> as

$$\mathcal{H}\_2 = \left[\frac{d}{d\mathbf{x}} + \mathcal{W}\_1(\mathbf{x})\right] \left[-\frac{d}{d\mathbf{x}} + \mathcal{W}\_1(\mathbf{x})\right] \tag{3.6}$$

so

6

where

However, once *ψ*<sup>2</sup>

The energy, *E*<sup>1</sup>

the form

<sup>1</sup>, of *<sup>ψ</sup>*<sup>1</sup>

**3. Computational examples**

which satisfies the equation

anharmonic oscillation we define *W*(*x*) to be

which obviously yields a potential for the first sector of

H<sup>1</sup> =

and possesses an analytic ground state wave function of

H<sup>2</sup> =

 − *d*

We remark that the ground state, *ψ*<sup>1</sup>

separate term in W. This is to say that,

*Q*1*Q*† 1*ψ*<sup>2</sup>

<sup>0</sup>(*x*) = *<sup>E</sup>*<sup>2</sup>

*Q*† 1*ψ*<sup>2</sup> <sup>0</sup>(*x*) =

*ψ*1

can be added to *<sup>W</sup>*(*x*) in Equation (2.30), provided only that *<sup>e</sup>*<sup>−</sup> *<sup>x</sup>*

polynomic anharmonic potentials can be treated but many others.

Where *x* is defined on the domain −∞ < *x* < ∞. We can thus define H<sup>1</sup> as:

*dx* <sup>+</sup> *<sup>W</sup>*1(*x*)

H1*ψ*(*x*)

<sup>0</sup> <sup>=</sup> *Ne*−( *<sup>x</sup>*<sup>4</sup>

*<sup>ψ</sup>*(1)

*dx* <sup>+</sup> *<sup>W</sup>*1(*x*)

To get the second Hamiltonian in the hierarchy we next define H<sup>2</sup> as

*d*

(1)

 − *d*

<sup>0</sup>(*x*) = *N*

*djdj*� *<sup>x</sup>*2(*j*+*<sup>j</sup>*

<sup>1</sup>(*x*) is, of course, equal to *<sup>E</sup>*<sup>2</sup>

0*ψ*<sup>2</sup>

*E*1

<sup>0</sup>(*x*) is known, one can generate the first excited state *<sup>ψ</sup>*<sup>1</sup>

 *E*2 0*ψ*1

0.

*J* ∏ *j* = 0 *e dj <sup>x</sup>*2*j*+<sup>2</sup>

, *j* �= *j*

*d*

<sup>4</sup> +*x*<sup>2</sup>)

where N is the normalization constant. This is true even though *V*1(*x*) contains cross terms of

� +1)

In fact, even more general anharmonic oscillators can be dealt with. Thus, any function, *g*(*x*)

In the following section we will explore the computational aspects of our SUSY-QM approach using two example anharmonic oscillator systems. To illustrate this approach to polynomic

<sup>0</sup>(*x*) = *<sup>E</sup>*<sup>1</sup>

1*ψ*<sup>2</sup>

<sup>0</sup>(*x*) (2.35)

<sup>1</sup>(*x*) according to

<sup>1</sup> �= 0 (2.36)

<sup>0</sup>(*x*) is equal to the product of the ground states for each

<sup>0</sup> *g*(*x*� ) *dx*�

*W*1(*x*) = *x*<sup>3</sup> + 2*x*, (3.1)

<sup>0</sup> = 0, (3.4)

. (3.5)

*<sup>V</sup>*1(*x*) = *<sup>x</sup>*<sup>6</sup> <sup>+</sup> <sup>4</sup>*x*<sup>4</sup> <sup>+</sup> *<sup>x</sup>*<sup>2</sup> <sup>−</sup> 2. (3.2)

*dx* <sup>+</sup> *<sup>W</sup>*1(*x*)

*dx* <sup>+</sup> *<sup>W</sup>*1(*x*)

<sup>0</sup>(*x*) (2.37)

(2*j*+2) , (2.38)

� (2.39)

is L2. Thus, not only

(3.3)

(3.6)

$$V\_2 = \mathbf{x}^6 + 4\mathbf{x}^4 + 7\mathbf{x}^2 + 2\tag{3.7}$$

and we must solve the equation

$$
\mathcal{H}\_2 \psi\_0^{(2)} = E\_0^{(2)} \psi\_0^{(2)}.\tag{3.8}
$$

Possessing *<sup>ψ</sup>*(2) <sup>0</sup> , we may develop the next Hamiltonian in the hierarchy. To do so, we begin by expressing H<sup>2</sup> in the following form

$$\mathcal{H}\_2 = \left[ -\frac{d}{d\mathbf{x}} + W\_2(\mathbf{x}) \right] \left[ \frac{d}{d\mathbf{x}} + W\_2(\mathbf{x}) \right] + E\_0^{(2)} \tag{3.9}$$

where

$$\mathcal{W}\_2(\mathbf{x}) = -\frac{d}{d\mathbf{x}} \ln \psi\_0^{(2)}.\tag{3.10}$$

with

$$
\psi\_0^{(3)} = \frac{Q\_2 \psi\_1^{(2)}}{\sqrt{E\_1^{(2)} - E\_0^{(2)}}}.\tag{3.11}
$$

From this point, one can obviously generate as many Hamiltonians as needed. It should also be noted that the excited state wave functions can be obtained by using the charge operators we have previously defined. We now turn to the proof of principle for this approach as a computational scheme to obtain improved excited state energies and wave functions in the Rayleigh-Ritz variational method. We should note that these results can be generalized to any system where a hierarchy of hamiltonians can be generated because of the nature of the Rayleigh-Ritz scheme. In the standard approach one calculates the energies and wave functions variationally, relying on the Hylleraas-Undheim theorem for convergence Hylleraas & Undheim (1930). This, however, is unattractive for higher energy states because they require a much larger basis to converge to the same error. We stress that this is true regardless of the specific basis set used. Of course, some bases will be more efficient than others but it is generally true that for a given basis, the Rayleigh-Ritz result is less accurate for excited states. We address this situation by always solving for ground states in the variational part of the problem.

To demonstrate our computational scheme, we investigate the first example system from the previous section. For this potential Equation (3.2), exact solutions are known for all states of H1. We use the exact results to assess the accuracy of the variational calculations. For our first variational calculations, we use the harmonic oscillator basis functions where:

$$\phi\_n(\mathbf{x}) = \frac{1}{\sqrt{2^n n! \sqrt{\pi}}} \mathcal{H}\_n(\mathbf{x}) e^{-\frac{\mathbf{x}^2}{2}},\tag{3.12}$$

with each matrix element determined using

$$
\langle \phi\_{n'} | \mathcal{H}\_i | \phi\_n \rangle \tag{3.13}
$$

Using the hierarchy of hamiltonians, we present the converged eigenvalues in Table 1. In Table 1, all energies were obtained for each of the Hamiltonians, H<sup>1</sup> and H2, by standard variational calculations using basis set sizes to achieve an accuracy of 10−6. It is easily seen that the ground state of H<sup>2</sup> is degenerate with the first excited state of H1. More interesting is the behavior of the excited state wave functions. Using the Cauchy criterion to measure

with

and the analytical ground state is

The ground state satisfies

Then the second sector Hamiltonian is

which must be solved numerically.

gained by the charge operator approach.

with n-DVR basis functions) and *E*<sup>2</sup>

corresponding to a 100 DVR points,

For any given basis size, *�*<sup>2</sup>

*�*1

*�*2

illustrates that our conclusion does not depend on the basis set used.

<sup>0</sup> <sup>&</sup>lt; *�*<sup>1</sup>

<sup>1</sup>(*n*) = log10 <sup>|</sup>*E*<sup>1</sup>

<sup>0</sup>(*n*) = log10 <sup>|</sup>*E*<sup>2</sup>

comparing *E*<sup>1</sup>

Likewise,

accurate than *E*<sup>1</sup>

H<sup>2</sup> =

*d*

analytically, we use the Cauchy convergence criterion

<sup>H</sup>1*ψ*<sup>1</sup>

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 109

<sup>0</sup> <sup>=</sup> *Ne*−( *<sup>x</sup>*<sup>4</sup>

<sup>4</sup> <sup>+</sup> *<sup>x</sup>*<sup>2</sup> <sup>2</sup> +*e<sup>x</sup>* )

 − *d*

<sup>0</sup> <sup>=</sup> *<sup>E</sup>*<sup>2</sup> 0*ψ*<sup>2</sup>

We performed the Rayleigh-Ritz calculations and found similar results for the second system described by *W* = *x*<sup>3</sup> + *x* + *ex*. Because the excited states of this oscillator are not known


where N is the basis size. In Table 4, we give the converged energy levels (to 5 significant figures) obtained by standard variational calculations applied to H<sup>1</sup> and H2. In Table 5, we show the basis set sizes needed in standard variational calculations to converge the wave functions for H<sup>1</sup> and H<sup>2</sup> (again, the Cauchy criterion of convergence was used.) Finally, in Table 6, we show the results for excited states obtained using the charge operators applied to the ground state wave functions of H2. Again, Δ*N* shows the reduction in the basis size

Finally, we compared the numerical accuracy of the first excitation energy of the anharmonic oscillator described by Equation (3.2), but now using a *n*-point discrete variable representation (DVR) based upon the Tchebychev polynomials to compute the eigenspectra of the first and second sectors. In Figure 2 we show the numerical error in the first excitation energy by

computed with n-DVR basis functions) from an *n* point DVR to the numerically "exact" value

energy computed using the second sector's ground state is between 10 and 100 times more

one can systematically improve upon the accuracy of a given variational calculation. It also

<sup>1</sup>(*n*) <sup>−</sup> *<sup>E</sup>*<sup>1</sup>

<sup>0</sup>(*n*) <sup>−</sup> *<sup>E</sup>*<sup>1</sup>

<sup>1</sup>(*n*). This effectively reiterates our point that by using the SUSY hierarchy,

<sup>1</sup>(*n*) (the first excited state energy from the standard variational calculation

<sup>0</sup>(*n*) (the ground state of the sector two Hamiltonian

<sup>1</sup>(*exact*)|.

<sup>1</sup>(*exact*)|.

<sup>1</sup>. Moreover, over a range of 15 < *n* < 40 points, the excitation

*<sup>ψ</sup>*(1)

*dx* <sup>+</sup> *<sup>x</sup>*<sup>3</sup> <sup>+</sup> *<sup>x</sup>* <sup>+</sup> *<sup>e</sup><sup>x</sup>*

 ∞ −∞ <sup>H</sup>2*ψ*<sup>2</sup>

<sup>0</sup> = 0 (3.17)

*dx* <sup>+</sup> *<sup>x</sup>*<sup>3</sup> <sup>+</sup> *<sup>x</sup>* <sup>+</sup> *<sup>e</sup><sup>x</sup>*

. (3.18)

(3.19)

<sup>0</sup> (3.20)

<sup>2</sup> *dx*, (3.21)

convergence, we show the basis set size (N) needed in a standard variational approach to obtain various eigenstates to the accuracy shown in Table 2. Clearly, excited state wave functions require substantially larger basis sets to achieve a high degree of accuracy. In Table 3 we show the results obtained for the same excited state wave functions obtained by applying the charge operator to the ground state wave function for H2. Again, N denotes the basis set required, and Δ*N* is the reduction of basis set achieved by use of the charge operators.


Table 1. Energies for the Anharmonic Polynomic Oscillator using Hierarchy of Hamiltonians.


Table 2. Wave function errors for the Anharmonic Polynomic Oscillator using the standard variational method for each hierarchy Hamiltonian. Each value has six significant figures.

$$\begin{array}{cccc}\hline n \, \Delta N \, \_N \psi\_0^n & \mathcal{L}\_2 & \mathcal{L}\_\infty\\\hline 1 & 4 & \_{74} \psi\_1^1 \, 4.083823 \text{e-} 07 \, 2.086041 \text{e-} 16 \\\hline \end{array}$$

Table 3. Wave function errors for the Anharmonic Polynomic Oscillator using Charge Operators to find excited states. Each value has six significant figures.

To find the solutions we used LAPACK routines to find these eigenvalues and vectors and GSL routines for numerical integration. Clearly, the use of the hierarchy of hamiltonians and charge operators provides more rapid convergence, which provides us with better methods to calculate the excited states.

The second example results from taking

$$\mathcal{W}\_1(\mathbf{x}) = \|\mathbf{x}^3 + \mathbf{x} + e^\mathbf{x}\|. \tag{3.14}$$

In this case,

$$\begin{aligned} V\_1(\mathbf{x}) &= \mathbf{x}^6 + 2\mathbf{x}^4 + 2\mathbf{x}^3 e^{\mathbf{x}} + \\ 2\mathbf{x}e^{\mathbf{x}} + \mathbf{x}^2 + e^{2\mathbf{x}} - 3\mathbf{x}^2 - e^{\mathbf{x}} - 1 \end{aligned} \tag{3.15}$$

Then

$$\mathcal{H}\_1 = \left[ -\frac{d}{d\mathbf{x}} + \mathbf{x}^3 + \mathbf{x} + e^{\mathbf{x}} \right] \left[ \frac{d}{d\mathbf{x}} + \mathbf{x}^3 + \mathbf{x} + e^{\mathbf{x}} \right] \tag{3.16}$$

with

8

convergence, we show the basis set size (N) needed in a standard variational approach to obtain various eigenstates to the accuracy shown in Table 2. Clearly, excited state wave functions require substantially larger basis sets to achieve a high degree of accuracy. In Table 3 we show the results obtained for the same excited state wave functions obtained by applying the charge operator to the ground state wave function for H2. Again, N denotes the basis set required, and Δ*N* is the reduction of basis set achieved by use of the charge operators.

<sup>1</sup> H<sup>2</sup> Δ N

<sup>0</sup> 5.024450 18

<sup>0</sup> 11.696825 20

<sup>0</sup> 19.497666 -

<sup>0</sup> <sup>H</sup><sup>1</sup> *<sup>N</sup>ψ<sup>n</sup>*

<sup>0</sup> 6.9441187e-07 <sup>34</sup>*ψ*<sup>1</sup>

<sup>1</sup> 5.024449 <sup>42</sup>*ψ*<sup>1</sup>

<sup>2</sup> 11.696820 <sup>50</sup>*ψ*<sup>1</sup>

Table 1. Energies for the Anharmonic Polynomic Oscillator using Hierarchy of Hamiltonians.

0 L2 L∞

Table 2. Wave function errors for the Anharmonic Polynomic Oscillator using the standard variational method for each hierarchy Hamiltonian. Each value has six significant figures.

Table 3. Wave function errors for the Anharmonic Polynomic Oscillator using Charge

To find the solutions we used LAPACK routines to find these eigenvalues and vectors and GSL routines for numerical integration. Clearly, the use of the hierarchy of hamiltonians and charge operators provides more rapid convergence, which provides us with better methods

*V*1(*x*) = *x*<sup>6</sup> + 2*x*<sup>4</sup> + 2*x*3*ex*+

<sup>2</sup>*<sup>x</sup>* <sup>−</sup> <sup>3</sup>*x*<sup>2</sup> <sup>−</sup> *<sup>e</sup><sup>x</sup>* <sup>−</sup> <sup>1</sup>

*d*

2*xe<sup>x</sup>* + *x*<sup>2</sup> + *e*

*dx* <sup>+</sup> *<sup>x</sup>*<sup>3</sup> <sup>+</sup> *<sup>x</sup>* <sup>+</sup> *<sup>e</sup><sup>x</sup>*

<sup>0</sup> 5.835283e-07 1.110223-16

<sup>1</sup> 1.975656e-07 4.019723e-16

<sup>0</sup> 2.303928e-07 2.220446e-16

0 L2 L∞

<sup>1</sup> 4.083823e-07 2.086041e-16

*W*1(*x*) = *x*<sup>3</sup> + *x* + *ex*. (3.14)

*dx* <sup>+</sup> *<sup>x</sup>*<sup>3</sup> <sup>+</sup> *<sup>x</sup>* <sup>+</sup> *<sup>e</sup><sup>x</sup>*

(3.15)

(3.16)

*n <sup>N</sup>ψ<sup>n</sup>*

0 <sup>44</sup>*ψ*<sup>1</sup>

1 <sup>52</sup>*ψ*<sup>1</sup>

2 <sup>62</sup>*ψ*<sup>1</sup>

*n <sup>N</sup>ψ<sup>n</sup>*

0 <sup>56</sup>*ψ*<sup>1</sup>

1 <sup>78</sup>*ψ*<sup>1</sup>

0 <sup>68</sup>*ψ*<sup>2</sup>

*n* Δ*N <sup>N</sup>ψ<sup>n</sup>*

1 4 <sup>74</sup>*ψ*<sup>1</sup>

Operators to find excited states. Each value has six significant figures.

to calculate the excited states.

In this case,

Then

The second example results from taking

H<sup>1</sup> =

 − *d*

$$\mathcal{H}\_1 \psi\_0^1 = \begin{array}{c} 0 \end{array} \tag{3.17}$$

and the analytical ground state is

$$
\psi\_0^{(1)} = N e^{-\left(\frac{\mathfrak{s}^4}{4} + \frac{\mathfrak{s}^2}{2} + \mathfrak{e}^x\right)}.\tag{3.18}
$$

Then the second sector Hamiltonian is

$$\mathcal{H}\_2 = \left[ \frac{d}{dx} + \mathbf{x}^3 + \mathbf{x} + \boldsymbol{\varepsilon}^x \right] \left[ -\frac{d}{dx} + \mathbf{x}^3 + \mathbf{x} + \boldsymbol{\varepsilon}^x \right] \tag{3.19}$$

The ground state satisfies

$$\mathcal{H}\_2 \psi\_0^2 = E\_0^2 \psi\_0^2 \tag{3.20}$$

which must be solved numerically.

We performed the Rayleigh-Ritz calculations and found similar results for the second system described by *W* = *x*<sup>3</sup> + *x* + *ex*. Because the excited states of this oscillator are not known analytically, we use the Cauchy convergence criterion

$$\int\_{-\infty}^{\infty} \left| \,\_N\psi\_n - \,\_{N-1}\psi\_n \right|^2 d\mathfrak{x}\_{\prime} \tag{3.21}$$

where N is the basis size. In Table 4, we give the converged energy levels (to 5 significant figures) obtained by standard variational calculations applied to H<sup>1</sup> and H2. In Table 5, we show the basis set sizes needed in standard variational calculations to converge the wave functions for H<sup>1</sup> and H<sup>2</sup> (again, the Cauchy criterion of convergence was used.) Finally, in Table 6, we show the results for excited states obtained using the charge operators applied to the ground state wave functions of H2. Again, Δ*N* shows the reduction in the basis size gained by the charge operator approach.

Finally, we compared the numerical accuracy of the first excitation energy of the anharmonic oscillator described by Equation (3.2), but now using a *n*-point discrete variable representation (DVR) based upon the Tchebychev polynomials to compute the eigenspectra of the first and second sectors. In Figure 2 we show the numerical error in the first excitation energy by comparing *E*<sup>1</sup> <sup>1</sup>(*n*) (the first excited state energy from the standard variational calculation with n-DVR basis functions) and *E*<sup>2</sup> <sup>0</sup>(*n*) (the ground state of the sector two Hamiltonian computed with n-DVR basis functions) from an *n* point DVR to the numerically "exact" value corresponding to a 100 DVR points,

$$
\epsilon\_1^1(n) = \log\_{10}|E\_1^1(n) - E\_1^1(e\mathbf{x}act)|.
$$

Likewise,

$$
\epsilon\_0^2(n) = \log\_{10}|E\_0^2(n) - E\_1^1(e\mathbf{x}act)|.
$$

For any given basis size, *�*<sup>2</sup> <sup>0</sup> <sup>&</sup>lt; *�*<sup>1</sup> <sup>1</sup>. Moreover, over a range of 15 < *n* < 40 points, the excitation energy computed using the second sector's ground state is between 10 and 100 times more accurate than *E*<sup>1</sup> <sup>1</sup>(*n*). This effectively reiterates our point that by using the SUSY hierarchy, one can systematically improve upon the accuracy of a given variational calculation. It also illustrates that our conclusion does not depend on the basis set used.

Another way to approximate excited state energies and wave functions of bound quantum systems is to take advantage of the Ricatti substitution for the purpose of constructing dynamically-adapted, system-specific coherent states. Perhaps the simplest procedure for creating an overcomplete set of such coherent states is to follow the work of Kouri, et al. Klauder & Skagerstam (1985); Kouri et al. (2003). In their approach, it was observed that the

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 111

*e* <sup>−</sup>*x*<sup>0</sup> *<sup>d</sup>*

*e*

) *dx*�

2*π D*

where *α* = *x*<sup>0</sup> + *ik*<sup>0</sup> is a point in the phase space Klauder & Skagerstam (1985) which completely describes the coherent state and the de Broglie relation tells us that �*p*ˆ*x*� = *k*0. Using the set of coherent states defined above, we can select a finite subset which remains overcomplete by discretizing the otherwise continuous label *α* = *q* + *ik* and setting up a von Neumann lattice in phase space with an appropriate density, *D*. We define an overcomplete

*<sup>ψ</sup>*(*α*|*x*) = *Neik*0(*x*−*x*0)

*e*

*m*Δ*x*

and *i* being a joint index consisting of *m* and *n* Andersson (2001).

<sup>−</sup> *<sup>x</sup>*−*qi* <sup>0</sup> *<sup>W</sup>*(*x*�

2*π <sup>D</sup>* , *<sup>n</sup>* Δ*x*

Due to the fact that the ground state of the system of interest solves the time-independent Schrödinger equation for the corresponding Hamiltonian, the coherent states defined above build in the dynamics of the system under investigation. This property leads to the expectation that these system-specific coherent states will prove more rapidly convergent in the approximation of excited state energies of bound quantum systems using variational

For *<sup>W</sup>*(*x*) = *<sup>x</sup>*3, and thus *<sup>V</sup>*(*x*) = *<sup>x</sup>*<sup>6</sup> <sup>−</sup> <sup>3</sup>*x*2, we carried out a variational calculation using the system-specific coherent states defined above and compared the accuracy in the approximation of the first three excited state energy eigenvalues with that achieved using the standard harmonic oscillator basis and the harmonic oscillator coherent states. To evaluate the accuracy of each method, we compare the results with a Chebyshev polynomial DVR (Discrete Variable Representation) calcuation using 1000 points Littlejohn (2002). The number of decimal places reported in Tables 7-10 correspond to the number of decimal places of agreement with the DVR plus an additional significant figure which is either rounded up

*E*<sup>0</sup> *E*<sup>1</sup> *E*<sup>2</sup> *E*<sup>3</sup>

0.000000000 1.935482104 6.298495901 11.680970886

where *M* is the number of basis functions and the phase space grid points are given by

= *Neik*0(*x*−*x*0)

<sup>0</sup>(*x*), could be translated an amount *x*<sup>0</sup> by application of the

*dx ψ*(0|*x*)

) *dx*� ,

: 1 ≤ *i* ≤ *M*, *M* ∈ **N**}, (3.23)

, *m*, *n* ∈ **Z**, (3.24)

(3.22)

<sup>−</sup> *<sup>x</sup>*−*x*<sup>0</sup> <sup>0</sup> *<sup>W</sup>*(*x*�

"fiducial" function, �*x*|*ψ*(0)� <sup>=</sup> *<sup>ψ</sup>*<sup>1</sup>

*dx*

{*φ*(*αi*|*x*) = *Neiki*(*x*−*qi*)

{(*qi*, *ki*)} =

shift operator *e*−*x*<sup>0</sup> *<sup>d</sup>*

basis of coherent states

methods.

or down.

Table 7. DVR Comparison.


Table 4. Energies for the Anharmonic Non-Polynomic Oscillator using Hierarchy of Hamiltonians, determined variationally.


Table 5. Errors for the Anharmonic Non-Polynomic Oscillator wave functions using Hierarchy of Hamiltonians all determined variationally.


Table 6. Errors for the Anharmonic Non-Polynomic Oscillator using wave functions Charge Operators to find excited states by applying the correct charge operator to the appropriate ground state

Fig. 2. Convergence of first excitation energy *E*<sup>1</sup> <sup>1</sup> for model potential *<sup>V</sup>*<sup>1</sup> <sup>=</sup> *<sup>x</sup>*<sup>6</sup> <sup>+</sup> <sup>4</sup>*x*<sup>4</sup> <sup>+</sup> *<sup>x</sup>*<sup>2</sup> <sup>−</sup> <sup>2</sup> using a *n*-point discrete variable representation (DVR). Gray squares: *�* <sup>=</sup> log10 <sup>|</sup>*E*<sup>1</sup> <sup>1</sup>(*n*) <sup>−</sup> *<sup>E</sup>*<sup>1</sup> <sup>1</sup>(*exact*)|, Black squares: *�* <sup>=</sup> log10 <sup>|</sup>*E*<sup>2</sup> <sup>0</sup>(*n*) <sup>−</sup> *<sup>E</sup>*<sup>1</sup> <sup>1</sup>(*exact*)|. Dashed lines are linear fits.

10

ground state

*�* <sup>=</sup> log10 <sup>|</sup>*E*<sup>1</sup>

linear fits.

*n <sup>N</sup>ψ<sup>n</sup>*

0 <sup>50</sup>*ψ*<sup>1</sup>

1 <sup>60</sup>*ψ*<sup>1</sup>

2 <sup>64</sup>*ψ*<sup>1</sup>

*n <sup>N</sup>ψ<sup>n</sup>*

0 <sup>70</sup>*ψ*<sup>1</sup>

1 <sup>88</sup>*ψ*<sup>1</sup>

0 <sup>76</sup>*ψ*<sup>2</sup>

*n* Δ*N <sup>N</sup>ψ<sup>n</sup>*

1 14 <sup>74</sup>*ψ*<sup>1</sup>

 

using a *n*-point discrete variable representation (DVR). Gray squares:

<sup>1</sup>(*exact*)|, Black squares: *�* <sup>=</sup> log10 <sup>|</sup>*E*<sup>2</sup>

Hierarchy of Hamiltonians all determined variationally.

 

�10

Fig. 2. Convergence of first excitation energy *E*<sup>1</sup>

<sup>1</sup>(*n*) <sup>−</sup> *<sup>E</sup>*<sup>1</sup>

�5

 

Ε

Hamiltonians, determined variationally.

<sup>0</sup> <sup>H</sup><sup>1</sup> *<sup>N</sup>ψ<sup>n</sup>*

<sup>0</sup> 2.703955e-06 <sup>44</sup>*ψ*<sup>1</sup>

<sup>1</sup> 5.263075 <sup>56</sup>*ψ*<sup>1</sup>

<sup>2</sup> 12.109717 <sup>66</sup>*ψ*<sup>1</sup>

0 L2 L∞

<sup>0</sup> 3.7158761e-07 2.220446e-16

<sup>1</sup> 6.477328e-08 1.221245e-15

<sup>0</sup> 5.659010e-07 2.109424e-15

0 L2 L∞

Table 6. Errors for the Anharmonic Non-Polynomic Oscillator using wave functions Charge Operators to find excited states by applying the correct charge operator to the appropriate

<sup>1</sup> 9.750546e-07 3.181791e-16

<sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> *<sup>n</sup>*DVR

<sup>1</sup> for model potential *<sup>V</sup>*<sup>1</sup> <sup>=</sup> *<sup>x</sup>*<sup>6</sup> <sup>+</sup> <sup>4</sup>*x*<sup>4</sup> <sup>+</sup> *<sup>x</sup>*<sup>2</sup> <sup>−</sup> <sup>2</sup>

<sup>1</sup>(*exact*)|. Dashed lines are

<sup>0</sup>(*n*) <sup>−</sup> *<sup>E</sup>*<sup>1</sup>

Table 4. Energies for the Anharmonic Non-Polynomic Oscillator using Hierarchy of

Table 5. Errors for the Anharmonic Non-Polynomic Oscillator wave functions using

1 H2

<sup>0</sup> 5.263075

<sup>0</sup> 12.109712

<sup>0</sup> 20.186019

Another way to approximate excited state energies and wave functions of bound quantum systems is to take advantage of the Ricatti substitution for the purpose of constructing dynamically-adapted, system-specific coherent states. Perhaps the simplest procedure for creating an overcomplete set of such coherent states is to follow the work of Kouri, et al. Klauder & Skagerstam (1985); Kouri et al. (2003). In their approach, it was observed that the "fiducial" function, �*x*|*ψ*(0)� <sup>=</sup> *<sup>ψ</sup>*<sup>1</sup> <sup>0</sup>(*x*), could be translated an amount *x*<sup>0</sup> by application of the shift operator *e*−*x*<sup>0</sup> *<sup>d</sup> dx*

$$\begin{split} \psi(\mathfrak{a}|\mathfrak{x}) &= N e^{ik\_0(\mathbf{x}-\mathbf{x}\_0)} e^{-\mathbf{x}\_0 \frac{d}{d\mathbf{x}}} \psi(\mathfrak{0}|\mathfrak{x}) \\ &= N e^{ik\_0(\mathbf{x}-\mathbf{x}\_0)} e^{-\int\_0^{\mathbf{x}-\mathbf{x}} \mathfrak{W}(\mathbf{x}') \, d\mathbf{x}'} \end{split} \tag{3.22}$$

where *α* = *x*<sup>0</sup> + *ik*<sup>0</sup> is a point in the phase space Klauder & Skagerstam (1985) which completely describes the coherent state and the de Broglie relation tells us that �*p*ˆ*x*� = *k*0. Using the set of coherent states defined above, we can select a finite subset which remains overcomplete by discretizing the otherwise continuous label *α* = *q* + *ik* and setting up a von Neumann lattice in phase space with an appropriate density, *D*. We define an overcomplete basis of coherent states

$$\{\phi(\mathfrak{a}\_i|\mathbf{x}) = \mathrm{Ne}^{ik\_l(\mathbf{x}-\boldsymbol{\eta})} e^{-\int\_0^{\mathbf{x}-\boldsymbol{\eta}\_i} W(\mathbf{x'})d\mathbf{x'}} \\ \colon \mathbf{1} \le \mathrm{i} \le M, \ \mathbf{M} \in \mathbf{N}\},\tag{3.23}$$

where *M* is the number of basis functions and the phase space grid points are given by

$$\{(q\_{i\prime}k\_{i})\} = \left\{ \left( m\Delta x \sqrt{\frac{2\pi}{D}}, \frac{n}{\Delta x} \sqrt{\frac{2\pi}{D}} \right) \right\} \;/\; m, n \in \mathbb{Z} \tag{3.24}$$

and *i* being a joint index consisting of *m* and *n* Andersson (2001).

Due to the fact that the ground state of the system of interest solves the time-independent Schrödinger equation for the corresponding Hamiltonian, the coherent states defined above build in the dynamics of the system under investigation. This property leads to the expectation that these system-specific coherent states will prove more rapidly convergent in the approximation of excited state energies of bound quantum systems using variational methods.

For *<sup>W</sup>*(*x*) = *<sup>x</sup>*3, and thus *<sup>V</sup>*(*x*) = *<sup>x</sup>*<sup>6</sup> <sup>−</sup> <sup>3</sup>*x*2, we carried out a variational calculation using the system-specific coherent states defined above and compared the accuracy in the approximation of the first three excited state energy eigenvalues with that achieved using the standard harmonic oscillator basis and the harmonic oscillator coherent states. To evaluate the accuracy of each method, we compare the results with a Chebyshev polynomial DVR (Discrete Variable Representation) calcuation using 1000 points Littlejohn (2002). The number of decimal places reported in Tables 7-10 correspond to the number of decimal places of agreement with the DVR plus an additional significant figure which is either rounded up or down.

$$E\_0 \qquad \qquad \qquad E\_1 \qquad \qquad \qquad E\_2 \qquad \qquad \qquad E\_3$$

#### 0.000000000 1.935482104 6.298495901 11.680970886

Table 7. DVR Comparison.

where

(2010b).

Schrödinger equation,

which is to say

*<sup>W</sup>* )*ψ*(1)

*E*(2) <sup>0</sup> <sup>=</sup> *<sup>E</sup>*(1)

<sup>0</sup> <sup>≡</sup> *<sup>Q</sup> <sup>ψ</sup>*(1)

That is to say, using Einstein notation,

It then follows that *<sup>Q</sup> <sup>ψ</sup>*(1)

<sup>1</sup> <sup>−</sup> *<sup>E</sup>*(1)

It is also clear that *<sup>Q</sup> <sup>ψ</sup>*(1)

<sup>∇</sup> <sup>=</sup> ∑ *j �j ∂ ∂uj*

and *�<sup>j</sup>* · *�<sup>k</sup>* = *δjk*. The subscript "1" indicates this is the "sector one" Hamiltonian. For simplicity we take the masses of the particles to be equal and use units such that ¯*h*2/2*m* = 1. For the development here we assume a Cartesian coordinate space, but have provided an extension to more general curvilinear coordinates in a previous publication Kouri et al.

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 113

As per usual in quantum mechanics, the ground-state wave function is a solution of the

<sup>0</sup> <sup>=</sup> *<sup>E</sup>*(1)

*<sup>W</sup>* <sup>=</sup> −∇ ln *<sup>ψ</sup>*(1)

3*n* ∑ *j*=1 *�j ∂ ∂uj*

<sup>0</sup> )=(−∇ <sup>+</sup> *<sup>W</sup>* ) · (+<sup>∇</sup> <sup>+</sup> *<sup>W</sup>* ) = (−*∂<sup>i</sup>* + *Wi*)(*∂<sup>i</sup>* + *Wi*)

<sup>0</sup> )*ψ*(1)

<sup>1</sup> <sup>−</sup> *<sup>E</sup>*(1)

<sup>1</sup> <sup>−</sup> *<sup>E</sup>*(1)

<sup>1</sup> <sup>−</sup> *<sup>E</sup>*(1)

<sup>0</sup> cannot generate a lower energy eigenstate of ←→H <sup>2</sup> since *<sup>Q</sup> <sup>ψ</sup>*(1)

<sup>0</sup> )*ψ*(1)

<sup>0</sup> )*<sup>Q</sup> <sup>ψ</sup>*(1)

<sup>0</sup> )*Qiψ*(1)

<sup>1</sup> is an eigenstate of the tensor Hamiltonian ←→H <sup>2</sup> = (*<sup>Q</sup> <sup>Q</sup>* †) with energy

*�jWj* = −

= *Q*† *<sup>i</sup>* · *Qi*,

where, according to the Einstein convention, we sum over repeated indices. Since (<sup>∇</sup> <sup>+</sup>

<sup>1</sup> = (*E*(1)

<sup>1</sup> = (*E*(1)

<sup>1</sup> = (*E*(1)

<sup>0</sup> . Since we are free to set the energy origin, taking *<sup>E</sup>*(1)

<sup>0</sup> *<sup>ψ</sup>*(1)

<sup>0</sup> , is nodeless.

ln *<sup>ψ</sup>*(1)

<sup>0</sup> . (4.3)

<sup>0</sup> , (4.4)

<sup>0</sup> = 0 as required. We can now define

<sup>0</sup> ), it is isospectral with H1.

<sup>1</sup> . (4.7)

<sup>1</sup> . (4.8)

<sup>1</sup> . (4.9)

<sup>0</sup> <sup>=</sup> 0 gives *<sup>E</sup>*(2)

<sup>0</sup> <sup>=</sup> *<sup>E</sup>*(1) <sup>1</sup> .

<sup>0</sup> <sup>=</sup>0, so

<sup>0</sup> . (4.5)

H1*ψ*(1)

We also emphasize that the lowest energy state, *<sup>ψ</sup>*(1)

*W* =

It is straightforward to see that one can write <sup>H</sup><sup>1</sup> in terms of *<sup>W</sup>* as

(H<sup>1</sup> <sup>−</sup> *<sup>E</sup>*(1)

<sup>0</sup> <sup>≡</sup> 0, it is clear that (H<sup>1</sup> <sup>−</sup> *<sup>E</sup>*(1)

*Q*†

(*QiQ*†

the sector two Hamiltonian such that, above the ground-state (*E*(1)

We do this as follows: for the first excited state in sector one we can write

*<sup>i</sup>* · *Qiψ*(1)

We then form the tensor product by operating on the left with *Q* so that

(*<sup>Q</sup> <sup>Q</sup>* †) · *<sup>Q</sup> <sup>ψ</sup>*(1)

*<sup>j</sup>* )*Qjψ*(1)

3*n* ∑ *j*=1

We now define a vector superpotential, *W* , as

(4.2)

(4.6)


9 0.000000000 1.9355 6.3 11.69 15 0.000000000 1.93548218 6.2985 11.681

Table 8. System-Specific Coherent States.

*M E*<sup>0</sup> *E*<sup>1</sup> *E*<sup>2</sup> *E*<sup>3</sup> 9 0.2 2.0 15.0 43.0 15 0.2 2.0 9.0 43.0

Table 9. Harmonic Oscillator Coherent States.


Table 10. Harmonic Oscillator Basis.

The results indicate that, in fact, the system-specific coherent states provide more accurate approximations of the excited state energies for the anharmonic oscillator given by *V*(*x*) = *<sup>x</sup>*<sup>6</sup> <sup>−</sup> <sup>3</sup>*x*<sup>2</sup> when compared with other bases. Namely, they give seven decimal places of agreement with the DVR when 15 basis functions are used. The same number of harmonic oscillator basis functions only provides one decimal point of agreement, and 15 harmonic oscillator coherent states fails to give agreement in the ones place with the DVR. Despite the accuracy achieved using a small number of system-specific coherent states, the non-orthogonality and complex-valued nature of the basis necessitates the calculation of a complex overlap matrix, whose elements must be computed by numerical integration, which is computationally demanding. In order to eliminate numerical integration from the calculation, one can expand the ground state wave function used in the construction of the coherent states into an incomplete set of scaled Gaussians centered about the {*qi* : 1 ≤ *i* ≤ *M*}. This expansion would allow us to compute the overlap matrix elements analytically, replacing numerical integration with function evaluation. In particular, we propose to use the Levenberg-Marquardt least-squares curve-fitting algorithm to build an arbitrarily-accurate approximation of each system-specific coherent state in the following manner:

$$\phi(\mathfrak{a}\_i|\mathbf{x}) \approx N e^{i\mathbf{k}\_i(\mathbf{x}-q\_l)} \sum\_{j=1}^{\tilde{M}} \mathfrak{c}\_j e^{-(\mathbf{x}-q\_l)^2/\sigma\_l} \,. \tag{3.25}$$

#### **4. Generalization to multi-dimensions**

In our generalization, we make use of a vectorial approach that simultaneously treats more than one dimension and any number of distinguishable particles. We consider, therefore, a system of *n*-particles in three-dimensional space. We denote the coordinates of particle *i* by (*xi*, *yi*, *zi*). We then define an orthogonal hyperspace of dimension 3*n*. We take the Hamiltonian for this system to be given by

$$\mathcal{H}\_1 = -\nabla^2 + V\_1 \tag{4.1}$$

where

12

Table 8. System-Specific Coherent States.

Table 9. Harmonic Oscillator Coherent States.

Table 10. Harmonic Oscillator Basis.

**4. Generalization to multi-dimensions**

Hamiltonian for this system to be given by

*M E*<sup>0</sup> *E*<sup>1</sup> *E*<sup>2</sup> *E*<sup>3</sup>

9 0.000000000 1.9355 6.3 11.69 15 0.000000000 1.93548218 6.2985 11.681

*M E*<sup>0</sup> *E*<sup>1</sup> *E*<sup>2</sup> *E*<sup>3</sup>

9 0.2 2.0 15.0 43.0 15 0.2 2.0 9.0 43.0

*M E*<sup>0</sup> *E*<sup>1</sup> *E*<sup>2</sup> *E*<sup>3</sup>

9 0.07 2.0 6.9 23.0 15 0.01 1.99 6.5 12.0

The results indicate that, in fact, the system-specific coherent states provide more accurate approximations of the excited state energies for the anharmonic oscillator given by *V*(*x*) = *<sup>x</sup>*<sup>6</sup> <sup>−</sup> <sup>3</sup>*x*<sup>2</sup> when compared with other bases. Namely, they give seven decimal places of agreement with the DVR when 15 basis functions are used. The same number of harmonic oscillator basis functions only provides one decimal point of agreement, and 15 harmonic oscillator coherent states fails to give agreement in the ones place with the DVR. Despite the accuracy achieved using a small number of system-specific coherent states, the non-orthogonality and complex-valued nature of the basis necessitates the calculation of a complex overlap matrix, whose elements must be computed by numerical integration, which is computationally demanding. In order to eliminate numerical integration from the calculation, one can expand the ground state wave function used in the construction of the coherent states into an incomplete set of scaled Gaussians centered about the {*qi* : 1 ≤ *i* ≤ *M*}. This expansion would allow us to compute the overlap matrix elements analytically, replacing numerical integration with function evaluation. In particular, we propose to use the Levenberg-Marquardt least-squares curve-fitting algorithm to build an arbitrarily-accurate

approximation of each system-specific coherent state in the following manner:

*M*˜ ∑ *j*=1 *cje*

In our generalization, we make use of a vectorial approach that simultaneously treats more than one dimension and any number of distinguishable particles. We consider, therefore, a system of *n*-particles in three-dimensional space. We denote the coordinates of particle *i* by (*xi*, *yi*, *zi*). We then define an orthogonal hyperspace of dimension 3*n*. We take the

<sup>−</sup>(*x*−*qi*)2/*σ<sup>j</sup>* . (3.25)

<sup>H</sup><sup>1</sup> <sup>=</sup> −∇<sup>2</sup> <sup>+</sup> *<sup>V</sup>*<sup>1</sup> (4.1)

*<sup>φ</sup>*(*αi*|*x*) <sup>≈</sup> *Neiki*(*x*−*qi*)

$$\vec{\nabla} = \sum\_{j} \vec{\varepsilon}\_{j} \frac{\partial}{\partial u\_{j}} \tag{4.2}$$

and *�<sup>j</sup>* · *�<sup>k</sup>* = *δjk*. The subscript "1" indicates this is the "sector one" Hamiltonian. For simplicity we take the masses of the particles to be equal and use units such that ¯*h*2/2*m* = 1. For the development here we assume a Cartesian coordinate space, but have provided an extension to more general curvilinear coordinates in a previous publication Kouri et al. (2010b).

As per usual in quantum mechanics, the ground-state wave function is a solution of the Schrödinger equation,

$$\mathcal{H}\_1 \boldsymbol{\psi}\_0^{(1)} = E\_0^{(1)} \boldsymbol{\psi}\_0^{(1)}.\tag{4.3}$$

We also emphasize that the lowest energy state, *<sup>ψ</sup>*(1) <sup>0</sup> , is nodeless. We now define a vector superpotential, *W* , as

$$
\vec{W} = -\vec{\nabla} \ln \psi\_0^{(1)} \,\tag{4.4}
$$

which is to say

$$\vec{\mathcal{W}} = \sum\_{j=1}^{3n} \vec{\epsilon}\_{j} \mathcal{W}\_{j} = -\sum\_{j=1}^{3n} \vec{\epsilon}\_{j} \frac{\partial}{\partial u\_{j}} \ln \psi\_{0}^{(1)}.\tag{4.5}$$

It is straightforward to see that one can write <sup>H</sup><sup>1</sup> in terms of *<sup>W</sup>* as

$$\begin{aligned} \left( \mathcal{H}\_1 - E\_0^{(1)} \right) &= (-\vec{\nabla} + \vec{\mathcal{W}}) \cdot (+\vec{\nabla} + \vec{\mathcal{W}}) \\ &= (-\partial\_{\vec{i}} + \mathcal{W}\_{\vec{i}}) (\partial\_{\vec{i}} + \mathcal{W}\_{\vec{i}}) \\ &= Q\_{\vec{i}}^{\dagger} \cdot Q\_{\vec{i}\nu} \end{aligned} \tag{4.6}$$

where, according to the Einstein convention, we sum over repeated indices. Since (<sup>∇</sup> <sup>+</sup> *<sup>W</sup>* )*ψ*(1) <sup>0</sup> <sup>≡</sup> *<sup>Q</sup> <sup>ψ</sup>*(1) <sup>0</sup> <sup>≡</sup> 0, it is clear that (H<sup>1</sup> <sup>−</sup> *<sup>E</sup>*(1) <sup>0</sup> )*ψ*(1) <sup>0</sup> = 0 as required. We can now define the sector two Hamiltonian such that, above the ground-state (*E*(1) <sup>0</sup> ), it is isospectral with H1. We do this as follows: for the first excited state in sector one we can write

$$Q\_i^\dagger \cdot Q\_i \psi\_1^{(1)} = (E\_1^{(1)} - E\_0^{(1)}) \psi\_1^{(1)}.\tag{4.7}$$

We then form the tensor product by operating on the left with *Q* so that

$$(\vec{Q}\vec{Q}^\dagger)\cdot\vec{Q}\psi\_1^{(1)} = (E\_1^{(1)} - E\_0^{(1)})\vec{Q}\psi\_1^{(1)}.\tag{4.8}$$

That is to say, using Einstein notation,

$$(Q\_i Q\_j^\dagger) Q\_j \psi\_1^{(1)} = (E\_1^{(1)} - E\_0^{(1)}) Q\_i \psi\_1^{(1)}.\tag{4.9}$$

It then follows that *<sup>Q</sup> <sup>ψ</sup>*(1) <sup>1</sup> is an eigenstate of the tensor Hamiltonian ←→H <sup>2</sup> = (*<sup>Q</sup> <sup>Q</sup>* †) with energy *E*(2) <sup>0</sup> <sup>=</sup> *<sup>E</sup>*(1) <sup>1</sup> <sup>−</sup> *<sup>E</sup>*(1) <sup>0</sup> . Since we are free to set the energy origin, taking *<sup>E</sup>*(1) <sup>0</sup> <sup>=</sup> 0 gives *<sup>E</sup>*(2) <sup>0</sup> <sup>=</sup> *<sup>E</sup>*(1) <sup>1</sup> . It is also clear that *<sup>Q</sup> <sup>ψ</sup>*(1) <sup>0</sup> cannot generate a lower energy eigenstate of ←→H <sup>2</sup> since *<sup>Q</sup> <sup>ψ</sup>*(1) <sup>0</sup> <sup>=</sup>0, so

We construct the rigorous sector two Hamiltonian as

*∂u*<sup>2</sup> 1 + *u*<sup>2</sup>

*∂u*1*∂u*<sup>2</sup>

←→H <sup>2</sup> <sup>=</sup> *�*ˆ1*�*ˆ1[<sup>−</sup> *<sup>∂</sup>*<sup>2</sup>

The eigenvalue equation is

with eigenstates

non-degenerate.

*Q* † <sup>1</sup> · *<sup>ψ</sup>* (2)

same energy.

<sup>0</sup>(1) = 2*u*1*e*

the charge operator, *Q* †

and

that

Then

<sup>+</sup> *�*ˆ2*�*ˆ1[<sup>−</sup> *<sup>∂</sup>*<sup>2</sup>

←→H <sup>2</sup> <sup>=</sup> *<sup>Q</sup>* <sup>1</sup>*<sup>Q</sup>* †

<sup>1</sup> + 2

which is a second rank tensor in this case. The Hamiltonian ←→*<sup>H</sup>* <sup>2</sup> is then given by

+ *u*1*u*<sup>2</sup> − *u*<sup>2</sup>

←→H <sup>2</sup> · *<sup>ψ</sup>* (2)

(*n*) <sup>=</sup> *�*ˆ1*ψ*(2)

It is not difficult to show that there are two degenerate ground state solutions given by

*<sup>ψ</sup>* (2)

*<sup>ψ</sup>* (2) (0)<sup>1</sup> = *�*ˆ1*e*

*<sup>ψ</sup>* (2) (0)<sup>2</sup> = *�*ˆ2*e*

*Q* †

<sup>−</sup>(*u*<sup>2</sup> 1+*u*<sup>2</sup>

<sup>1</sup> <sup>=</sup> *�*ˆ1(<sup>−</sup> *<sup>∂</sup>*

<sup>2</sup>)/2 <sup>∝</sup> *<sup>ψ</sup>*(1)

*∂u*<sup>1</sup>

<sup>1</sup> <sup>+</sup> <sup>3</sup>] + *�*ˆ1*�*ˆ2[<sup>−</sup> *<sup>∂</sup>*<sup>2</sup>

*∂ ∂u*<sup>1</sup>

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 115

(*n*) <sup>=</sup> *<sup>E</sup>*(2) (*n*) *<sup>ψ</sup>* (2)

(*n*)<sup>1</sup> <sup>+</sup> *�*ˆ2*ψ*(2)

<sup>−</sup>(*u*<sup>2</sup> 1+*u*<sup>2</sup>

<sup>−</sup>(*u*<sup>2</sup> 1+*u*<sup>2</sup>

respectively. This is extremely interesting and in contrast to the usual situation in quantum mechanics. For most systems (excluding spin effects) the ground state is unique, i.e.,

We shall see that the degenerate states, Equation (4.23)-(4.24), are exactly what is required for

<sup>+</sup> *<sup>u</sup>*1) + *�*ˆ2(<sup>−</sup> *<sup>∂</sup>*

*∂u*<sup>2</sup>

<sup>0</sup>(2) = 2*u*2*e*

<sup>−</sup>(*u*<sup>2</sup> 1+*u*<sup>2</sup>

<sup>1</sup> · *<sup>ψ</sup>* (2)

<sup>1</sup> to produce the doubly degenerate states *<sup>ψ</sup>*(1)

(1,0) and *<sup>Q</sup>* †

Our results Equation (4.23)-(4.24) possess a remarkable property. Only one component is nonzero! We shall see that this is indicative of an extremely interesting property that we observe in the non-separable examples that we consider next. Indeed, we recall that in relativistic quantum mechanics, one obtains a tensor Hamiltonian and the solutions are characterized by large and small components. In the present case, the small component is exactly zero. In the degenerate pair of solutions, which component is zero changes. We stress, however, that any linear combination of the two degenerate solutions is also a solution of the

*∂u*1*∂u*<sup>2</sup>

+ *u*<sup>1</sup> *∂ ∂u*<sup>2</sup>

←→1 , ←→<sup>1</sup> <sup>=</sup> *�*ˆ1*�*ˆ1 <sup>+</sup> *�*ˆ2*�*ˆ2, (4.19)

*∂ ∂u*<sup>2</sup>

*∂u*<sup>2</sup> 2 + *u*<sup>2</sup>

(*n*) (4.21)

(*n*)2. (4.22)

<sup>2</sup>)/2 (4.23)

<sup>2</sup>)/2, (4.24)

(1,0) and *<sup>ψ</sup>*(1)

(0,1)

+ *u*2) (4.25)

<sup>2</sup>)/2 <sup>∝</sup> *<sup>ψ</sup>*(1)

. Thus, recall

(0,1) (4.26)

+ *u*<sup>2</sup> *∂ ∂u*<sup>1</sup> ]

<sup>2</sup> + 3]. (4.20)

+ *u*1*u*<sup>2</sup> − *u*<sup>1</sup>

] + *�*ˆ2*�*ˆ2[<sup>−</sup> *<sup>∂</sup>*<sup>2</sup>

that *<sup>Q</sup> <sup>ψ</sup>*(1) <sup>1</sup> is indeed proportional to the ground state of ←→H 2. The precise relation for obtaining the sector one state from a sector two state is given by

$$
\psi\_{n+1}^{(1)} = \frac{1}{\sqrt{E\_{n+1}^{(1)} - E\_0^{(1)}}} \vec{\mathcal{Q}}^{\dagger} \cdot \vec{\psi}\_n^{(2)}.\tag{4.10}
$$

It is very instructive to illustrate this by considering a simple two dimensional separable harmonic oscillator model problem. This is because we can learn something of how our SUSY formalism works with an exactly soluble problem. We therefore consider a system described by the Hamiltonian

$$\mathcal{H} = -\frac{\partial^2}{\partial u\_1^2} - \frac{\partial^2}{\partial u\_2^2} + u\_1^2 + u\_1^2 \tag{4.11}$$

where again, we set ¯*h*2/2*m*<sup>1</sup> = *h*¯ 2/2*m*<sup>2</sup> = 1. The solution of the Schrödinger equation is well known to be the product of one dimensional harmonic oscillator states,

$$
\boldsymbol{\psi}\_{\left(\boldsymbol{\eta}\_{1},\boldsymbol{\eta}\_{2}\right)}^{\left(1\right)} = N\_{\boldsymbol{\eta}\_{1},\boldsymbol{\eta}\_{2}} \mathcal{H}\_{\boldsymbol{\eta}\_{1}}(\boldsymbol{u}\_{1}) \mathcal{H}\_{\boldsymbol{\eta}\_{2}}(\boldsymbol{u}\_{2}) \boldsymbol{e}^{-\left(\boldsymbol{u}\_{1}^{2} + \boldsymbol{u}\_{2}^{2}\right)/2} \tag{4.12}
$$

where *Nn*1,*n*<sup>2</sup> is the normalization constant and <sup>H</sup>*<sup>n</sup>* denotes the *<sup>n</sup>th* Hermite polynomial. The ground state is

$$
\psi\_{(0,0)}^{(1)} = \aleph\_{0,0} e^{-(u\_1^2 + u\_2^2)/2} \tag{4.13}
$$

with the zero point energy in this case equals to 2. We next generate the vector superpotential, *W* 1, as

$$\vec{W}\_1 = -\vec{\nabla} \ln \psi\_{(0,0)}^{(1)} = \mu\_1 \pounds\_1 + \mu\_2 \pounds\_2 \tag{4.14}$$

where

$$
\vec{\nabla} = \hat{\varepsilon}\_1 \frac{\partial}{\partial u\_1} + \hat{\varepsilon}\_2 \frac{\partial}{\partial u\_2}. \tag{4.15}
$$

We consider

$$(-\vec{\nabla} + \vec{W}\_1) \cdot (\vec{\nabla} + \vec{W}\_1) = -\nabla^2 + \vec{W}\_1 \cdot \vec{W}\_1 - \vec{\nabla} \cdot \vec{W}\_1 \tag{4.16}$$

we see that *<sup>W</sup>* <sup>1</sup> · *<sup>W</sup>* <sup>1</sup> <sup>−</sup> ∇ · *<sup>W</sup>* <sup>1</sup> <sup>=</sup> *<sup>u</sup>*<sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>u</sup>*<sup>2</sup> <sup>2</sup> − 2 = *V* − 2, so that

$$\mathcal{H}\_1 = \vec{Q}\_1^\dagger \cdot \vec{Q}\_1 + 2\tag{4.17}$$

It is easily verified that

$$\mathcal{H}\_1 \mathfrak{\psi}\_{(0,0)}^{(1)} = 2 \mathfrak{\psi}\_{(0,0)'}^{(1)} \tag{4.18}$$

as required. The first excited states of <sup>H</sup><sup>1</sup> are doubly degenerate with energy *<sup>E</sup>*(1) (1,0) <sup>=</sup> *<sup>E</sup>*(1) (0,1) = 3 and denoted by *<sup>ψ</sup>*(1) (1,0) and *<sup>ψ</sup>*(1) (0,1) . The next excited state, *<sup>ψ</sup>*(1) (1,1) is degenerate with *<sup>ψ</sup>*(1) (0,2) and *<sup>ψ</sup>*(1) (2,0) with energy *<sup>E</sup>*(1) (1,1) <sup>=</sup> *<sup>E</sup>*(1) (2,0) <sup>=</sup> *<sup>E</sup>*(1) (0,2) = 4.

We construct the rigorous sector two Hamiltonian as

$$
\overleftrightarrow{\mathcal{H}}\_{2} = \vec{Q}\_{1}\vec{Q}\_{1}^{\dagger} + 2\stackrel{\longleftrightarrow}{1}, \quad \overleftrightarrow{1}^{\prime} = \mathfrak{E}\_{1}\mathfrak{E}\_{1} + \mathfrak{E}\_{2}\mathfrak{E}\_{2} \tag{4.19}
$$

which is a second rank tensor in this case. The Hamiltonian ←→*<sup>H</sup>* <sup>2</sup> is then given by

$$
\begin{split}
\mathring{\mathcal{H}}\_{2} &= \mathring{\varepsilon}\_{1}\mathring{\varepsilon}\_{1}[-\frac{\partial^{2}}{\partial u\_{1}^{2}} + u\_{1}^{2} + 3] + \mathring{\varepsilon}\_{1}\mathring{\varepsilon}\_{2}[-\frac{\partial^{2}}{\partial u\_{1}\partial u\_{2}} + u\_{1}u\_{2} - u\_{1}\frac{\partial}{\partial u\_{2}} + u\_{2}\frac{\partial}{\partial u\_{1}}] \\ &+ \mathring{\varepsilon}\_{2}\mathring{\varepsilon}\_{1}[-\frac{\partial^{2}}{\partial u\_{1}\partial u\_{2}} + u\_{1}u\_{2} - u\_{2}\frac{\partial}{\partial u\_{1}} + u\_{1}\frac{\partial}{\partial u\_{2}}] + \mathring{\varepsilon}\_{2}\mathring{\varepsilon}\_{2}[-\frac{\partial^{2}}{\partial u\_{2}^{2}} + u\_{2}^{2} + 3].
\end{split}
\tag{4.20}
$$

The eigenvalue equation is

$$
\overleftrightarrow{\mathcal{H}}\_{2} \cdot \vec{\psi}\_{(n)}^{(2)} = E\_{(n)}^{(2)} \vec{\psi}\_{(n)}^{(2)} \tag{4.21}
$$

with eigenstates

$$
\vec{\psi}\_{(n)}^{(2)} = \pounds\_1 \psi\_{(n)1}^{(2)} + \pounds\_2 \psi\_{(n)2}^{(2)}.\tag{4.22}
$$

It is not difficult to show that there are two degenerate ground state solutions given by

$$
\vec{\psi}\_{(0)1}^{(2)} = \mathfrak{e}\_1 e^{-(u\_1^2 + u\_2^2)/2} \tag{4.23}
$$

and

14

that *<sup>Q</sup> <sup>ψ</sup>*(1)

by the Hamiltonian

ground state is

*W* 1, as

where

We consider

we see that *<sup>W</sup>* <sup>1</sup> · *<sup>W</sup>* <sup>1</sup> <sup>−</sup> ∇ ·

It is easily verified that

3 and denoted by *<sup>ψ</sup>*(1)

(2,0) with energy *<sup>E</sup>*(1)

*<sup>ψ</sup>*(1)

the sector one state from a sector two state is given by

*<sup>ψ</sup>*(1)

*W* <sup>1</sup> = *u*<sup>2</sup>

(1,0) and *<sup>ψ</sup>*(1)

(1,1) <sup>=</sup> *<sup>E</sup>*(1)

(0,1)

(2,0) <sup>=</sup> *<sup>E</sup>*(1)

*<sup>ψ</sup>*(1)

*<sup>n</sup>*+<sup>1</sup> <sup>=</sup> <sup>1</sup> *E*(1) *<sup>n</sup>*+<sup>1</sup> <sup>−</sup> *<sup>E</sup>*(1) 0

<sup>H</sup> <sup>=</sup> <sup>−</sup> *<sup>∂</sup>*<sup>2</sup> *∂u*<sup>2</sup> 1

known to be the product of one dimensional harmonic oscillator states,

*<sup>ψ</sup>*(1)

*<sup>W</sup>* <sup>1</sup> <sup>=</sup> −∇ ln *<sup>ψ</sup>*(1)

<sup>∇</sup> <sup>=</sup> *�*ˆ1

<sup>1</sup> <sup>+</sup> *<sup>u</sup>*<sup>2</sup>

<sup>1</sup> is indeed proportional to the ground state of ←→H 2. The precise relation for obtaining

It is very instructive to illustrate this by considering a simple two dimensional separable harmonic oscillator model problem. This is because we can learn something of how our SUSY formalism works with an exactly soluble problem. We therefore consider a system described

> <sup>−</sup> *<sup>∂</sup>*<sup>2</sup> *∂u*<sup>2</sup> 2 + *u*<sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>u</sup>*<sup>2</sup>

where again, we set ¯*h*2/2*m*<sup>1</sup> = *h*¯ 2/2*m*<sup>2</sup> = 1. The solution of the Schrödinger equation is well

where *Nn*1,*n*<sup>2</sup> is the normalization constant and <sup>H</sup>*<sup>n</sup>* denotes the *<sup>n</sup>th* Hermite polynomial. The

with the zero point energy in this case equals to 2. We next generate the vector superpotential,

*∂ ∂u*<sup>1</sup>

(−∇ <sup>+</sup> *<sup>W</sup>* <sup>1</sup>) · (<sup>∇</sup> <sup>+</sup> *<sup>W</sup>* <sup>1</sup>) = −∇<sup>2</sup> <sup>+</sup> *<sup>W</sup>* <sup>1</sup> · *<sup>W</sup>* <sup>1</sup> <sup>−</sup> ∇ ·

<sup>H</sup><sup>1</sup> <sup>=</sup> *<sup>Q</sup>* †

H1*ψ*(1)

as required. The first excited states of <sup>H</sup><sup>1</sup> are doubly degenerate with energy *<sup>E</sup>*(1)

(0,2) = 4.

+ *�*ˆ2 *∂ ∂u*<sup>2</sup>

<sup>2</sup> − 2 = *V* − 2, so that

(0,0) <sup>=</sup> <sup>2</sup>*ψ*(1)

. The next excited state, *<sup>ψ</sup>*(1)

(0,0)

<sup>−</sup>(*u*<sup>2</sup> 1+*u*<sup>2</sup> <sup>−</sup>(*u*<sup>2</sup> 1+*u*<sup>2</sup>

(*n*1,*n*2) = *Nn*1,*n*2H*n*<sup>1</sup> (*u*1)H*n*<sup>2</sup> (*u*2)*<sup>e</sup>*

(0,0) = *N*0,0*e*

*<sup>Q</sup>* † · *<sup>ψ</sup>* (2) *<sup>n</sup>* . (4.10)

<sup>1</sup> (4.11)

<sup>2</sup>)/2 (4.12)

<sup>2</sup>)/2 (4.13)

. (4.15)

*W* <sup>1</sup> (4.16)

(1,0) <sup>=</sup> *<sup>E</sup>*(1)

(0,1) =

(0,2) and

(0,0) = *u*1*�*ˆ1 + *u*2*�*ˆ2 (4.14)

<sup>1</sup> · *<sup>Q</sup>* <sup>1</sup> <sup>+</sup> <sup>2</sup> (4.17)

, (4.18)

(1,1) is degenerate with *<sup>ψ</sup>*(1)

$$
\vec{\psi}\_{(0)2}^{(2)} = \pounds\_2 e^{-(u\_1^2 + u\_2^2)/2} \,\text{.}\tag{4.24}
$$

respectively. This is extremely interesting and in contrast to the usual situation in quantum mechanics. For most systems (excluding spin effects) the ground state is unique, i.e., non-degenerate.

We shall see that the degenerate states, Equation (4.23)-(4.24), are exactly what is required for the charge operator, *Q* † <sup>1</sup> to produce the doubly degenerate states *<sup>ψ</sup>*(1) (1,0) and *<sup>ψ</sup>*(1) (0,1) . Thus, recall that

$$\vec{Q}\_1^\dagger = \hat{\mathbf{e}}\_1(-\frac{\partial}{\partial u\_1} + u\_1) + \hat{\mathbf{e}}\_2(-\frac{\partial}{\partial u\_2} + u\_2) \tag{4.25}$$

Then

$$
\vec{Q}\_1^\dagger \cdot \vec{\psi}\_{0(1)}^{(2)} = 2u\_1 e^{-(u\_1^2 + u\_2^2)/2} \propto \psi\_{(1,0)}^{(1)} \text{ and } \,\, \vec{Q}\_1^\dagger \cdot \vec{\psi}\_{0(2)}^{(2)} = 2u\_2 e^{-(u\_1^2 + u\_2^2)/2} \propto \psi\_{(0,1)}^{(1)} \tag{4.26}
$$

Our results Equation (4.23)-(4.24) possess a remarkable property. Only one component is nonzero! We shall see that this is indicative of an extremely interesting property that we observe in the non-separable examples that we consider next. Indeed, we recall that in relativistic quantum mechanics, one obtains a tensor Hamiltonian and the solutions are characterized by large and small components. In the present case, the small component is exactly zero. In the degenerate pair of solutions, which component is zero changes. We stress, however, that any linear combination of the two degenerate solutions is also a solution of the same energy.

where the braces indicate the anticommutator bracket. According to Wess and Bagger's text on supersymmetry, these are the necessary conditions for a superalgebra Wess & Bagger

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 117

sector Hamiltonians <sup>H</sup><sup>1</sup> and ←→H 2. An important consequence of the algebra is the existence of "inter-twining" relations. These are of fundamental importance because they underlie the isospectral property and in addition, they can be used to establish the unique correspondence between the eigenstates of sectors 1 and 2. Indeed, they are responsible for establishing the completeness of the eigenvectors {*ψ*� (2) *<sup>n</sup>* } of ←→H 2. It is of interest to note that inter-twining relations are essential to the fact that in ordinary quantum scattering, the continua of the full Hamiltonian, H, and the unperturbed Hamiltonian H<sup>0</sup> ( where H = H<sup>0</sup> + *V*, with *V* the perturbation responsible for scattering) coincide. In that case, the inter-twining relation is

<sup>1</sup> and the

*<sup>i</sup>*H*t*/¯*h*Ω+. (4.38)

<sup>0</sup> (4.40)

<sup>0</sup> <sup>≡</sup>�<sup>0</sup> (4.41)

<sup>0</sup> )*ψ*(1) *<sup>n</sup>* <sup>=</sup> *<sup>E</sup>*(1) *<sup>n</sup> <sup>ψ</sup>*(1) *<sup>n</sup>* (4.42)

<sup>0</sup> )*ψ*(1) *<sup>n</sup>* <sup>=</sup> *<sup>E</sup>*(1) *<sup>n</sup> <sup>Q</sup>*� <sup>1</sup>*ψ*(1) *<sup>n</sup>* (4.43)

<sup>H</sup>1*ψ*(1) *<sup>n</sup>* <sup>=</sup> *<sup>E</sup>*(1) *<sup>n</sup> <sup>ψ</sup>*(1) *<sup>n</sup>* , (4.39)

We now consider in more detail the degeneracy between the two sectors, *Q*� <sup>1</sup> and *Q*� †

Ω+*e*

It is useful to derive the SUSY inter-twining relations explicitly. We have

set {*ψ*(1) *<sup>n</sup>* } is complete on the physically allowed space of state vectors.

(*Q*� †

(*Q*� <sup>1</sup>*Q*� †

<sup>H</sup><sup>1</sup> <sup>=</sup> *<sup>Q</sup>*� †

<sup>1</sup> · *<sup>Q</sup>*� <sup>1</sup> <sup>+</sup> *<sup>E</sup>*(1)

<sup>1</sup> · *<sup>Q</sup>*� <sup>1</sup> <sup>+</sup> *<sup>E</sup>*(1)

*<sup>Q</sup>*� <sup>1</sup>*ψ*(1)

where we now assume that *<sup>n</sup>* <sup>&</sup>gt; 0; i.e., *<sup>ψ</sup>*(1) *<sup>n</sup>* is an excited state of <sup>H</sup>1. We apply *<sup>Q</sup>*� <sup>1</sup> to Equation

factors H1) is such that

The general sector 1 Schrödinger equation is

where

(4.42) to find

*<sup>i</sup>*H0*t*/¯*<sup>h</sup>* = *e*

where H<sup>1</sup> is a standard Schrödinger operator (comprised of a Laplacian for the kinetic energy and a Hermitian potential *V*1). One result of this fact is that the ground state of H<sup>1</sup> is nodeless. In addition, H<sup>1</sup> is Hermitian and a well known postulate of quantum mechanics asserts that its eigenstates are complete. Essentially from a physical standpoint (as opposed to pure mathematics) H<sup>1</sup> is required to be Hermitian because (a) it represents an observable, implying only real eigenvalues (b) quantum mechanics further asserts that these eigenvalues are the only values that can be obtained when measuring the energy for the physical system represented by H1. This implies that any physically realizable state, *ψ*, of the system must be a superposition (in general) of these and only these eigenvectors. This then implies that the

To derive the inter-twining relation, we again recall that the charge operator (which exactly

<sup>1</sup> · *<sup>Q</sup>*� <sup>1</sup> <sup>+</sup> *<sup>E</sup>*(1)

(1992).

With a view toward the next section, where we consider a two dimensional, nonseparable anharmonic oscillator(or equivalently a pair of one dimensional coupled oscillators), we form the equivalent degenerate solutions

$$\vec{\phi}\_{(0)1}^{(2)} = N e^{-(\mu\_1^2 + \mu\_2^2)/2} [\pounds\_1 + \pounds\_2] \text{ and } \ e^{-(\mu\_1^2 + \mu\_2^2)/2} [\pounds\_1 - \pounds\_2] \tag{4.27}$$

In this case, both components of the 2-degenerate solutions are non-zero, of the same magnitude and of definite sign. In dealing with the two dimensional separable harmonic oscillator, the most convenient form is given by Equation (4.26).

A major concern is whether our approach satisfies the supersymmetric algebra which we will consider here. It is clear that we can define our Hamiltonian operator by

$$\mathcal{H} = \begin{pmatrix} \vec{\mathbb{Q}}^{\dagger} \cdot \vec{\mathbb{Q}} & 0\\ 0 & \vec{\mathbb{Q}}\vec{\mathbb{Q}}^{\dagger} \cdot \end{pmatrix} = \begin{pmatrix} \mathcal{H}\_1 \ 0\\ 0 \ \overleftrightarrow{\mathcal{H}\_2} \end{pmatrix} \tag{4.28}$$

where the zero in the upper right is a row vector and the zero in the lower left is a column vector. This Hamiltonian will act on the state

$$
\vec{\psi} = \begin{pmatrix} \psi\_n^{(1)} \\ \vec{\psi}\_{n-1}^{(2)} \end{pmatrix}. \tag{4.29}
$$

Then, we can define a "super-charge" operator as

$$\mathcal{Q} = \begin{pmatrix} 0 \ 0 \\ \vec{\mathcal{Q}} \ 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \ 0 \\ Q\_1 \ 0 \ 0 \\ Q\_2 \ 0 \ 0 \end{pmatrix} \tag{4.30}$$

with the adjoint being

$$\mathcal{Q}^{\dagger} = \begin{pmatrix} 0 \ \vec{Q}^{\dagger} \\ 0 \ 0 \end{pmatrix} = \begin{pmatrix} 0 \ Q\_1^{\dagger} \ Q\_2^{\dagger} \\ 0 \ 0 \ 0 \\ 0 \ 0 \ 0 \end{pmatrix}. \tag{4.31}$$

If we take the product of <sup>Q</sup>† and <sup>Q</sup>, we find that

$$
\begin{aligned}
\dagger \dddot{\mathcal{H}}\_1 &= \begin{pmatrix} \vec{\mathcal{Q}}^\dagger \cdot \vec{\mathcal{Q}} \, 0 \\ 0 & 0 \end{pmatrix} \end{aligned} \tag{4.32}
$$

and similarly, if we take the product of <sup>Q</sup> and <sup>Q</sup>†, we find that

$$
\stackrel{\leftrightarrow}{\mathcal{H}}\_2 = \begin{pmatrix} 0 & 0 \\ 0 \ \vec{Q} \vec{Q}^\dagger \cdot \end{pmatrix} . \tag{4.33}
$$

It's straightforward to show that

$$\left[\mathcal{Q}, \mathcal{H}\right] = 0,\tag{4.34}$$

$$[\mathcal{Q}^{\dagger}\mathcal{Q}, \mathcal{Q}\mathcal{Q}^{\dagger}] = 0,\tag{4.35}$$

$$
\mathcal{Q}\mathcal{Q} = \mathcal{Q}^\dagger \mathcal{Q}^\dagger = 0,\tag{4.36}
$$

$$
\begin{array}{c}
\text{and} \\
\overleftrightarrow{\mathcal{H}} = \{\mathcal{Q}, \mathcal{Q}^{\dagger}\}
\end{array}
\tag{4.37}
$$

where the braces indicate the anticommutator bracket. According to Wess and Bagger's text on supersymmetry, these are the necessary conditions for a superalgebra Wess & Bagger (1992).

We now consider in more detail the degeneracy between the two sectors, *Q*� <sup>1</sup> and *Q*� † <sup>1</sup> and the sector Hamiltonians <sup>H</sup><sup>1</sup> and ←→H 2. An important consequence of the algebra is the existence of "inter-twining" relations. These are of fundamental importance because they underlie the isospectral property and in addition, they can be used to establish the unique correspondence between the eigenstates of sectors 1 and 2. Indeed, they are responsible for establishing the completeness of the eigenvectors {*ψ*� (2) *<sup>n</sup>* } of ←→H 2. It is of interest to note that inter-twining relations are essential to the fact that in ordinary quantum scattering, the continua of the full Hamiltonian, H, and the unperturbed Hamiltonian H<sup>0</sup> ( where H = H<sup>0</sup> + *V*, with *V* the perturbation responsible for scattering) coincide. In that case, the inter-twining relation is

$$
\Omega^+ e^{i\mathcal{H}\_0 t/\hbar} = e^{i\mathcal{H}t/\hbar} \Omega^+.\tag{4.38}
$$

It is useful to derive the SUSY inter-twining relations explicitly. We have

$$\mathcal{H}\_1 \psi\_n^{(1)} = E\_n^{(1)} \psi\_n^{(1)} \,\,\,\,\,\,\tag{4.39}$$

where H<sup>1</sup> is a standard Schrödinger operator (comprised of a Laplacian for the kinetic energy and a Hermitian potential *V*1). One result of this fact is that the ground state of H<sup>1</sup> is nodeless. In addition, H<sup>1</sup> is Hermitian and a well known postulate of quantum mechanics asserts that its eigenstates are complete. Essentially from a physical standpoint (as opposed to pure mathematics) H<sup>1</sup> is required to be Hermitian because (a) it represents an observable, implying only real eigenvalues (b) quantum mechanics further asserts that these eigenvalues are the only values that can be obtained when measuring the energy for the physical system represented by H1. This implies that any physically realizable state, *ψ*, of the system must be a superposition (in general) of these and only these eigenvectors. This then implies that the set {*ψ*(1) *<sup>n</sup>* } is complete on the physically allowed space of state vectors.

To derive the inter-twining relation, we again recall that the charge operator (which exactly factors H1) is such that

$$\mathcal{H}\_1 = \vec{Q}\_1^\dagger \cdot \vec{Q}\_1 + E\_0^{(1)} \tag{4.40}$$

where

16

the equivalent degenerate solutions

 *<sup>φ</sup>*(2)

(0)<sup>1</sup> <sup>=</sup> *Ne*−(*u*<sup>2</sup>

1+*u*<sup>2</sup>

consider here. It is clear that we can define our Hamiltonian operator by

*<sup>Q</sup>* † · *<sup>Q</sup>* <sup>0</sup> <sup>0</sup> *<sup>Q</sup> <sup>Q</sup>* †·

> *ψ* = � *<sup>ψ</sup>*(1) *<sup>n</sup> <sup>ψ</sup>* (2) *n*−1

� 0 0 *Q* 0 � = ⎛ ⎝

� 0 *Q* † 0 0

←→H <sup>1</sup> <sup>=</sup>

←→H <sup>2</sup> <sup>=</sup>

� = ⎛ ⎝

�

Q =

<sup>Q</sup>† <sup>=</sup>

and similarly, if we take the product of <sup>Q</sup> and <sup>Q</sup>†, we find that

oscillator, the most convenient form is given by Equation (4.26).

H =

vector. This Hamiltonian will act on the state

Then, we can define a "super-charge" operator as

If we take the product of <sup>Q</sup>† and <sup>Q</sup>, we find that

with the adjoint being

It's straightforward to show that

With a view toward the next section, where we consider a two dimensional, nonseparable anharmonic oscillator(or equivalently a pair of one dimensional coupled oscillators), we form

<sup>2</sup>)/2[*�*ˆ1 + *�*ˆ2] and *e*

In this case, both components of the 2-degenerate solutions are non-zero, of the same magnitude and of definite sign. In dealing with the two dimensional separable harmonic

A major concern is whether our approach satisfies the supersymmetric algebra which we will

where the zero in the upper right is a row vector and the zero in the lower left is a column

� =

�

�

�

� *<sup>Q</sup>* † · *<sup>Q</sup>* <sup>0</sup> 0 0

� 0 0 <sup>0</sup> *<sup>Q</sup> <sup>Q</sup>* †·

and ←→H <sup>=</sup> {Q, <sup>Q</sup>†} (4.37)

⎞

⎞

<sup>−</sup>(*u*<sup>2</sup> 1+*u*<sup>2</sup>

� <sup>H</sup><sup>1</sup> <sup>0</sup> 0 ←→H2 �

<sup>2</sup>)/2[*�*ˆ1 <sup>−</sup> *�*ˆ2] (4.27)

. (4.29)

⎠ (4.30)

⎠ . (4.31)

. (4.33)

[Q, H] = 0, (4.34) [Q†Q, QQ†] = 0, (4.35) QQ <sup>=</sup> <sup>Q</sup>†Q† <sup>=</sup> 0, (4.36)

(4.28)

(4.32)

$$
\vec{Q}\_1 \psi\_0^{(1)} \equiv \vec{0} \tag{4.41}
$$

The general sector 1 Schrödinger equation is

$$(\vec{Q}\_1^\dagger \cdot \vec{Q}\_1 + E\_0^{(1)})\psi\_n^{(1)} = E\_n^{(1)}\psi\_n^{(1)}\tag{4.42}$$

where we now assume that *<sup>n</sup>* <sup>&</sup>gt; 0; i.e., *<sup>ψ</sup>*(1) *<sup>n</sup>* is an excited state of <sup>H</sup>1. We apply *<sup>Q</sup>*� <sup>1</sup> to Equation (4.42) to find

$$(\vec{Q}\_1 \vec{Q}\_1^\dagger \cdot \vec{Q}\_1 + E\_0^{(1)}) \psi\_n^{(1)} = E\_n^{(1)} \vec{Q}\_1 \psi\_n^{(1)} \tag{4.43}$$

**5. Clusters of distinguishable particles**

having the components *W*<sup>11</sup> = (2*u*1*u*<sup>2</sup>

trial wave function for sector one is

and that for sector two is

are

Hamiltonian for sector one of the following form

<sup>H</sup><sup>1</sup> <sup>=</sup> −∇<sup>2</sup> <sup>+</sup> *<sup>V</sup>*1(*u*1, *<sup>u</sup>*2) = <sup>−</sup> *<sup>∂</sup>*<sup>2</sup>

In this case, the exact ground state energy is *E*(1)

*<sup>ψ</sup>*(1) (*trial*)

*<sup>ψ</sup>*(2)

*<sup>ψ</sup>*(2)

 H(2) <sup>11</sup> <sup>H</sup>(2) 12

H(2) <sup>21</sup> <sup>H</sup>(2) 22

*<sup>ψ</sup>*(1) (0)

We next consider a model non-separable two dimensional anharmonic oscillator system for sector one for which the ground state energy is zero and the ground state wave function is

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 119

1*u*2 <sup>2</sup> <sup>−</sup> *<sup>u</sup>*<sup>2</sup>

(0)

using these components we can generate the model potential for sector one. Thus we get the

<sup>−</sup> *<sup>∂</sup>*<sup>2</sup> *∂u*<sup>2</sup> 2

<sup>1</sup> <sup>+</sup> <sup>1</sup>) <sup>−</sup> <sup>2</sup>(*u*<sup>2</sup>

can be generated with <sup>∇</sup> and *<sup>W</sup>* 1. The calculation for sector one and sector two eigenvalues and eigenfunctions is done variationally by diagonalizing each sector Hamiltonian in an approximate truncated basis. We choose to employ a basis of the direct product of the

eigenstates of a harmonic oscillator in each dimension, each with frequency *ω* = 2

*m*,*n*

*m*,*n C*(2) 1*m*,*<sup>n</sup>*

*m*,*n C*(2) 2*m*,*<sup>n</sup>*

 *<sup>C</sup>*(2) 1 *C*(2) 2

 = *E C*(2) 1 *C*(2) 2

at the Hermitian eigenvalue equation for both sectors. For sector one the form is

Using these trial wave functions and treating the *Cm*,*<sup>n</sup>* as a variational parameters, we arrive

where *<sup>α</sup>* <sup>=</sup> <sup>√</sup>*mω*/¯*h*. Similarly for the sector two the trial wave functions for each component

<sup>2</sup> <sup>+</sup> <sup>2</sup>*u*1) and *<sup>W</sup>*<sup>12</sup> = (2*u*<sup>2</sup>

+ (2*u*1*u*<sup>2</sup>

<sup>1</sup> <sup>−</sup> *<sup>u</sup>*<sup>2</sup>

<sup>2</sup>) (5.1)

<sup>1</sup>*u*<sup>2</sup> + 2*u*2), respectively. Now

<sup>1</sup>*u*<sup>2</sup> <sup>+</sup> <sup>2</sup>*u*2)<sup>2</sup>

<sup>√</sup>2. The

(5.7)

(*u*1, *u*2) (5.2)

<sup>2</sup> + 1) (5.3)

<sup>0</sup> = 0. The sector two tensor Hamiltonian

*<sup>C</sup>*(1) *<sup>m</sup>*,*nφm*(*α*, *<sup>u</sup>*1)*φn*(*α*, *<sup>u</sup>*2) (5.4)

*φm*(*α*, *u*1)*φn*(*α*, *u*2) (5.5)

<sup>H</sup>1*C*(1) <sup>=</sup> *EC*(1) (5.6)

*φm*(*α*, *u*1)*φn*(*α*, *u*2)

<sup>2</sup> <sup>+</sup> <sup>2</sup>*u*1)<sup>2</sup> + (2*u*<sup>2</sup>

(*u*1, *<sup>u</sup>*2) = *<sup>N</sup>* exp(−*u*<sup>2</sup>

*<sup>W</sup>* <sup>1</sup> <sup>=</sup> −∇ ln *<sup>ψ</sup>*(1)

*∂u*<sup>2</sup> 1

<sup>−</sup> <sup>2</sup>(*u*<sup>2</sup>

(*u*1, *u*2) = ∑

(*trial*)1(*u*1, *<sup>u</sup>*2) = ∑

(*trial*)2(*u*1, *<sup>u</sup>*2) = ∑

We can generate the superpotential corresponding to this ground state as

**5.1 Degenerate case**

exactly given by

We define ←→H <sup>2</sup> as

$$
\overleftrightarrow{\mathcal{H}}\_2 \equiv \vec{Q}\_1 \vec{Q}\_1^\dagger + E\_0^{(1)} \tag{4.44}
$$

so that Equation (4.43) yields

$$
\overrightarrow{Q}\_1 \mathcal{H}\_1 = \overleftrightarrow{\mathcal{H}}\_2 \cdot \overrightarrow{Q}\_1. \tag{4.45}
$$

*This is the inter-twining relation*. Let us explore inter-twining consequences further. Suppose we consider an eigenstate , *<sup>ψ</sup>*(1) *<sup>n</sup>* of <sup>H</sup>1. It follows from Equation (4.45) that there is also a unique eigenstate of ←→H 2, *<sup>Q</sup>* <sup>1</sup>*ψ*(1) *<sup>n</sup>* with the same energy. Next, assume that ←→H <sup>2</sup> possesses an eigenvalue *E*(2) *<sup>λ</sup>* that differs from all of the *<sup>E</sup>*(1) *<sup>n</sup>* . Then we have

$$
\overleftrightarrow{\mathcal{H}}\_2 \cdot \vec{\psi}\_{\lambda}^{(2)} = E\_{\lambda}^{(2)} \vec{\psi}\_{\lambda}^{(2)} \tag{4.46}
$$

Now ←→H <sup>2</sup> and <sup>H</sup><sup>1</sup> are manifestly Hermitian. Taking the adjoint of Equation (4.45) yields

$$
\mathcal{H}\_1 \vec{Q}\_1^\dagger = \vec{Q}\_1^\dagger \cdot \overleftarrow{\mathcal{H}}\_2 \tag{4.47}
$$

which is again an inter-twining relation. We then take the scalar product of Equation (4.46) with *Q* † 1

$$
\vec{Q}\_1^\dagger \cdot \overleftrightarrow{\mathcal{H}}\_2 \cdot \vec{\psi}\_\lambda^{(2)} = E\_\lambda^{(2)} \vec{Q}\_1^\dagger \cdot \vec{\psi}\_\lambda^{(2)}\tag{4.48}
$$

But by the adjoint inter-twining relation, we have

$$\mathcal{H}\_1 \vec{Q}\_1^\dagger \cdot \vec{\psi}\_\lambda^{(2)} = E\_\lambda^{(2)} \vec{Q}\_1^\dagger \cdot \vec{\psi}\_\lambda^{(2)}\tag{4.49}$$

Thus, we find that H<sup>1</sup> also has the scalar eigenvector

$$
\psi\_{
\lambda}^{(1)} \approx \vec{Q}\_1^{\dagger} \cdot \vec{\psi}\_{
\lambda}^{(2)} \tag{4.50}
$$

and its eigenvalue is equal to *E*(2) *<sup>λ</sup>* . This violates our initial assertion that H<sup>1</sup> did not have the eigenvalue *E*(2) *<sup>λ</sup>* . We conclude that for eigenvectors *<sup>ψ</sup>* (2) *<sup>n</sup>* , there corresponds a unique eigenvector *<sup>ψ</sup>*(1) *<sup>n</sup>*� , where *<sup>n</sup>*� <sup>≡</sup> *<sup>n</sup>* <sup>+</sup> 1. That is, *<sup>ψ</sup>*<sup>0</sup> (2) must have the same energy as the first excited state *<sup>ψ</sup>*(1) <sup>1</sup> . It can not be lower than *<sup>E</sup>*(1) <sup>1</sup> because it is the lowest eigenvalue of ←→H <sup>2</sup> and it cannot equal *E*(1) <sup>0</sup> . In fact, the inter-twining relation is sufficient to establish that ←→H <sup>2</sup> is a Schrödinger operator and as such, its eigenvectors must be complete on the space *ψ* . Note that we are not saying that the *<sup>ψ</sup>* (2) *<sup>n</sup>* span the the space generated by <sup>H</sup>1. They are completely separate vector spaces arising from two distinct Hermitian Hamiltonians. All of the above can be made mathematically rigorous but our purpose here is to supply a physically reasonable argument for the properties of the tensor sector. Finally, at no point in this discussion have we imposed a condition that the spectra of <sup>H</sup><sup>1</sup> (and ←→H <sup>2</sup>) are strictly discrete. The inter-twining relations hold for systems with a mixed discrete and continuous spectra and even for systems with a purely continuous spectrum.

#### **5. Clusters of distinguishable particles**

#### **5.1 Degenerate case**

18

We define ←→H <sup>2</sup> as

eigenvalue *E*(2)

Now

with *Q* † 1

so that Equation (4.43) yields

←→H <sup>2</sup> <sup>≡</sup> *<sup>Q</sup>* <sup>1</sup>*<sup>Q</sup>* †

*<sup>λ</sup>* that differs from all of the *<sup>E</sup>*(1) *<sup>n</sup>* . Then we have

←→H <sup>2</sup> · *<sup>ψ</sup>* (2)

<sup>H</sup>1*<sup>Q</sup>* †

←→H <sup>2</sup> · *<sup>ψ</sup>* (2)

<sup>1</sup> · *<sup>ψ</sup>* (2)

*<sup>ψ</sup>*(1) *<sup>λ</sup>* <sup>∝</sup> *<sup>Q</sup>* †

*Q* † 1 ·

*<sup>n</sup>*� , where *<sup>n</sup>*� <sup>≡</sup> *<sup>n</sup>* <sup>+</sup> 1. That is, *<sup>ψ</sup>*<sup>0</sup>

<sup>1</sup> . It can not be lower than *<sup>E</sup>*(1)

<sup>H</sup>1*<sup>Q</sup>* †

But by the adjoint inter-twining relation, we have

Thus, we find that H<sup>1</sup> also has the scalar eigenvector

and its eigenvalue is equal to *E*(2)

with a purely continuous spectrum.

the eigenvalue *E*(2)

eigenvector *<sup>ψ</sup>*(1)

excited state *<sup>ψ</sup>*(1)

it cannot equal *E*(1)

*This is the inter-twining relation*. Let us explore inter-twining consequences further. Suppose we consider an eigenstate , *<sup>ψ</sup>*(1) *<sup>n</sup>* of <sup>H</sup>1. It follows from Equation (4.45) that there is also a unique eigenstate of ←→H 2, *<sup>Q</sup>* <sup>1</sup>*ψ*(1) *<sup>n</sup>* with the same energy. Next, assume that ←→H <sup>2</sup> possesses an

*<sup>λ</sup>* <sup>=</sup> *<sup>E</sup>*(2)

←→H <sup>2</sup> and <sup>H</sup><sup>1</sup> are manifestly Hermitian. Taking the adjoint of Equation (4.45) yields

which is again an inter-twining relation. We then take the scalar product of Equation (4.46)

*<sup>λ</sup>* <sup>=</sup> *<sup>E</sup>*(2)

*<sup>λ</sup>* <sup>=</sup> *<sup>E</sup>*(2)

Schrödinger operator and as such, its eigenvectors must be complete on the space *ψ* . Note that we are not saying that the *<sup>ψ</sup>* (2) *<sup>n</sup>* span the the space generated by <sup>H</sup>1. They are completely separate vector spaces arising from two distinct Hermitian Hamiltonians. All of the above can be made mathematically rigorous but our purpose here is to supply a physically reasonable argument for the properties of the tensor sector. Finally, at no point in this discussion have we imposed a condition that the spectra of <sup>H</sup><sup>1</sup> (and ←→H <sup>2</sup>) are strictly discrete. The inter-twining relations hold for systems with a mixed discrete and continuous spectra and even for systems

*<sup>λ</sup> <sup>Q</sup>* †

*<sup>λ</sup> <sup>Q</sup>* †

<sup>1</sup> · *<sup>ψ</sup>* (2)

<sup>1</sup> · *<sup>ψ</sup>* (2)

<sup>1</sup> · *<sup>ψ</sup>* (2)

*<sup>λ</sup>* . We conclude that for eigenvectors *<sup>ψ</sup>* (2) *<sup>n</sup>* , there corresponds a unique

<sup>0</sup> . In fact, the inter-twining relation is sufficient to establish that ←→H <sup>2</sup> is a

*<sup>λ</sup>* . This violates our initial assertion that H<sup>1</sup> did not have

<sup>1</sup> <sup>=</sup> *<sup>Q</sup>* † 1 ·

*<sup>λ</sup> <sup>ψ</sup>* (2)

<sup>1</sup> <sup>+</sup> *<sup>E</sup>*(1)

<sup>0</sup> (4.44)

*<sup>λ</sup>* (4.46)

←→H <sup>2</sup> (4.47)

*<sup>λ</sup>* (4.48)

*<sup>λ</sup>* (4.49)

*<sup>λ</sup>* (4.50)

(2) must have the same energy as the first

<sup>1</sup> because it is the lowest eigenvalue of ←→H <sup>2</sup> and

*<sup>Q</sup>* <sup>1</sup>H<sup>1</sup> <sup>=</sup> ←→H <sup>2</sup> · *<sup>Q</sup>* 1. (4.45)

We next consider a model non-separable two dimensional anharmonic oscillator system for sector one for which the ground state energy is zero and the ground state wave function is exactly given by

$$
\psi\_{(0)}^{(1)}(\boldsymbol{\mu}\_1, \boldsymbol{\mu}\_2) = N \exp(-\boldsymbol{\mu}\_1^2 \boldsymbol{\mu}\_2^2 - \boldsymbol{\mu}\_1^2 - \boldsymbol{\mu}\_2^2) \tag{5.1}
$$

We can generate the superpotential corresponding to this ground state as

$$\vec{\mathcal{W}}\_1 = -\vec{\nabla} \ln \psi\_{(0)}^{(1)}(\mu\_1, \mu\_2) \tag{5.2}$$

having the components *W*<sup>11</sup> = (2*u*1*u*<sup>2</sup> <sup>2</sup> <sup>+</sup> <sup>2</sup>*u*1) and *<sup>W</sup>*<sup>12</sup> = (2*u*<sup>2</sup> <sup>1</sup>*u*<sup>2</sup> + 2*u*2), respectively. Now using these components we can generate the model potential for sector one. Thus we get the Hamiltonian for sector one of the following form

$$\mathcal{H}\_1 = -\nabla^2 + V\_1(\mu\_1, \mu\_2) = -\frac{\partial^2}{\partial u\_1^2} - \frac{\partial^2}{\partial u\_2^2} + (2\mu\_1 u\_2^2 + 2u\_1)^2 + (2u\_1^2 u\_2 + 2u\_2)^2$$

$$-2(u\_1^2 + 1) - 2(u\_2^2 + 1)\tag{5.3}$$

In this case, the exact ground state energy is *E*(1) <sup>0</sup> = 0. The sector two tensor Hamiltonian can be generated with <sup>∇</sup> and *<sup>W</sup>* 1. The calculation for sector one and sector two eigenvalues and eigenfunctions is done variationally by diagonalizing each sector Hamiltonian in an approximate truncated basis. We choose to employ a basis of the direct product of the eigenstates of a harmonic oscillator in each dimension, each with frequency *ω* = 2 <sup>√</sup>2. The trial wave function for sector one is

$$
\psi^{(1)}\_{(trial)}(\boldsymbol{\mu}\_1, \boldsymbol{\mu}\_2) = \sum\_{m,n} \mathbb{C}^{(1)}\_{m,n} \phi\_m(\boldsymbol{\alpha}, \boldsymbol{\mu}\_1) \phi\_n(\boldsymbol{\alpha}, \boldsymbol{\mu}\_2) \tag{5.4}
$$

where *<sup>α</sup>* <sup>=</sup> <sup>√</sup>*mω*/¯*h*. Similarly for the sector two the trial wave functions for each component are

$$
\psi^{(2)}\_{(trial)1}(\boldsymbol{\mu}\_1, \boldsymbol{\mu}\_2) = \sum\_{m,n} \mathbb{C}^{(2)}\_{1\_{\boldsymbol{\mu},\boldsymbol{\mu}}} \phi\_{\boldsymbol{m}}(\boldsymbol{\alpha}, \boldsymbol{\mu}\_1) \phi\_{\boldsymbol{\mu}}(\boldsymbol{\alpha}, \boldsymbol{\mu}\_2)
$$

$$
\psi^{(2)}\_{(trial)2}(\boldsymbol{\mu}\_1, \boldsymbol{\mu}\_2) = \sum\_{m,n} \mathbb{C}^{(2)}\_{2\_{\boldsymbol{\mu},\boldsymbol{\mu}}} \phi\_{\boldsymbol{m}}(\boldsymbol{\alpha}, \boldsymbol{\mu}\_1) \phi\_{\boldsymbol{\mu}}(\boldsymbol{\alpha}, \boldsymbol{\mu}\_2) \tag{5.5}
$$

Using these trial wave functions and treating the *Cm*,*<sup>n</sup>* as a variational parameters, we arrive at the Hermitian eigenvalue equation for both sectors. For sector one the form is

$$\mathcal{H}\_1 \mathbb{C}^{(1)} = E \mathbb{C}^{(1)} \tag{5.6}$$

and that for sector two is

$$
\begin{pmatrix}
\mathcal{H}\_{11}^{(2)} \,\mathcal{H}\_{12}^{(2)} \\
\mathcal{H}\_{21}^{(2)} \,\mathcal{H}\_{22}^{(2)}
\end{pmatrix}
\begin{pmatrix}
\mathcal{C}\_{1}^{(2)} \\
\mathcal{C}\_{2}^{(2)}
\end{pmatrix} = E \begin{pmatrix}
\mathcal{C}\_{1}^{(2)} \\
\mathcal{C}\_{2}^{(2)}
\end{pmatrix} \tag{5.7}
$$

The L<sup>2</sup> error is defined by

the SUSY relation *<sup>ψ</sup>*(1)

L<sup>2</sup> =

(1,0) <sup>=</sup> *<sup>Q</sup>* †

number of basis functions (*Nu*<sup>1</sup> , *Nu*<sup>2</sup> ).

(0)<sup>1</sup> and *<sup>ψ</sup>*(2)

components *<sup>ψ</sup>*(2)

∞

*du*<sup>1</sup> ∞

−∞

*du*2|*ψ*(1)

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 121

In the first column of Table 12 we show the difference in the number of basis states used (in each degree of freedom) and the maximum, *Nu*<sup>1</sup> = *Nu*<sup>2</sup> = 60, used for the reference result. Since L<sup>2</sup> and *L*<sup>∞</sup> are computed relative to the *Nu*<sup>1</sup> , *Nu*<sup>2</sup> = 60, 60 basis, they measure the degree of convergence of the calculations. It is clear from Tables 13 and 13 that the state obtained from

from the variational solution for sector one. We note that the same level of convergence is obtained for both of the degenerate wave functions. Since the analytical solution for the ground state wave function of the sector one is known, we also have calculated the L<sup>2</sup> and L<sup>∞</sup> error for this wave function, comparing the analytical and variational wave function of sector one for different numbers of basis states to determine a basis size which gives a satisfactory convergence. The results are given in Table 14. It is again clear that the variational results for

sector one sector two sector one sector two sector one sector two *Nu*<sup>1</sup> , *Nu*<sup>2</sup> = 10 *Nu*<sup>1</sup> , *Nu*<sup>2</sup> = 10 *Nu*<sup>1</sup> , *Nu*<sup>2</sup> = 40 *Nu*<sup>1</sup> , *Nu*<sup>2</sup> = 40 *Nu*<sup>1</sup> , *Nu*<sup>2</sup> = 60 *Nu*<sup>1</sup> , *Nu*<sup>2</sup> = 60 (in a.u.) (in a.u.) (in a.u.) (in a.u.) (in a.u.) (in a.u.) 9.0×10−<sup>3</sup> - 4.0×10−<sup>7</sup> - 5.0×10−<sup>9</sup> -

4.6 4.5849 4.58473 4.5847275 4.5847275 4.58472742 8.3 8.005 8.00007 8.0000005 8.000001 8.000000005

(1,0) <sup>=</sup> *<sup>Q</sup>* †

40 = 60 - 20 <sup>L</sup><sup>∞</sup> 1.1×10−<sup>4</sup> 4.9×10−<sup>4</sup> 30 = 60 - 30 <sup>L</sup><sup>∞</sup> 2.2×10−<sup>5</sup> 8.5×10−<sup>5</sup> 20 = 60 - 40 <sup>L</sup><sup>∞</sup> 5.3×10−<sup>6</sup> 1.9×10−<sup>5</sup> 10 = 60 - 50 <sup>L</sup><sup>∞</sup> 1.6×10−<sup>6</sup> 3.9×10−<sup>6</sup>

Table 12. Comparison between wave-function L<sup>∞</sup> -Error for the doubly-degenerate sector one excited state, (1, 0) generated by standard variational calculation of the sector one and variational SUSY calculation for sector two ground state, followed by application of the SUSY Charge Operator for different number of basis(*n* = *Nu*<sup>1</sup> , *Nu*<sup>2</sup> ). (*Nref* = (*Nu*<sup>1</sup> = 60, *Nu*<sup>2</sup> = 60)). In Figures 3(a-b) we show the two components of one of the degenerate sector two ground state wave functions and in Figures 3(c-d) we show the two components for the other degenerate sector two ground state wave function. It may seem problematic that the

We shall see below that these nodes can be eliminated in a very simple manner. However, we stress that for each of the degenerate sector two ground state wave functions, there is a large and small component. Unlike the two dimensional separable harmonic oscillator case,

<sup>1</sup>.*<sup>ψ</sup>* (2)

(0�)<sup>2</sup> for the pair of sector two ground state wave functions have nodes.

(0) *<sup>ψ</sup>*(1) (1,0)

Table 11. Comparison of energy eigenvalues of sector one and sector two for different

(∞) <sup>−</sup> *<sup>ψ</sup>*(1)

(*n*)| 2.

(0) converges more rapidly than the result obtained directly

−∞

<sup>1</sup>.*<sup>ψ</sup>* (2)

the sector one ground state wave function are very well converged.

<sup>Δ</sup>*<sup>N</sup>* <sup>=</sup> *Nref* <sup>−</sup> *<sup>n</sup>* Error *<sup>ψ</sup>*(1)

Each term of the Hamiltonian matrix can be calculated analytically in the harmonic oscillator basis.

We have calculated energies and wave functions of the Hamiltonian in Equation (5.3) for sectors one and two using the variational approach we just described. In all calculations, we use the exact *<sup>ψ</sup>*(1) <sup>0</sup> to generate *<sup>W</sup>* <sup>1</sup> and ←→H <sup>2</sup> exactly. In Table 11 we compare sector one and sector two energies for different harmonic oscillator basis set sizes. The notation *Nu*<sup>1</sup> , *Nu*<sup>2</sup> gives the number of basis functions for the variable *u*1, *u*2, respectively. The first row gives the approximate results for *E*(1) <sup>0</sup> . The next is the doubly degenerate first excited state energy, *E*(1) <sup>1</sup> followed by the sector two ground state energy, *<sup>E</sup>*(2) <sup>0</sup> . The third row contains *E*(1) <sup>2</sup> and *<sup>E</sup>*(2) <sup>1</sup> , for different basis sets. It is easily seen that the doubly degenerate ground state of sector two is also isospectral with the doubly degenerate first excited state of sector one. This correspondence is clearly in accordance with the general SUSY prediction about the eigenstates for the two supersymmetric partner potentials. For the higher excited states this precise correspondence between the two sectors breaks down when we use a small number of basis functions (*i.e.*, there appear some "spurious" solutions) but it is gradually restored by increasing the basis size. We attribute this apparent breakdown of the SUSY-correspondence for higher states to the error that arises in the calculation due to the truncation of an infinite basis to a finite one. Essentially, some "spurious" eigenvalues appear in the SUSY-QM sector two spectrum, but they disappear as the basis size is increased. This may raise a question regarding the precise nature of the Hylleraas-Undheim theorem for the SUSY sector two tensor Hamiltonian.

The accuracy of the variational results are known for the ground state of sector one, since we know the exact energy is *E*(1) <sup>0</sup> <sup>=</sup> 0. Thus, the (10, 10) basis gives an error of 9 <sup>×</sup> <sup>10</sup>−<sup>3</sup> while the (60, 60) basis gives an error of 4.9 <sup>×</sup> <sup>10</sup>−9. In the case of the first excited state of sector one, the error for the (10, 10) basis (computed relative to (60, 60) basis result) is 0.0634. By contrast, the error in the (10, 10) basis result for the sector two ground state (again, relative to the (60, 60) basis result) is 2.2 <sup>×</sup> <sup>10</sup>−4. Consequently, the use of the sector two Hamiltonian for a ground state calculation enables us to obtain much improved accuracy for the first excitation energy of sector one. Basically, we estimate an increase in accuracy (defined as the ratio of the accuracy of the sector one result to that of the sector two result) to be a factor of 280. Our exploratory calculation thus clearly reveals that for the calculation of excited state energies, the SUSY-variational method requires a smaller number of basis functions to achieve the same order of accuracy. Of course, this level of accuracy resulted in part because we have used the exact ←→H 2.

As this model problem has no analytical solution for the excited states, we have taken the results of the (60, 60) basis set calculation as the reference result for both sectors in order to check the convergence in wave functions. In Tables 12 and 13 we compare the L<sup>∞</sup> and L<sup>2</sup> error of the first excited states of the sector one that we have obtained by the SUSY-variational calculation and the simple variational calculation. The L<sup>∞</sup> error is defined as the absolute maximum difference between the solution computed with an infinite basis set (*ψ*(1)(∞)) which we approximate with the (60,60) basis, and a smaller finite (*n*, *n*) basis set (*ψ*(1)(*n*))

$$\mathcal{L}\_{\infty} = \text{Max}\{ |\psi^{(1)}(\infty) - \psi^{(1)}(n)| \}.$$

The L<sup>2</sup> error is defined by

20

basis.

we use the exact *<sup>ψ</sup>*(1)

energy, *E*(1)

<sup>2</sup> and *<sup>E</sup>*(2)

tensor Hamiltonian.

know the exact energy is *E*(1)

*E*(1)

exact

←→H 2.

gives the approximate results for *E*(1)

Each term of the Hamiltonian matrix can be calculated analytically in the harmonic oscillator

We have calculated energies and wave functions of the Hamiltonian in Equation (5.3) for sectors one and two using the variational approach we just described. In all calculations,

and sector two energies for different harmonic oscillator basis set sizes. The notation *Nu*<sup>1</sup> , *Nu*<sup>2</sup> gives the number of basis functions for the variable *u*1, *u*2, respectively. The first row

state of sector two is also isospectral with the doubly degenerate first excited state of sector one. This correspondence is clearly in accordance with the general SUSY prediction about the eigenstates for the two supersymmetric partner potentials. For the higher excited states this precise correspondence between the two sectors breaks down when we use a small number of basis functions (*i.e.*, there appear some "spurious" solutions) but it is gradually restored by increasing the basis size. We attribute this apparent breakdown of the SUSY-correspondence for higher states to the error that arises in the calculation due to the truncation of an infinite basis to a finite one. Essentially, some "spurious" eigenvalues appear in the SUSY-QM sector two spectrum, but they disappear as the basis size is increased. This may raise a question regarding the precise nature of the Hylleraas-Undheim theorem for the SUSY sector two

The accuracy of the variational results are known for the ground state of sector one, since we

the (60, 60) basis gives an error of 4.9 <sup>×</sup> <sup>10</sup>−9. In the case of the first excited state of sector one, the error for the (10, 10) basis (computed relative to (60, 60) basis result) is 0.0634. By contrast, the error in the (10, 10) basis result for the sector two ground state (again, relative to the (60, 60) basis result) is 2.2 <sup>×</sup> <sup>10</sup>−4. Consequently, the use of the sector two Hamiltonian for a ground state calculation enables us to obtain much improved accuracy for the first excitation energy of sector one. Basically, we estimate an increase in accuracy (defined as the ratio of the accuracy of the sector one result to that of the sector two result) to be a factor of 280. Our exploratory calculation thus clearly reveals that for the calculation of excited state energies, the SUSY-variational method requires a smaller number of basis functions to achieve the same order of accuracy. Of course, this level of accuracy resulted in part because we have used the

As this model problem has no analytical solution for the excited states, we have taken the results of the (60, 60) basis set calculation as the reference result for both sectors in order to check the convergence in wave functions. In Tables 12 and 13 we compare the L<sup>∞</sup> and L<sup>2</sup> error of the first excited states of the sector one that we have obtained by the SUSY-variational calculation and the simple variational calculation. The L<sup>∞</sup> error is defined as the absolute maximum difference between the solution computed with an infinite basis set (*ψ*(1)(∞)) which we approximate with the (60,60) basis, and a smaller finite (*n*, *n*) basis set (*ψ*(1)(*n*))

(∞) <sup>−</sup> *<sup>ψ</sup>*(1)

(*n*)|}.

<sup>L</sup><sup>∞</sup> <sup>=</sup> *Max*{|*ψ*(1)

<sup>1</sup> followed by the sector two ground state energy, *<sup>E</sup>*(2)

<sup>0</sup> to generate *<sup>W</sup>* <sup>1</sup> and ←→H <sup>2</sup> exactly. In Table 11 we compare sector one

<sup>1</sup> , for different basis sets. It is easily seen that the doubly degenerate ground

<sup>0</sup> . The next is the doubly degenerate first excited state

<sup>0</sup> <sup>=</sup> 0. Thus, the (10, 10) basis gives an error of 9 <sup>×</sup> <sup>10</sup>−<sup>3</sup> while

<sup>0</sup> . The third row contains

$$\mathcal{L}\_2 = \int\_{-\infty}^{\infty} du\_1 \int\_{-\infty}^{\infty} du\_2 |\psi^{(1)}(\infty) - \psi^{(1)}(n)|^2.$$

In the first column of Table 12 we show the difference in the number of basis states used (in each degree of freedom) and the maximum, *Nu*<sup>1</sup> = *Nu*<sup>2</sup> = 60, used for the reference result. Since L<sup>2</sup> and *L*<sup>∞</sup> are computed relative to the *Nu*<sup>1</sup> , *Nu*<sup>2</sup> = 60, 60 basis, they measure the degree of convergence of the calculations. It is clear from Tables 13 and 13 that the state obtained from the SUSY relation *<sup>ψ</sup>*(1) (1,0) <sup>=</sup> *<sup>Q</sup>* † <sup>1</sup>.*<sup>ψ</sup>* (2) (0) converges more rapidly than the result obtained directly from the variational solution for sector one. We note that the same level of convergence is obtained for both of the degenerate wave functions. Since the analytical solution for the ground state wave function of the sector one is known, we also have calculated the L<sup>2</sup> and L<sup>∞</sup> error for this wave function, comparing the analytical and variational wave function of sector one for different numbers of basis states to determine a basis size which gives a satisfactory convergence. The results are given in Table 14. It is again clear that the variational results for the sector one ground state wave function are very well converged.


Table 11. Comparison of energy eigenvalues of sector one and sector two for different number of basis functions (*Nu*<sup>1</sup> , *Nu*<sup>2</sup> ).


Table 12. Comparison between wave-function L<sup>∞</sup> -Error for the doubly-degenerate sector one excited state, (1, 0) generated by standard variational calculation of the sector one and variational SUSY calculation for sector two ground state, followed by application of the SUSY Charge Operator for different number of basis(*n* = *Nu*<sup>1</sup> , *Nu*<sup>2</sup> ). (*Nref* = (*Nu*<sup>1</sup> = 60, *Nu*<sup>2</sup> = 60)).

In Figures 3(a-b) we show the two components of one of the degenerate sector two ground state wave functions and in Figures 3(c-d) we show the two components for the other degenerate sector two ground state wave function. It may seem problematic that the components *<sup>ψ</sup>*(2) (0)<sup>1</sup> and *<sup>ψ</sup>*(2) (0�)<sup>2</sup> for the pair of sector two ground state wave functions have nodes. We shall see below that these nodes can be eliminated in a very simple manner. However, we stress that for each of the degenerate sector two ground state wave functions, there is a large and small component. Unlike the two dimensional separable harmonic oscillator case,

(a) *ψ*(2)

(c) *ψ*(2)

harmonic oscillator.

In Figures 5(a-d) we show the components of

(0)<sup>1</sup> (b) *<sup>ψ</sup>*(2)

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 123

(0�)<sup>1</sup> (d) *<sup>ψ</sup>*(2)

Fig. 3. (a-b) represent the two components of one of the degenerate sector two ground states. (c-d) represent the same for the other sector two ground state. Contour shading is such that red indicates positive amplitude and blue indicates negative amplitude. The prime on the

> *<sup>φ</sup>*(2) (0) and

to the results in Equation (4.27), obtained for the degenerate separable two dimensional

For sector two, in the case of a doubly degenerate first excited state of H<sup>1</sup> , we obtain a doubly degenerate ground state and the energies obtained by the Rayleigh-Ritz method are consistently lower for all the excited states of the sector one Hamiltonian, for the same basis size. In addition, the SUSY-QM sector two result for the first excited state energy is always several orders of magnitude more accurate than the Rayleigh-Ritz result for sector one for any given basis set size. Assessing the accuracy of the excited state wave functions is more

*<sup>φ</sup>*(2) (0�)

quantum number "0" denotes the second of the 2 degenerate ground states.

and have definite symmetry. We stress that the forms of the above

(0)2

(0�)2

. These combinations are nodeless

*<sup>φ</sup>*(2)

(0�) are analogous

*<sup>φ</sup>*(2) (0) and


Table 13. Comparison between wave-function L2-Error for the doubly-degenerate sector one Excited state, (1, 0) generated by standard variational calculation of the sector one and variational SUSY calculation for the sector two ground state, followed by application of the SUSY charge operator for different size basis(*n* = *Nu*<sup>1</sup> , *Nu*<sup>2</sup> ). (*Nref* = (*Nu*<sup>1</sup> = 60, *Nu*<sup>2</sup> = 60)).


Table 14. Comparison between wave-function <sup>L</sup><sup>2</sup> and *<sup>L</sup>*<sup>∞</sup> -Error for the 1*st* sector exact ground state wave function *<sup>ψ</sup>*(1) (0,0) (∞) and variationally calculated ground state wave function *<sup>ψ</sup>*(1) (0,0) (*n*) for different number of basis states(*n* = *Nu*<sup>1</sup> , *Nu*<sup>2</sup> ).

the small component is not only non-zero but it has nodes. It is roughly ten times smaller in magnitude than the large component. The two degenerate states are 90*<sup>o</sup>* out of phase so far as their signs. In Figures 4(a-b) we show the first excited states *<sup>ψ</sup>*(1) (0,1) and *<sup>ψ</sup>*(1) (1,0) of the sector one that we have obtained after applying the SUSY charge operator to the sector two ground states and Figures 4(c-d) present the same states that were found variationally from the sector one Hamiltonian. The similarity of Figure 4a to 4c and 4b to 4d clearly reflects the correctness of our method. To eliminate the nodes in the components of *<sup>ψ</sup>* (2) (0) and *<sup>ψ</sup>* (2) (0�) , we note that since they are degenerate, any linear combination of them is also a valid wave function. Accordingly, in analogy to the separable two dimensional harmonic oscillator considered previously we can define *<sup>φ</sup>*(2) (0) and *<sup>φ</sup>*(2) (0�) by combining the components of *<sup>ψ</sup>* (2) (0) and *<sup>ψ</sup>* (2) (0�) according to

$$
\boldsymbol{\phi}^{(2)}\_{(0)1} = \boldsymbol{\psi}^{(2)}\_{(0)1} + \boldsymbol{\psi}^{(2)}\_{(0')1} \tag{5.8}
$$

$$
\boldsymbol{\phi}^{(2)}\_{(0)2} = \boldsymbol{\psi}^{(2)}\_{(0)2} + \boldsymbol{\psi}^{(2)}\_{(0')2} \tag{5.9}
$$

$$
\phi\_{(0')1}^{(2)} = \psi\_{(0)1}^{(2)} - \psi\_{(0')1}^{(2)} \tag{5.10}
$$

$$
\psi\_{(0')2}^{(2)} = \psi\_{(0)2}^{(2)} - \psi\_{(0')2}^{(2)}.\tag{5.11}
$$

22

ground state wave function *<sup>ψ</sup>*(1)

considered previously we can define

(0,0)

function *<sup>ψ</sup>*(1)

*<sup>ψ</sup>* (2) (0�)

*<sup>ψ</sup>* (2)

(0�) according to

<sup>Δ</sup>*<sup>N</sup>* <sup>=</sup> *Nref* <sup>−</sup> *<sup>n</sup>* Error *<sup>ψ</sup>*(1)

(1,0) <sup>=</sup> *<sup>Q</sup>* †

40 = 60 - 20 <sup>L</sup><sup>2</sup> 3.0×10−<sup>6</sup> 1.8×10−<sup>5</sup> 30 = 60 - 30 <sup>L</sup><sup>2</sup> 1.1×10−<sup>7</sup> 5.7×10−<sup>7</sup> 20 = 60 - 40 <sup>L</sup><sup>2</sup> 6.2×10−<sup>9</sup> 3.0×10−<sup>8</sup> 10 = 60 - 50 <sup>L</sup><sup>2</sup> 3.5×10−<sup>10</sup> 1.5×10−<sup>9</sup>

Table 13. Comparison between wave-function L2-Error for the doubly-degenerate sector one Excited state, (1, 0) generated by standard variational calculation of the sector one and variational SUSY calculation for the sector two ground state, followed by application of the SUSY charge operator for different size basis(*n* = *Nu*<sup>1</sup> , *Nu*<sup>2</sup> ). (*Nref* = (*Nu*<sup>1</sup> = 60, *Nu*<sup>2</sup> = 60)).

> *Nu*<sup>1</sup> , *Nu*<sup>2</sup> L<sup>∞</sup> L<sup>2</sup> 20, 20 9.3×10−<sup>5</sup> 1.5 <sup>×</sup>10−<sup>6</sup> 30, 30 1.5×10−<sup>5</sup> 3.9×10−<sup>8</sup> 40, 40 3.4×10−<sup>6</sup> 2.0×10−<sup>9</sup> 50, 50 9.1×10−<sup>7</sup> 1.4×10−<sup>10</sup> 60, 60 2.8×10−<sup>7</sup> 1.3×10−<sup>11</sup>

Table 14. Comparison between wave-function <sup>L</sup><sup>2</sup> and *<sup>L</sup>*<sup>∞</sup> -Error for the 1*st* sector exact

the small component is not only non-zero but it has nodes. It is roughly ten times smaller in magnitude than the large component. The two degenerate states are 90*<sup>o</sup>* out of phase

the sector one that we have obtained after applying the SUSY charge operator to the sector two ground states and Figures 4(c-d) present the same states that were found variationally from the sector one Hamiltonian. The similarity of Figure 4a to 4c and 4b to 4d clearly reflects the correctness of our method. To eliminate the nodes in the components of *<sup>ψ</sup>* (2)

, we note that since they are degenerate, any linear combination of them is also a valid wave function. Accordingly, in analogy to the separable two dimensional harmonic oscillator

*<sup>φ</sup>*(2)

(0)<sup>1</sup> <sup>+</sup> *<sup>ψ</sup>*(2)

(0)<sup>2</sup> <sup>+</sup> *<sup>ψ</sup>*(2)

(0)1) <sup>−</sup> *<sup>ψ</sup>*(2)

(0)<sup>2</sup> <sup>−</sup> *<sup>ψ</sup>*(2)

(*n*) for different number of basis states(*n* = *Nu*<sup>1</sup> , *Nu*<sup>2</sup> ).

so far as their signs. In Figures 4(a-b) we show the first excited states *<sup>ψ</sup>*(1)

*<sup>φ</sup>*(2) (0) and

*<sup>φ</sup>*(2)

*<sup>φ</sup>*(2)

*<sup>φ</sup>*(2)

*<sup>φ</sup>*(2)

(0)<sup>1</sup> <sup>=</sup> *<sup>ψ</sup>*(2)

(0)<sup>2</sup> <sup>=</sup> *<sup>ψ</sup>*(2)

(0�)<sup>1</sup> <sup>=</sup> *<sup>ψ</sup>*(2)

(0�)<sup>2</sup> <sup>=</sup> *<sup>ψ</sup>*(2)

(0,0)

<sup>1</sup>.*<sup>ψ</sup>* (2)

(∞) and variationally calculated ground state wave

(0�) by combining the components of *<sup>ψ</sup>* (2)

(0�)<sup>1</sup> (5.8)

(0�)<sup>2</sup> (5.9)

(0�)<sup>1</sup> (5.10)

(0�)2. (5.11)

(0,1) and *<sup>ψ</sup>*(1)

(1,0) of

(0) and

(0) and

(0) *<sup>ψ</sup>*(1) (1,0)

Fig. 3. (a-b) represent the two components of one of the degenerate sector two ground states. (c-d) represent the same for the other sector two ground state. Contour shading is such that red indicates positive amplitude and blue indicates negative amplitude. The prime on the quantum number "0" denotes the second of the 2 degenerate ground states.

In Figures 5(a-d) we show the components of *<sup>φ</sup>*(2) (0) and *<sup>φ</sup>*(2) (0�) . These combinations are nodeless and have definite symmetry. We stress that the forms of the above *<sup>φ</sup>*(2) (0) and *<sup>φ</sup>*(2) (0�) are analogous to the results in Equation (4.27), obtained for the degenerate separable two dimensional harmonic oscillator.

For sector two, in the case of a doubly degenerate first excited state of H<sup>1</sup> , we obtain a doubly degenerate ground state and the energies obtained by the Rayleigh-Ritz method are consistently lower for all the excited states of the sector one Hamiltonian, for the same basis size. In addition, the SUSY-QM sector two result for the first excited state energy is always several orders of magnitude more accurate than the Rayleigh-Ritz result for sector one for any given basis set size. Assessing the accuracy of the excited state wave functions is more

(a) *φ*(2)

(c) *φ*(2)

**5.2 Non-degenerate case**

generate an approximate

(5.1) to the form

two.

(0�)<sup>1</sup> <sup>=</sup> *<sup>ψ</sup>*(2)

(0)<sup>1</sup> <sup>−</sup> *<sup>ψ</sup>*(2)

*<sup>ψ</sup>*(1) (0)

(0)<sup>1</sup> <sup>=</sup> *<sup>ψ</sup>*(2)

(0)<sup>1</sup> <sup>+</sup> *<sup>ψ</sup>*(2)

(0�)<sup>1</sup> (b) *<sup>φ</sup>*(2)

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 125

(0�)<sup>1</sup> (d) *<sup>φ</sup>*(2)

Fig. 5. Components of linear combinations of the two degenerate ground states of the sector

For completeness of our presentation, we also consider a non-degenerate, two dimensional anharmonic oscillator model. To generate such a Hamiltonian, we modify the ground state in

Then the exact Hamiltonians <sup>H</sup><sup>1</sup> and ←→H <sup>2</sup> are readily generated. However, we also shall

wave function. The formal structure of the equations is the same as that above. In the case

1*u*2 <sup>2</sup> <sup>−</sup> *<sup>u</sup>*<sup>2</sup>

<sup>1</sup> <sup>−</sup> <sup>√</sup> 2*u*<sup>2</sup>

←→H <sup>2</sup> using the variationally determined, approximate ground state

(*u*1, *<sup>u</sup>*2) = *<sup>N</sup>* exp(−2*u*<sup>2</sup>

(0)<sup>2</sup> <sup>=</sup> *<sup>ψ</sup>*(2)

(0�)<sup>2</sup> <sup>=</sup> *<sup>ψ</sup>*(2)

(0)<sup>2</sup> <sup>−</sup> *<sup>ψ</sup>*(2) (0�)2

<sup>2</sup>) (5.12)

(0)<sup>2</sup> <sup>+</sup> *<sup>ψ</sup>*(2) (0�)2

Fig. 4. (a-b) show the doubly degenerate 1*st* excited states of the sector one. The two states are generated by the SUSY -variational method. (c-d) the corresponding states of the sector one Hamiltonian, which are generated variationally.

difficult. We chose to do this in terms of convergence of the wave functions relative to the largest basis set results. However, we are able to assess the accuracy of our variational results quantitatively in the case of the sector one ground state , since it is exactly known. We report our results in terms of L<sup>2</sup> and L<sup>∞</sup> measures, as is typical for assessing convergence and accuracy of functions in a Hilbert space. We find that the L<sup>2</sup> and L<sup>∞</sup> accuracies of the SUSY-QM results are consistently better than the Rayleigh-Ritz results for excited states of sector one. As a further proof of this, we also consider the accuracy for the ground state of sector one ( where we have the exact wave function), with the variational result. In fact, we find that the convergence of the sector two ground state is consistently better than the convergence of the variationally obtained ground state wave function for sector one.

24

(a) *Q* †

(c) *ψ*(1)

one Hamiltonian, which are generated variationally.

<sup>1</sup> · *<sup>ψ</sup>* (2)

(0) (b) *<sup>Q</sup>* †

(0,1) (d) *<sup>ψ</sup>*(1)

Fig. 4. (a-b) show the doubly degenerate 1*st* excited states of the sector one. The two states are generated by the SUSY -variational method. (c-d) the corresponding states of the sector

difficult. We chose to do this in terms of convergence of the wave functions relative to the largest basis set results. However, we are able to assess the accuracy of our variational results quantitatively in the case of the sector one ground state , since it is exactly known. We report our results in terms of L<sup>2</sup> and L<sup>∞</sup> measures, as is typical for assessing convergence and accuracy of functions in a Hilbert space. We find that the L<sup>2</sup> and L<sup>∞</sup> accuracies of the SUSY-QM results are consistently better than the Rayleigh-Ritz results for excited states of sector one. As a further proof of this, we also consider the accuracy for the ground state of sector one ( where we have the exact wave function), with the variational result. In fact, we find that the convergence of the sector two ground state is consistently better than the

convergence of the variationally obtained ground state wave function for sector one.

<sup>1</sup> · *<sup>ψ</sup>* (2) (0�)

(1,0)

Fig. 5. Components of linear combinations of the two degenerate ground states of the sector two.

#### **5.2 Non-degenerate case**

For completeness of our presentation, we also consider a non-degenerate, two dimensional anharmonic oscillator model. To generate such a Hamiltonian, we modify the ground state in (5.1) to the form

$$
\psi\_{(0)}^{(1)}(\boldsymbol{u}\_1, \boldsymbol{u}\_2) = N \exp(-2\boldsymbol{u}\_1^2 \boldsymbol{u}\_2^2 - \boldsymbol{u}\_1^2 - \sqrt{2}\boldsymbol{u}\_2^2) \tag{5.12}
$$

Then the exact Hamiltonians <sup>H</sup><sup>1</sup> and ←→H <sup>2</sup> are readily generated. However, we also shall generate an approximate ←→H <sup>2</sup> using the variationally determined, approximate ground state wave function. The formal structure of the equations is the same as that above. In the case

<sup>Δ</sup>*<sup>N</sup>* <sup>=</sup> *Nref* <sup>−</sup> *<sup>n</sup>* Error *<sup>ψ</sup>*(1)

<sup>Δ</sup>*<sup>N</sup>* <sup>=</sup> *Nref* <sup>−</sup> *<sup>n</sup>* Error *<sup>ψ</sup>*(1)

sector one and sector two Hamiltonians are used.

sector one and sector two Hamiltonians are used.

<sup>1</sup> *<sup>E</sup>*(2)

the SUSY charge operator to the sector two ground (*<sup>ψ</sup>* (2)

*n E*(1)

degrees of approximation for *<sup>ψ</sup>*(1)

<sup>1</sup> ) and second excited state (*ψ*(1)

(*ψ*(1)

case.

(1) <sup>=</sup> *<sup>Q</sup>* †

(1) <sup>=</sup> *<sup>Q</sup>* †

40 = 60 - 20 <sup>L</sup><sup>2</sup> 1.5×10−<sup>6</sup> 8.9×10−<sup>6</sup> 30 = 60 - 30 <sup>L</sup><sup>2</sup> 5.8×10−<sup>8</sup> 2.7×10−<sup>7</sup> 20 = 60 - 40 <sup>L</sup><sup>2</sup> 3.2×10−<sup>9</sup> 1.5×10−<sup>8</sup> 10 = 60 - 50 <sup>L</sup><sup>2</sup> 1.8×10−<sup>10</sup> 1.1×10−<sup>9</sup>

Table 17. Comparison between wave-function L<sup>2</sup> -Error for the sector one excited state, generated by standard variational calculation of the sector one and variational SUSY calculation for sector two ground state, followed by application of the SUSY Charge Operator for different number of basis(*n* = *Nu*<sup>1</sup> , *Nu*<sup>2</sup> ). (*Nref* = (*Nu*<sup>1</sup> = 60, *Nu*<sup>2</sup> = 60)). Exact

(0),*cuto f f <sup>ψ</sup>*(1)

10, 10 4.80 4.752 4.794 4.791 4.78 4.756 20, 20 4.7532 4.75181 4.75317 4.75313 4.7526 4.75181 30, 30 4.752 4.751808 4.75187 4.75187 4.75186 4.751808

case ( Figures 3(a-b) and (c-d))! This suggests to us that the non-degenerate and degenerate cases are very similar so far as the wave functions are concerned. Again, in all cases the large component is nodeless and the small component has nodes. In Figures 7(a-b) we show the first

the Figures 7(c-d) presents the same states that were found variationally from the sector one Hamiltonian. These results also reflect the similarity between degenerate and non-degenerate

Table 18. Comparison among 1*st* excited state energy of sector one (calculated using analytical *W* 1), ground state energy of sector two (calculated using analytical *W* 1) and

different sector two ground state energy that we obtained using *W approx*

*<sup>ψ</sup>*(1)

<sup>1</sup>.*<sup>ψ</sup>* (2)

(0),*cuto f f <sup>ψ</sup>*(1)

10−<sup>10</sup> 10−<sup>5</sup> 10−<sup>3</sup> 10−<sup>2</sup>

<sup>0</sup> = = = =

0,*approx* for different basis size (*n* = *Nu*<sup>1</sup> , *Nu*<sup>2</sup> ).

(0),*cuto f f <sup>ψ</sup>*(1)

<sup>2</sup> ) , of the sector one that we have obtained after applying

(0),*cuto f f*

<sup>1</sup> with different

<sup>1</sup> ) and

<sup>0</sup> ) and first excited state (*<sup>ψ</sup>* (2)

(0) *<sup>ψ</sup>*(1) (1)

40 = 60 - 20 <sup>L</sup><sup>∞</sup> 9.5×10−<sup>5</sup> 4.5×10−<sup>4</sup> 30 = 60 - 30 <sup>L</sup><sup>∞</sup> 1.9×10−<sup>5</sup> 7.5×10−<sup>5</sup> 20 = 60 - 40 <sup>L</sup><sup>∞</sup> 4.5×10−<sup>6</sup> 1.7×10−<sup>5</sup> 10 = 60 - 50 <sup>L</sup><sup>∞</sup> 9.9×10−<sup>7</sup> 4.3×10−<sup>6</sup>

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 127

Table 16. Comparison between wave-function L<sup>∞</sup> -Error for the sector one excited state, generated by standard variational calculation of the sector one and variational SUSY calculation for sector two ground state, followed by application of the SUSY Charge Operator for different number of basis(*n* = *Nu*<sup>1</sup> , *Nu*<sup>2</sup> ). (*Nref* = (*Nu*<sup>1</sup> = 60, *Nu*<sup>2</sup> = 60)). Exact

<sup>1</sup>.*<sup>ψ</sup>* (2)

(0) *<sup>ψ</sup>*(1) (1)

where *<sup>ψ</sup>*(1) 0,*approx* is used to develop *<sup>W</sup>* <sup>1</sup> and ←→H 2, we shall see that its accuracy is an extremely important consideration.

In the non-degenerate example, we have performed two distinct calculations. First we used the exact *<sup>W</sup>* <sup>1</sup> to construct the exact ←→H 2. The results are given in Tables 15 - 17. In Table 15, we see basically the same behavior as was obtained in the non-separable degenerate two dimensional example. The errors in the ground state energy are of similar size for both the degenerate and non-degenerate cases, with the same variation with basis set size. This behavior extends also to the first and second excited state energies. We conclude that the presence or absence of degeneracy does not affect the performance of our SUSY approach when the exact *W* <sup>1</sup> is used.

In the case of the <sup>L</sup><sup>∞</sup> and <sup>L</sup><sup>2</sup> errors obtained when using the exact *<sup>W</sup>* 1, we again see the same basic behavior with regard to the convergence of the wave functions.

However, the situation is more interesting when we use the variationally obtained approximate ground state, *<sup>ψ</sup>*(1) 0,*approx* to generate *<sup>W</sup> approx* <sup>1</sup> and thereby ←→*<sup>H</sup>* 2,*approx*. These results are shown in Table 18 and are compared to the exact *<sup>W</sup>* <sup>1</sup> results (the columns labeled *<sup>E</sup>*(1) 1 and *E*(2) <sup>0</sup> ). Results are shown for three different basis set sizes, (10,10), (20,20) , and (30,30). Now because we are using an approximate *<sup>ψ</sup>*(1) <sup>0</sup> to generate *<sup>W</sup> approx* <sup>1</sup> , it is important to note how the accuracy depends not only on basis size (which affects the accuracy of *<sup>ψ</sup>*(1) <sup>0</sup> ) but also how the accuracy of *W approx* <sup>1</sup> is affected by errors in *<sup>ψ</sup>*(1) <sup>0</sup> . We have found that *<sup>W</sup> approx* <sup>1</sup> is most sensitive to errors in region where *<sup>ψ</sup>*(1) <sup>0</sup> is small in magnitude. This is reasonable since *W approx* <sup>1</sup> <sup>=</sup> −∇ ln *<sup>ψ</sup>*(1) 0,*approx* and we expect that (*∂ψ*(1) <sup>0</sup> /*∂uj*)/*ψ*(1) <sup>0</sup> to be most sensitive to errors in regions where *<sup>ψ</sup>*(1) <sup>0</sup> is smallest in magnitude. In view of this, we have introduced *ψcuto f f* levels at which we cease calculating *<sup>W</sup>* 1. These correspond to cutoff values of *<sup>ψ</sup>*(1) 0,*cuto f f* = 10−10, 10−5, 10−<sup>3</sup> and 10−2. Those results are in columns 4 - 7 in Table 18. It is clear that the SUSY result is always better than the sector one variational result, although this is only marginally the case with the very small cutoff values (i.e., *<sup>ψ</sup>*(1) 0,*cuto f f* <sup>≤</sup> <sup>10</sup>−5). The best results are obtained with the 10−<sup>2</sup> cutoff value. While obviously, this is a single computational example, it is encouraging. However, additional careful studies are underway.


Table 15. Comparison of energy eigenvalues of sector one and sector two for different number of basis functions (*Nu*<sup>1</sup> , *Nu*<sup>2</sup> ). Exact sector one and sector two Hamiltonians are used. Finally, in Figures 6(a-b), we give the two components of the (non-degenerate) ground state, *<sup>ψ</sup>*(2) (0)<sup>1</sup> and *<sup>ψ</sup>*(2) (0)2. In Figures 6(c-d), we display the components of the first excited state, *<sup>ψ</sup>*(2) (1)1 and *<sup>ψ</sup>*(2) (1)2. We note that they are qualitatively similar to the results obtained for the degenerate

26

where *<sup>ψ</sup>*(1)

and *E*(2)

*W approx*

*<sup>ψ</sup>*(2)

and *<sup>ψ</sup>*(2)

(0)<sup>1</sup> and *<sup>ψ</sup>*(2)

important consideration.

when the exact *W* <sup>1</sup> is used.

approximate ground state, *<sup>ψ</sup>*(1)

how the accuracy of *W approx*

<sup>1</sup> <sup>=</sup> −∇ ln *<sup>ψ</sup>*(1)

in regions where *<sup>ψ</sup>*(1)

Now because we are using an approximate *<sup>ψ</sup>*(1)

most sensitive to errors in region where *<sup>ψ</sup>*(1)

the case with the very small cutoff values (i.e., *<sup>ψ</sup>*(1)

encouraging. However, additional careful studies are underway.

0,*approx* is used to develop *<sup>W</sup>* <sup>1</sup> and ←→H 2, we shall see that its accuracy is an extremely

In the non-degenerate example, we have performed two distinct calculations. First we used the exact *<sup>W</sup>* <sup>1</sup> to construct the exact ←→H 2. The results are given in Tables 15 - 17. In Table 15, we see basically the same behavior as was obtained in the non-separable degenerate two dimensional example. The errors in the ground state energy are of similar size for both the degenerate and non-degenerate cases, with the same variation with basis set size. This behavior extends also to the first and second excited state energies. We conclude that the presence or absence of degeneracy does not affect the performance of our SUSY approach

In the case of the <sup>L</sup><sup>∞</sup> and <sup>L</sup><sup>2</sup> errors obtained when using the exact *<sup>W</sup>* 1, we again see the same

However, the situation is more interesting when we use the variationally obtained

are shown in Table 18 and are compared to the exact *<sup>W</sup>* <sup>1</sup> results (the columns labeled *<sup>E</sup>*(1)

10−5, 10−<sup>3</sup> and 10−2. Those results are in columns 4 - 7 in Table 18. It is clear that the SUSY result is always better than the sector one variational result, although this is only marginally

with the 10−<sup>2</sup> cutoff value. While obviously, this is a single computational example, it is

sector one sector two sector one sector two sector one sector two *Nu*<sup>1</sup> , *Nu*<sup>2</sup> = 10 *Nu*<sup>1</sup> , *Nu*<sup>2</sup> = 10 *Nu*<sup>1</sup> , *Nu*<sup>2</sup> = 40 *Nu*<sup>1</sup> , *Nu*<sup>2</sup> = 40 *Nu*<sup>1</sup> , *Nu*<sup>2</sup> = 60 *Nu*<sup>1</sup> , *Nu*<sup>2</sup> = 60 (in a.u.) (in a.u.) (in a.u.) (in a.u.) (in a.u.) (in a.u.) 6.4×10−<sup>3</sup> - 3.4×10−<sup>7</sup> - 3.6×10−<sup>9</sup> - 4.80 4.752 4.75181 4.75180771 4.75180778 4.75180770 6.70 6.65 6.64636 6.64634938 6.6463495 6.64634937

Table 15. Comparison of energy eigenvalues of sector one and sector two for different

number of basis functions (*Nu*<sup>1</sup> , *Nu*<sup>2</sup> ). Exact sector one and sector two Hamiltonians are used. Finally, in Figures 6(a-b), we give the two components of the (non-degenerate) ground state,

(0)2. In Figures 6(c-d), we display the components of the first excited state, *<sup>ψ</sup>*(2)

(1)2. We note that they are qualitatively similar to the results obtained for the degenerate

<sup>0</sup> ). Results are shown for three different basis set sizes, (10,10), (20,20) , and (30,30).

<sup>0</sup> to generate *<sup>W</sup> approx*

<sup>0</sup> /*∂uj*)/*ψ*(1)

<sup>0</sup> is smallest in magnitude. In view of this, we have introduced *ψcuto f f*

<sup>1</sup> and thereby ←→*<sup>H</sup>* 2,*approx*. These results

<sup>0</sup> . We have found that *<sup>W</sup> approx*

<sup>0</sup> is small in magnitude. This is reasonable since

0,*cuto f f* <sup>≤</sup> <sup>10</sup>−5). The best results are obtained

<sup>1</sup> , it is important to note

<sup>0</sup> to be most sensitive to errors

1

<sup>0</sup> ) but also

0,*cuto f f* = 10−10,

<sup>1</sup> is

(1)1

0,*approx* to generate *<sup>W</sup> approx*

how the accuracy depends not only on basis size (which affects the accuracy of *<sup>ψ</sup>*(1)

<sup>1</sup> is affected by errors in *<sup>ψ</sup>*(1)

levels at which we cease calculating *<sup>W</sup>* 1. These correspond to cutoff values of *<sup>ψ</sup>*(1)

0,*approx* and we expect that (*∂ψ*(1)

basic behavior with regard to the convergence of the wave functions.


Table 16. Comparison between wave-function L<sup>∞</sup> -Error for the sector one excited state, generated by standard variational calculation of the sector one and variational SUSY calculation for sector two ground state, followed by application of the SUSY Charge Operator for different number of basis(*n* = *Nu*<sup>1</sup> , *Nu*<sup>2</sup> ). (*Nref* = (*Nu*<sup>1</sup> = 60, *Nu*<sup>2</sup> = 60)). Exact sector one and sector two Hamiltonians are used.


Table 17. Comparison between wave-function L<sup>2</sup> -Error for the sector one excited state, generated by standard variational calculation of the sector one and variational SUSY calculation for sector two ground state, followed by application of the SUSY Charge Operator for different number of basis(*n* = *Nu*<sup>1</sup> , *Nu*<sup>2</sup> ). (*Nref* = (*Nu*<sup>1</sup> = 60, *Nu*<sup>2</sup> = 60)). Exact sector one and sector two Hamiltonians are used.


Table 18. Comparison among 1*st* excited state energy of sector one (calculated using analytical *W* 1), ground state energy of sector two (calculated using analytical *W* 1) and different sector two ground state energy that we obtained using *W approx* <sup>1</sup> with different degrees of approximation for *<sup>ψ</sup>*(1) 0,*approx* for different basis size (*n* = *Nu*<sup>1</sup> , *Nu*<sup>2</sup> ).

case ( Figures 3(a-b) and (c-d))! This suggests to us that the non-degenerate and degenerate cases are very similar so far as the wave functions are concerned. Again, in all cases the large component is nodeless and the small component has nodes. In Figures 7(a-b) we show the first (*ψ*(1) <sup>1</sup> ) and second excited state (*ψ*(1) <sup>2</sup> ) , of the sector one that we have obtained after applying the SUSY charge operator to the sector two ground (*<sup>ψ</sup>* (2) <sup>0</sup> ) and first excited state (*<sup>ψ</sup>* (2) <sup>1</sup> ) and the Figures 7(c-d) presents the same states that were found variationally from the sector one Hamiltonian. These results also reflect the similarity between degenerate and non-degenerate case.

(a) *Q* †

(c) *ψ*(1)

least computational effort.

<sup>1</sup> · *<sup>ψ</sup>* (2)

(0) (b) *<sup>Q</sup>* †

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 129

<sup>1</sup> (d) *<sup>ψ</sup>*(1)

Fig. 7. (a-b) show the 1*st* and 2*nd* excited states of the sector one. The two states are generated

for the quantum Monte Carlo method. Thus, the implementation of the quantum Monte Carlo method in the non-degenerate case appears to require further consideration. We are currently

It is important to stress that our basic strategy is to use *only* the ground state results of the higher SUSY sectors. We believe that this will allow us to obtain the best results for both excited state energies and wave functions of the sector one Hamiltonian, while requiring the

Our upcoming computational studies will be to apply the present approach to more interesting, non-separable higher dimensional systems such as rare-gas atomic clusters where the structure and thermodynamics seem to require a fully quantum many-body treatment.

by the SUSY -variational method. (c-d) the corresponding states of the sector one

Hamiltonian, which are generated variationally. The exact *W* <sup>1</sup> is used.

exploring this aspect of our multi-dimensional SUSY approach.

<sup>1</sup> · *<sup>ψ</sup>* (2) (1)

2

Fig. 6. (a-b) display the two components of the sector two ground state. (c-d) represent the two components for the sector two first excited state. Contour shading is such that red indicates positive amplitude and blue indicates negative amplitude.

For comparison, we display in Figures 8(a-b) the sum of *<sup>ψ</sup>* (2) <sup>0</sup> and *<sup>ψ</sup>* (2) <sup>1</sup> and in Figures 8(c-d) the difference of *<sup>ψ</sup>* (2) <sup>0</sup> and *<sup>ψ</sup>* (2) <sup>1</sup> . The results are qualitatively the same as those in the two dimensional separable and two dimensional non-separable degenerate cases. That is, both linear combinations are nodeless and of definite sign.

The two dimensional non-seperable, non-degenerate case is interesting in that it appears that there is a similar relationship between *<sup>ψ</sup>* (2) <sup>0</sup> and *<sup>ψ</sup>* (2) <sup>1</sup> to that which was seen for the degenerate states *<sup>ψ</sup>* (2) (0) and *<sup>ψ</sup>* (2) (<sup>0</sup>) . That is, one component is nodeless and large and the second component has nodes and is smaller in magnitude. As in the degenerate case, sums and differences yield states with both components being nodeless. In this case, however, the non degenerate character of the states precludes simply using two different nodeless, orthogonal *<sup>φ</sup>*(2) *trial* states 28

(a) *ψ*(2)

(c) *ψ*(2)

the difference of *<sup>ψ</sup>* (2)

(0) and *<sup>ψ</sup>* (2)

states *<sup>ψ</sup>* (2)

(0)<sup>1</sup> (b) *<sup>ψ</sup>*(2)

(1)<sup>1</sup> (d) *<sup>ψ</sup>*(2)

Fig. 6. (a-b) display the two components of the sector two ground state. (c-d) represent the two components for the sector two first excited state. Contour shading is such that red

dimensional separable and two dimensional non-separable degenerate cases. That is, both

The two dimensional non-seperable, non-degenerate case is interesting in that it appears that

has nodes and is smaller in magnitude. As in the degenerate case, sums and differences yield states with both components being nodeless. In this case, however, the non degenerate

character of the states precludes simply using two different nodeless, orthogonal

<sup>0</sup> and *<sup>ψ</sup>* (2)

indicates positive amplitude and blue indicates negative amplitude.

For comparison, we display in Figures 8(a-b) the sum of *<sup>ψ</sup>* (2)

<sup>0</sup> and *<sup>ψ</sup>* (2)

there is a similar relationship between *<sup>ψ</sup>* (2)

linear combinations are nodeless and of definite sign.

(0)2

(1)2

<sup>1</sup> to that which was seen for the degenerate

<sup>1</sup> and in Figures 8(c-d)

*<sup>φ</sup>*(2) *trial* states

<sup>0</sup> and *<sup>ψ</sup>* (2)

<sup>1</sup> . The results are qualitatively the same as those in the two

(<sup>0</sup>) . That is, one component is nodeless and large and the second component

Fig. 7. (a-b) show the 1*st* and 2*nd* excited states of the sector one. The two states are generated by the SUSY -variational method. (c-d) the corresponding states of the sector one Hamiltonian, which are generated variationally. The exact *W* <sup>1</sup> is used.

for the quantum Monte Carlo method. Thus, the implementation of the quantum Monte Carlo method in the non-degenerate case appears to require further consideration. We are currently exploring this aspect of our multi-dimensional SUSY approach.

It is important to stress that our basic strategy is to use *only* the ground state results of the higher SUSY sectors. We believe that this will allow us to obtain the best results for both excited state energies and wave functions of the sector one Hamiltonian, while requiring the least computational effort.

Our upcoming computational studies will be to apply the present approach to more interesting, non-separable higher dimensional systems such as rare-gas atomic clusters where the structure and thermodynamics seem to require a fully quantum many-body treatment.

**6. Electronic structure of atoms: Hydrogen and helium**

accurate excited state energies and wave functions.

full three-dimensional nature of the hydrogen atom.

**6.1 SUSY-QM for the three-dimensional hydrogen atom**

*l*(*l* + 1)). The precise form for the ground state (*l* = 0) is

*W* = *�<sup>x</sup>*

*x r* + *�<sup>y</sup> y r* + *�<sup>z</sup> z*

Then, the vector superpotential is given by

In three dimensions, we have

functions of atoms.

given by

In the previous sections, we have provided a generalization of SUSY-QM to treat any number of dimensions or particles with a focus on its usefulness as a computational tool for calculating

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 131

Because of the significant analytical and computational ramifications, and motivated by the future study of more complex electronic systems, we here apply our multi-dimensional generalization of SUSY-QM to the hydrogen atom in full three-dimensional detail. This is of interest because, until now, the standard application of SUSY-QM to the hydrogen atom required that we first separate out the angular degrees of freedom – effectively reducing the problem to a one-dimensional treatment [Kirchberg et al. (2003); Lahiri et al. (1987); Tangerman & Tjon (1993)]. With our vector superpotential approach, one can deal with the

Our approach provides, for the first time, a SUSY-QM framework that can be employed to treat non-hydrogenic atoms. For example, the standard SUSY-QM treatment of the hydrogen atom cannot be extended readily to the helium atom because it is impossible reduce it to a one-dimensional system. In addition, the form of the three-dimensional vector superpotential for the hydrogen atom is of interest in its own right. It is quite different from the radial superpotential obtained in earlier SUSY-QM studies of the hydrogen atom. The present study thus lays the groundwork for a systematic SUSY-QM study of excited state energies and wave

We now consider the hydrogen atom. We begin by noting that the ground state is exactly

*<sup>ψ</sup>*1,0,0 <sup>=</sup> *<sup>e</sup>*−*<sup>r</sup>*

where we have set the Bohr radius equal to 1. The Hamiltonian (in atomic units) is simply

where *r*ˆ is a unit vector in the direction of *r*. This is an extremely interesting result. First, we see that the superpotential for the Coulomb interaction is, itself, non-singular. Second, in the standard approach, because the angular degrees of freedom have already been separated out, the superpotential is a scalar and it depends on the angular momentum squared (*i.e.* on

The magnitude of *W* is equal to the radial superpotential, as one expects, but the individual components are radically different. Note that these components can also be written solely

<sup>∇</sup><sup>2</sup> <sup>−</sup> <sup>1</sup> *r*

<sup>H</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> 2 <sup>√</sup>*<sup>π</sup>* , (6.1)

*<sup>W</sup>* <sup>=</sup> −∇ ln *<sup>ψ</sup>*1,0,0 <sup>=</sup> *<sup>r</sup>*ˆ, (6.3)

*Wradial* = 1 (6.4)

. (6.2)

*<sup>r</sup>* <sup>=</sup> *<sup>r</sup>*ˆ. (6.5)

Fig. 8. Components of linear combinations of non-degenerate ground and first excited state.

Chakravarty (1995a;b); Derrickson & Bittner (2006; 2007); Franke et al. (1993); Lynden-Bell & Wales (1994); Rick et al. (1991); Schmidt et al. (2001); Wales & Doye (1997) For systems composed of a single type atom or molecule, we expect to encounter degeneracies. Thus, we expect the situation to mirror the present two dimensional non-separable degenerate case.

In dealing with such systems, we anticipate that as the number of particles is increased, we will find that a Monte Carlo based approach may be preferred.

Finally, we stress that this is the first formulation of a general SUSY approach for multi-dimensional and/or multi-particle systems. There remain many formal and computational questions, which we are continuing to explore. Our main conclusion is that there is sufficient promise that such studies are justified.

30

(a) *φ*(2)

(c) *φ*(2)

(1)<sup>1</sup> <sup>=</sup> *<sup>ψ</sup>*(2)

(0)<sup>1</sup> <sup>−</sup> *<sup>ψ</sup>*(2)

will find that a Monte Carlo based approach may be preferred.

there is sufficient promise that such studies are justified.

(0)<sup>1</sup> <sup>=</sup> *<sup>ψ</sup>*(2)

(0)<sup>1</sup> <sup>+</sup> *<sup>ψ</sup>*(2)

(1)<sup>1</sup> (b) *<sup>φ</sup>*(2)

(1)<sup>1</sup> (d) *<sup>φ</sup>*(2)

Fig. 8. Components of linear combinations of non-degenerate ground and first excited state.

Chakravarty (1995a;b); Derrickson & Bittner (2006; 2007); Franke et al. (1993); Lynden-Bell & Wales (1994); Rick et al. (1991); Schmidt et al. (2001); Wales & Doye (1997) For systems composed of a single type atom or molecule, we expect to encounter degeneracies. Thus, we expect the situation to mirror the present two dimensional non-separable degenerate case. In dealing with such systems, we anticipate that as the number of particles is increased, we

Finally, we stress that this is the first formulation of a general SUSY approach for multi-dimensional and/or multi-particle systems. There remain many formal and computational questions, which we are continuing to explore. Our main conclusion is that

(0)<sup>2</sup> <sup>=</sup> *<sup>ψ</sup>*(2)

(1)<sup>2</sup> <sup>=</sup> *<sup>ψ</sup>*(2)

(0)<sup>2</sup> <sup>−</sup> *<sup>ψ</sup>*(2) (1)2

(0)<sup>2</sup> <sup>+</sup> *<sup>ψ</sup>*(2) (1)2

#### **6. Electronic structure of atoms: Hydrogen and helium**

In the previous sections, we have provided a generalization of SUSY-QM to treat any number of dimensions or particles with a focus on its usefulness as a computational tool for calculating accurate excited state energies and wave functions.

Because of the significant analytical and computational ramifications, and motivated by the future study of more complex electronic systems, we here apply our multi-dimensional generalization of SUSY-QM to the hydrogen atom in full three-dimensional detail. This is of interest because, until now, the standard application of SUSY-QM to the hydrogen atom required that we first separate out the angular degrees of freedom – effectively reducing the problem to a one-dimensional treatment [Kirchberg et al. (2003); Lahiri et al. (1987); Tangerman & Tjon (1993)]. With our vector superpotential approach, one can deal with the full three-dimensional nature of the hydrogen atom.

Our approach provides, for the first time, a SUSY-QM framework that can be employed to treat non-hydrogenic atoms. For example, the standard SUSY-QM treatment of the hydrogen atom cannot be extended readily to the helium atom because it is impossible reduce it to a one-dimensional system. In addition, the form of the three-dimensional vector superpotential for the hydrogen atom is of interest in its own right. It is quite different from the radial superpotential obtained in earlier SUSY-QM studies of the hydrogen atom. The present study thus lays the groundwork for a systematic SUSY-QM study of excited state energies and wave functions of atoms.

#### **6.1 SUSY-QM for the three-dimensional hydrogen atom**

We now consider the hydrogen atom. We begin by noting that the ground state is exactly given by

$$
\psi\_{1,0,0} = \frac{e^{-r}}{\sqrt{\pi}} \,\tag{6.1}
$$

where we have set the Bohr radius equal to 1. The Hamiltonian (in atomic units) is simply

$$\mathcal{H} = -\frac{1}{2}\nabla^2 - \frac{1}{r}.\tag{6.2}$$

Then, the vector superpotential is given by

$$
\vec{W} = -\nabla \ln \psi\_{1,0,0} = \mathbf{f},
\tag{6.3}
$$

where *r*ˆ is a unit vector in the direction of *r*. This is an extremely interesting result. First, we see that the superpotential for the Coulomb interaction is, itself, non-singular. Second, in the standard approach, because the angular degrees of freedom have already been separated out, the superpotential is a scalar and it depends on the angular momentum squared (*i.e.* on *l*(*l* + 1)). The precise form for the ground state (*l* = 0) is

$$W\_{radial} = 1\tag{6.4}$$

In three dimensions, we have

$$
\vec{\mathcal{W}} = \vec{\epsilon\_x}\frac{\mathbf{x}}{r} + \vec{\epsilon\_y}\frac{\mathbf{y}}{r} + \vec{\epsilon\_z}\frac{z}{r} = \text{ft.}\tag{6.5}
$$

The magnitude of *W* is equal to the radial superpotential, as one expects, but the individual components are radically different. Note that these components can also be written solely

Fig. 9. The three components of the wave function for *<sup>ψ</sup>* (2)

Fig. 10. The three components of the wave function for *<sup>ψ</sup>* (2)

**6.2 An approximate superpotential for the helium atom**

Jastrow trial wave function Umrigar & Wilson (1988):

interactions. The approximate *W* is generated from

*<sup>ψ</sup>*(1) *<sup>T</sup>*,*<sup>α</sup>* = *e*

with the optimum *<sup>α</sup>* given by *<sup>α</sup>* = 0.353. This yields an energy of *<sup>E</sup>*(1)

positive values and red to negative.

positive values and red to negative.

1,*<sup>S</sup>* . Here, blue corresponds to

. Here, blue corresponds to

*<sup>r</sup>*<sup>12</sup> <sup>2</sup>(1+*αr*12) , (6.16)

*<sup>T</sup>*,*<sup>α</sup>* (6.17)

<sup>1</sup> ≈ 2.878, which is

1,*px*

It is of interest to begin exploring how our approach to multidimensional SUSY-QM would deal with a two electron atom. It is clear that the usual radial (one-dimensional) hydrogen atom SUSY-QM treatment is not readily generalizable to deal with helium. We have carried out a Quantum Monte Carlo study of the sector one ground state of helium using the Padè

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 133

<sup>−</sup>2*r*<sup>1</sup> *e*

in error by about 1%. This error is reasonable for a simple treatment neglecting relativistic

*<sup>W</sup>* (*<sup>r</sup>*1,*<sup>r</sup>*<sup>2</sup>) = −∇ ln *<sup>ψ</sup>*(1)

<sup>−</sup>2*r*<sup>2</sup> *e*

in terms of angular functions (the direction cosines of *r*). To obtain the atomic potential for hydrogen, we form

$$
\vec{W} \cdot \vec{W} - \nabla \cdot \vec{W} = 1 - \left(\frac{3}{r} - \frac{x^2 + y^2 + z^2}{r^3}\right) \tag{6.6}
$$

$$1 = 1 - \frac{2}{r} = -2E\_0 - \frac{2}{r}.\tag{6.7}$$

Now we recall that

$$H\psi\_{m\_l} = E\_n\psi\_{m\_l} \tag{6.8}$$

and

$$-\frac{1}{2}\nabla^2\psi\_{m\_l} = \left[E\_n + \frac{1}{r}\right]\psi\_{m\_l} \tag{6.9}$$

yields

$$
\nabla^2 \psi\_{1,0,0} = -\left[2E\_0 + \frac{2}{r}\right] \psi\_{1,0,0}.\tag{6.10}
$$

Since the ground state energy of hydrogen in atomic units is -1/2, we find that Equations (6.7) and (6.10) are consistent and we have obtained the correct vector superpotential. Of great interest is the wave equation for the sector two problem. This Hamiltonian is given by

$$
\overleftrightarrow{\mathcal{H}}\_2 = -\frac{1}{2}\nabla\nabla + \frac{1}{2}\left[\vec{\mathcal{W}}\vec{\mathcal{W}} + \nabla\vec{\mathcal{W}}\right].\tag{6.11}
$$

In the case of the hydrogen atom, because we have exact analytical expressions for the excited states of H1, it is a simple matter to generate analytical expressions for all the states of the sector two Hamiltonian. It is convenient to label the sector two states with an index indicating the *nth* energy state (i.e., we use the principle quantum number n = 1,2,...) along with the quantum numbers of the sector one excited state from which they are obtained. Thus, the four degenerate ground states of ←→H <sup>2</sup> will be denoted by *<sup>ψ</sup>* (2) 1,2*px* , *<sup>ψ</sup>* (2) 1,2*py* , *<sup>ψ</sup>* (2) 1,2*pz* , *<sup>ψ</sup>* (2) 1,2*s*. We choose here to use the real states rather than those labeled by *ml* = ±1 and *ml* = 0 values. We find that these solutions are given by

$$\vec{\psi}\_{1,2p\_x}^{(2)} = N \left[ \hat{i} e^{-r/2} + \frac{\mathfrak{x}\mathfrak{f}}{2} e^{-r/2} \right],\tag{6.12}$$

$$\vec{\psi}\_{1,2p\_y}^{(2)} = N \left[ \hat{j} e^{-r/2} + \frac{y\hat{\mathcal{H}}}{2} e^{-r/2} \right],\tag{6.13}$$

$$
\vec{\psi}\_{1,2p\_z}^{(2)} = N \left[ \hat{k} e^{-r/2} + \frac{z\hbar}{2} e^{-r/2} \right],
\tag{6.14}
$$

$$
\vec{\psi}\_{1,2s}^{(2)} = -N\frac{\vec{r}}{2}e^{-r/2}.\tag{6.15}
$$

These equations can be verified by simply applying *Q* to the first excited state wave functions of sector one. It is also easily verified that *<sup>Q</sup>* † acting on these states regenerates the *<sup>ψ</sup>*(1) <sup>2</sup>*<sup>p</sup>* and *<sup>ψ</sup>*(1) <sup>2</sup>*<sup>s</sup>* states. Furthermore, in Figures 9 and 10, we provide plots of the *<sup>ψ</sup>* (2) 1,2*<sup>s</sup>* and *<sup>ψ</sup>* (2) 1,2*px* . It is straight forward to see that *<sup>ψ</sup>* (2) 1,2*py* and *<sup>ψ</sup>* (2) 1,2*pz* are both similar to *<sup>ψ</sup>* (2) 1,2*px* .

32

and

yields

*<sup>ψ</sup>*(1)

hydrogen, we form

Now we recall that

in terms of angular functions (the direction cosines of *r*). To obtain the atomic potential for

3

*<sup>r</sup>* <sup>=</sup> <sup>−</sup>2*E*<sup>0</sup> <sup>−</sup> <sup>2</sup>

 *En* + 1 *r* 

 2*E*<sup>0</sup> + 2 *r* 

Since the ground state energy of hydrogen in atomic units is -1/2, we find that Equations (6.7) and (6.10) are consistent and we have obtained the correct vector superpotential. Of great interest is the wave equation for the sector two problem. This Hamiltonian is given by

> 1 2

In the case of the hydrogen atom, because we have exact analytical expressions for the excited states of H1, it is a simple matter to generate analytical expressions for all the states of the sector two Hamiltonian. It is convenient to label the sector two states with an index indicating the *nth* energy state (i.e., we use the principle quantum number n = 1,2,...) along with the quantum numbers of the sector one excited state from which they are obtained. Thus, the four

to use the real states rather than those labeled by *ml* = ±1 and *ml* = 0 values. We find that

ˆ*ie*−*r*/2 +

<sup>ˆ</sup>*je*−*r*/2 <sup>+</sup> *yr*<sup>ˆ</sup>

2 *e*

 ˆ *ke*−*r*/2 +

1,2*<sup>s</sup>* <sup>=</sup> <sup>−</sup>*<sup>N</sup><sup>r</sup>*

These equations can be verified by simply applying *Q* to the first excited state wave functions of sector one. It is also easily verified that *<sup>Q</sup>* † acting on these states regenerates the *<sup>ψ</sup>*(1)

*<sup>ψ</sup>* (2)

<sup>2</sup>*<sup>s</sup>* states. Furthermore, in Figures 9 and 10, we provide plots of the *<sup>ψ</sup>* (2)

1,2*py* and *<sup>ψ</sup>* (2)

*<sup>r</sup>* <sup>−</sup> *<sup>x</sup>*<sup>2</sup> <sup>+</sup> *<sup>y</sup>*<sup>2</sup> <sup>+</sup> *<sup>z</sup>*<sup>2</sup> *r*3

*r*

*<sup>W</sup> <sup>W</sup>* <sup>+</sup> <sup>∇</sup>*<sup>W</sup>*

1,2*px*

*xr*ˆ 2 *e* −*r*/2 

2 *e* −*r*/2 

*zr*ˆ 2 *e* −*r*/2 

1,2*pz* are both similar to *<sup>ψ</sup>* (2)

, *<sup>ψ</sup>* (2) 1,2*py* , *<sup>ψ</sup>* (2) 1,2*pz* , *<sup>ψ</sup>* (2)

, (6.12)

, (6.13)

, (6.14)

1,2*<sup>s</sup>* and *<sup>ψ</sup>* (2)

<sup>−</sup>*<sup>r</sup>*/2. (6.15)

1,2*px* .

*Hψml* = *Enψml* (6.8)

. (6.7)

*ψml* (6.9)

*ψ*1,0,0. (6.10)

. (6.11)

1,2*s*. We choose here

<sup>2</sup>*<sup>p</sup>* and

. It is

1,2*px*

(6.6)

*<sup>W</sup>* · *<sup>W</sup>* −∇· *<sup>W</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup>

− 1 2

←→H <sup>2</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

degenerate ground states of ←→H <sup>2</sup> will be denoted by *<sup>ψ</sup>* (2)

*<sup>ψ</sup>* (2) 1,2*px* = *N*

*<sup>ψ</sup>* (2) 1,2*py* = *N*

*<sup>ψ</sup>* (2) 1,2*pz* = *N*

these solutions are given by

straight forward to see that *<sup>ψ</sup>* (2)

<sup>∇</sup>2*ψ*1,0,0 <sup>=</sup> <sup>−</sup>

2 ∇∇ +

<sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>2</sup>

<sup>∇</sup>2*ψml* <sup>=</sup>

Fig. 9. The three components of the wave function for *<sup>ψ</sup>* (2) 1,*<sup>S</sup>* . Here, blue corresponds to positive values and red to negative.

Fig. 10. The three components of the wave function for *<sup>ψ</sup>* (2) 1,*px* . Here, blue corresponds to positive values and red to negative.

#### **6.2 An approximate superpotential for the helium atom**

It is of interest to begin exploring how our approach to multidimensional SUSY-QM would deal with a two electron atom. It is clear that the usual radial (one-dimensional) hydrogen atom SUSY-QM treatment is not readily generalizable to deal with helium. We have carried out a Quantum Monte Carlo study of the sector one ground state of helium using the Padè Jastrow trial wave function Umrigar & Wilson (1988):

$$
\psi\_{T,\mathfrak{a}}^{(1)} = e^{-2r\_1} e^{-2r\_2} e^{\frac{r\_{12}}{2(1+ar\_{12})}},\tag{6.16}
$$

with the optimum *<sup>α</sup>* given by *<sup>α</sup>* = 0.353. This yields an energy of *<sup>E</sup>*(1) <sup>1</sup> ≈ 2.878, which is in error by about 1%. This error is reasonable for a simple treatment neglecting relativistic interactions. The approximate *W* is generated from

$$\vec{\mathcal{W}}(\vec{r\_1}, \vec{r\_2}) = -\vec{\nabla} \ln \psi\_{T, \mathfrak{a}}^{(1)} \tag{6.17}$$

And similarly, we can find the second sector first excited singlet state by simply adding the

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 135

appropriate spatial wave functions. This is to say that, to within a multiplicative constant, we

From this, we observe that the aufbau principle in the second sector is remarkably simple for

However, this basis doesn't include the correlation. To do this, we can multiply our antisymmetrized second sector wave function by a correlation function, given by the Padé-Jastrow function which only depends on *r*12. It is clear, then, that because our correlation function is only a function of *r*12, its symmetry will not be affected by the application of *Q* and, thus, we can simply multiply it by our second sector state of interest (where the minus

<sup>−</sup>2*r*2−*r*<sup>1</sup> [2*r*ˆ2 (<sup>1</sup> <sup>−</sup> *<sup>r</sup>*1) <sup>+</sup> *<sup>r</sup>*ˆ1]

Higher accuracy will result if we insert additional variational parameters (*e.g.*, effective

In this chapter, we began by presenting our computational approach to one dimensional systems. We showed with the anharmonic oscillator that we were able to achieve significant computational gains in a robust fashion, permitting more exact numerical solutions of one dimensional problems. Although anharmonic oscillator models are useful for a wide variety of problems in both chemistry and physics, it should be clear that other systems should show similar behaviour. The SUSY-QM approach enabled us to develop a hierarchy of isospectral Hamiltonians. This also led to the introduction of charge operators that transform wave functions between the various sectors, and the energies are always determined in a ground state setting. Because these are most easily and accurately obtained by the variational method, we realize a significant reduction in the basis size needed to yield accurate excited state wave functions. We then considered 2 specific examples of anharmonic oscillators. We concluded that using the SUSY hierarchy of hamiltonians and charge operators, provided faster convergence to the same level of accuracy and thus, provides a better method than the

*<sup>φ</sup>*(2)

<sup>−</sup>2*r*1−*r*<sup>2</sup> [2*r*ˆ1 (<sup>1</sup> <sup>−</sup> *<sup>r</sup>*2) <sup>+</sup> *<sup>r</sup>*ˆ2] <sup>−</sup>

*<sup>φ</sup>*(2)

<sup>−</sup>2*r*1−*r*<sup>2</sup> [2*r*ˆ1 (<sup>1</sup> <sup>−</sup> *<sup>r</sup>*2) <sup>+</sup> *<sup>r</sup>*ˆ2] <sup>∓</sup>

 . (7.5)

(7.8)

1,*singlet* give the

1,*triplet* and *<sup>ψ</sup>* (2)

<sup>1</sup> and antisymmetrize or symmetrize

<sup>1</sup> , is neither symmetric nor antisymmetric

<sup>1</sup> = *α*(*r*1)*β*(*r*2) − *α*(*r*2)*β*(*r*1) (7.6)

<sup>2</sup> = *α*(*r*1)*β*(*r*2) + *α*(*r*2)*β*(*r*1). (7.7)

<sup>−</sup>2*r*2−*r*<sup>1</sup> [2*r*ˆ2 (<sup>1</sup> <sup>−</sup> *<sup>r</sup>*1) <sup>+</sup> *<sup>r</sup>*ˆ1] .

two building blocks as given below.

get that

and

*<sup>ψ</sup>* (2)

*<sup>Q</sup>* † · *<sup>ψ</sup>* (2)

*<sup>Q</sup>* † · *<sup>ψ</sup>* (2)

Helium. We merely need to take the building block

corresponds to the triplet and the plus to the singlet):

*<sup>ψ</sup>* (2) 1,*triplet* =*e*

charges, *etc.*) when doing computations.

**8. Conclusions and perspectives**

appropriately. Of course our "building block",

under particle exchange.

1,*singlet* = − *e*

Indeed, by taking the scalar product with *<sup>Q</sup>* †, we can verify that *<sup>ψ</sup>* (2)

*e*

1,*triplet* <sup>=</sup> *<sup>ψ</sup>*(1)

1,*singlet* <sup>=</sup> *<sup>ψ</sup>*(1)

*<sup>r</sup>*<sup>12</sup> <sup>2</sup>(1+*δr*12) −*e*

*e*

$$=-\vec{\nabla}\left[-2r\_1 - 2r\_2 + \frac{r\_{12}}{2(1+\alpha r\_{12})}\right].\tag{6.18}$$

Here,

$$\vec{\nabla} = \varepsilon \hat{\mathbf{1}\_{1x}} \frac{\partial}{\partial \mathbf{x}\_{1}} + \varepsilon \hat{\mathbf{1}\_{1y}} \frac{\partial}{\partial y\_{1}} + \varepsilon \hat{\mathbf{1}\_{2z}} \frac{\partial}{\partial \mathbf{z}\_{1}} + \varepsilon \hat{\mathbf{2}\_{1}} \frac{\partial}{\partial \mathbf{x}\_{2}} + \varepsilon \hat{\mathbf{2}\_{y}} \frac{\partial}{\partial y\_{2}} + \varepsilon \hat{\mathbf{2}\_{2z}} \frac{\partial}{\partial \mathbf{z}\_{2}},\tag{6.19}$$

where the {*�*ˆ*ij*} are orthonormal vectors. The resulting vector superpotential for the Padè-Jastrow trial function is readily found to be

$$\vec{W}\_{He(PI)} = 2\mathfrak{k}\_1 + 2\mathfrak{k}\_2 - \mathfrak{k}\_{12} \left[ 1 - \frac{a}{(1 + ar\_{12})} \right]. \tag{6.20}$$

Thus, the structure of *<sup>W</sup> He*(*P J*) is analogous to *<sup>W</sup> <sup>H</sup>* in that Coulomb interactions generate vector superpotentials that involve unit vectors anti-parallel to the direction of the forces. This is true in general for Coulombic interactions. This emphasizes the important distinction between our three-dimensional SUSY-QM treatment of an atom and the standard hydrogen atom one-dimensional radial SUSY-QM.

#### **7. Aufbau approach for excited states**

For multielectron atoms, it becomes necessary to consider how the aufbau principle acts in the second sector to permit efficient calculations of sector one excited states. This is because we can use this principle to design reasonable trial wave functions for a variational approach to the sector two ground state. In this section, we consider a simple aufbau description of the sector one helium excited states in order to design an approximate sector two ground state of helium. We assume that in the first excited state of sector one, we have one electron in the 1*S* orbital, given by wave function *α*, and one electron in the 2*S* orbital, given by wave function *β* where

$$\alpha(r) = \frac{e^{-2r}}{\sqrt{\pi}} \text{ and } \beta(r) = \frac{e^{-r}}{4\sqrt{2\pi}}(1-r). \tag{7.1}$$

Then, it is of interest to take the product of these states such that we have *α*(*r*1)*β*(*r*2), to which we can apply our *Q* to find

$$\vec{Q}\left(\mathfrak{a}(r\_1)\mathfrak{z}(r\_2)\right) = -e^{-2r\_1 - r\_2} \left[2\hat{r}\_1\left(1 - r\_2\right) + \hat{r}\_2\right] \equiv \vec{\phi}\_1^{(2)}.\tag{7.2}$$

The first excited sector one state of Helium is a triplet so the wave function is anti-symmetric under spatial electron exchange. It is clear that we require the second sector ground state also be anti-symmetric when we interchange labels 1 and 2. To obtain this, we apply *P*<sup>12</sup> to Equation (7.2) to get

$$\vec{Q}\left(a(r\_2)\beta(r\_1)\right) = -e^{-2r\_2 - r\_1} \left[2\mathfrak{H}\_2\left(1 - r\_1\right) + \mathfrak{H}\_2\right] \equiv P\_{12}\vec{\phi}\_1^{(2)},\tag{7.3}$$

where *P*<sup>12</sup> exchanges the electron labels. Then, we can use Equations (7.2) and (7.3) as a "building blocks" to construct our ground state in the second sector by subtracting the first building block from the second. This gives us a second sector result of

$$\begin{split} \vec{\psi}\_{1,triplet}^{(2)} &= -e^{-2r\_1 - r\_2} \left[ 2\mathfrak{H}\_1 \left( 1 - r\_2 \right) + \mathfrak{H}\_2 \right] + \\ &e^{-2r\_2 - r\_1} \left[ 2\mathfrak{H}\_2 \left( 1 - r\_1 \right) + \mathfrak{H}\_1 \right]. \end{split} \tag{7.4}$$

And similarly, we can find the second sector first excited singlet state by simply adding the two building blocks as given below.

$$\begin{split} \vec{\psi}\_{1,singlet}^{(2)} &= -e^{-2r\_1 - r\_2} \left[ 2\mathfrak{f}\_1 \left( 1 - r\_2 \right) + \mathfrak{f}\_2 \right] - \\ &e^{-2r\_2 - r\_1} \left[ 2\mathfrak{f}\_2 \left( 1 - r\_1 \right) + \mathfrak{f}\_1 \right]. \end{split} \tag{7.5}$$

Indeed, by taking the scalar product with *<sup>Q</sup>* †, we can verify that *<sup>ψ</sup>* (2) 1,*triplet* and *<sup>ψ</sup>* (2) 1,*singlet* give the appropriate spatial wave functions. This is to say that, to within a multiplicative constant, we get that

$$
\vec{Q}^{\dagger} \cdot \vec{\psi}\_{1,triplet}^{(2)} = \psi\_1^{(1)} = \mathfrak{a}(r\_1)\mathfrak{f}(r\_2) - \mathfrak{a}(r\_2)\mathfrak{f}(r\_1) \tag{7.6}
$$

and

34

Here,

*β* where

we can apply our *Q* to find

Equation (7.2) to get

<sup>=</sup> −∇ 

+ *�*ˆ 1*y ∂ ∂y*<sup>1</sup>

*<sup>α</sup>*(*r*) = *<sup>e</sup>*−2*<sup>r</sup>* <sup>√</sup>*<sup>π</sup>*

*<sup>Q</sup>* (*α*(*r*1)*β*(*r*2)) <sup>=</sup> <sup>−</sup>*<sup>e</sup>*

*<sup>Q</sup>* (*α*(*r*2)*β*(*r*1)) <sup>=</sup> <sup>−</sup>*<sup>e</sup>*

*<sup>ψ</sup>* (2)

building block from the second. This gives us a second sector result of

1,*triplet* = − *e*

*e*

<sup>∇</sup> <sup>=</sup> *�*<sup>ˆ</sup> 1*x ∂ ∂x*<sup>1</sup>

atom one-dimensional radial SUSY-QM.

**7. Aufbau approach for excited states**

Padè-Jastrow trial function is readily found to be

−2*r*<sup>1</sup> − 2*r*<sup>2</sup> +

+ *�*ˆ 2*x ∂ ∂x*<sup>2</sup>

where the {*�*ˆ*ij*} are orthonormal vectors. The resulting vector superpotential for the

Thus, the structure of *<sup>W</sup> He*(*P J*) is analogous to *<sup>W</sup> <sup>H</sup>* in that Coulomb interactions generate vector superpotentials that involve unit vectors anti-parallel to the direction of the forces. This is true in general for Coulombic interactions. This emphasizes the important distinction between our three-dimensional SUSY-QM treatment of an atom and the standard hydrogen

For multielectron atoms, it becomes necessary to consider how the aufbau principle acts in the second sector to permit efficient calculations of sector one excited states. This is because we can use this principle to design reasonable trial wave functions for a variational approach to the sector two ground state. In this section, we consider a simple aufbau description of the sector one helium excited states in order to design an approximate sector two ground state of helium. We assume that in the first excited state of sector one, we have one electron in the 1*S* orbital, given by wave function *α*, and one electron in the 2*S* orbital, given by wave function

and *<sup>β</sup>*(*r*) = *<sup>e</sup>*−*<sup>r</sup>*

Then, it is of interest to take the product of these states such that we have *α*(*r*1)*β*(*r*2), to which

The first excited sector one state of Helium is a triplet so the wave function is anti-symmetric under spatial electron exchange. It is clear that we require the second sector ground state also be anti-symmetric when we interchange labels 1 and 2. To obtain this, we apply *P*<sup>12</sup> to

where *P*<sup>12</sup> exchanges the electron labels. Then, we can use Equations (7.2) and (7.3) as a "building blocks" to construct our ground state in the second sector by subtracting the first

4 <sup>√</sup>2*<sup>π</sup>*

<sup>−</sup>2*r*1−*r*<sup>2</sup> [2*r*ˆ1 (<sup>1</sup> <sup>−</sup> *<sup>r</sup>*2) <sup>+</sup> *<sup>r</sup>*ˆ2] <sup>≡</sup>

<sup>−</sup>2*r*2−*r*<sup>1</sup> [2*r*ˆ2 (<sup>1</sup> <sup>−</sup> *<sup>r</sup>*1) <sup>+</sup> *<sup>r</sup>*ˆ2] <sup>≡</sup> *<sup>P</sup>*<sup>12</sup>

<sup>−</sup>2*r*1−*r*<sup>2</sup> [2*r*ˆ1 (<sup>1</sup> <sup>−</sup> *<sup>r</sup>*2) <sup>+</sup> *<sup>r</sup>*ˆ2] <sup>+</sup>

<sup>−</sup>2*r*2−*r*<sup>1</sup> [2*r*ˆ2 (<sup>1</sup> <sup>−</sup> *<sup>r</sup>*1) <sup>+</sup> *<sup>r</sup>*ˆ1] .

+ *�*ˆ 1*z ∂ ∂z*<sup>1</sup>

*<sup>W</sup> He*(*P J*) <sup>=</sup> <sup>2</sup>*r*ˆ1 <sup>+</sup> <sup>2</sup>*r*ˆ2 <sup>−</sup> *<sup>r</sup>*ˆ12

*r*<sup>12</sup> 2(1 + *αr*12)

<sup>1</sup> <sup>−</sup> *<sup>α</sup>* (1 + *αr*12)

+ *�*ˆ 2*z ∂ ∂z*<sup>2</sup>

+ *�*ˆ 2*y ∂ ∂y*<sup>2</sup>

. (6.18)

. (6.20)

(1 − *r*). (7.1)

<sup>1</sup> . (7.2)

<sup>1</sup> , (7.3)

(7.4)

*<sup>φ</sup>*(2)

*<sup>φ</sup>*(2)

, (6.19)

$$
\vec{Q}^\dagger \cdot \vec{\psi}\_{1,singlet}^{(2)} = \psi\_2^{(1)} = \mathfrak{a}(r\_1)\mathfrak{z}(r\_2) + \mathfrak{a}(r\_2)\mathfrak{z}(r\_1). \tag{7.7}
$$

From this, we observe that the aufbau principle in the second sector is remarkably simple for Helium. We merely need to take the building block *<sup>φ</sup>*(2) <sup>1</sup> and antisymmetrize or symmetrize appropriately. Of course our "building block", *<sup>φ</sup>*(2) <sup>1</sup> , is neither symmetric nor antisymmetric under particle exchange.

However, this basis doesn't include the correlation. To do this, we can multiply our antisymmetrized second sector wave function by a correlation function, given by the Padé-Jastrow function which only depends on *r*12. It is clear, then, that because our correlation function is only a function of *r*12, its symmetry will not be affected by the application of *Q* and, thus, we can simply multiply it by our second sector state of interest (where the minus corresponds to the triplet and the plus to the singlet):

$$\begin{split} \left| \overline{\psi}\_{1,triplet}^{(2)} \right| &= e^{\frac{r\_1}{2(1+dr\_1)}} \left( -e^{-2r\_1-r\_2} \left[ 2\mathfrak{H}\_1 \left( 1-r\_2 \right) + \mathfrak{H}\_2 \right] \mp \\ & \qquad e^{-2r\_2-r\_1} \left[ 2\mathfrak{H}\_2 \left( 1-r\_1 \right) + \mathfrak{H}\_1 \right] \right) . \end{split} \tag{7.8}$$

Higher accuracy will result if we insert additional variational parameters (*e.g.*, effective charges, *etc.*) when doing computations.

#### **8. Conclusions and perspectives**

In this chapter, we began by presenting our computational approach to one dimensional systems. We showed with the anharmonic oscillator that we were able to achieve significant computational gains in a robust fashion, permitting more exact numerical solutions of one dimensional problems. Although anharmonic oscillator models are useful for a wide variety of problems in both chemistry and physics, it should be clear that other systems should show similar behaviour. The SUSY-QM approach enabled us to develop a hierarchy of isospectral Hamiltonians. This also led to the introduction of charge operators that transform wave functions between the various sectors, and the energies are always determined in a ground state setting. Because these are most easily and accurately obtained by the variational method, we realize a significant reduction in the basis size needed to yield accurate excited state wave functions. We then considered 2 specific examples of anharmonic oscillators. We concluded that using the SUSY hierarchy of hamiltonians and charge operators, provided faster convergence to the same level of accuracy and thus, provides a better method than the

Finally, we have shown how our multi-dimensional generalization of SUSY-QM can be applied to the hydrogen atom. Previously, most detailed attempts to treat the hydrogen atom first separated the angular degrees of freedom, leaving a one-dimensional radial wave equation. It was then possible to obtain the SUSY-QM factorization, yielding a scalar superpotential that, for the *l* = 0 states, is simply *W* = 1. While these results are interesting, the one-dimensional radial SUSY-QM approach is not readily generalizable to treat even the

Generalized Non-Relativistic Supersymmetric Quantum Mechanics 137

In our approach, the full three-dimensional character of the hydrogen atom is considered, with the result being a vector-valued superpotential, *W* , which for the hydrogen atom, is *W* = *r*ˆ. That is, the vector superpotential points in the opposite direction of the attractive Coulomb force between the electron and the nucleus. This is interesting also because, although the Coulomb potential is singular, its vector superpotential is not. It is important to note that such a superpotential was also obtained earlier by Stedman Stedman (1985). However, his sector two Hamiltonian differs from ours and produces "extra" states that are not degenerate with

The fact that *W* for the three-dimensional hydrogen atom is a vector does not, in any way, modify the sector one dynamical equation. However, the sector two situation is radically affected! In the one-dimensional SUSY-QM case, there is no significant change in the basic mathematical structure of the sector two partner Hamiltonian. In the multi-dimensional case, the sector two Hamiltonian is a tensor. However, we have shown in previous studies, that many of the standard computational techniques remain valid. Of particular interest is the Dirac-Frankel-McLachlan Variational Method, since this is known to deal better with

In the case of the hydrogen atom, it is straight forward to generate all the sector two eigenstates. This is a consequence of the fact that exact analytical eigenstates of the three-dimensional hydrogen atom are known. It is then easy to apply the charge operator, *Q* , to the excited hydrogen atom states and obtain sector two eigenstates. (We note that because of the four fold degeneracy for the sector two ground state, the resulting eigenstates can be super-posed in any manner convenient for the study at hand). It is of considerable interest to begin exploring how our multi-dimensional SUSY-QM treatment can be applied to the helium atom. In this case, the exact sector one ground state is, of course, unavailable. In our previous one and two-dimensional studies we have considered other systems for which an exact *W* was not possible. In the case of helium, we chose to examine an accurate Padè-Jastrow approximation to the sector one ground state. In this case, it is easy to obtain an analytical (albeit approximate) *W* that displays very reasonable intuitive character. In direct analogy with the exact hydrogen atom *<sup>W</sup>* , we find that the *<sup>W</sup> He*(*P J*) vector superpotential consists of a combination of unit vectors that again, are anti-parallel to the Coulomb forces associated with the helium atom potential energy. The next step in our study will consist of computations of a sector two ground state, which will allow us to obtain an approximate helium atom sector

Future studies will explore extending the approach to more than two electron atoms. There, the issue will be taking account of the electrons' spin degrees of freedom. Our current plan is to employ the "spin-free" techniques of Matsen Matsen (1964; 1966; 1970); Matsen & Cantu (1968; 1969); Matsen et al. (1966); Matsen & Ellzey (1969); Matsen & Junker (1971); Matsen &

We have also generalized the aufbau principle to work in the second sector Hamiltonian, demonstrating that we are able to produce reasonable forms of excited states by simply using

helium atom.

sector one.

higher-dimensional systemsRaab (2000).

one first excited state energy and wave function.

Klein (1969; 1971); Matsen et al. (1971) and othersPauncz (1995).

standard variational approach. In most cases, only half as many basis functions were needed to generate the ground state wave function as were required for the first excited state of the same sector. As a result, the computational time for molecular models using anharmonic potentials will be significantly reduced, without sacrificing accuracy.

We also stress that our results do not depend on precisely what basis set is used for the calculations. Rather we are capitalizing on the general behavior of the Rayleigh-Ritz variational method with regard to accuracy and convergence rate for ground versus excited states of a given Hamiltonian.

We then presented our approach to generalizing SUSY-QM to deal with more than one dimension and more than one (distinguishable) particle. In general, previous attempts to do this have typically introduced Pauli spin matrices and so far as we are aware, none of these has been proved useful for the general case. Andrianov et al. (1985); Andrianov, Borisov & Ioffe (1984a;b;c); Andrianov et al. (1986); Andrianov, Borisov, Ioffe & Eides (1984); Andrianov & Ioffe (1988); Andrianov et al. (2002); Cannata et al. (2002); Das & Pernice (1996) One principle difficulty is that while the coordinates of different particles are independent variables, they are not defined relative to orthogonal axes. That is, there are only 3 independent, physical axes along which all particle positions are measured. Our approach introduces a higher dimensional vector space in which there is an orthonormal basis vector associated with each independent particle coordinate. This is analogous to the relativistic situation where each particle has its own coordinate system(and, of course, in the relativistic case, its own "proper time"). Here, however the device is a mathematical convenience (so far as we are currently aware) and it is, of course, non-relativistic. That is, we assume Gallilean transformations. The most striking consequence similar to relativistic quantum mechanics is that our second sector Hamiltonian becomes a tensor in the expanded space. This does *not* increase the number of independent variables (i.e., the wave function for the second sector is a vector in the new hyperspace). It is shown that this tensor character is then absent from the 3*rd* sector Hamiltonian (which is once again a scalar operator). One in general obtains an alternating series of scalar and tensor Hamiltonians. *The occurrence of a tensor sector Hamiltonian is, of course, an added computational cost to the approach*. This is mitigated , to some degree, by the fact that we *never* must calculate an accurate wave function and energy except for *ground states* . It is this feature that makes the SUSY-QM approach attractive, since ground state energies and wave functions are the least computationally demanding of all and typically are obtained with the highest accuracy. Thus the computational effort of obtaining the second excited state energy and wave function again involves solving an equation comparable to that generated by the original *H*1. A complication, however, arises due to the observed fact that for the ground state sector 2 wave function in both the degenerate and non-degenerate cases, one of the two components possesses nodes while the other component is nodeless. This is mitigated (in terms of accuracy of the sector 2 ground state calculation) by the fact that the component containing the node is (in the present computational examples) an order of magnitude smaller that the nodeless component. This appears to enable the variational evaluation of the ground state of the sector 2 to yield better accuracy for the first excited state energy and wave function than a comparable basis set calculation applied to *H*1.

An extremely important question is, however, raised by the fact that the small component of *<sup>ψ</sup>* (2) <sup>0</sup> has nodes. This is whether the presence of nodes will prevent us from applying a simple variational quantum Monte-Carlo method to obtain *<sup>ψ</sup>* (2) <sup>0</sup> . We are currently exploring this question. However, in the present context, it does not appear to create difficulties for the variational approach.

36

states of a given Hamiltonian.

standard variational approach. In most cases, only half as many basis functions were needed to generate the ground state wave function as were required for the first excited state of the same sector. As a result, the computational time for molecular models using anharmonic

We also stress that our results do not depend on precisely what basis set is used for the calculations. Rather we are capitalizing on the general behavior of the Rayleigh-Ritz variational method with regard to accuracy and convergence rate for ground versus excited

We then presented our approach to generalizing SUSY-QM to deal with more than one dimension and more than one (distinguishable) particle. In general, previous attempts to do this have typically introduced Pauli spin matrices and so far as we are aware, none of these has been proved useful for the general case. Andrianov et al. (1985); Andrianov, Borisov & Ioffe (1984a;b;c); Andrianov et al. (1986); Andrianov, Borisov, Ioffe & Eides (1984); Andrianov & Ioffe (1988); Andrianov et al. (2002); Cannata et al. (2002); Das & Pernice (1996) One principle difficulty is that while the coordinates of different particles are independent variables, they are not defined relative to orthogonal axes. That is, there are only 3 independent, physical axes along which all particle positions are measured. Our approach introduces a higher dimensional vector space in which there is an orthonormal basis vector associated with each independent particle coordinate. This is analogous to the relativistic situation where each particle has its own coordinate system(and, of course, in the relativistic case, its own "proper time"). Here, however the device is a mathematical convenience (so far as we are currently aware) and it is, of course, non-relativistic. That is, we assume Gallilean transformations. The most striking consequence similar to relativistic quantum mechanics is that our second sector Hamiltonian becomes a tensor in the expanded space. This does *not* increase the number of independent variables (i.e., the wave function for the second sector is a vector in the new hyperspace). It is shown that this tensor character is then absent from the 3*rd* sector Hamiltonian (which is once again a scalar operator). One in general obtains an alternating series of scalar and tensor Hamiltonians. *The occurrence of a tensor sector Hamiltonian is, of course, an added computational cost to the approach*. This is mitigated , to some degree, by the fact that we *never* must calculate an accurate wave function and energy except for *ground states* . It is this feature that makes the SUSY-QM approach attractive, since ground state energies and wave functions are the least computationally demanding of all and typically are obtained with the highest accuracy. Thus the computational effort of obtaining the second excited state energy and wave function again involves solving an equation comparable to that generated by the original *H*1. A complication, however, arises due to the observed fact that for the ground state sector 2 wave function in both the degenerate and non-degenerate cases, one of the two components possesses nodes while the other component is nodeless. This is mitigated (in terms of accuracy of the sector 2 ground state calculation) by the fact that the component containing the node is (in the present computational examples) an order of magnitude smaller that the nodeless component. This appears to enable the variational evaluation of the ground state of the sector 2 to yield better accuracy for the first excited state energy and wave function

An extremely important question is, however, raised by the fact that the small component

this question. However, in the present context, it does not appear to create difficulties for the

<sup>0</sup> has nodes. This is whether the presence of nodes will prevent us from applying a

<sup>0</sup> . We are currently exploring

potentials will be significantly reduced, without sacrificing accuracy.

than a comparable basis set calculation applied to *H*1.

simple variational quantum Monte-Carlo method to obtain *<sup>ψ</sup>* (2)

of *<sup>ψ</sup>* (2)

variational approach.

Finally, we have shown how our multi-dimensional generalization of SUSY-QM can be applied to the hydrogen atom. Previously, most detailed attempts to treat the hydrogen atom first separated the angular degrees of freedom, leaving a one-dimensional radial wave equation. It was then possible to obtain the SUSY-QM factorization, yielding a scalar superpotential that, for the *l* = 0 states, is simply *W* = 1. While these results are interesting, the one-dimensional radial SUSY-QM approach is not readily generalizable to treat even the helium atom.

In our approach, the full three-dimensional character of the hydrogen atom is considered, with the result being a vector-valued superpotential, *W* , which for the hydrogen atom, is *W* = *r*ˆ. That is, the vector superpotential points in the opposite direction of the attractive Coulomb force between the electron and the nucleus. This is interesting also because, although the Coulomb potential is singular, its vector superpotential is not. It is important to note that such a superpotential was also obtained earlier by Stedman Stedman (1985). However, his sector two Hamiltonian differs from ours and produces "extra" states that are not degenerate with sector one.

The fact that *W* for the three-dimensional hydrogen atom is a vector does not, in any way, modify the sector one dynamical equation. However, the sector two situation is radically affected! In the one-dimensional SUSY-QM case, there is no significant change in the basic mathematical structure of the sector two partner Hamiltonian. In the multi-dimensional case, the sector two Hamiltonian is a tensor. However, we have shown in previous studies, that many of the standard computational techniques remain valid. Of particular interest is the Dirac-Frankel-McLachlan Variational Method, since this is known to deal better with higher-dimensional systemsRaab (2000).

In the case of the hydrogen atom, it is straight forward to generate all the sector two eigenstates. This is a consequence of the fact that exact analytical eigenstates of the three-dimensional hydrogen atom are known. It is then easy to apply the charge operator, *Q* , to the excited hydrogen atom states and obtain sector two eigenstates. (We note that because of the four fold degeneracy for the sector two ground state, the resulting eigenstates can be super-posed in any manner convenient for the study at hand). It is of considerable interest to begin exploring how our multi-dimensional SUSY-QM treatment can be applied to the helium atom. In this case, the exact sector one ground state is, of course, unavailable. In our previous one and two-dimensional studies we have considered other systems for which an exact *W* was not possible. In the case of helium, we chose to examine an accurate Padè-Jastrow approximation to the sector one ground state. In this case, it is easy to obtain an analytical (albeit approximate) *W* that displays very reasonable intuitive character. In direct analogy with the exact hydrogen atom *<sup>W</sup>* , we find that the *<sup>W</sup> He*(*P J*) vector superpotential consists of a combination of unit vectors that again, are anti-parallel to the Coulomb forces associated with the helium atom potential energy. The next step in our study will consist of computations of a sector two ground state, which will allow us to obtain an approximate helium atom sector one first excited state energy and wave function.

Future studies will explore extending the approach to more than two electron atoms. There, the issue will be taking account of the electrons' spin degrees of freedom. Our current plan is to employ the "spin-free" techniques of Matsen Matsen (1964; 1966; 1970); Matsen & Cantu (1968; 1969); Matsen et al. (1966); Matsen & Ellzey (1969); Matsen & Junker (1971); Matsen & Klein (1969; 1971); Matsen et al. (1971) and othersPauncz (1995).

We have also generalized the aufbau principle to work in the second sector Hamiltonian, demonstrating that we are able to produce reasonable forms of excited states by simply using

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92(12): 3525–3538.

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Algebra on the Circle and a Related Quantum Mechanical Model for Hindered

Algebra on the Circle and a Related Quantum Mechanical Model for Hindered

uncertainty wavelets and their relations to the harmonic oscillator and the coherent

hydrogenic orbitals. The equations have a reasonable structure but variational computations are necessary. We shall report these results later.

#### **9. References**


URL: *http://www.iop.org/EJ/ref/-target=inspec/0305-4470/35/6/305/20*


38

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**0**

**6**

Ulf Klein

*Austria*

**Quantum Theory**

, (1)

*Universitity of Linz, Institute for Theoretical Physics*

**A Statistical Derivation of Non-Relativistic**

Quantum theory (QT) may either be defined by a set of axioms or otherwise be 'derived' from classical physics by using certain assumptions. Today, QT is frequently identified with a set of axioms defining a Hilbert space structure. This mathematical structure has been created (by von Neumann) by abstraction from the linear solution space of the central equation of QT, the Schrödinger equation. Thus, deriving Schrödinger's equation is basically the same as deriving QT. To derive the most general version of the time-dependent Schrödinger equation, describing *N* particles with spin in an external gauge field, means to derive essentially the

The second way of proceeding is sometimes called 'quantization'. In the standard (canonical) quantization method one starts from a classical Hamiltonian whose basic variables are then

, *E* → −

into operators. Then, all relevant classical observables may be rewritten as operators acting on states of a Hilbert space etc; the details of the 'derivation' of Schrödinger's equation along this lines may be found in many textbooks. There are formal problems with this approach which have been identified many years ago, and can be expressed e.g. in terms of Groenewold's theorem, see Groenewold (1946), Gotay (1999). Even more seriously, there is no satisfactory *explanation* for this 'metamorphosis' of observables into operators. This quantization method (as well as several other mathematically more sophisticated versions of it) is just a *recipe* or, depending on one's taste, "black magic", Hall (2005). Note that the enormous success of this recipe in various contexts - including field quantization - is no substitute for an explanation. The choice of a particular quantization procedure will be strongly influenced by the preferred interpretation of the quantum theoretical formalism. If QT is interpreted as a theory describing individual events, then the Hamiltonian of classical mechanics becomes a natural starting point. This 'individuality assumption' is an essential part of the dominating 'conventional', or 'Copenhagen', interpretation (CI) of QT. It is well-known, that QT becomes a source of mysteries and paradoxes1 whenever it is interpreted in the sense of CI, as a (complete) theory for individual events. Thus, the canonical quantization method and the CI are similar in

<sup>1</sup> I cannot report here a list, all the less a description, of all the quantum mechanical paradoxes found in

*h*¯ *ı* d d*t*

'transformed', by means of well-known correspondence rules,

*p* → *h*¯ *ı* d d*x*

**1. Introduction**

whole of non-relativistic QT.

the last eighty years.


URL: *http://link.aps.org/doi/10.1103/PhysRevLett.60.1719*


URL: *http://books.google.com/books?id=jQbEcDDqGb8C*


### **A Statistical Derivation of Non-Relativistic Quantum Theory**

### Ulf Klein

*Universitity of Linz, Institute for Theoretical Physics Austria*

#### **1. Introduction**

40

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calculations, *Physical Review Letters* .

*European Journal of Physics* 6(4): 225–231. URL: *http://iopscience.iop.org/0143-0807/6/4/002*

Quantum theory (QT) may either be defined by a set of axioms or otherwise be 'derived' from classical physics by using certain assumptions. Today, QT is frequently identified with a set of axioms defining a Hilbert space structure. This mathematical structure has been created (by von Neumann) by abstraction from the linear solution space of the central equation of QT, the Schrödinger equation. Thus, deriving Schrödinger's equation is basically the same as deriving QT. To derive the most general version of the time-dependent Schrödinger equation, describing *N* particles with spin in an external gauge field, means to derive essentially the whole of non-relativistic QT.

The second way of proceeding is sometimes called 'quantization'. In the standard (canonical) quantization method one starts from a classical Hamiltonian whose basic variables are then 'transformed', by means of well-known correspondence rules,

$$p \rightarrow \frac{\hbar}{n} \frac{\mathbf{d}}{\mathbf{d} \mathbf{x}'} , \quad \mathbf{E} \rightarrow -\frac{\hbar}{n} \frac{\mathbf{d}}{\mathbf{d} \mathbf{t}'} \tag{1}$$

into operators. Then, all relevant classical observables may be rewritten as operators acting on states of a Hilbert space etc; the details of the 'derivation' of Schrödinger's equation along this lines may be found in many textbooks. There are formal problems with this approach which have been identified many years ago, and can be expressed e.g. in terms of Groenewold's theorem, see Groenewold (1946), Gotay (1999). Even more seriously, there is no satisfactory *explanation* for this 'metamorphosis' of observables into operators. This quantization method (as well as several other mathematically more sophisticated versions of it) is just a *recipe* or, depending on one's taste, "black magic", Hall (2005). Note that the enormous success of this recipe in various contexts - including field quantization - is no substitute for an explanation. The choice of a particular quantization procedure will be strongly influenced by the preferred interpretation of the quantum theoretical formalism. If QT is interpreted as a theory describing individual events, then the Hamiltonian of classical mechanics becomes a natural starting point. This 'individuality assumption' is an essential part of the dominating 'conventional', or 'Copenhagen', interpretation (CI) of QT. It is well-known, that QT becomes a source of mysteries and paradoxes1 whenever it is interpreted in the sense of CI, as a (complete) theory for individual events. Thus, the canonical quantization method and the CI are similar in

<sup>1</sup> I cannot report here a list, all the less a description, of all the quantum mechanical paradoxes found in the last eighty years.

These general ideas lead to the mathematical assumptions which represent the basis for the treatment reported in I. This work was restricted to a one-dimensional configuration space (a single particle ensemble with a single spatial degree of freedom). The present work generalizes the treatment of I to a 3*N*−dimensional configuration space ( ensembles representing an arbitrary number *N* of particles allowed to move in three-dimensional space), gauge-coupling, and spin. In a first step the generalization to three spatial dimensions is performed; the properly generalized basic relations are reported in section 2. This section

A Statistical Derivation of Non-Relativistic Quantum Theory 143

In section 3 we make use of a mathematical freedom, which is already contained in our basic assumptions, namely the multi-valuedness of the variable *S*. This leads to the appearance of potentials in statistical relations replacing the local forces of single-event (mechanical) theories. The mysterious non-local action of the vector potential (in effects of the Aharonov-Bohm type) is explained as a consequence of the statistical nature of QT. In section 4 we discuss a related question: Which constraints on admissible forces exist for the present class of statistical theories ? The answer is that only macroscopic (elementary) forces of the form of the Lorentz force can occur in nature, because only these survive the transition to QT . These forces are statistically represented by potentials, i.e. by the familiar gauge coupling terms in matter field equations. The present statistical approach provides a natural explanation for the long-standing question why potentials play an indispensable role

In section 5 it is shown that among all statistical theories only the time-dependent Schrödinger equation follows the logical requirement of maximal disorder or minimal Fisher information. Spin one-half is introduced, in section 6, as the property of a statistical ensemble to respond to an external gauge field in two different ways. A generalized calculation, reported in sections 6 and 7, leads to Pauli's (single-particle) equation. In section 8 an alternative derivation, following Arunsalam (1970), and Gould (1995) is reported, which is particularly convenient for the generalization to arbitrary *N*. The latter is performed in section 9, which completes

In section 10 the classical limit of QT is studied and it is stressed that the classical limit of QT is not classical mechanics but a classical statistical theory. In section 11 various questions related to the present approach, including the role of potentials and the interpretation of QT, are discussed. The final section 12 contains a short summary and mentions a possible direction

In I three different types of theories have been defined which differ from each other with regard to the role of probability. We give a short review of the defining properties and supply

The dogma underlying *theories of type 1* is determinism with regard to single events; probability does not play any role. If nature behaves according to this dogma, then measurements on identically prepared individual systems yield identical results. Classical mechanics is obviously such a deterministic type 1 theory. We shall use below (as a 'template' for the dynamics of our statistical theories) the following version of Newton's law, where the particle momentum *pk*(*t*) plays the role of a second dynamic variable besides the spatial

> *<sup>m</sup>* , <sup>d</sup> d*t*

*pk*(*t*) = *Fk*(*x*, *p*, *t*). (2)

contains also a review of the fundamental ideas.

in the field equations of physics.

for future research.

coordinate *xk*(*t*):

our statistical derivation of non-relativistic QT.

**2. Basic equations for a class of statistical theories**

some additional comments characterizing these theories.

d d*t* *xk*(*t*) = *pk*(*t*)

two respects: both rely heavily on the concept of individual particles and both are rather mysterious.

This situation confronts us with a fundamental alternative. Should we accept the mysteries and paradoxes as inherent attributes of reality or should we not, instead, critically reconsider our assumptions, in particular the 'individuality assumption'. As a matter of fact, the dynamical numerical output of quantum mechanics consists of *probabilities*. A probability is a "deterministic" prediction which can be verified in a statistical sense only, i.e. by performing experiments on a large number of identically prepared individual systems, see Belinfante (1978), Margenau (1963). Therefore, the very structure of QT tells us that it is a theory about statistical ensembles only, see Ballentine (1970). If dogmatic or philosophical reasons 'force' us to interpret QT as a theory about individual events, we have to create complicated intellectual constructs, which are not part of the physical formalism, and lead to unsolved problems and contradictions.

The present author believes, like several other physicists [see e.g. Ali (2009); Ballentine (1970); Belinfante (1975); Blokhintsev (1964); Einstein (1936); Kemble (1929); Krüger (2004); Margenau (1935); Newman (1980); Pippard (1986); Ross-Bonney (1975); Toyozawa (1992); Tschudi (1987); Young (1980)] that QT is a purely statistical theory whose predictions can only be used to describe the behavior of statistical ensembles and not of individual particles. This statistical interpretation (SI) of QT eliminates all mysteries and paradoxes - and this shows that the mysteries and paradoxes are not part of QT itself but rather the result of a particular (mis)interpretation of QT. In view of the similarity discussed above, we adopt the statistical point of view, not only for the interpretation of QT itself, but also in our search for *quantization conditions*. The general strategy is to find a new set of (as) simple (as possible) statistical assumptions which can be understood in physical terms and imply QT. Such an approach would also provide an explanation for the correspondence rules (1).

The present paper belongs to a series of works aimed at such an explanation. Quite generally, the present work continues a long tradition of attempts, see Frieden (1989; 2004); Hall & Reginatto (2002a); Lee & Zhu (1999); Motz (1962); Rosen (1964); Schiller (1962a); Schrödinger (1926), to characterize QT by mathematical relations which can be understood in physical terms2 (in contrast to the axiomatic approach). More specifically, it continues previous attempts to derive Schrödinger's equation with the help of statistical concepts, see Hall & Reginatto (2002b), Reginatto (1998a); Syska (2007), Klein (2009). These works, being quite different in detail, share the common feature that a statistical ensemble and not a particle Hamiltonian is used as a starting point for quantization. Finally, in a previous work, Klein (2011), of the present author an attempt has been undertaken to construct a complete statistical approach to QT with the help of a small number of very simple (statistical) assumptions. This work will be referred to as I. The present paper is a continuation and extension of I. The quantization method reported in I is based on the following general ideas:


matter whether quantum or classical.

<sup>2</sup> The listing given here is far from complete.

2 Will-be-set-by-IN-TECH

two respects: both rely heavily on the concept of individual particles and both are rather

This situation confronts us with a fundamental alternative. Should we accept the mysteries and paradoxes as inherent attributes of reality or should we not, instead, critically reconsider our assumptions, in particular the 'individuality assumption'. As a matter of fact, the dynamical numerical output of quantum mechanics consists of *probabilities*. A probability is a "deterministic" prediction which can be verified in a statistical sense only, i.e. by performing experiments on a large number of identically prepared individual systems, see Belinfante (1978), Margenau (1963). Therefore, the very structure of QT tells us that it is a theory about statistical ensembles only, see Ballentine (1970). If dogmatic or philosophical reasons 'force' us to interpret QT as a theory about individual events, we have to create complicated intellectual constructs, which are not part of the physical formalism, and lead to unsolved problems and

The present author believes, like several other physicists [see e.g. Ali (2009); Ballentine (1970); Belinfante (1975); Blokhintsev (1964); Einstein (1936); Kemble (1929); Krüger (2004); Margenau (1935); Newman (1980); Pippard (1986); Ross-Bonney (1975); Toyozawa (1992); Tschudi (1987); Young (1980)] that QT is a purely statistical theory whose predictions can only be used to describe the behavior of statistical ensembles and not of individual particles. This statistical interpretation (SI) of QT eliminates all mysteries and paradoxes - and this shows that the mysteries and paradoxes are not part of QT itself but rather the result of a particular (mis)interpretation of QT. In view of the similarity discussed above, we adopt the statistical point of view, not only for the interpretation of QT itself, but also in our search for *quantization conditions*. The general strategy is to find a new set of (as) simple (as possible) statistical assumptions which can be understood in physical terms and imply QT. Such an approach

The present paper belongs to a series of works aimed at such an explanation. Quite generally, the present work continues a long tradition of attempts, see Frieden (1989; 2004); Hall & Reginatto (2002a); Lee & Zhu (1999); Motz (1962); Rosen (1964); Schiller (1962a); Schrödinger (1926), to characterize QT by mathematical relations which can be understood in physical terms2 (in contrast to the axiomatic approach). More specifically, it continues previous attempts to derive Schrödinger's equation with the help of statistical concepts, see Hall & Reginatto (2002b), Reginatto (1998a); Syska (2007), Klein (2009). These works, being quite different in detail, share the common feature that a statistical ensemble and not a particle Hamiltonian is used as a starting point for quantization. Finally, in a previous work, Klein (2011), of the present author an attempt has been undertaken to construct a complete statistical approach to QT with the help of a small number of very simple (statistical) assumptions. This

work will be referred to as I. The present paper is a continuation and extension of I. The quantization method reported in I is based on the following general ideas: • QT should be a probabilistic theory in configuration space (not in phase space).

• QT should fullfil abstract versions of (i) a conservation law for probability (continuity equation) , and (ii) Ehrenfest's theorem. Such relations hold in all statistical theories no

• There are no laws for particle trajectories in QT anymore. This arbitrariness, which represents a crucial difference between QT and classical statistics, should be handled by a statistical principle analogous to the principle of maximal entropy in classical statistics.

would also provide an explanation for the correspondence rules (1).

matter whether quantum or classical.

<sup>2</sup> The listing given here is far from complete.

mysterious.

contradictions.

These general ideas lead to the mathematical assumptions which represent the basis for the treatment reported in I. This work was restricted to a one-dimensional configuration space (a single particle ensemble with a single spatial degree of freedom). The present work generalizes the treatment of I to a 3*N*−dimensional configuration space ( ensembles representing an arbitrary number *N* of particles allowed to move in three-dimensional space), gauge-coupling, and spin. In a first step the generalization to three spatial dimensions is performed; the properly generalized basic relations are reported in section 2. This section contains also a review of the fundamental ideas.

In section 3 we make use of a mathematical freedom, which is already contained in our basic assumptions, namely the multi-valuedness of the variable *S*. This leads to the appearance of potentials in statistical relations replacing the local forces of single-event (mechanical) theories. The mysterious non-local action of the vector potential (in effects of the Aharonov-Bohm type) is explained as a consequence of the statistical nature of QT. In section 4 we discuss a related question: Which constraints on admissible forces exist for the present class of statistical theories ? The answer is that only macroscopic (elementary) forces of the form of the Lorentz force can occur in nature, because only these survive the transition to QT . These forces are statistically represented by potentials, i.e. by the familiar gauge coupling terms in matter field equations. The present statistical approach provides a natural explanation for the long-standing question why potentials play an indispensable role in the field equations of physics.

In section 5 it is shown that among all statistical theories only the time-dependent Schrödinger equation follows the logical requirement of maximal disorder or minimal Fisher information. Spin one-half is introduced, in section 6, as the property of a statistical ensemble to respond to an external gauge field in two different ways. A generalized calculation, reported in sections 6 and 7, leads to Pauli's (single-particle) equation. In section 8 an alternative derivation, following Arunsalam (1970), and Gould (1995) is reported, which is particularly convenient for the generalization to arbitrary *N*. The latter is performed in section 9, which completes our statistical derivation of non-relativistic QT.

In section 10 the classical limit of QT is studied and it is stressed that the classical limit of QT is not classical mechanics but a classical statistical theory. In section 11 various questions related to the present approach, including the role of potentials and the interpretation of QT, are discussed. The final section 12 contains a short summary and mentions a possible direction for future research.

#### **2. Basic equations for a class of statistical theories**

In I three different types of theories have been defined which differ from each other with regard to the role of probability. We give a short review of the defining properties and supply some additional comments characterizing these theories.

The dogma underlying *theories of type 1* is determinism with regard to single events; probability does not play any role. If nature behaves according to this dogma, then measurements on identically prepared individual systems yield identical results. Classical mechanics is obviously such a deterministic type 1 theory. We shall use below (as a 'template' for the dynamics of our statistical theories) the following version of Newton's law, where the particle momentum *pk*(*t*) plays the role of a second dynamic variable besides the spatial coordinate *xk*(*t*):

$$\frac{\mathbf{d}}{\mathbf{d}t}\mathbf{x}\_k(t) = \frac{p\_k(t)}{m}, \qquad \frac{\mathbf{d}}{\mathbf{d}t}p\_k(t) = F\_k(\mathbf{x}, p, t). \tag{2}$$

Let us now generalize the basic equations of I (see section 3 of I) with respect to the number of spatial dimensions and with respect to gauge freedom. The continuity equation takes the

A Statistical Derivation of Non-Relativistic Quantum Theory 145

*ρ*(*x*, *t*) *m*

We use the summation convention, indices *i*, *k*, ... run from 1 to 3 and are omitted if the corresponding variable occurs in the argument of a function. The existence of a local conservation law for the probability density *ρ*(*x*, *t*) is a necessity for a probabilistic theory. The same is true for the fact that the probability current takes the form *jk*(*x*, *t*) = *ρ*(*x*, *t*)*p*˜*k*(*x*, *t*)/*m*, where *p*˜*k*(*x*, *t*) is the *k*−th component of the momentum probability density. The only non-trivial assumption contained in (3), is the fact that *p*˜*k*(*x*, *t*) takes the form of the gradient,

*<sup>p</sup>*˜*k*(*x*, *<sup>t</sup>*) = *<sup>∂</sup>S*˜(*x*, *<sup>t</sup>*)

of a function *S*˜(*x*, *t*). In order to gain a feeling for the physical meaning of (4) we could refer to the fact that a similar relation may be found in the Hamilton-Jacobi formulation of classical mechanics Synge (1960); alternatively we could also refer to the fact that this condition characterizes 'irrotational flow' in fluid mechanics. Relation (4) could also be justified by means of the principle of simplicity; a gradient is the simplest way to represent a vector field,

In contrast to I we allow now for *multi-valued* functions *S*˜(*x*, *t*). At first sight this seems strange since a multi-valued quantity cannot be an observable and should, consequently, not appear in equations bearing a physical meaning. However, only derivatives of *S*˜(*x*, *t*) occur in our basic equations. Thus, this freedom is possible without any additional postulate; we just have

> *∂t* , *∂S*˜ *∂xk*

(the quantity *p*˜ defined in (4) is not multi-valued; this notation is used to indicate that this quantity has been defined with the help of a multi-valued *S*˜). As discussed in more detail in section 3 this new 'degree of freedom' is intimately related to the existence of gauge fields. In contrast to *S*˜, the second dynamic variable *ρ* is a physical observable (in the statistical sense)

The necessary and sufficient condition for single-valuedness of a function *S*˜(*x*, *t*) (in a subspace G⊆R4) is that all second order derivatives of *<sup>S</sup>*˜(*x*, *<sup>t</sup>*) with respect to *xk* and *<sup>t</sup>* commute with each other (in G) [see e.g. Kaempfer (1965)]. As a consequence, the order of two derivatives of *S*˜ with respect to anyone of the variables *xk*, *t* must not be changed. We

> , *S*˜

The second of the assumptions listed above has been referred to in I as 'statistical conditions'. For the present three-dimensional theory these are obtained in the same way as in I,

[0,*k*] =

 *∂*2*S*˜ *∂t∂xk* <sup>−</sup> *<sup>∂</sup>*2*S*˜ *∂xk∂t* (6)

*∂xk*

*∂S*˜(*x*, *t*) *∂xk*

= 0. (3)

, (4)

single-valued. (5)

*∂ ∂xk*

*∂ρ*(*x*, *t*) *<sup>∂</sup><sup>t</sup>* <sup>+</sup>

because it can be derived from a single scalar function.

and is treated as a single-valued function.

introduce the (single-valued) quantities

 *∂*2*S*˜ *∂xj∂xk*

*S*˜ [*j*,*k*] =

*<sup>S</sup>*˜(*x*, *<sup>t</sup>*) multi-valued, *<sup>∂</sup>S*˜

<sup>−</sup> *<sup>∂</sup>*2*S*˜ *∂xk∂xj*

in order to describe the non-commuting derivatives in the following calculations.

form

to require that

In classical mechanics there is no restriction as regards the admissible forces. Thus, *Fk* is an arbitrary function of *xk*, *pk*, *t*; it is, in particular, not required that it be derivable from a potential. Note that Eqs. (2) do *not* hold in the present theory; these relations are just used to establish a correspondence between classical mechanics and associated statistical theories.

Experimental data from atomic systems, recorded since the beginning of the last century, indicate that nature does not behave according to this single-event deterministic dogma. A simple but somewhat unfamiliar idea is, to construct a theory which is deterministic only in a statistical sense. This means that measurements on identically prepared individual systems do not yield identical results (no determinism with regard to single events) but repeated measurements on ensembles [consisting each time of a large (infinite) number of measurements on individual systems] yield identical results. In this case we have 'determinism' with regard to ensembles (expectation values, or probabilities).

Note that such a theory is far from chaotic even if our macroscopic anticipation of (single-event) determinism is not satisfied. Note also that there is no reason to assume that such a statistical theory for microscopic events is incompatible with macroscopic determinism. It is a frequently observed (but not always completely understood) phenomenon in nature that systems with many (microscopic) degrees of freedom can be described by a much smaller number of variables. During this process of elimination of variables the details of the corresponding microscopic theory for the individual constituents are generally lost. In other words, there is no reason to assume that a fundamental statistical law for individual atoms and a deterministic law for a piece of matter consisting of, say, 10<sup>23</sup> atoms should not be compatible with each other. This way of characterizing the relation between two physical theories is completely different from the common reductionistic point of view. Convincing arguments in favor of the former may, however, be found in Anderson (1972), Laughlin (2005). As discussed in I two types (referred to as type 2 and type 3) of indeterministic theories may be identified. In *type 2 theories* laws for individual particles exist (roughly speaking the individuality of particles remains intact) but the initial values are unknown and are described by probabilities only. An example for such a (classical-statistical) type 2 theory is statistical thermodynamics. On the other hand, in *type 3 theories* the amount of uncertainty is still greater, insofar as no dynamic laws for individual particles exist any more. A possible candidate for this 'extreme' type of indeterministic theory is quantum mechanics.

The method used in I to construct statistical theories was based on the following three assumptions,


These (properly generalized) assumptions represent also the formal basis of the present work. The first and second of these cover type 2 as well as type 3 theories, while it will be shown that the third - the requirement of maximal disorder - does only hold for a single type 3 theory, namely quantum mechanics. In this sense quantum mechanics may be considered as the *most reasonable* theory among all statistical theories defined by the first two assumptions. There is obviously an analogy between quantum mechanics and the principle of minimal Fisher information on the one hand and classical statistical mechanics and the principle of maximal entropy on the other hand; both theories are realizations of the principle of maximal disorder. 4 Will-be-set-by-IN-TECH

In classical mechanics there is no restriction as regards the admissible forces. Thus, *Fk* is an arbitrary function of *xk*, *pk*, *t*; it is, in particular, not required that it be derivable from a potential. Note that Eqs. (2) do *not* hold in the present theory; these relations are just used to establish a correspondence between classical mechanics and associated statistical theories. Experimental data from atomic systems, recorded since the beginning of the last century, indicate that nature does not behave according to this single-event deterministic dogma. A simple but somewhat unfamiliar idea is, to construct a theory which is deterministic only in a statistical sense. This means that measurements on identically prepared individual systems do not yield identical results (no determinism with regard to single events) but repeated measurements on ensembles [consisting each time of a large (infinite) number of measurements on individual systems] yield identical results. In this case we have

Note that such a theory is far from chaotic even if our macroscopic anticipation of (single-event) determinism is not satisfied. Note also that there is no reason to assume that such a statistical theory for microscopic events is incompatible with macroscopic determinism. It is a frequently observed (but not always completely understood) phenomenon in nature that systems with many (microscopic) degrees of freedom can be described by a much smaller number of variables. During this process of elimination of variables the details of the corresponding microscopic theory for the individual constituents are generally lost. In other words, there is no reason to assume that a fundamental statistical law for individual atoms and a deterministic law for a piece of matter consisting of, say, 10<sup>23</sup> atoms should not be compatible with each other. This way of characterizing the relation between two physical theories is completely different from the common reductionistic point of view. Convincing arguments in favor of the former may, however, be found in Anderson (1972), Laughlin (2005). As discussed in I two types (referred to as type 2 and type 3) of indeterministic theories may be identified. In *type 2 theories* laws for individual particles exist (roughly speaking the individuality of particles remains intact) but the initial values are unknown and are described by probabilities only. An example for such a (classical-statistical) type 2 theory is statistical thermodynamics. On the other hand, in *type 3 theories* the amount of uncertainty is still greater, insofar as no dynamic laws for individual particles exist any more. A possible candidate for

The method used in I to construct statistical theories was based on the following three

• A local conservation law of probability with a particular form of the probability current. • Two differential equations which are similar in structure to the canonical equations (2) but

• A differential version (minimal Fisher information) of the statistical principle of maximal

These (properly generalized) assumptions represent also the formal basis of the present work. The first and second of these cover type 2 as well as type 3 theories, while it will be shown that the third - the requirement of maximal disorder - does only hold for a single type 3 theory, namely quantum mechanics. In this sense quantum mechanics may be considered as the *most reasonable* theory among all statistical theories defined by the first two assumptions. There is obviously an analogy between quantum mechanics and the principle of minimal Fisher information on the one hand and classical statistical mechanics and the principle of maximal entropy on the other hand; both theories are realizations of the principle of maximal disorder.

'determinism' with regard to ensembles (expectation values, or probabilities).

this 'extreme' type of indeterministic theory is quantum mechanics.

with observables replaced by expectation values.

assumptions,

disorder.

Let us now generalize the basic equations of I (see section 3 of I) with respect to the number of spatial dimensions and with respect to gauge freedom. The continuity equation takes the form

$$\frac{\partial \rho(\mathbf{x},t)}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_k} \frac{\rho(\mathbf{x},t)}{m} \frac{\partial \bar{S}(\mathbf{x},t)}{\partial \mathbf{x}\_k} = 0. \tag{3}$$

We use the summation convention, indices *i*, *k*, ... run from 1 to 3 and are omitted if the corresponding variable occurs in the argument of a function. The existence of a local conservation law for the probability density *ρ*(*x*, *t*) is a necessity for a probabilistic theory. The same is true for the fact that the probability current takes the form *jk*(*x*, *t*) = *ρ*(*x*, *t*)*p*˜*k*(*x*, *t*)/*m*, where *p*˜*k*(*x*, *t*) is the *k*−th component of the momentum probability density. The only non-trivial assumption contained in (3), is the fact that *p*˜*k*(*x*, *t*) takes the form of the gradient,

$$
\tilde{p}\_k(\mathbf{x}, t) = \frac{\partial \tilde{S}(\mathbf{x}, t)}{\partial \mathbf{x}\_k} \,\mathrm{}\tag{4}
$$

of a function *S*˜(*x*, *t*). In order to gain a feeling for the physical meaning of (4) we could refer to the fact that a similar relation may be found in the Hamilton-Jacobi formulation of classical mechanics Synge (1960); alternatively we could also refer to the fact that this condition characterizes 'irrotational flow' in fluid mechanics. Relation (4) could also be justified by means of the principle of simplicity; a gradient is the simplest way to represent a vector field, because it can be derived from a single scalar function.

In contrast to I we allow now for *multi-valued* functions *S*˜(*x*, *t*). At first sight this seems strange since a multi-valued quantity cannot be an observable and should, consequently, not appear in equations bearing a physical meaning. However, only derivatives of *S*˜(*x*, *t*) occur in our basic equations. Thus, this freedom is possible without any additional postulate; we just have to require that

$$
\tilde{S}(\mathbf{x},t) \quad \text{multi-valued}, \quad \frac{\partial \tilde{S}}{\partial t}, \frac{\partial \tilde{S}}{\partial \mathbf{x}\_k} \quad \text{single-valued.} \tag{5}
$$

(the quantity *p*˜ defined in (4) is not multi-valued; this notation is used to indicate that this quantity has been defined with the help of a multi-valued *S*˜). As discussed in more detail in section 3 this new 'degree of freedom' is intimately related to the existence of gauge fields. In contrast to *S*˜, the second dynamic variable *ρ* is a physical observable (in the statistical sense) and is treated as a single-valued function.

The necessary and sufficient condition for single-valuedness of a function *S*˜(*x*, *t*) (in a subspace G⊆R4) is that all second order derivatives of *<sup>S</sup>*˜(*x*, *<sup>t</sup>*) with respect to *xk* and *<sup>t</sup>* commute with each other (in G) [see e.g. Kaempfer (1965)]. As a consequence, the order of two derivatives of *S*˜ with respect to anyone of the variables *xk*, *t* must not be changed. We introduce the (single-valued) quantities

$$\tilde{S}\_{[j,k]} = \left[\frac{\partial^2 \tilde{S}}{\partial \mathbf{x}\_j \partial \mathbf{x}\_k} - \frac{\partial^2 \tilde{S}}{\partial \mathbf{x}\_k \partial \mathbf{x}\_j}\right], \quad \tilde{S}\_{[0,k]} = \left[\frac{\partial^2 \tilde{S}}{\partial t \partial \mathbf{x}\_k} - \frac{\partial^2 \tilde{S}}{\partial \mathbf{x}\_k \partial t}\right] \tag{6}$$

in order to describe the non-commuting derivatives in the following calculations.

The second of the assumptions listed above has been referred to in I as 'statistical conditions'. For the present three-dimensional theory these are obtained in the same way as in I,

Using the continuity equation (3) and the statistical conditions (7) and (8) the present generalization of the integral equation Eq. (24) of I may be derived. The steps leading to this result are very similar to the corresponding steps in I and may be skipped. The essential difference to the one-dimensional treatment is - apart from the number of space dimensions - the non-commutativity of the second order derivatives of *S*˜(*x*, *t*) leading to non-vanishing

A Statistical Derivation of Non-Relativistic Quantum Theory 147

1 <sup>2</sup>*<sup>m</sup>* ∑ *j*

In the course of the calculation leading to (14) it has been assumed that the macroscopic force *Fk*(*x*, *p*, *t*) entering the second statistical condition (8) may be written as a sum of two

Comparing Eq. (14) with the corresponding formula obtained in I [see Eq. (24) of I] we see that two new terms appear now, the expectation value of the *p*−dependent force on the r.h.s., and the second term on the l.h.s. of Eq. (14). The latter is a direct consequence of our assumption of a multi-valued variable *S*˜. In section 4 it will be shown that for vanishing multi-valuedness Eq. (14) has to agree with the three-dimensional generalization of the corresponding result [Eq. (24) of I] obtained in I. This means that the *p*−dependent term on the r.h.s. has to vanish too in this limit and indicates a relation between multi-valuedness of *<sup>S</sup>*˜ and *<sup>p</sup>*−dependence of

In this section we study the consequences of the multi-valuedness [London (1927), Weyl (1929), Dirac (1931)] of the quantity *S*˜(*x*, *t*) in the present theory. We assume that *S*˜(*x*, *t*) may be written as a sum of a single-valued part *S*(*x*, *t*) and a multi-valued part *N*˜ . Then, given

*∂xk*

where the four functions Φ and *Ak* are proportional to the derivatives of *N*˜ with respect to *t* and *xk* respectively (Note the change in sign of Φ and *Ak* in comparison to Klein (2009); this is due to the fact that the multi-valued phase is now denoted by *S*˜). The physical motivations for introducing the pre-factors *e* and *c* in Eq. (16) have been extensively discussed elsewhere, see Kaempfer (1965), Klein (2009), in an electrodynamical context. In agreement with Eq. (16),

<sup>=</sup> *<sup>∂</sup><sup>S</sup> ∂xk* − *e c*

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>e</sup>*Φ, *<sup>∂</sup>S*˜

*c* � *x*,*t x*0,*t*0;C � d*x*� *kAk*(*x*� , *t* � ) − *c*d*t* � *φ*(*x*� , *t* � ) �

*<sup>k</sup>* (*x*, *<sup>t</sup>*) + *<sup>F</sup>*(*e*)

*<sup>k</sup>* (*x*, *t*) takes the form of a negative gradient of a scalar function *V*(*x*, *t*) (mechanical

� *∂S*˜ *∂xj*

[0,*k*] � �<sup>2</sup>

= *F*(*e*)

+ *V* ⎤ ⎦

*<sup>k</sup>* (*x*, *p*, *t*)

*<sup>k</sup>* (*x*, *p*, *t*), (15)

*Ak*, (16)

, (17)

, (14)

[0,*k*] defined in Eq. (6). The result takes the form

⎡ ⎣ *∂S*˜ *<sup>∂</sup><sup>t</sup>* <sup>+</sup>

quantities *S*˜

contributions, *F*(*m*)

potential) and *F*(*e*)

the external force.

where *F*(*m*)

[*j*,*k*], *<sup>S</sup>*˜

− � ∞ −∞

+ � ∞ −∞

*<sup>k</sup>* (*x*, *<sup>t</sup>*) and *<sup>F</sup>*(*e*)

d3*<sup>x</sup> ∂ρ ∂xk*

d3*xρ* � 1 *m ∂S*˜ *∂xj S*˜ [*j*,*k*] + *<sup>S</sup>*˜

*<sup>k</sup>* (*x*, *p*, *t*),

*Fk*(*x*, *<sup>p</sup>*, *<sup>t</sup>*) = *<sup>F</sup>*(*m*)

**3. Gauge coupling as a consequence of a multi-valued phase**

that (5) holds, the derivatives of *S*˜(*x*, *t*) may be written in the form

*∂S*˜ *<sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>∂</sup><sup>S</sup>*

*S*˜ may be written in the form [Kaempfer (1965), Klein (2009)]

*<sup>S</sup>*˜(*x*, *<sup>t</sup>*; <sup>C</sup>) = *<sup>S</sup>*(*x*, *<sup>t</sup>*) <sup>−</sup> *<sup>e</sup>*

*<sup>k</sup>* (*x*, *p*, *t*) is the remaining *p*−dependent part.

by replacing the observables *xk*(*t*), *pk*(*t*) and the force field *Fk*(*x*(*t*), *p*(*t*), *t*) of the type 1 theory (2) by averages *xk*, *pk* and *Fk*. This leads to the relations

$$\frac{\mathbf{d}}{\mathbf{d}t}\overline{\mathbf{x}\_k} = \frac{\overline{p\_k}}{m} \tag{7}$$

$$\frac{\mathbf{d}}{\mathbf{d}t}\overline{p\_k} = \overline{F\_k(\mathbf{x}, p, t)},\tag{8}$$

where the averages are given by the following integrals over the random variables *xk*, *pk* (which should be clearly distinguished from the type I observables *xk*(*t*), *pk*(*t*) which will not be used any more):

$$\overline{\mathbf{x}\_k} = \int\_{-\infty}^{\infty} \mathbf{d}^3 \mathbf{x} \,\rho(\mathbf{x}, t) \,\mathbf{x}\_k \tag{9}$$

$$
\overline{p\_k} = \int\_{-\infty}^{\infty} \mathbf{d}^3 p \, w(p\_\prime t) \, p\_k \tag{10}
$$

$$\overline{F\_{\mathbf{k}}(\mathbf{x},\mathbf{p},t)} = \int\_{-\infty}^{\infty} \mathbf{d}^3 \mathbf{x} \, \mathbf{d}^3 p \, \mathcal{W}(\mathbf{x},\mathbf{p},t) F\_{\mathbf{k}}(\mathbf{x},\mathbf{p},t) . \tag{11}$$

The time-dependent probability densities *W*, *ρ*, *w* should be positive semidefinite and normalized to unity, i.e. they should fulfill the conditions

$$\int\_{-\infty}^{\infty} \mathbf{d}^3 \mathbf{x} \,\rho(\mathbf{x}, t) = \int\_{-\infty}^{\infty} \mathbf{d}^3 p \, w(p, t) = \int\_{-\infty}^{\infty} \mathbf{d}^3 \mathbf{x} \,\mathbf{d}^3 p \, W(\mathbf{x}, p, t) = 1 \tag{12}$$

The densities *ρ* and *w* may be derived from the fundamental probability density *W* by means of the relations

$$\rho(\mathbf{x},t) = \int\_{-\infty}^{\infty} \mathbf{d}^3 p \, W(\mathbf{x}, p, t); \qquad \qquad w(p, t) = \int\_{-\infty}^{\infty} \mathbf{d}^3 \mathbf{x} \, W(\mathbf{x}, p, t). \tag{13}$$

The present construction of the statistical conditions (7) and (8) from the type 1 theory (2) shows two differences as compared to the treatment in I. The first is that we allow now for a *p*−dependent external force. This leads to a more complicated probability density *W*(*x*, *p*, *t*) as compared to the two decoupled densities *ρ*(*x*, *t*) and *w*(*p*, *t*) of I. The second difference, which is in fact related to the first, is the use of a multi-valued *S*˜(*x*, *t*).

Note, that the *p*−dependent probability densities *w*(*p*, *t*) and *W*(*x*, *p*, *t*) have been introduced in the above relations in a purely formal way. We defined an expectation value *pk* [via Eq. (7)] and assumed [in Eq. (10) ] that a random variable *pk* and a corresponding probability density *w*(*p*, *t*) exist. But the validity of this assumption is not guaranteed . There is no compelling conceptual basis for the existence of these quantities in a pure configuration-space theory. If they exist, they must be defined with the help of additional considerations (see section 6 of I). The deeper reason for this problem is that the concept of measurement of momentum (which is proportional to the time derivative of position) is ill-defined in a theory whose observables are defined in terms of a large number of experiments at *one and the same* instant of time (measurement of a derivative requires measurements at different times). Fortunately, these considerations, which have been discussed in more detail in I, play not a prominent role [apart from the choice of *W*(*x*, *p*, *t*) discussed in section 4], for the derivation of Schrödinger's equation reported in the present paper3.

<sup>3</sup> These considerations seem relevant for attempts to define phase-space densities, e.g. of the Wigner type, in QT

6 Will-be-set-by-IN-TECH

by replacing the observables *xk*(*t*), *pk*(*t*) and the force field *Fk*(*x*(*t*), *p*(*t*), *t*) of the type 1

*xk* <sup>=</sup> *pk*

where the averages are given by the following integrals over the random variables *xk*, *pk* (which should be clearly distinguished from the type I observables *xk*(*t*), *pk*(*t*) which will not

The time-dependent probability densities *W*, *ρ*, *w* should be positive semidefinite and

The densities *ρ* and *w* may be derived from the fundamental probability density *W* by means

The present construction of the statistical conditions (7) and (8) from the type 1 theory (2) shows two differences as compared to the treatment in I. The first is that we allow now for a *p*−dependent external force. This leads to a more complicated probability density *W*(*x*, *p*, *t*) as compared to the two decoupled densities *ρ*(*x*, *t*) and *w*(*p*, *t*) of I. The second difference,

Note, that the *p*−dependent probability densities *w*(*p*, *t*) and *W*(*x*, *p*, *t*) have been introduced in the above relations in a purely formal way. We defined an expectation value *pk* [via Eq. (7)] and assumed [in Eq. (10) ] that a random variable *pk* and a corresponding probability density *w*(*p*, *t*) exist. But the validity of this assumption is not guaranteed . There is no compelling conceptual basis for the existence of these quantities in a pure configuration-space theory. If they exist, they must be defined with the help of additional considerations (see section 6 of I). The deeper reason for this problem is that the concept of measurement of momentum (which is proportional to the time derivative of position) is ill-defined in a theory whose observables are defined in terms of a large number of experiments at *one and the same* instant of time (measurement of a derivative requires measurements at different times). Fortunately, these considerations, which have been discussed in more detail in I, play not a prominent role [apart from the choice of *W*(*x*, *p*, *t*) discussed in section 4], for the derivation of Schrödinger's

<sup>3</sup> These considerations seem relevant for attempts to define phase-space densities, e.g. of the Wigner

 ∞ −∞

d3 *p w*(*p*, *t*) =

d<sup>3</sup> *p W*(*x*, *p*, *t*); *w*(*p*, *t*) =

*<sup>m</sup>* (7)

*pk* = *Fk*(*x*, *p*, *t*), (8)

<sup>d</sup>3*<sup>x</sup> <sup>ρ</sup>*(*x*, *<sup>t</sup>*) *xk* (9)

d3 *p w*(*p*, *<sup>t</sup>*) *pk* (10)

d3*x* d<sup>3</sup> *p W*(*x*, *p*, *t*) = 1 (12)

d3*x W*(*x*, *p*, *t*). (13)

d3*<sup>x</sup>* <sup>d</sup><sup>3</sup> *p W*(*x*, *<sup>p</sup>*, *<sup>t</sup>*)*Fk*(*x*, *<sup>p</sup>*, *<sup>t</sup>*). (11)

 ∞ −∞

d d*t*

 ∞ −∞

 ∞ −∞

d d*t*

*xk* =

*pk* =

 ∞ −∞

which is in fact related to the first, is the use of a multi-valued *S*˜(*x*, *t*).

 ∞ −∞

*Fk*(*x*, *p*, *t*) =

normalized to unity, i.e. they should fulfill the conditions

d3*x ρ*(*x*, *t*) =

 ∞ −∞

 ∞ −∞

*ρ*(*x*, *t*) =

equation reported in the present paper3.

theory (2) by averages *xk*, *pk* and *Fk*. This leads to the relations

be used any more):

of the relations

type, in QT

Using the continuity equation (3) and the statistical conditions (7) and (8) the present generalization of the integral equation Eq. (24) of I may be derived. The steps leading to this result are very similar to the corresponding steps in I and may be skipped. The essential difference to the one-dimensional treatment is - apart from the number of space dimensions - the non-commutativity of the second order derivatives of *S*˜(*x*, *t*) leading to non-vanishing quantities *S*˜ [*j*,*k*], *<sup>S</sup>*˜ [0,*k*] defined in Eq. (6). The result takes the form

$$\begin{aligned} & -\int\_{-\infty}^{\infty} \mathbf{d}^3 \mathbf{x} \frac{\partial \rho}{\partial \mathbf{x}\_k} \left[ \frac{\partial \tilde{\mathbf{S}}}{\partial t} + \frac{1}{2m} \sum\_{j} \left( \frac{\partial \tilde{\mathbf{S}}}{\partial \mathbf{x}\_j} \right)^2 + V \right] \\ & + \int\_{-\infty}^{\infty} \mathbf{d}^3 \mathbf{x} \rho \left[ \frac{1}{m} \frac{\partial \tilde{\mathbf{S}}}{\partial \mathbf{x}\_j} \tilde{\mathbf{S}}\_{[j,k]} + \tilde{\mathbf{S}}\_{[0,k]} \right] = \overline{F\_k^{(e)}(\mathbf{x}\_\prime, p\_\prime, t)} \end{aligned} \tag{14}$$

In the course of the calculation leading to (14) it has been assumed that the macroscopic force *Fk*(*x*, *p*, *t*) entering the second statistical condition (8) may be written as a sum of two contributions, *F*(*m*) *<sup>k</sup>* (*x*, *<sup>t</sup>*) and *<sup>F</sup>*(*e*) *<sup>k</sup>* (*x*, *p*, *t*),

$$F\_k(\mathbf{x}, \; p, t) = F\_k^{(m)}(\mathbf{x}, t) + F\_k^{(\varepsilon)}(\mathbf{x}, \; p, t), \tag{15}$$

where *F*(*m*) *<sup>k</sup>* (*x*, *t*) takes the form of a negative gradient of a scalar function *V*(*x*, *t*) (mechanical potential) and *F*(*e*) *<sup>k</sup>* (*x*, *p*, *t*) is the remaining *p*−dependent part.

Comparing Eq. (14) with the corresponding formula obtained in I [see Eq. (24) of I] we see that two new terms appear now, the expectation value of the *p*−dependent force on the r.h.s., and the second term on the l.h.s. of Eq. (14). The latter is a direct consequence of our assumption of a multi-valued variable *S*˜. In section 4 it will be shown that for vanishing multi-valuedness Eq. (14) has to agree with the three-dimensional generalization of the corresponding result [Eq. (24) of I] obtained in I. This means that the *p*−dependent term on the r.h.s. has to vanish too in this limit and indicates a relation between multi-valuedness of *<sup>S</sup>*˜ and *<sup>p</sup>*−dependence of the external force.

#### **3. Gauge coupling as a consequence of a multi-valued phase**

In this section we study the consequences of the multi-valuedness [London (1927), Weyl (1929), Dirac (1931)] of the quantity *S*˜(*x*, *t*) in the present theory. We assume that *S*˜(*x*, *t*) may be written as a sum of a single-valued part *S*(*x*, *t*) and a multi-valued part *N*˜ . Then, given that (5) holds, the derivatives of *S*˜(*x*, *t*) may be written in the form

$$\frac{\partial \tilde{S}}{\partial t} = \frac{\partial S}{\partial t} + \varepsilon \Phi, \quad \frac{\partial \tilde{S}}{\partial \mathbf{x}\_k} = \frac{\partial S}{\partial \mathbf{x}\_k} - \frac{e}{c} A\_{k\prime} \tag{16}$$

where the four functions Φ and *Ak* are proportional to the derivatives of *N*˜ with respect to *t* and *xk* respectively (Note the change in sign of Φ and *Ak* in comparison to Klein (2009); this is due to the fact that the multi-valued phase is now denoted by *S*˜). The physical motivations for introducing the pre-factors *e* and *c* in Eq. (16) have been extensively discussed elsewhere, see Kaempfer (1965), Klein (2009), in an electrodynamical context. In agreement with Eq. (16), *S*˜ may be written in the form [Kaempfer (1965), Klein (2009)]

$$\tilde{S}(\mathbf{x},t;\mathcal{C}) = S(\mathbf{x},t) - \frac{e}{c} \int\_{\mathbf{x}\_0,t\_0;\mathcal{C}}^{\mathbf{x},t} \left[ \mathbf{d} \mathbf{x}\_k' A\_k(\mathbf{x}',t') - c \mathbf{d}t' \phi(\mathbf{x}',t') \right],\tag{17}$$

identically prepared individual particles). The statistical effect of a force field on an ensemble may obviously *differ* from the local effect of the same force field on individual particles; thus

A Statistical Derivation of Non-Relativistic Quantum Theory 149

common problem with the interpretation of the 'potentials' Φ and *Ak* is their non-uniqueness. It is hard to understand that a quantity ruling the behavior of individual particles should not be uniquely defined. In contrast, this non-uniqueness is much easier to accept if Φ and *Ak* rule the behavior of ensembles instead of individual particles. We have no problem to accept the fact that a function that represents a global (integral) effect may have many different local

It seems that this interpretation of the potentials Φ and *Ak* is highly relevant for the interpretation of the effect found by Aharonov & Bohm (1959). If QT is interpreted as a theory about individual particles, the Aharonov-Bohm effects imply that a charged particle may be influenced in a nonlocal way by electromagnetic fields in inaccessible regions. This paradoxical prediction, which is however in strict agreement with QT, led even to a discussion about the reality of these effects, see Bocchieri & Loinger (1978), Roy (1980), Klein (1981), Peshkin & Tonomura (1989). A statistical interpretation of the potentials has apparently never been suggested, neither in the vast literature about the Aharonov-Bohm effect nor in papers promoting the statistical interpretation of QT; most physicists discuss this nonlocal 'paradox' from the point of view of 'the wave function of a single electron'. Further comments

macroscopic forces whose functional form is still unknown. Both the potentials and these local forces represent an external influence, and it is reasonable to assume that the (nonlocal) potentials are the statistical representatives of the local forces on the r.h.s. of Eq. (14). The latter have to be determined by the potentials but must be uniquely defined at each space-time

fulfill these requirements. As a consequence of the defining relations (23) they obey

In a next step we rewrite the second term on the l.h.s. of Eq. (14). The commutator terms (6)

As a consequence, they may be expressed in terms of the local fields (23), which have been introduced above for reasons of gauge-invariance. Using (24), (23) and the relation (19) for the

> 1 <sup>2</sup>*<sup>m</sup>* ∑ *j*

*�kijv*˜*iBj* + *eEk*

with a velocity field defined by *v*˜*<sup>i</sup>* = *p*˜*i*/*m*. Thus, the new terms on the l.h.s. of (25) - stemming from the multi-valuedness of *<sup>S</sup>*˜ - take the form of an expectation value (with <sup>R</sup><sup>3</sup> as sample

*∂*Φ *∂xk*

⎡ ⎣ *∂S*˜ *<sup>∂</sup><sup>t</sup>* <sup>+</sup>

� , *S*˜ *E* and

*<sup>k</sup>* (*x*, *p*, *t*) on the right hand side of (14) is to be calculated using local,

*∂Aj ∂xi*

�*∂Aj ∂xk*

�<sup>2</sup>

+ *V* ⎤ ⎦

*<sup>k</sup>* (*x*, *p*, *t*)

<sup>−</sup> *<sup>∂</sup>Ak ∂xj*

�

, *Bk* = *�kij*

[*j*,*k*] <sup>=</sup> *<sup>e</sup> c*

> � *∂S*˜ *∂xj*

� = *F*(*e*) *B* is no surprise. The second

, (23)

. (24)

, (25)

the very existence of fields Φ and *Ak* different from

on this point may be found in section 11.

*Ek* <sup>=</sup> <sup>−</sup><sup>1</sup> *c ∂Ak <sup>∂</sup><sup>t</sup>* <sup>−</sup> *<sup>∂</sup>*<sup>Φ</sup> *∂xk*

�1 *c ∂Ak <sup>∂</sup><sup>t</sup>* <sup>+</sup>

> d3*<sup>x</sup> ∂ρ ∂xk*

d3*xρ* � *e c*

automatically the homogeneous Maxwell equations.

[0,*k*] = −*e*

− � ∞ −∞

+ � ∞ −∞

The expectation value *F*(*e*)

take the form

point. The gauge-invariant fields

*S*˜

momentum field, Eq. (14) takes the form

realizations.

as a path-integral performed along an arbitrary path C in four-dimensional space; the multi-valuedness of *<sup>S</sup>*˜ simply means that it depends not only on *<sup>x</sup>*, *<sup>t</sup>* but also on the path <sup>C</sup> connecting the points *x*0, *t*<sup>0</sup> and *x*, *t*.

The quantity *S*˜ cannot be a physical observable because of its multi-valuedness. The fundamental physical quantities to be determined by our (future) theory are the four derivatives of *<sup>S</sup>*˜ which will be rewritten here as two observable fields <sup>−</sup>*E*˜(*x*, *<sup>t</sup>*), *<sup>p</sup>*˜*k*(*x*, *<sup>t</sup>*),

$$-\tilde{E}(\mathbf{x},t) = \frac{\partial S(\mathbf{x},t)}{\partial t} + e\Phi(\mathbf{x},t),\tag{18}$$

$$
\mathfrak{p}\_k(\mathbf{x}, t) = \frac{\partial S(\mathbf{x}, t)}{\partial \mathbf{x}\_k} - \frac{e}{c} A\_k(\mathbf{x}, t), \tag{19}
$$

with dimensions of energy and momentum respectively.

We encounter a somewhat unusual situation in Eqs. (18), (19): On the one hand the left hand sides are observables of our theory, on the other hand we cannot solve our (future) differential equations for these quantities because of the peculiar multi-valued structure of *S*˜. We have to use instead the decompositions as given by the right hand sides of (18) and (19). The latter eight terms (the four derivatives of *S* and the four scalar functions Φ and *Ak*) are single-valued (in the mathematical sense) but need not be unique because only the left hand sides are uniquely determined by the physical situation. We tentatively assume that the fields Φ and *Ak* are 'given' quantities in the sense that they represent an external influence (of 'external forces') on the considered statistical situation. An actual calculation has to be performed in such a way that fixed fields Φ and *Ak* are chosen and then the differential equations are solved for *S* (and *ρ*). However, as mentioned already, what is actually uniquely determined by the physical situation is the *sum* of the two terms on the right hand sides of (18) and (19). Consequently, a different set of fixed fields Φ� and *A*� *<sup>k</sup>* may lead to a physically equivalent, but mathematically different, solution *S* � in such a way that the sum of the new terms [on the right hand sides of (18) and (19)] is the *same* as the sum of the old terms. We assume here, that the formalism restores the values of the physically relevant terms. This implies that the relation between the old and new terms is given by

$$\boldsymbol{S}^{'}(\mathbf{x},t) = \mathbf{S}(\mathbf{x},t) + \boldsymbol{\varphi}(\mathbf{x},t) \tag{20}$$

$$\boldsymbol{\Phi}^{\prime}(\mathbf{x},t) = \boldsymbol{\Phi}(\mathbf{x},t) - \frac{1}{e} \frac{\partial \boldsymbol{\phi}(\mathbf{x},t)}{\partial t} \tag{21}$$

$$A\_k^{'}(\mathbf{x},t) = A\_k(\mathbf{x},t) + \frac{c}{e} \frac{\partial \varphi(\mathbf{x},t)}{\partial \mathbf{x}\_k},\tag{22}$$

where *ϕ*(*x*, *t*) is an arbitrary, single-valued function of *xk*, *t*. Consequently, all 'theories' (differential equations for *S* and *ρ* defined by the assumptions listed in section 2) should be form-invariant under the transformations (20)-(22). These invariance transformations, predicted here from general considerations, are (using an arbitrary function *χ* = *cϕ*/*e* instead of *ϕ*) denoted as 'gauge transformations of the second kind'.

The fields Φ(*x*, *t*) and *Ak*(*x*, *t*) describe an external influence but their numerical value is undefined; their value at *x*, *t* may be changed according to (21) and (22) without changing their physical effect. Thus, these fields *cannot play a local role* in space and time like forces and fields in classical mechanics and electrodynamics. What, then, is the physical meaning of these fields ? An explanation which seems obvious in the present context is the following: They describe the *statistical effect* of an external influence on the considered system (ensemble of 8 Will-be-set-by-IN-TECH

as a path-integral performed along an arbitrary path C in four-dimensional space; the multi-valuedness of *<sup>S</sup>*˜ simply means that it depends not only on *<sup>x</sup>*, *<sup>t</sup>* but also on the path <sup>C</sup>

The quantity *S*˜ cannot be a physical observable because of its multi-valuedness. The fundamental physical quantities to be determined by our (future) theory are the four derivatives of *<sup>S</sup>*˜ which will be rewritten here as two observable fields <sup>−</sup>*E*˜(*x*, *<sup>t</sup>*), *<sup>p</sup>*˜*k*(*x*, *<sup>t</sup>*),

*∂xk*

We encounter a somewhat unusual situation in Eqs. (18), (19): On the one hand the left hand sides are observables of our theory, on the other hand we cannot solve our (future) differential equations for these quantities because of the peculiar multi-valued structure of *S*˜. We have to use instead the decompositions as given by the right hand sides of (18) and (19). The latter eight terms (the four derivatives of *S* and the four scalar functions Φ and *Ak*) are single-valued (in the mathematical sense) but need not be unique because only the left hand sides are uniquely determined by the physical situation. We tentatively assume that the fields Φ and *Ak* are 'given' quantities in the sense that they represent an external influence (of 'external forces') on the considered statistical situation. An actual calculation has to be performed in such a way that fixed fields Φ and *Ak* are chosen and then the differential equations are solved for *S* (and *ρ*). However, as mentioned already, what is actually uniquely determined by the physical situation is the *sum* of the two terms on the right hand sides of (18) and (19). Consequently, a

of (18) and (19)] is the *same* as the sum of the old terms. We assume here, that the formalism restores the values of the physically relevant terms. This implies that the relation between the

(*x*, *<sup>t</sup>*) = <sup>Φ</sup>(*x*, *<sup>t</sup>*) <sup>−</sup> <sup>1</sup>

*<sup>k</sup>*(*x*, *<sup>t</sup>*) = *Ak*(*x*, *<sup>t</sup>*) + *<sup>c</sup>*

where *ϕ*(*x*, *t*) is an arbitrary, single-valued function of *xk*, *t*. Consequently, all 'theories' (differential equations for *S* and *ρ* defined by the assumptions listed in section 2) should be form-invariant under the transformations (20)-(22). These invariance transformations, predicted here from general considerations, are (using an arbitrary function *χ* = *cϕ*/*e* instead

The fields Φ(*x*, *t*) and *Ak*(*x*, *t*) describe an external influence but their numerical value is undefined; their value at *x*, *t* may be changed according to (21) and (22) without changing their physical effect. Thus, these fields *cannot play a local role* in space and time like forces and fields in classical mechanics and electrodynamics. What, then, is the physical meaning of these fields ? An explanation which seems obvious in the present context is the following: They describe the *statistical effect* of an external influence on the considered system (ensemble of

− *e c*

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>e</sup>*Φ(*x*, *<sup>t</sup>*), (18)

*<sup>k</sup>* may lead to a physically equivalent, but mathematically

(*x*, *t*) = *S*(*x*, *t*) + *ϕ*(*x*, *t*) (20)

*<sup>∂</sup><sup>t</sup>* (21)

, (22)

in such a way that the sum of the new terms [on the right hand sides

*∂ϕ*(*x*, *t*)

*∂ϕ*(*x*, *t*) *∂xk*

*e*

*e*

*Ak*(*x*, *t*), (19)

<sup>−</sup>*E*˜(*x*, *<sup>t</sup>*) = *<sup>∂</sup>S*(*x*, *<sup>t</sup>*)

*<sup>p</sup>*˜*k*(*x*, *<sup>t</sup>*) = *<sup>∂</sup>S*(*x*, *<sup>t</sup>*)

with dimensions of energy and momentum respectively.

and *A*�

*S* �

Φ�

*A*�

of *ϕ*) denoted as 'gauge transformations of the second kind'.

connecting the points *x*0, *t*<sup>0</sup> and *x*, *t*.

different set of fixed fields Φ�

old and new terms is given by

�

different, solution *S*

identically prepared individual particles). The statistical effect of a force field on an ensemble may obviously *differ* from the local effect of the same force field on individual particles; thus the very existence of fields Φ and *Ak* different from *E* and *B* is no surprise. The second common problem with the interpretation of the 'potentials' Φ and *Ak* is their non-uniqueness. It is hard to understand that a quantity ruling the behavior of individual particles should not be uniquely defined. In contrast, this non-uniqueness is much easier to accept if Φ and *Ak* rule the behavior of ensembles instead of individual particles. We have no problem to accept the fact that a function that represents a global (integral) effect may have many different local realizations.

It seems that this interpretation of the potentials Φ and *Ak* is highly relevant for the interpretation of the effect found by Aharonov & Bohm (1959). If QT is interpreted as a theory about individual particles, the Aharonov-Bohm effects imply that a charged particle may be influenced in a nonlocal way by electromagnetic fields in inaccessible regions. This paradoxical prediction, which is however in strict agreement with QT, led even to a discussion about the reality of these effects, see Bocchieri & Loinger (1978), Roy (1980), Klein (1981), Peshkin & Tonomura (1989). A statistical interpretation of the potentials has apparently never been suggested, neither in the vast literature about the Aharonov-Bohm effect nor in papers promoting the statistical interpretation of QT; most physicists discuss this nonlocal 'paradox' from the point of view of 'the wave function of a single electron'. Further comments on this point may be found in section 11.

The expectation value *F*(*e*) *<sup>k</sup>* (*x*, *p*, *t*) on the right hand side of (14) is to be calculated using local, macroscopic forces whose functional form is still unknown. Both the potentials and these local forces represent an external influence, and it is reasonable to assume that the (nonlocal) potentials are the statistical representatives of the local forces on the r.h.s. of Eq. (14). The latter have to be determined by the potentials but must be uniquely defined at each space-time point. The gauge-invariant fields

$$E\_k = -\frac{1}{c} \frac{\partial A\_k}{\partial t} - \frac{\partial \Phi}{\partial \mathbf{x}\_k} \prime \qquad B\_k = \varepsilon\_{kij} \frac{\partial A\_j}{\partial \mathbf{x}\_i} \prime \tag{23}$$

fulfill these requirements. As a consequence of the defining relations (23) they obey automatically the homogeneous Maxwell equations.

In a next step we rewrite the second term on the l.h.s. of Eq. (14). The commutator terms (6) take the form

$$\tilde{S}\_{[0,k]} = -e \left( \frac{1}{c} \frac{\partial A\_k}{\partial t} + \frac{\partial \Phi}{\partial \mathbf{x}\_k} \right), \quad \tilde{S}\_{[j,k]} = \frac{e}{c} \left( \frac{\partial A\_j}{\partial \mathbf{x}\_k} - \frac{\partial A\_k}{\partial \mathbf{x}\_j} \right). \tag{24}$$

As a consequence, they may be expressed in terms of the local fields (23), which have been introduced above for reasons of gauge-invariance. Using (24), (23) and the relation (19) for the momentum field, Eq. (14) takes the form

$$\begin{aligned} & -\int\_{-\infty}^{\infty} \mathbf{d}^3 \mathbf{x} \frac{\partial \rho}{\partial \mathbf{x}\_k} \left[ \frac{\partial \tilde{S}}{\partial t} + \frac{1}{2m} \sum\_{j} \left( \frac{\partial \tilde{S}}{\partial \mathbf{x}\_j} \right)^2 + V \right] \\ & + \int\_{-\infty}^{\infty} \mathbf{d}^3 \mathbf{x} \rho \left[ \frac{\varepsilon}{c} \varepsilon\_{kij} \tilde{v}\_i B\_j + e E\_k \right] = \overline{F\_k^{(\varepsilon)}} (\mathbf{x}, \ p, t) \end{aligned} \tag{25}$$

with a velocity field defined by *v*˜*<sup>i</sup>* = *p*˜*i*/*m*. Thus, the new terms on the l.h.s. of (25) - stemming from the multi-valuedness of *<sup>S</sup>*˜ - take the form of an expectation value (with <sup>R</sup><sup>3</sup> as sample

For *Fk* → 0 Eq. (30) leads to the relation

partial integration, we see that a relation

the real existing (gauge) interactions of nature.

*F*(*e*)

*Hk*(*x*,

<sup>d</sup>3*<sup>x</sup> ∂ρ ∂xk*

*hk*(*x*,

*∂S*˜ *∂x*

*∂S*˜ *∂x*

⎡ ⎣ *∂S*˜ *<sup>∂</sup><sup>t</sup>* <sup>+</sup>

the form

by writing

− � ∞ −∞

*S*˜. This leads to the condition

�

*Fk* <sup>=</sup> <sup>−</sup> *<sup>∂</sup><sup>V</sup> ∂xk*

(see the discussion in section 4 of I) and that the variable *p* in *F*(*e*)

*<sup>k</sup>* (*x*, *<sup>p</sup>*, *<sup>t</sup>*) = � <sup>∞</sup>

, *<sup>t</sup>*) = *<sup>e</sup> c �kij* 1 *m ∂S*˜ *∂xi*

d3*<sup>x</sup> ∂ρ ∂xk*

+ *sk*,

which remains true for finite forces because *L*<sup>0</sup> does not depend on *Fk*. Finally, performing a

A Statistical Derivation of Non-Relativistic Quantum Theory 151

exists between *Fk* and *V*, with a vanishing expectation value of the (statistically irrelevant) functions *sk*. This example shows that the restriction to gradient fields, made above and in I, is actually not necessary. We may *admit* force fields which are arbitrary functions of *x* and *t*; the statistical conditions (which play now the role of a 'statistical constraint') eliminate automatically all forces that cannot be written after statistical averaging as gradient fields. This is very interesting and indicates the possibility that the present statistical assumptions leading to Schrödinger's equation may also be responsible, at least partly, for the structure of

Does this statistical constraint also work in the present *p*−dependent case ? We assume that the force in (25) is a standard random variable with the configuration space as sample space

be replaced by the field *p*˜(*x*, *t*) [see (19)]. Then, the expectation value on the r.h.s. of (25) takes

The second term on the l.h.s. of (25) has the *same* form. Therefore, the latter may be eliminated

�<sup>2</sup>

On a first look this condition for the allowed forces looks similar to the *p*−independent case [see (27)]. But the dependence of *hk* on *x*, *t* cannot be considered as 'given' (externally controlled), as in the *p*−independent case, because it contains now the unknown *x*, *t*-dependence of the derivatives of *S*˜. We may nevertheless try to incorporate the r.h.s by adding a term *T*˜ to the bracket which depends on the derivatives of the multivalued quantity

> *<sup>∂</sup><sup>x</sup>* , *t*) *∂xk*

But this relation cannot be fulfilled for nontrivial *hk*, *T*˜ because the derivatives of *S*˜ cannot be subject to further constraints beyond those given by the differential equation; on top of that the derivatives with regard to *x* on the r.h.s. create higher order derivatives of *S*˜ which are

+ *sk*,

� ∞ −∞

+ *V* ⎤ ⎦ =

d3*xρ*(*x*, *<sup>t</sup>*)*Hk*(*x*,

*Bj* + *eEk* + *hk*(*x*,

*∂S*˜(*x*, *t*)

� ∞ −∞ *∂S*˜ *∂x*

d3*xρhk*(*x*,

−∞

with *hk*(*x*, *p*, *t*) as our new unknown functions. They obey the simpler relations

� *∂S*˜ *∂xj*

1 <sup>2</sup>*<sup>m</sup>* ∑ *j*

, *<sup>t</sup>*) = <sup>−</sup>*∂T*˜(*x*, *<sup>∂</sup>S*˜

� ∞ −∞

*L*<sup>0</sup> = 0, (31)

<sup>d</sup>3*<sup>x</sup> <sup>ρ</sup>sk* = 0, (32)

*<sup>k</sup>* (*x*, *p*, *t*) may consequently

*<sup>∂</sup><sup>x</sup>* , *<sup>t</sup>*). (33)

*∂S*˜ *∂x*

d3*<sup>x</sup> <sup>ρ</sup>sk* = 0. (36)

, *t*), (34)

, *t*). (35)

space) of the Lorentz force field

$$
\vec{F}\_L(\mathbf{x}, t) = e\vec{E}(\mathbf{x}, t) + \frac{e}{c}\vec{v}(\mathbf{x}, t) \times \vec{B}(\mathbf{x}, t), \tag{26}
$$

if the particle velocity is identified with the velocity field *v*˜(*x*, *t*).

The above steps imply a relation between potentials and local fields. From the present statistical (nonlocal) point of view the potentials are more fundamental than the local fields. In contrast, considered from the point of view of macroscopic physics, the local fields are the physical quantities of primary importance and the potentials may (or may not) be introduced for mathematical convenience.

#### **4. A constraint for forces in statistical theories**

Let us discuss now the nature of the macroscopic forces *F*(*e*) *<sup>k</sup>* (*x*, *p*, *t*) entering the expectation value on the r.h.s. of Eq. (25). In our type I parent theory, classical mechanics, there are no constraints for the possible functional form of *F*(*e*) *<sup>k</sup>* (*x*, *p*, *t*). However, this need not be true in the present statistical framework. As a matter of fact, the way the mechanical potential *V*(*x*, *t*) entered the differential equation for *S* (in the previous work I) indicates already that such constraints do actually exist. Let us recall that in I we tacitly restricted the class of forces to those derivable from a potential *V*(*x*, *t*). If we eliminate this restriction and admit arbitrary forces, with components *Fk*(*x*, *t*), we obtain instead of the above relation (25) the simpler relation [Eq. (24) of I, generalized to three dimensions and arbitrary forces of the form *Fk*(*x*, *t*)]

$$-\int\_{-\infty}^{\infty} \mathbf{d}^3 \mathbf{x} \, \frac{\partial \rho}{\partial \mathbf{x}\_k} \left[\frac{1}{2m} \sum\_{\dot{j}} \left(\frac{\partial \mathbf{S}}{\partial \mathbf{x}\_{\dot{j}}}\right)^2 + \frac{\partial \mathbf{S}}{\partial t}\right] = \int\_{-\infty}^{\infty} \mathbf{d} \mathbf{x} \rho \mathbf{F}\_k(\mathbf{x}, t). \tag{27}$$

This is a rather complicated integro-differential equation for our variables *ρ*(*x*, *t*) and *S*(*x*, *t*). We assume now, using mathematical simplicity as a guideline, that Eq. (27) can be written in the common form of a local differential equation. This assumption is of course not evident; in principle the laws of physics could be integro-differential equations or delay differential equations or take an even more complicated mathematical form. Nevertheless, this assumption seems rather weak considering the fact that all fundamental laws of physics take this 'simple' form. Thus, we postulate that Eq. (27) is equivalent to a differential equation

$$\frac{1}{2m}\sum\_{\dot{j}} \left(\frac{\partial \mathcal{S}}{\partial \mathbf{x}\_{\dot{j}}}\right)^2 + \frac{\partial \mathcal{S}}{\partial t} + T = 0,\tag{28}$$

where the unknown term *T* describes the influence of the force *Fk* but may also contain other contributions. Let us write

$$T = -L\_0 + V\_\prime \tag{29}$$

where *L*<sup>0</sup> does not depend on *Fk*, while *V* depends on it and vanishes for *Fk* → 0. Inserting (28) and (29) in (27) yields

$$
\int \mathrm{d}^3 \mathbf{x} \, \frac{\partial \rho}{\partial \mathbf{x}\_k} \, (-L\_0 + V) = \int \mathrm{d}^3 \mathbf{x} \, \rho F\_k(\mathbf{x}, t). \tag{30}
$$

For *Fk* → 0 Eq. (30) leads to the relation

10 Will-be-set-by-IN-TECH

*c*

The above steps imply a relation between potentials and local fields. From the present statistical (nonlocal) point of view the potentials are more fundamental than the local fields. In contrast, considered from the point of view of macroscopic physics, the local fields are the physical quantities of primary importance and the potentials may (or may not) be introduced

value on the r.h.s. of Eq. (25). In our type I parent theory, classical mechanics, there are no

in the present statistical framework. As a matter of fact, the way the mechanical potential *V*(*x*, *t*) entered the differential equation for *S* (in the previous work I) indicates already that such constraints do actually exist. Let us recall that in I we tacitly restricted the class of forces to those derivable from a potential *V*(*x*, *t*). If we eliminate this restriction and admit arbitrary forces, with components *Fk*(*x*, *t*), we obtain instead of the above relation (25) the simpler relation [Eq. (24) of I, generalized to three dimensions and arbitrary forces of the form

> �<sup>2</sup> + *∂S ∂t*

This is a rather complicated integro-differential equation for our variables *ρ*(*x*, *t*) and *S*(*x*, *t*). We assume now, using mathematical simplicity as a guideline, that Eq. (27) can be written in the common form of a local differential equation. This assumption is of course not evident; in principle the laws of physics could be integro-differential equations or delay differential equations or take an even more complicated mathematical form. Nevertheless, this assumption seems rather weak considering the fact that all fundamental laws of physics take this 'simple' form. Thus, we postulate that Eq. (27) is equivalent to a differential equation

> �<sup>2</sup> + *∂S*

where the unknown term *T* describes the influence of the force *Fk* but may also contain other

where *L*<sup>0</sup> does not depend on *Fk*, while *V* depends on it and vanishes for *Fk* → 0. Inserting (28)

�

(−*L*<sup>0</sup> + *V*) =

⎤ ⎦ =

� ∞ −∞

� *∂S ∂xj* *<sup>v</sup>*˜(*x*, *<sup>t</sup>*) <sup>×</sup>

*B*(*x*, *t*), (26)

*<sup>k</sup>* (*x*, *p*, *t*) entering the expectation

d*xρFk*(*x*, *t*). (27)

*<sup>k</sup>* (*x*, *p*, *t*). However, this need not be true

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>T</sup>* <sup>=</sup> 0, (28)

<sup>d</sup>3*<sup>x</sup> <sup>ρ</sup>Fk*(*x*, *<sup>t</sup>*). (30)

*T* = −*L*<sup>0</sup> + *V*, (29)

*<sup>E</sup>*(*x*, *<sup>t</sup>*) + *<sup>e</sup>*

space) of the Lorentz force field

for mathematical convenience.

− � ∞ −∞

contributions. Let us write

and (29) in (27) yields

*Fk*(*x*, *t*)]

**4. A constraint for forces in statistical theories**

constraints for the possible functional form of *F*(*e*)

<sup>d</sup>3*<sup>x</sup> ∂ρ ∂xk* ⎡ ⎣ 1 <sup>2</sup>*<sup>m</sup>* ∑ *j*

1 <sup>2</sup>*<sup>m</sup>* ∑ *j*

�

d3*<sup>x</sup> ∂ρ ∂xk* � *∂S ∂xj*

Let us discuss now the nature of the macroscopic forces *F*(*e*)

*FL*(*x*, *t*) = *e*

if the particle velocity is identified with the velocity field *v*˜(*x*, *t*).

$$\int \mathrm{d}^3 \mathbf{x} \, \frac{\partial \rho}{\partial \mathbf{x}\_k} \, L\_0 = \mathbf{0},\tag{31}$$

which remains true for finite forces because *L*<sup>0</sup> does not depend on *Fk*. Finally, performing a partial integration, we see that a relation

$$F\_k = -\frac{\partial V}{\partial \mathbf{x}\_k} + s\_{k\prime} \quad \int\_{-\infty}^{\infty} \mathbf{d}^3 \mathbf{x} \,\rho s\_k = 0,\tag{32}$$

exists between *Fk* and *V*, with a vanishing expectation value of the (statistically irrelevant) functions *sk*. This example shows that the restriction to gradient fields, made above and in I, is actually not necessary. We may *admit* force fields which are arbitrary functions of *x* and *t*; the statistical conditions (which play now the role of a 'statistical constraint') eliminate automatically all forces that cannot be written after statistical averaging as gradient fields.

This is very interesting and indicates the possibility that the present statistical assumptions leading to Schrödinger's equation may also be responsible, at least partly, for the structure of the real existing (gauge) interactions of nature.

Does this statistical constraint also work in the present *p*−dependent case ? We assume that the force in (25) is a standard random variable with the configuration space as sample space (see the discussion in section 4 of I) and that the variable *p* in *F*(*e*) *<sup>k</sup>* (*x*, *p*, *t*) may consequently be replaced by the field *p*˜(*x*, *t*) [see (19)]. Then, the expectation value on the r.h.s. of (25) takes the form

$$\overline{F\_k^{(\varepsilon)}(\mathbf{x},\boldsymbol{p},\mathbf{t})} = \int\_{-\infty}^{\infty} \mathbf{d}^3 \mathbf{x} \rho(\mathbf{x},\boldsymbol{t}) H\_k(\mathbf{x}, \frac{\partial \tilde{\mathcal{S}}(\mathbf{x}, \boldsymbol{t})}{\partial \mathbf{x}}, \mathbf{t}). \tag{33}$$

The second term on the l.h.s. of (25) has the *same* form. Therefore, the latter may be eliminated by writing

$$H\_k(\mathbf{x}, \frac{\partial \tilde{S}}{\partial \mathbf{x}'}, t) = \frac{e}{c} \varepsilon\_{kij} \frac{1}{m} \frac{\partial \tilde{S}}{\partial \mathbf{x}\_i} B\_j + eE\_k + h\_k(\mathbf{x}, \frac{\partial \tilde{S}}{\partial \mathbf{x}'}, t)\_\prime \tag{34}$$

with *hk*(*x*, *p*, *t*) as our new unknown functions. They obey the simpler relations

$$-\int\_{-\infty}^{\infty} \mathbf{d}^3 \mathbf{x} \frac{\partial \rho}{\partial \mathbf{x}\_k} \left[ \frac{\partial \tilde{\mathbf{S}}}{\partial t} + \frac{1}{2m} \sum\_{\dot{j}} \left( \frac{\partial \tilde{\mathbf{S}}}{\partial \mathbf{x}\_{\dot{j}}} \right)^2 + V \right] = \int\_{-\infty}^{\infty} \mathbf{d}^3 \mathbf{x} \rho h\_k(\mathbf{x}, \frac{\partial \tilde{\mathbf{S}}}{\partial \mathbf{x}}, t). \tag{35}$$

On a first look this condition for the allowed forces looks similar to the *p*−independent case [see (27)]. But the dependence of *hk* on *x*, *t* cannot be considered as 'given' (externally controlled), as in the *p*−independent case, because it contains now the unknown *x*, *t*-dependence of the derivatives of *S*˜. We may nevertheless try to incorporate the r.h.s by adding a term *T*˜ to the bracket which depends on the derivatives of the multivalued quantity *S*˜. This leads to the condition

$$h\_k(\mathbf{x}, \frac{\partial \tilde{\mathbf{S}}}{\partial \mathbf{x}}, t) = -\frac{\partial \tilde{T}(\mathbf{x}, \frac{\partial \tilde{\mathbf{S}}}{\partial \mathbf{x}'}, t)}{\partial \mathbf{x}\_k} + \mathbf{s}\_{k\prime} \quad \int\_{-\infty}^{\infty} \mathbf{d}^3 \mathbf{x} \, \rho \mathbf{s}\_k = 0. \tag{36}$$

But this relation cannot be fulfilled for nontrivial *hk*, *T*˜ because the derivatives of *S*˜ cannot be subject to further constraints beyond those given by the differential equation; on top of that the derivatives with regard to *x* on the r.h.s. create higher order derivatives of *S*˜ which are

Then, our problem is to find a function *L*<sup>0</sup> which fulfills the differential equation

variational problem and to the following conditions for our unknown function *L*0:

*<sup>L</sup>*¯(*x*, *<sup>t</sup>*) <sup>−</sup> *<sup>L</sup>*<sup>0</sup>

� <sup>+</sup>

� *ρ*, *∂ρ ∂x*

*<sup>L</sup>*¯(*x*, *<sup>t</sup>*) <sup>−</sup> *<sup>L</sup>*<sup>0</sup>

*∂β*

one-dimensional relation [equation (68) of I] to three spatial dimensions.

terms) has to vanish separately and (45) can be replaced by the two equations

*∂ ∂xi*

*∂β*

*∂* � *∂*2*ρ ∂xk∂xi*

*∂ ∂xk*

> *∂* � *∂ρ ∂xk*

*∂ ∂xk*

*L*<sup>0</sup> = *B*<sup>0</sup>

⎡ <sup>⎣</sup><sup>−</sup> <sup>1</sup> <sup>2</sup>*ρ*<sup>2</sup> ∑ *j*

*δ* � d3*xρ* �

problem (44) lead to the following differential equation

*∂ ∂xi*

*∂* � *∂*2*ρ ∂xk∂xi*

<sup>−</sup> *<sup>∂</sup> ∂xk*

this solution is given by

in I.

and condition (31). The method used in I for a one-dimensional situation, to determine *L*<sup>0</sup> from the requirement of minimal Fisher information, remains essentially unchanged in the present three-dimensional case. The reader is referred to the detailed explanations reported

A Statistical Derivation of Non-Relativistic Quantum Theory 153

In I it has been shown that this principle of maximal disorder leads to an anomalous

� *ρ*, *∂ρ ∂x*

where *L*<sup>0</sup> contains only derivatives of *ρ* up to second order and does not explicitely depend on *x*, *t*. If Eq. (43) is taken into account, the Euler-Lagrange equations of the variational

> *∂ ∂xk*

for the variable *β* = *ρL*0. Eq. (45) is a straightforward generalization of the corresponding

Besides (45) a further (consistency) condition exists, which leads to a simplification of the problem. The function *L*<sup>0</sup> may depend on second order derivatives of *ρ* but this dependence must be of a special form not leading to any terms in the Euler-Lagrange equations [according to (43) our final differential equation for *S* and *ρ* must not contain higher than second order derivatives of *ρ*]. Consequently, the first term in Eq. (45) (as well as the sum of the remaining

*∂β*

� <sup>−</sup> *∂β*

In I a new derivation of Fisher's functional has been obtained, using the general solution of the one-dimensional version of (45), as well as the so-called composition law. In the present three-dimensional situation we set ourselves a less ambitious aim. We know that Fisher's functional describes the maximal amount of disorder. If we are able to find a solution of (46), (47) that agrees with this functional (besides 'null-terms' giving no contribution to the Euler-Lagrange equations) then we will accept it as our correct solution. It is easy to see that

> � *∂ρ ∂xj*

*∂ρ* <sup>+</sup> *<sup>β</sup>*

�<sup>2</sup> + 1 *<sup>ρ</sup>* ∑ *j*

*∂*2*ρ ∂x*<sup>2</sup> *j*

⎤

*∂* � *∂ρ ∂xk*

, *<sup>∂</sup>*2*<sup>ρ</sup> ∂x∂x*

*∂β*

�

, *<sup>∂</sup>*2*<sup>ρ</sup> ∂x∂x*

� <sup>−</sup> *∂β*

*∂ρ* <sup>+</sup> *<sup>β</sup>*

*<sup>L</sup>*¯(*x*, *<sup>t</sup>*) <sup>−</sup> *<sup>L</sup>*<sup>0</sup> <sup>=</sup> 0, (42)

= 0 (43)

�� <sup>=</sup> 0, (44)

*<sup>ρ</sup>* <sup>=</sup> <sup>0</sup> (45)

� <sup>=</sup> <sup>0</sup> (46)

*<sup>ρ</sup>* <sup>=</sup> 0. (47)

⎦ , (48)

not present at the l.h.s. of Eq. (36). The only possibility to fulfill this relation is for constant *∂S*˜ *<sup>∂</sup><sup>x</sup>* , a special case which has in fact already be taken into account by adding the mechanical potential *V*. We conclude that the statistical constraint leads to *hk* = *T*˜ = 0 and that the statistical condition (35) takes the form

$$-\int \mathrm{d}^3 \mathbf{x} \frac{\partial \rho}{\partial \mathbf{x}\_k} \left[ \frac{\partial \tilde{\mathbf{S}}}{\partial t} + \frac{1}{2m} \sum\_j \left( \frac{\partial \tilde{\mathbf{S}}}{\partial \mathbf{x}\_j} \right)^2 + V \right] = 0. \tag{37}$$

Thus, only a mechanical potential and the four electrodynamic potentials are compatible with the statistical constraint and will consequently - assuming that the present statistical approach reflects a fundamental principle of nature - be realized in nature. As is well known all existing interactions follow (sometimes in a generalized form) the gauge coupling scheme derived above. The statistical conditions imply not only Schrödinger's equation but also the form of the (gauge) coupling to external influences and the form of the corresponding local force, the Lorentz force,

$$
\vec{F}\_L = e\vec{E} + \frac{e}{c}\vec{v} \times \vec{B}\_\prime \tag{38}
$$

if the particle velocity *v* is identified with the velocity field *v*˜(*x*, *t*).

In the present derivation the usual order of proceeding is just inverted. In the conventional deterministic treatment the form of the local forces (Lorentz force), as taken from experiment, is used as a starting point. The potentials are introduced afterwards, in the course of a transition to a different formal framework (Lagrange formalism). In the present approach the fundamental assumptions are the statistical conditions. Then, taking into account an existing mathematical freedom (multi-valuedness of a variable) leads to the introduction of potentials. From these, the shape of the macroscopic (Lorentz) force can be derived, using the validity of the statistical conditions as a constraint.

#### **5. Fisher information as the hallmark of quantum theory**

The remaining nontrivial task is the derivation of a local differential equation for *S* and *ρ* from the integral equation (37). As our essential constraint we will use, besides general principles of simplicity (like homogeneity and isotropy of space) the principle of maximal disorder, as realized by the requirement of minimal Fisher information. Using the abbreviation

$$\bar{L}(\mathbf{x},t) = \frac{\partial \tilde{\mathbf{S}}}{\partial t} + \frac{1}{2m} \left( \frac{\partial \tilde{\mathbf{S}}(\mathbf{x},t)}{\partial \mathbf{x}} \right)^2 + V(\mathbf{x},t), \tag{39}$$

the general solution of (37) may be written in the form

$$\frac{\partial \rho}{\partial \mathbf{x}\_k} \bar{L}(\mathbf{x}, t) = G\_k(\mathbf{x}, t), \tag{40}$$

where the three functions *Gk*(*x*, *<sup>t</sup>*) have to vanish upon integration over <sup>R</sup><sup>3</sup> and are otherwise arbitrary. If we restrict ourselves to an isotropic law, we may write

$$G\_k(\mathbf{x}, t) = \frac{\partial \rho}{\partial \mathbf{x}\_k} L\_0. \tag{41}$$

12 Will-be-set-by-IN-TECH

not present at the l.h.s. of Eq. (36). The only possibility to fulfill this relation is for constant

*<sup>∂</sup><sup>x</sup>* , a special case which has in fact already be taken into account by adding the mechanical potential *V*. We conclude that the statistical constraint leads to *hk* = *T*˜ = 0 and that the

> 1 <sup>2</sup>*<sup>m</sup>* ∑ *j*

Thus, only a mechanical potential and the four electrodynamic potentials are compatible with the statistical constraint and will consequently - assuming that the present statistical approach reflects a fundamental principle of nature - be realized in nature. As is well known all existing interactions follow (sometimes in a generalized form) the gauge coupling scheme derived above. The statistical conditions imply not only Schrödinger's equation but also the form of the (gauge) coupling to external influences and the form of the corresponding local force, the

In the present derivation the usual order of proceeding is just inverted. In the conventional deterministic treatment the form of the local forces (Lorentz force), as taken from experiment, is used as a starting point. The potentials are introduced afterwards, in the course of a transition to a different formal framework (Lagrange formalism). In the present approach the fundamental assumptions are the statistical conditions. Then, taking into account an existing mathematical freedom (multi-valuedness of a variable) leads to the introduction of potentials. From these, the shape of the macroscopic (Lorentz) force can be derived, using the validity of

The remaining nontrivial task is the derivation of a local differential equation for *S* and *ρ* from the integral equation (37). As our essential constraint we will use, besides general principles of simplicity (like homogeneity and isotropy of space) the principle of maximal disorder, as

where the three functions *Gk*(*x*, *<sup>t</sup>*) have to vanish upon integration over <sup>R</sup><sup>3</sup> and are otherwise

*∂xk*

*Gk*(*x*, *<sup>t</sup>*) = *∂ρ*

� *∂S*˜(*x*, *t*) *∂x*

�2

realized by the requirement of minimal Fisher information. Using the abbreviation

1 2*m*

*<sup>∂</sup><sup>t</sup>* <sup>+</sup>

*∂ρ ∂xk* � *∂S*˜ *∂xj* �<sup>2</sup>

+ *V* ⎤

⎦ = 0. (37)

*B*, (38)

+ *V*(*x*, *t*), (39)

*L*0. (41)

*L*¯(*x*, *t*) = *Gk*(*x*, *t*), (40)

*∂S*˜

Lorentz force,

statistical condition (35) takes the form

the statistical conditions as a constraint.

− � d3*<sup>x</sup> ∂ρ ∂xk* ⎡ ⎣ *∂S*˜ *<sup>∂</sup><sup>t</sup>* <sup>+</sup>

> *FL* = *e E* + *e c <sup>v</sup>* <sup>×</sup>

if the particle velocity *v* is identified with the velocity field *v*˜(*x*, *t*).

**5. Fisher information as the hallmark of quantum theory**

*<sup>L</sup>*¯(*x*, *<sup>t</sup>*) = *<sup>∂</sup>S*˜

arbitrary. If we restrict ourselves to an isotropic law, we may write

the general solution of (37) may be written in the form

Then, our problem is to find a function *L*<sup>0</sup> which fulfills the differential equation

$$
\bar{L}(\mathbf{x}, t) - L\_0 = \mathbf{0},\tag{42}
$$

and condition (31). The method used in I for a one-dimensional situation, to determine *L*<sup>0</sup> from the requirement of minimal Fisher information, remains essentially unchanged in the present three-dimensional case. The reader is referred to the detailed explanations reported in I.

In I it has been shown that this principle of maximal disorder leads to an anomalous variational problem and to the following conditions for our unknown function *L*0:

$$\left(\bar{L}(\mathbf{x},t) - L\_0\left(\rho, \frac{\partial \rho}{\partial \mathbf{x}'}, \frac{\partial^2 \rho}{\partial \mathbf{x} \partial \mathbf{x}}\right) = 0\tag{43}$$

$$
\delta \int \mathrm{d}^3 \mathbf{x} \rho \left[ \bar{L}(\mathbf{x}, t) - L\_0 \left( \rho, \frac{\partial \rho}{\partial \mathbf{x}'}, \frac{\partial^2 \rho}{\partial \mathbf{x} \partial \mathbf{x}} \right) \right] = 0,\tag{44}$$

where *L*<sup>0</sup> contains only derivatives of *ρ* up to second order and does not explicitely depend on *x*, *t*. If Eq. (43) is taken into account, the Euler-Lagrange equations of the variational problem (44) lead to the following differential equation

$$-\frac{\partial}{\partial \mathbf{x}\_k} \frac{\partial}{\partial \mathbf{x}\_l} \frac{\partial \beta}{\partial \left(\frac{\partial^2 \rho}{\partial \mathbf{x}\_k \partial \mathbf{x}\_l}\right)} + \frac{\partial}{\partial \mathbf{x}\_k} \frac{\partial \beta}{\partial \left(\frac{\partial \rho}{\partial \mathbf{x}\_k}\right)} - \frac{\partial \beta}{\partial \rho} + \frac{\rho}{\rho} = 0 \tag{45}$$

for the variable *β* = *ρL*0. Eq. (45) is a straightforward generalization of the corresponding one-dimensional relation [equation (68) of I] to three spatial dimensions.

Besides (45) a further (consistency) condition exists, which leads to a simplification of the problem. The function *L*<sup>0</sup> may depend on second order derivatives of *ρ* but this dependence must be of a special form not leading to any terms in the Euler-Lagrange equations [according to (43) our final differential equation for *S* and *ρ* must not contain higher than second order derivatives of *ρ*]. Consequently, the first term in Eq. (45) (as well as the sum of the remaining terms) has to vanish separately and (45) can be replaced by the two equations

$$\frac{\partial}{\partial \mathbf{x}\_k} \frac{\partial}{\partial \mathbf{x}\_i} \frac{\partial \beta}{\partial \left(\frac{\partial^2 \rho}{\partial \mathbf{x}\_k \partial \mathbf{x}\_i}\right)} = 0 \tag{46}$$

$$\frac{\partial}{\partial \mathbf{x}\_k} \frac{\partial \beta}{\partial \left(\frac{\partial \rho}{\partial \mathbf{x}\_k}\right)} - \frac{\partial \beta}{\partial \rho} + \frac{\mathcal{B}}{\rho} = 0. \tag{47}$$

In I a new derivation of Fisher's functional has been obtained, using the general solution of the one-dimensional version of (45), as well as the so-called composition law. In the present three-dimensional situation we set ourselves a less ambitious aim. We know that Fisher's functional describes the maximal amount of disorder. If we are able to find a solution of (46), (47) that agrees with this functional (besides 'null-terms' giving no contribution to the Euler-Lagrange equations) then we will accept it as our correct solution. It is easy to see that this solution is given by

$$L\_0 = B\_0 \left[ -\frac{1}{2\rho^2} \sum\_{\vec{j}} \left( \frac{\partial \rho}{\partial \mathbf{x}\_{\vec{j}}} \right)^2 + \frac{1}{\rho} \sum\_{\vec{j}} \frac{\partial^2 \rho}{\partial \mathbf{x}\_{\vec{j}}^2} \right],\tag{48}$$

**6. Spin as a statistical degree of freedom**

*ρ*2, *S*<sup>2</sup> - then such a degree of freedom could exist in nature.

*∂*(*ρ*<sup>1</sup> + *ρ*2) *<sup>∂</sup><sup>t</sup>* <sup>+</sup>

in this and the next two sections.

equation is given by

equation

*F*(*R*)

Eq. (58) takes the form

d d*t pk* = d3*x ∂ρ*1 *∂t*

> <sup>−</sup> *∂ρ*<sup>1</sup> *∂xk*

*∂S*˜ 1 *<sup>∂</sup><sup>t</sup>* <sup>−</sup> *∂ρ*<sup>2</sup> *∂xk*

indicated by the notation *S*˜

Spin is generally believed to be a phenomenon of quantum-theoretic origin. For a long period of time, following Dirac's derivation of his relativistic equation, it was also believed to be essentially of relativistic origin. This has changed since the work of Schiller (1962b), Levy-Leblond (1967), Arunsalam (1970), Gould (1995), Reginatto (1998b) and others, who showed that spin may be derived entirely in the framework of non-relativistic QT without using any relativistic concepts. Thus, a new derivation of non-relativistic QT like the present one should also include a derivation of the phenomenon of spin. This will be done

A Statistical Derivation of Non-Relativistic Quantum Theory 155

A simple idea to extend the present theory is to assume that sometimes - under certain external conditions to be identified later - a situation occurs where the behavior of our statistical ensemble of particles cannot longer be described by *ρ*, *S* alone but requires, e.g., the double number of field variables; let us denote these by *ρ*1, *S*1, *ρ*2, *S*<sup>2</sup> (we restrict ourselves here to spin one-half). The relations defining this generalized theory should be formulated in such a way that the previous relations are obtained in the appropriate limits. One could say that we undertake an attempt to introduce a new (discrete) degree of freedom for the ensemble. If we are able to derive a non-trivial set of differential equations - with coupling between *ρ*1, *S*<sup>1</sup> and

Using these guidelines, the basic equations of the generalized theory can be easily formulated. The probability density and probability current take the form *ρ* = *ρ*<sup>1</sup> + *ρ*<sup>2</sup> and*j* =*j*<sup>1</sup> +*j*2, with *ji* (*i* = 1, 2) defined in terms of *ρi*, *Si* exactly as before (see section 2). Then, the continuity

> *ρ*<sup>1</sup> *m ∂S*˜ 1 *∂xl*

where we took the possibility of multi-valuedness of the "phases" already into account, as

*xk* <sup>=</sup> *pk*

which are similar to the relations used previously (in section 2 and in I), and by an additional

which is required as a consequence of our larger number of dynamic variables. Eq. (59) is

*<sup>k</sup>* (*x*, *p*, *t*) on the r.h.s. of (58) and (59) are again subject to the "statistical constraint", which

Performing mathematical manipulations similar to the ones reported in section 2, the l.h.s. of

*∂S*˜ 2 *<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>ρ</sup>*1*S*˜(1)

<sup>+</sup> *∂ρ*<sup>2</sup> *∂t*

*∂S*˜ 2 *∂xk*

[0,*k*] <sup>+</sup> *<sup>ρ</sup>*2*S*˜(2)

[0,*k*] ,

*∂S*˜ 1 *∂xk*

<sup>+</sup> *<sup>ρ</sup>*<sup>2</sup> *m ∂S*˜ 2 *∂xl*

*<sup>i</sup>*. The statistical conditions are given by the two relations

= 0, (56)

*<sup>k</sup>* (*x*, *p*, *t*) and

(60)

*<sup>m</sup>* (57)

*<sup>k</sup>* (*x*, *p*, *t*), (58)

*<sup>k</sup>* (*x*, *p*, *t*), (59)

*∂ ∂xl*

> d d*t*

*pk* <sup>=</sup> *<sup>F</sup>*(*T*)

*sk* <sup>=</sup> *<sup>F</sup>*(*R*)

d d*t*

d d*t*

best explained later; it is written down here for completeness. The forces *F*(*T*)

has been defined in section 3. The expectation values are defined as in (9)-(11).

where *B*<sup>0</sup> is an arbitrary constant. Eq. (48) presents again the three-dimensional (and isotropic) generalization of the one-dimensional result obtained in I. By means of the identity

$$
\frac{\partial}{\partial \mathbf{x}\_i} \frac{\partial \sqrt{\rho}}{\partial \mathbf{x}\_i} \frac{\partial \sqrt{\rho}}{\partial \mathbf{x}\_k} = \frac{\partial \sqrt{\rho}}{\partial \mathbf{x}\_k} \frac{\partial}{\partial \mathbf{x}\_i} \frac{\partial \sqrt{\rho}}{\partial \mathbf{x}\_i} + \frac{1}{2} \frac{\partial}{\partial \mathbf{x}\_k} \frac{\partial \sqrt{\rho}}{\partial \mathbf{x}\_i} \frac{\partial \sqrt{\rho}}{\partial \mathbf{x}\_i},\tag{49}
$$

it is easily verified that the solution (48) obeys also condition (31). Using the decomposition (16) and renaming *B* according to *B* = *h*¯ 2/4*m*, the continuity equation (3) and the second differential equation (43) respectively, take the form

$$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_k} \frac{\rho}{m} \left( \frac{\partial S}{\partial \mathbf{x}\_k} - \frac{e}{c} A\_k \right) = \mathbf{0},\tag{50}$$

$$\frac{\partial S}{\partial t} + e\phi + \frac{1}{2m} \sum\_{k} \left( \frac{\partial S}{\partial \mathbf{x}\_k} - \frac{e}{c} A\_k \right)^2 + V - \frac{\hbar^2}{2m} \frac{\triangle \sqrt{\rho}}{\sqrt{\rho}} = 0. \tag{51}$$

The function *S* occurring in (50), (51) is single-valued but not unique (not gauge-invariant). If now the complex-valued variable

$$
\psi = \sqrt{\rho} \mathbf{e}^{\mathbf{l} \frac{\mathbf{\tilde{s}}}{\hbar}},\tag{52}
$$

is introduced, the two equations (50), (51) may be written in compact form as real and imaginary parts of the linear differential equation

$$\left(\frac{\hbar}{\iota}\frac{\partial}{\partial t} + e\phi\right)\psi + \frac{1}{2m}\left(\frac{\hbar}{\iota}\frac{\partial}{\partial \vec{x}} - \frac{e}{c}\vec{A}\right)^2\psi + V\psi = 0,\tag{53}$$

which completes our derivation of Schrödinger's equation in the presence of a gauge field. Eq. (53) is in manifest gauge-invariant form. The gauge-invariant derivatives of *S*˜ with respect to *t* and *x* correspond to the two brackets in (53). In particular, the canonical momentum *∂S*/*∂x* corresponds to the momentum operator proportional to *∂*/*∂x*. Very frequently, Eq. (53) is written in the form

$$-\frac{\hbar}{\hbar}\frac{\partial}{\partial t}\psi = H\psi\_{\prime\prime} \tag{54}$$

with the Hamilton operator

$$H = \frac{1}{2m} \left( \frac{\hbar}{\iota} \frac{\partial}{\partial \vec{X}} - \frac{e}{c} \vec{A} \right)^2 + V + e\phi,\tag{55}$$

Our final result, Eqs. (54), (55), agrees with the result of the conventional quantization procedure. In its simplest form, the latter starts from the classical relation *H*(*x*, *p*) = *E*, where *H*(*x*, *p*) is the Hamiltonian of a classical particle in a conservative force field, and *E* is its energy. To perform a "canonical quantization" means to replace *p* and *E* by differential expressions according to (1) and let then act both sides of the equation *H*(*x*, *p*) = *E* on states *ψ* of a function space. The 'black magic' involved in this process has been eliminated, or at least dramatically reduced, in the present approach, where Eqs. (54), (55) have been derived from a set of assumptions which can all be interpreted in physical terms.

The Hamiltonian (55) depends on the potentials Φ and *A* and is consequently a non-unique (not gauge-invariant) mathematical object. The same is true for the time-development operator *U*(*H*) which is an operator function of *H*, see e.g. Kobe & Yang (1985). This non-uniqueness is a problem if *U*(*H*) is interpreted as a quantity ruling the time-evolution of a single particle. It is no problem from the point of view of the SI where *H* and *U*(*H*) are primarily convenient mathematical objects which occur in a natural way if the time-dependence of statistically relevant (uniquely defined) quantities, like expectation values and transition probabilities, is to be calculated.

#### **6. Spin as a statistical degree of freedom**

14 Will-be-set-by-IN-TECH

where *B*<sup>0</sup> is an arbitrary constant. Eq. (48) presents again the three-dimensional (and isotropic)

*∂ ∂xi ∂* √*ρ ∂xi* + 1 2 *∂ ∂xk ∂* √*ρ ∂xi*

it is easily verified that the solution (48) obeys also condition (31). Using the decomposition (16) and renaming *B* according to *B* = *h*¯ 2/4*m*, the continuity equation (3)

> *∂S ∂xk*

− *e c Ak* <sup>2</sup>

The function *S* occurring in (50), (51) is single-valued but not unique (not gauge-invariant). If

*<sup>ψ</sup>* = √*ρ*e*<sup>ı</sup> <sup>S</sup>*

is introduced, the two equations (50), (51) may be written in compact form as real and

which completes our derivation of Schrödinger's equation in the presence of a gauge field. Eq. (53) is in manifest gauge-invariant form. The gauge-invariant derivatives of *S*˜ with respect to *t* and *x* correspond to the two brackets in (53). In particular, the canonical momentum *∂S*/*∂x* corresponds to the momentum operator proportional to *∂*/*∂x*. Very frequently, Eq. (53)

Our final result, Eqs. (54), (55), agrees with the result of the conventional quantization procedure. In its simplest form, the latter starts from the classical relation *H*(*x*, *p*) = *E*, where *H*(*x*, *p*) is the Hamiltonian of a classical particle in a conservative force field, and *E* is its energy. To perform a "canonical quantization" means to replace *p* and *E* by differential expressions according to (1) and let then act both sides of the equation *H*(*x*, *p*) = *E* on states *ψ* of a function space. The 'black magic' involved in this process has been eliminated, or at least dramatically reduced, in the present approach, where Eqs. (54), (55) have been derived from

The Hamiltonian (55) depends on the potentials Φ and *A* and is consequently a non-unique (not gauge-invariant) mathematical object. The same is true for the time-development operator *U*(*H*) which is an operator function of *H*, see e.g. Kobe & Yang (1985). This non-uniqueness is a problem if *U*(*H*) is interpreted as a quantity ruling the time-evolution of a single particle. It is no problem from the point of view of the SI where *H* and *U*(*H*) are primarily convenient mathematical objects which occur in a natural way if the time-dependence of statistically relevant (uniquely defined) quantities, like expectation values

− *h*¯ *ı ∂ ∂t*

*<sup>H</sup>* <sup>=</sup> <sup>1</sup> 2*m h*¯ *ı ∂ <sup>∂</sup><sup>x</sup>* <sup>−</sup> *<sup>e</sup> c*

a set of assumptions which can all be interpreted in physical terms.

and transition probabilities, is to be calculated.

− *e c Ak* 

> <sup>+</sup> *<sup>V</sup>* <sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup> 2*m*

*∂* √*ρ ∂xi*

�√*<sup>ρ</sup>*

, (49)

= 0, (50)

*<sup>h</sup>*¯ , (52)

*ψ* = *Hψ*, , (54)

*<sup>A</sup>* <sup>2</sup> + *<sup>V</sup>* + *<sup>e</sup>φ*, (55)

*ψ* + *Vψ* = 0, (53)

<sup>√</sup>*<sup>ρ</sup>* <sup>=</sup> 0. (51)

generalization of the one-dimensional result obtained in I. By means of the identity

= *∂* √*ρ ∂xk*

*∂ ∂xk ρ m*

 *∂S ∂xk*

*∂ ∂xi ∂* √*ρ ∂xi*

*∂S <sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>e</sup><sup>φ</sup>* <sup>+</sup>

imaginary parts of the linear differential equation *h*¯ *ı ∂ ∂t* + *eφ ψ* + 1 2*m h*¯ *ı ∂ <sup>∂</sup><sup>x</sup>* <sup>−</sup> *<sup>e</sup> c <sup>A</sup>* <sup>2</sup>

now the complex-valued variable

is written in the form

with the Hamilton operator

*∂* √*ρ ∂xk*

and the second differential equation (43) respectively, take the form *∂ρ <sup>∂</sup><sup>t</sup>* <sup>+</sup>

> 1 <sup>2</sup>*<sup>m</sup>* ∑ *k*

Spin is generally believed to be a phenomenon of quantum-theoretic origin. For a long period of time, following Dirac's derivation of his relativistic equation, it was also believed to be essentially of relativistic origin. This has changed since the work of Schiller (1962b), Levy-Leblond (1967), Arunsalam (1970), Gould (1995), Reginatto (1998b) and others, who showed that spin may be derived entirely in the framework of non-relativistic QT without using any relativistic concepts. Thus, a new derivation of non-relativistic QT like the present one should also include a derivation of the phenomenon of spin. This will be done in this and the next two sections.

A simple idea to extend the present theory is to assume that sometimes - under certain external conditions to be identified later - a situation occurs where the behavior of our statistical ensemble of particles cannot longer be described by *ρ*, *S* alone but requires, e.g., the double number of field variables; let us denote these by *ρ*1, *S*1, *ρ*2, *S*<sup>2</sup> (we restrict ourselves here to spin one-half). The relations defining this generalized theory should be formulated in such a way that the previous relations are obtained in the appropriate limits. One could say that we undertake an attempt to introduce a new (discrete) degree of freedom for the ensemble. If we are able to derive a non-trivial set of differential equations - with coupling between *ρ*1, *S*<sup>1</sup> and *ρ*2, *S*<sup>2</sup> - then such a degree of freedom could exist in nature.

Using these guidelines, the basic equations of the generalized theory can be easily formulated. The probability density and probability current take the form *ρ* = *ρ*<sup>1</sup> + *ρ*<sup>2</sup> and*j* =*j*<sup>1</sup> +*j*2, with *ji* (*i* = 1, 2) defined in terms of *ρi*, *Si* exactly as before (see section 2). Then, the continuity equation is given by

$$\frac{\partial(\rho\_1 + \rho\_2)}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_l} \left( \frac{\rho\_1}{m} \frac{\partial \tilde{\mathbf{S}}\_1}{\partial \mathbf{x}\_l} + \frac{\rho\_2}{m} \frac{\partial \tilde{\mathbf{S}}\_2}{\partial \mathbf{x}\_l} \right) = \mathbf{0},\tag{56}$$

where we took the possibility of multi-valuedness of the "phases" already into account, as indicated by the notation *S*˜ *<sup>i</sup>*. The statistical conditions are given by the two relations

$$\frac{\mathbf{d}}{\mathbf{d}t}\overline{\mathbf{x}\_k} = \frac{\overline{p\_k}}{m} \tag{57}$$

$$\frac{\mathbf{d}}{\mathbf{d}t}\overline{p\_k} = \overline{F\_k^{(T)}(\mathbf{x}, \; p, t)},\tag{58}$$

which are similar to the relations used previously (in section 2 and in I), and by an additional equation

$$\frac{\mathbf{d}}{\mathbf{d}t}\overline{\mathbf{s}\_k} = \overline{F\_k^{(R)}(\mathbf{x}, p, t)} \, \tag{59}$$

which is required as a consequence of our larger number of dynamic variables. Eq. (59) is best explained later; it is written down here for completeness. The forces *F*(*T*) *<sup>k</sup>* (*x*, *p*, *t*) and *F*(*R*) *<sup>k</sup>* (*x*, *p*, *t*) on the r.h.s. of (58) and (59) are again subject to the "statistical constraint", which has been defined in section 3. The expectation values are defined as in (9)-(11).

Performing mathematical manipulations similar to the ones reported in section 2, the l.h.s. of Eq. (58) takes the form

$$\begin{split} \frac{\mathbf{d}}{\mathbf{d}t} \overline{p\_{k}} &= \int \mathbf{d}^{3}x \left[ \frac{\partial \rho\_{1}}{\partial t} \frac{\partial \tilde{S}\_{1}}{\partial \mathbf{x}\_{k}} + \frac{\partial \rho\_{2}}{\partial t} \frac{\partial \tilde{S}\_{2}}{\partial \mathbf{x}\_{k}} \\ &- \frac{\partial \rho\_{1}}{\partial \mathbf{x}\_{k}} \frac{\partial \tilde{S}\_{1}}{\partial t} - \frac{\partial \rho\_{2}}{\partial \mathbf{x}\_{k}} \frac{\partial \tilde{S}\_{2}}{\partial t} + \rho\_{1} \tilde{S}\_{[0,k]}^{(1)} + \rho\_{2} \tilde{S}\_{[0,k]}^{(2)} \right] \end{split} \tag{60}$$

If now fields *El*, *Bl* and *<sup>E</sup>*(*s*)

of (65) may be written in the form

manner as in section 3 by writing

Eq. (65) in the form

of constant length *<sup>h</sup>*¯

The differential equation

classical force

set

where

*F*(*T*)

− 

+ d3*x h*¯ 2 *ρ* sin *ϑ*

= *F*(*I*)

system) determining the direction of a vector

*<sup>l</sup>* , *<sup>B</sup>*(*s*)

 d3*xρ e E* + *e c <sup>v</sup>* <sup>×</sup> *B <sup>l</sup>* <sup>+</sup> *e E*(*s*) + *e c <sup>v</sup>* <sup>×</sup> *B*(*s*) *l* 

*<sup>l</sup>* (*x*, *<sup>p</sup>*, *<sup>t</sup>*) =

d3*<sup>x</sup> ∂ρ ∂xl*

*<sup>l</sup>* (*x*, *<sup>p</sup>*, *<sup>t</sup>*) =

using (67), (68) and the definition (63) of the fields *A*(*s*)

*<sup>s</sup>* <sup>=</sup> *<sup>h</sup>*¯ 2 

been chosen to yield the correct *g*−factor of the electron.

*∂S*

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>e</sup>φ*<sup>ˆ</sup>

 *∂ϑ ∂xl ∂ϕ <sup>∂</sup><sup>t</sup>* <sup>+</sup> *vj*

 + 1 <sup>2</sup>*<sup>m</sup>* ∑ *j*

<sup>d</sup>3*xρF*(*I*)

*<sup>l</sup>* are introduced by relations analogous to (23), the second line

<sup>d</sup>3*xρF*(*I*)

*∂ϑ ∂xj*

*B* ×*s*, (71)

*B* ×*s* (72)

, (70)

(69)

*B* has

which shows that both types of fields, the external fields as well as the internal fields due to

A Statistical Derivation of Non-Relativistic Quantum Theory 157

The first, externally controlled Lorentz force in (67) may be eliminated in exactly the same

This means that one of the forces acting on the system as a whole is again given by a Lorentz force; there may be other nontrivial forces *F*(*I*) which are still to be determined. The second 'internal' Lorentz force in (67) can, of course, not be eliminated in this way. In order to proceed, the third statistical condition (59) must be implemented. To do that it is useful to rewrite

> *∂S ∂xj* − *e c A*ˆ *j* 2

*<sup>l</sup>* and *<sup>φ</sup>*(*s*).

<sup>2</sup> . As a consequence, ˙*<sup>s</sup>* and *<sup>s</sup>* are perpendicular to each other and the

*<sup>F</sup>*(*R*) in Eq. (59) should be of the form *<sup>D</sup>* <sup>×</sup>*<sup>s</sup>*, where *<sup>D</sup>* is an unknown field. In

*∂ϕ ∂xj* <sup>−</sup> *∂ϕ ∂xl ∂ϑ <sup>∂</sup><sup>t</sup>* <sup>+</sup> *vj*

*<sup>l</sup>* (*x*, *p*, *t*),

sin *ϑ* sin *ϕex* + sin *ϑ* cos *ϕey* + cos *ϑez*

We interpret the fields *ϕ* and *ϑ* as angles (with *ϕ* measured from the *y*−axis of our coordinate

contrast to the 'external force', we are unable to determine the complete form of this 'internal' force from the statistical constraint [an alternative treatment will be reported in section 8] and

*mc*

*B* is the external 'magnetic field', as defined by Eq. (23), and the factor in front of

*<sup>F</sup>*(*R*) <sup>=</sup> <sup>−</sup> *<sup>e</sup>*

*<sup>s</sup>* <sup>=</sup> <sup>−</sup> *<sup>e</sup> mc*

for particle variables *ϑ*(*t*), *ϕ*(*t*) describes the rotational state of a classical magnetic dipole in a magnetic field, see Schiller (1962b). Recall that we do *not* require that (72) is fulfilled in the

d d*t*

*ϑ*, *ϕ*, enter the theory in the same way, namely in the form of a Lorentz force.

d3*xρ e E* + *e c <sup>v</sup>* <sup>×</sup> *B l* +  , (67)

*<sup>l</sup>* (*x*, *p*, *t*). (68)

where the quantities *S*˜(*i*) [*j*,*k*] , *i* = 1, 2 are defined as above [see Eq. (6)] but with *S*˜ replaced by *S*˜ *i*. Let us write now *S*˜ in analogy to section 2 in the form *S*˜ *<sup>i</sup>* = *Si* + *N*˜*i*, as a sum of a single-valued part *Si* and a multi-valued part *N*˜*i*. If *N*˜ <sup>1</sup> and *N*˜ <sup>2</sup> are to represent an external influence, they must be identical and a single multi-valued part *N*˜ = *N*˜ <sup>1</sup> = *N*˜ <sup>2</sup> may be used instead. The derivatives of *N*˜ with respect to *t* and *xk* must be single-valued and we may write

$$\frac{\partial \tilde{S}\_{\dot{l}}}{\partial t} = \frac{\partial S\_{\dot{l}}}{\partial t} + e\Phi\_{\prime} \qquad \frac{\partial \tilde{S}\_{\dot{l}}}{\partial x\_{k}} = \frac{\partial S\_{\dot{l}}}{\partial x\_{k}} - \frac{e}{c}A\_{k\prime} \tag{61}$$

using the same familiar electrodynamic notation as in section 2. In this way we arrive at eight single-valued functions to describe the external conditions and the dynamical state of our system, namely Φ, *Ak* and *ρi*, *Si*.

In a next step we replace *ρi*, *Si* by new dynamic variables *ρ*, *S*, *ϑ*, *ϕ* defined by

$$\begin{aligned} \rho\_1 &= \rho \cos^2 \frac{\theta}{2}, & \quad & S\_1 = S + \frac{\hbar}{2} \rho, \\ \rho\_2 &= \rho \sin^2 \frac{\theta}{2}, & \quad & S\_2 = S - \frac{\hbar}{2} \rho. \end{aligned} \tag{62}$$

A transformation similar to Eq. (62) has been introduced by Takabayasi (1955) in his reformulation of Pauli's equation. Obviously, the variables *S*, *ρ* describe 'center of mass' properties (which are common to both states 1 and 2) while *ϑ*, *ϕ* describe relative (internal) properties of the system.

The dynamical variables *S*, *ρ* and *ϑ*, *ϕ* are not decoupled from each other. It turns out (see below) that the influence of *ϑ*, *ϕ* on *S*, *ρ* can be described in a (formally) similar way as the influence of an external electromagnetic field if a 'vector potential' *A* (*s*) and a 'scalar potential' *φ*(*s*), defined by

$$A\_{l}^{(s)} = -\frac{\hbar c}{2e} \cos \theta \frac{\partial \varphi}{\partial \mathbf{x}\_{l}} , \qquad \quad \quad \phi^{(s)} = \frac{\hbar}{2e} \cos \theta \frac{\partial \varphi}{\partial t} , \tag{63}$$

are introduced. Denoting these fields as 'potentials', we should bear in mind that they are not externally controlled but defined in terms of the internal dynamical variables. Using the abbreviations

$$
\hat{A}\_{l} = A\_{l} + A\_{l}^{(s)}, \qquad \hat{\phi} = \phi + \phi^{(s)}, \tag{64}
$$

the second statistical condition (58) can be written in the following compact form

$$\begin{aligned} & -\int \mathbf{d}^3 \mathbf{x} \frac{\partial \rho}{\partial \mathbf{x}\_l} \left[ \left( \frac{\partial S}{\partial t} + e \hat{\boldsymbol{\Phi}} \right) + \frac{1}{2m} \sum\_j \left( \frac{\partial S}{\partial \mathbf{x}\_j} - \frac{e}{c} \hat{A}\_j \right)^2 \right] \\ & + \int \mathbf{d}^3 \mathbf{x} \rho \left[ -\frac{e}{c} v\_j \left( \frac{\partial \hat{A}\_l}{\partial \mathbf{x}\_j} - \frac{\partial \hat{A}\_j}{\partial \mathbf{x}\_l} \right) - \frac{e}{c} \frac{\partial \hat{A}\_l}{\partial t} - e \frac{\partial \hat{\phi}}{\partial \mathbf{x}\_l} \right] \\ & = \overline{F\_l^{(T)}} (\mathbf{x}, \ p, t) = \int \mathbf{d}^3 x \rho F\_l^{(T)} (\mathbf{x}, \ p, t), \end{aligned} \tag{65}$$

which shows a formal similarity to the spinless case [see (14) and (24)]. The components of the velocity field in (65) are given by

$$v\_j = \frac{1}{m} \left( \frac{\partial \mathcal{S}}{\partial x\_j} - \frac{e}{c} \hat{A}\_j \right). \tag{66}$$

16 Will-be-set-by-IN-TECH

part *Si* and a multi-valued part *N*˜*i*. If *N*˜ <sup>1</sup> and *N*˜ <sup>2</sup> are to represent an external influence, they must be identical and a single multi-valued part *N*˜ = *N*˜ <sup>1</sup> = *N*˜ <sup>2</sup> may be used instead. The

using the same familiar electrodynamic notation as in section 2. In this way we arrive at eight single-valued functions to describe the external conditions and the dynamical state of our

A transformation similar to Eq. (62) has been introduced by Takabayasi (1955) in his reformulation of Pauli's equation. Obviously, the variables *S*, *ρ* describe 'center of mass' properties (which are common to both states 1 and 2) while *ϑ*, *ϕ* describe relative (internal)

The dynamical variables *S*, *ρ* and *ϑ*, *ϕ* are not decoupled from each other. It turns out (see below) that the influence of *ϑ*, *ϕ* on *S*, *ρ* can be described in a (formally) similar way as the influence of an external electromagnetic field if a 'vector potential' *A* (*s*) and a 'scalar potential'

are introduced. Denoting these fields as 'potentials', we should bear in mind that they are not externally controlled but defined in terms of the internal dynamical variables. Using the

> + 1 <sup>2</sup>*<sup>m</sup>* ∑ *j*

<sup>−</sup> *<sup>∂</sup>A*ˆ*<sup>j</sup> ∂xl*

d3*xρF*(*T*)

which shows a formal similarity to the spinless case [see (14) and (24)]. The components of

 *∂S ∂xj* − *e c A*ˆ *j* 

*i ∂xk*

, *S*<sup>1</sup> = *S* +

, *<sup>S</sup>*<sup>2</sup> <sup>=</sup> *<sup>S</sup>* <sup>−</sup> *<sup>h</sup>*¯

, *<sup>φ</sup>*(*s*) <sup>=</sup> *<sup>h</sup>*¯

*<sup>l</sup>* , *<sup>φ</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>φ</sup>* <sup>+</sup> *<sup>φ</sup>*(*s*)

 − *e c ∂A*ˆ *l <sup>∂</sup><sup>t</sup>* <sup>−</sup> *<sup>e</sup>*

*<sup>l</sup>* (*x*, *p*, *t*),

2*e*

 *∂S ∂xj* − *e c A*ˆ *j* <sup>2</sup>

cos *<sup>ϑ</sup> ∂ϕ ∂t*

> *∂φ*ˆ *∂xl*

<sup>=</sup> *<sup>∂</sup>Si ∂xk*

derivatives of *N*˜ with respect to *t* and *xk* must be single-valued and we may write

In a next step we replace *ρi*, *Si* by new dynamic variables *ρ*, *S*, *ϑ*, *ϕ* defined by

2

2

cos *<sup>ϑ</sup> ∂ϕ ∂xl*

*<sup>l</sup>* <sup>=</sup> *Al* <sup>+</sup> *<sup>A</sup>*(*s*)

*∂S*

the second statistical condition (58) can be written in the following compact form

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>e</sup>φ*<sup>ˆ</sup>

*vj* <sup>=</sup> <sup>1</sup> *m*

*<sup>ρ</sup>*<sup>1</sup> <sup>=</sup> *<sup>ρ</sup>* cos<sup>2</sup> *<sup>ϑ</sup>*

*<sup>ρ</sup>*<sup>2</sup> <sup>=</sup> *<sup>ρ</sup>* sin<sup>2</sup> *<sup>ϑ</sup>*

*A*(*s*) *<sup>l</sup>* <sup>=</sup> <sup>−</sup> *hc*¯ 2*e*

− 

+ d3*xρ* − *e c vj ∂A*ˆ *l ∂xj*

the velocity field in (65) are given by

= *F*(*T*)

*A*ˆ

<sup>d</sup>3*<sup>x</sup> ∂ρ ∂xl*

*<sup>l</sup>* (*x*, *p*, *t*) =

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>e</sup>*Φ, *<sup>∂</sup>S*˜

, *i* = 1, 2 are defined as above [see Eq. (6)] but with *S*˜ replaced by *S*˜

− *e c*

> *h*¯ 2 *ϕ*,

2 *ϕ*.

*<sup>i</sup>* = *Si* + *N*˜*i*, as a sum of a single-valued

*Ak*, (61)

, (63)

, (64)

. (66)

*i*.

(62)

(65)

where the quantities *S*˜(*i*)

system, namely Φ, *Ak* and *ρi*, *Si*.

properties of the system.

*φ*(*s*), defined by

abbreviations

[*j*,*k*]

Let us write now *S*˜ in analogy to section 2 in the form *S*˜

*∂S*˜ *i <sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>∂</sup>Si* If now fields *El*, *Bl* and *<sup>E</sup>*(*s*) *<sup>l</sup>* , *<sup>B</sup>*(*s*) *<sup>l</sup>* are introduced by relations analogous to (23), the second line of (65) may be written in the form

$$\int \mathbf{d}^3 \mathbf{x} \rho \left[ \left( e \vec{E} + \frac{e}{c} \vec{v} \times \vec{B} \right)\_I + \left( e E^{\vec{(s)}} + \frac{e}{c} \vec{v} \times \vec{B^{(s)}} \right)\_I \right] \tag{67}$$

which shows that both types of fields, the external fields as well as the internal fields due to *ϑ*, *ϕ*, enter the theory in the same way, namely in the form of a Lorentz force.

The first, externally controlled Lorentz force in (67) may be eliminated in exactly the same manner as in section 3 by writing

$$\overline{F\_l^{(T)}(\mathbf{x},\boldsymbol{p},\mathbf{t})} = \int \mathbf{d}^3 \mathbf{x} \rho \left( e \vec{\mathbf{E}} + \frac{e}{c} \vec{v} \times \vec{B} \right)\_l + \int \mathbf{d}^3 \mathbf{x} \rho F\_l^{(I)}(\mathbf{x},\boldsymbol{p},\mathbf{t}).\tag{68}$$

This means that one of the forces acting on the system as a whole is again given by a Lorentz force; there may be other nontrivial forces *F*(*I*) which are still to be determined. The second 'internal' Lorentz force in (67) can, of course, not be eliminated in this way. In order to proceed, the third statistical condition (59) must be implemented. To do that it is useful to rewrite Eq. (65) in the form

$$\begin{split} & -\int \mathrm{d}^3 \mathbf{x} \frac{\partial \rho}{\partial \mathbf{x}\_l} \left[ \left( \frac{\partial S}{\partial t} + e \boldsymbol{\delta} \right) + \frac{1}{2m} \sum\_{j} \left( \frac{\partial S}{\partial \mathbf{x}\_j} - \frac{e}{c} \boldsymbol{\hat{A}}\_j \right)^2 \right] \\ & + \int \mathrm{d}^3 \mathbf{x} \frac{\hbar}{2} \rho \sin \theta \left( \frac{\partial \theta}{\partial \mathbf{x}\_l} \left[ \frac{\partial \boldsymbol{\rho}}{\partial t} + \boldsymbol{v}\_j \frac{\partial \boldsymbol{\rho}}{\partial \mathbf{x}\_j} \right] - \frac{\partial \boldsymbol{\rho}}{\partial \mathbf{x}\_l} \left[ \frac{\partial \boldsymbol{\theta}}{\partial t} + \boldsymbol{v}\_j \frac{\partial \boldsymbol{\theta}}{\partial \mathbf{x}\_j} \right] \right) \\ & = \overline{F\_l^{(I)}} (\mathbf{x}, \boldsymbol{p}, t) = \int \mathbf{d}^3 \mathbf{x} \rho F\_l^{(I)} (\mathbf{x}, \boldsymbol{p}, t), \end{split} \tag{69}$$

using (67), (68) and the definition (63) of the fields *A*(*s*) *<sup>l</sup>* and *<sup>φ</sup>*(*s*).

We interpret the fields *ϕ* and *ϑ* as angles (with *ϕ* measured from the *y*−axis of our coordinate system) determining the direction of a vector

$$\vec{s} = \frac{\hbar}{2} \left( \sin \theta \sin \varphi \,\vec{e}\_x + \sin \theta \cos \varphi \,\vec{e}\_y + \cos \theta \,\vec{e}\_z \right) \tag{70}$$

of constant length *<sup>h</sup>*¯ <sup>2</sup> . As a consequence, ˙*<sup>s</sup>* and *<sup>s</sup>* are perpendicular to each other and the classical force *<sup>F</sup>*(*R*) in Eq. (59) should be of the form *<sup>D</sup>* <sup>×</sup>*<sup>s</sup>*, where *<sup>D</sup>* is an unknown field. In contrast to the 'external force', we are unable to determine the complete form of this 'internal' force from the statistical constraint [an alternative treatment will be reported in section 8] and set

$$\vec{F}^{(R)} = -\frac{e}{mc}\vec{B} \times \vec{s} \,\tag{71}$$

where *B* is the external 'magnetic field', as defined by Eq. (23), and the factor in front of *B* has been chosen to yield the correct *g*−factor of the electron. The differential equation

$$\frac{\mathbf{d}}{\mathbf{d}t}\vec{s} = -\frac{e}{mc}\vec{B} \times \vec{s} \tag{72}$$

for particle variables *ϑ*(*t*), *ϕ*(*t*) describes the rotational state of a classical magnetic dipole in a magnetic field, see Schiller (1962b). Recall that we do *not* require that (72) is fulfilled in the

where *L*�

*L*<sup>0</sup> = *L*�

<sup>0</sup> + Δ*L*0.

<sup>0</sup> is an unknown field depending on *G*1, *G*2, *G*3.

**7. 'Missing' quantum spin terms from Fisher information**

*∂ρ <sup>∂</sup><sup>t</sup>* <sup>+</sup>

> *δ* d*t*

continuity equation (56), which is given, in terms of the present variables, by

*∂ ∂xl*  *ρ m ∂S ∂xj* − *e c A*ˆ *j* 

*L*¯ :=

The unknown function *L*<sup>0</sup> must contain *L*�

principles has been discussed in detail in I.

written in the form

 *∂S <sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>e</sup>φ*<sup>ˆ</sup>

condition (69) takes the form of a generalized Hamilton-Jacobi equation:

 + 1 <sup>2</sup>*<sup>m</sup>* ∑ *j*

Collecting terms and restricting ourselves, as in section 5, to an isotropic law, the statistical

A Statistical Derivation of Non-Relativistic Quantum Theory 159

 *∂S ∂xj* − *e c A*ˆ *j* 2

Let us summarize at this point what has been achieved so far. We have four coupled differential equations for our dynamic field variables *ρ*, *S*, *ϑ*, *ϕ*. The first of these is the

The three other differential equations, the evolution equations (73), (74) and the generalized Hamilton-Jacobi equation (79), do not yet possess a definite mathematical form. They contain four unknown functions *Gi*, *L*<sup>0</sup> which are constrained, but not determined, by (75), (78). The simplest choice, from a formal point of view, is *Gi* = *L*<sup>0</sup> = 0. In this limit the present theory agrees with Schiller's field-theoretic (Hamilton-Jacobi) version, see Schiller (1962b), of the equations of motion of a classical dipole. This is a classical (statistical) theory despite the fact that it contains [see (63)] a number ¯*h*. But this classical theory is not realized in nature; at least not in the microscopic domain. The reason is that the simplest choice from a formal point of view is not the simplest choice from a physical point of view. The postulate of maximal simplicity (Ockham's razor) implies equal probabilities and the principle of maximal entropy in classical statistical physics. A similar principle which is able to 'explain' the nonexistence of classical physics (in the microscopic domain) is the principle of minimal Fisher information Frieden (2004). The relation between the two (classical and quantum-mechanical)

The mathematical formulation of the principle of minimal Fisher information for the present problem requires a generalization, as compared to I, because we have now several fields with coupled time-evolution equations. As a consequence, the spatial integral (spatial average) over *<sup>ρ</sup>*(*L*¯ <sup>−</sup> *<sup>L</sup>*0) in the variational problem (44) should be replaced by a space-time integral, and the variation should be performed with respect to all four variables. The problem can be

where *Ea* = 0 is a shorthand notation for the equations (80), (79), (74) (73). Eqs. (81), (82) require that the four Euler-Lagrange equations of the variational problem (81) agree with the differential equations (82). This imposes conditions for the unknown functions *L*0, *Gi*. If the *solutions* of (81), (82) for *L*0, *Gi* are inserted in the variational problem (81), the four

+ *μiBi* = *L*0. (79)

= 0. (80)

<sup>0</sup> but may also contain other terms, let us write

d3*x<sup>ρ</sup>* (*L*¯ <sup>−</sup> *<sup>L</sup>*0) <sup>=</sup> <sup>0</sup> (81)

*Ea* = 0, *a* = *S*, *ρ*, *ϑ*, *ϕ*, (82)

present theory. The present variables are the fields *ϑ*(*x*, *t*), *ϕ*(*x*, *t*) which may be thought of as describing a kind of 'rotational state' of the statistical ensemble as a whole, and have to fulfill the 'averaged version' (59) of (72).

Performing steps similar to the ones described in I (see also section 2), the third statistical condition (59) implies the following differential relations,

$$\begin{split} \dot{\varphi} + \upsilon\_{j} \frac{\partial \varphi}{\partial \mathbf{x}\_{j}} &= \frac{e}{mc} \frac{1}{\sin \theta} \left( B\_{\mathrm{z}} \sin \theta - B\_{\mathrm{y}} \cos \theta \cos \varphi - B\_{\mathrm{z}} \cos \theta \sin \varphi \right) \\ &+ \frac{\cos \varphi}{\sin \theta} G\_{1} - \frac{\sin \varphi}{\sin \theta} G\_{2\prime} \end{split} \tag{73}$$

$$+\frac{\frac{\partial G\_{j}}{\partial \sin \theta} G\_{1} - \frac{\partial G\_{j}}{\sin \theta} G\_{2'}}{\frac{\partial}{\partial \alpha\_{j}} - \frac{e}{mc} \left(B\_{X} \cos \varphi - B\_{Y} \sin \varphi \right) - \frac{G\_{3}}{\sin \theta'}}\tag{74}$$

$$\vartheta + v\_{j} \frac{\partial \vartheta}{\partial x\_{j}} = \frac{e}{mc} \left(B\_{X} \cos \varphi - B\_{Y} \sin \varphi \right) - \frac{G\_{3}}{\sin \theta'}\tag{75}$$

for the dynamic variables *ϑ* and *ϕ*. These equations contain three fields *Gi*(*x*, *t*), *i* = 1, 2, 3 which have to obey the conditions

$$
\int \mathrm{d}^3 \mathbf{x} \rho \, G\_{\vec{l}} = 0, \qquad \vec{\mathrm{G}} \vec{s} = 0,\tag{75}
$$

and are otherwise arbitrary. The 'total derivatives' of *ϕ* and *ϑ* in (69) may now be eliminated with the help of (73),(74) and the second line of Eq. (69) takes the form

$$\begin{split} &\int \mathbf{d}^3 \mathbf{x} \frac{\partial \rho}{\partial \mathbf{x}\_l} \frac{e}{mc} s\_j B\_j + \int \mathbf{d}^3 \mathbf{x} \rho \frac{e}{mc} s\_j \frac{\partial}{\partial \mathbf{x}\_l} B\_j \\ &+ \int \mathbf{d}^3 \mathbf{x} \rho \frac{\hbar}{2} \left( \cos \rho \frac{\partial \theta}{\partial \mathbf{x}\_l} G\_1 - \sin \rho \frac{\partial \theta}{\partial \mathbf{x}\_l} G\_2 + \frac{\partial \rho}{\partial \mathbf{x}\_l} G\_3 \right). \end{split} \tag{76}$$

The second term in (76) presents an external macroscopic force. It may be eliminated from (69) by writing

$$
\overline{F\_l^{(I)}(\mathbf{x}, \mathbf{p}, t)} = \int \mathbf{d}^3 \mathbf{x} \rho \left( -\mu\_j \frac{\partial}{\partial \mathbf{x}\_l} \mathcal{B}\_j \right) + \overline{F\_l^{(V)}(\mathbf{x}, \mathbf{p}, t)},
\tag{77}
$$

where the magnetic moment of the electron *μ<sup>i</sup>* = −(*e*/*mc*)*si* has been introduced. The first term on the r.h.s. of (77) is the expectation value of the well-known electrodynamical force exerted by an inhomogeneous magnetic field on the translational motion of a magnetic dipole; this classical force plays an important role in the standard interpretation of the quantum-mechanical Stern-Gerlach effect. It is satisfying that both translational forces, the Lorentz force as well as this dipole force, can be derived in the present approach. The remaining unknown force *F*(*V*) in (77) leads (in the same way as in section 3) to a mechanical potential *V*, which will be omitted for brevity.

The integrand of the first term in (76) is linear in the derivative of *ρ* with respect to *xl*. It may consequently be added to the first line of (69) which has the same structure. Therefore, it represents (see below) a contribution to the generalized Hamilton-Jacobi differential equation. The third term in (76) has the mathematical structure of a force term, but does not contain any externally controlled fields. Thus, it must also represent a contribution to the generalized Hamilton-Jacobi equation. This implies that this third term can be written as

$$\int \mathrm{d}^3 \mathbf{x} \rho \, \frac{\hbar}{2} \left( \cos \varphi \frac{\partial \theta}{\partial \mathbf{x}\_l} \mathbf{G}\_1 - \sin \varphi \frac{\partial \theta}{\partial \mathbf{x}\_l} \mathbf{G}\_2 + \frac{\partial \varphi}{\partial \mathbf{x}\_l} \mathbf{G}\_3 \right) = \int \mathrm{d}^3 \mathbf{x} \frac{\partial \rho}{\partial \mathbf{x}\_l} L'\_0 \tag{78}$$

18 Will-be-set-by-IN-TECH

present theory. The present variables are the fields *ϑ*(*x*, *t*), *ϕ*(*x*, *t*) which may be thought of as describing a kind of 'rotational state' of the statistical ensemble as a whole, and have to fulfill

Performing steps similar to the ones described in I (see also section 2), the third statistical

for the dynamic variables *ϑ* and *ϕ*. These equations contain three fields *Gi*(*x*, *t*), *i* = 1, 2, 3

and are otherwise arbitrary. The 'total derivatives' of *ϕ* and *ϑ* in (69) may now be eliminated

*G*<sup>1</sup> − sin *ϕ*

The second term in (76) presents an external macroscopic force. It may be eliminated from (69)

where the magnetic moment of the electron *μ<sup>i</sup>* = −(*e*/*mc*)*si* has been introduced. The first term on the r.h.s. of (77) is the expectation value of the well-known electrodynamical force exerted by an inhomogeneous magnetic field on the translational motion of a magnetic dipole; this classical force plays an important role in the standard interpretation of the quantum-mechanical Stern-Gerlach effect. It is satisfying that both translational forces, the Lorentz force as well as this dipole force, can be derived in the present approach. The

The integrand of the first term in (76) is linear in the derivative of *ρ* with respect to *xl*. It may consequently be added to the first line of (69) which has the same structure. Therefore, it represents (see below) a contribution to the generalized Hamilton-Jacobi differential equation. The third term in (76) has the mathematical structure of a force term, but does not contain any externally controlled fields. Thus, it must also represent a contribution to the generalized

> *∂ϑ ∂xl*

*<sup>G</sup>*<sup>2</sup> <sup>+</sup> *∂ϕ ∂xl G*3 = 

*∂ϑ ∂xl*

*Bj*) + *<sup>F</sup>*(*V*)

*F*(*V*) in (77) leads (in the same way as in section 3) to a mechanical

*<sup>G</sup>*<sup>2</sup> <sup>+</sup> *∂ϕ ∂xl G*3 .

*Bz* sin *ϑ* − *By* cos *ϑ* cos *ϕ* − *Bz* cos *ϑ* sin *ϕ*

 <sup>−</sup> *<sup>G</sup>*<sup>3</sup>

sin *<sup>ϑ</sup> <sup>G</sup>*2, (73)

d3*xρ Gi* = 0, *Gs* = 0, (75)

(76)

sin *<sup>ϑ</sup>* , (74)

*<sup>l</sup>* (*x*, *p*, *t*), (77)

d3*<sup>x</sup> ∂ρ ∂xl L*�

<sup>0</sup>, (78)

the 'averaged version' (59) of (72).

*ϕ*˙ + *vj*

*ϑ*˙ + *vj*

which have to obey the conditions

by writing

*∂ϕ ∂xj*

*∂ϑ ∂xj*

+ d3*xρ h*¯ 2 cos *ϕ ∂ϑ ∂xl*

*F*(*I*)

potential *V*, which will be omitted for brevity.

remaining unknown force

 d3*xρ h*¯ 2 cos *ϕ ∂ϑ ∂xl*

d3*<sup>x</sup> ∂ρ ∂xl*

*<sup>l</sup>* (*x*, *p*, *t*) =

condition (59) implies the following differential relations,

1 sin *ϑ* 

with the help of (73),(74) and the second line of Eq. (69) takes the form

*e mc*

> d3*xρ* − *μ<sup>j</sup> ∂ ∂xl*

Hamilton-Jacobi equation. This implies that this third term can be written as

*G*<sup>1</sup> − sin *ϕ*

*sjBj* + d3*xρ e mc sj ∂ ∂xl Bj*

sin *<sup>ϑ</sup> <sup>G</sup>*<sup>1</sup> <sup>−</sup> sin *<sup>ϕ</sup>*

*Bx* cos *ϕ* − *By* sin *ϕ*

<sup>=</sup> *<sup>e</sup> mc*

cos *ϕ*

+

<sup>=</sup> *<sup>e</sup> mc*  where *L*� <sup>0</sup> is an unknown field depending on *G*1, *G*2, *G*3.

Collecting terms and restricting ourselves, as in section 5, to an isotropic law, the statistical condition (69) takes the form of a generalized Hamilton-Jacobi equation:

$$\bar{L} := \left(\frac{\partial S}{\partial t} + e\hat{\phi}\right) + \frac{1}{2m} \sum\_{j} \left(\frac{\partial S}{\partial \mathbf{x}\_{j}} - \frac{e}{c}\hat{A}\_{j}\right)^{2} + \mu\_{i}B\_{i} = L\_{0}.\tag{79}$$

The unknown function *L*<sup>0</sup> must contain *L*� <sup>0</sup> but may also contain other terms, let us write *L*<sup>0</sup> = *L*� <sup>0</sup> + Δ*L*0.

#### **7. 'Missing' quantum spin terms from Fisher information**

Let us summarize at this point what has been achieved so far. We have four coupled differential equations for our dynamic field variables *ρ*, *S*, *ϑ*, *ϕ*. The first of these is the continuity equation (56), which is given, in terms of the present variables, by

$$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x\_l} \left[ \frac{\rho}{m} (\frac{\partial S}{\partial x\_j} - \frac{e}{c} \hat{A}\_j) \right] = 0. \tag{80}$$

The three other differential equations, the evolution equations (73), (74) and the generalized Hamilton-Jacobi equation (79), do not yet possess a definite mathematical form. They contain four unknown functions *Gi*, *L*<sup>0</sup> which are constrained, but not determined, by (75), (78).

The simplest choice, from a formal point of view, is *Gi* = *L*<sup>0</sup> = 0. In this limit the present theory agrees with Schiller's field-theoretic (Hamilton-Jacobi) version, see Schiller (1962b), of the equations of motion of a classical dipole. This is a classical (statistical) theory despite the fact that it contains [see (63)] a number ¯*h*. But this classical theory is not realized in nature; at least not in the microscopic domain. The reason is that the simplest choice from a formal point of view is not the simplest choice from a physical point of view. The postulate of maximal simplicity (Ockham's razor) implies equal probabilities and the principle of maximal entropy in classical statistical physics. A similar principle which is able to 'explain' the nonexistence of classical physics (in the microscopic domain) is the principle of minimal Fisher information Frieden (2004). The relation between the two (classical and quantum-mechanical) principles has been discussed in detail in I.

The mathematical formulation of the principle of minimal Fisher information for the present problem requires a generalization, as compared to I, because we have now several fields with coupled time-evolution equations. As a consequence, the spatial integral (spatial average) over *<sup>ρ</sup>*(*L*¯ <sup>−</sup> *<sup>L</sup>*0) in the variational problem (44) should be replaced by a space-time integral, and the variation should be performed with respect to all four variables. The problem can be written in the form

$$\delta \int \mathbf{d}t \int \mathbf{d}^3 \mathbf{x} \rho \left(\bar{L} - L\_0\right) = 0 \tag{81}$$

$$E\_{\mathfrak{a}} = 0, \quad \mathfrak{a} = S\_{\prime\prime} \mathfrak{o}\_{\prime} \mathfrak{d}\_{\prime} \mathfrak{d}\_{\prime} \mathfrak{d}\_{\prime} \tag{82}$$

where *Ea* = 0 is a shorthand notation for the equations (80), (79), (74) (73). Eqs. (81), (82) require that the four Euler-Lagrange equations of the variational problem (81) agree with the differential equations (82). This imposes conditions for the unknown functions *L*0, *Gi*. If the *solutions* of (81), (82) for *L*0, *Gi* are inserted in the variational problem (81), the four

Eqs. (88) and (89) show that the first condition listed in (75) is also satisfied. The last condition

A Statistical Derivation of Non-Relativistic Quantum Theory 161

motion while Δ*L*<sup>0</sup> is related to the probability density of the ensemble (as could have been guessed considering the mathematical form of these terms). The last term is the same as in the

The remaining task is to show that the above solution for *L*<sup>0</sup> does indeed lead to a (appropriately generalized) Fisher functional. This can be done in several ways. The simplest

> � d3*x* 3 ∑ *k*=1

The functions *ρ*(*j*) represent the probability that a particle is at space-time point *x*, *t* and *s* points into direction *j*. Inserting (86) the validity of (91) may easily be verified. The r.h.s. of Eq. (91) shows that the averaged value of *L*<sup>0</sup> represents indeed a Fisher functional, which

Summarizing, our assumption, that under certain external conditions four state variables instead of two may be required, led to a nontrivial result, namely the four coupled differential equations (80), (79), (74), (73) with *L*0, *Gi* given by (86), (89), (88). The external condition which stimulates this splitting is given by a gauge field; the most important case is a magnetic field

*B* but other possibilities do exist (see below). These four differential equations are equivalent

which is linear in the complex-valued two-component state variable *ψ*ˆ and is referred to as Pauli equation (the components of the vector *σ* are the three Pauli matrices and *μ<sup>B</sup>* =

−*eh*¯ /2*mc*). To see the equivalence one writes, see Takabayasi (1955), Holland (1995),

*ı h*¯ *S* ⎛

four differential equations (80), (79), (74) (73) and completes the present spin theory.

⎜⎝

and evaluates the real and imaginary parts of the two scalar equations (93). This leads to the

In terms of the real-valued functions *ρ*, *S*, *ϑ*, *ϕ* the quantum-mechanical solutions (86), (88), (89) for *L*0, *Gi* look complicated in comparison to the classical solutions *L*<sup>0</sup> = 0, *Gi* = 0. In terms of the variable *ψ*ˆ the situation changes to the contrary: The quantum-mechanical equation becomes simple (linear) and the classical equation, which has been derived by Schiller (1962b), becomes complicated (nonlinear). The simplicity of the

cos *<sup>ϑ</sup>* 2 e*ı ϕ* 2

*ı* sin *<sup>ϑ</sup>* <sup>2</sup> <sup>e</sup>−*<sup>ı</sup> ϕ* 2

*<sup>ψ</sup>*<sup>ˆ</sup> = √*<sup>ρ</sup>* <sup>e</sup>

�<sup>2</sup> �

, <sup>Δ</sup>*L*<sup>0</sup> <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup>

1 *ρ*(*j*)

*ψ*ˆ + *μ<sup>B</sup>σ*

⎞

<sup>2</sup> sin2 *<sup>ϕ</sup>*

� *∂ρ*(*j*) *∂xk*

�<sup>2</sup>

<sup>2</sup> , *<sup>ρ</sup>*(3) :<sup>=</sup> *<sup>ρ</sup>* cos<sup>2</sup> *<sup>ϑ</sup>*

2*m*

<sup>0</sup> is a quantum-mechanical contribution to the rotational

1 √*ρ ∂ ∂x ∂ ∂x*

√*ρ*, (90)

, (91)

. (92)

2

*Bψ*ˆ = 0, (93)

⎟⎠ , (94)

<sup>0</sup> + Δ*L*0, where

� *∂ϕ ∂x* �<sup>2</sup> − � *∂ϑ ∂x*

is also fulfilled: *L*<sup>0</sup> can be written as *L*�

<sup>0</sup> <sup>=</sup> <sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup> 8*m* � sin<sup>2</sup> *ϑ*

<sup>0</sup> fulfills (78). We see that *L*�

�

*<sup>ρ</sup>*(1) :<sup>=</sup> *<sup>ρ</sup>* sin<sup>2</sup> *<sup>ϑ</sup>*

to the much simpler differential equation

� *h*¯ *ı ∂ ∂t* + *eφ* � *ψ*ˆ + 1 2*m* � *h*¯ *ı ∂ <sup>∂</sup><sup>x</sup>* <sup>−</sup> *<sup>e</sup> c <sup>A</sup>* �<sup>2</sup>

is to use the following result due to Reginatto (1998b):

<sup>d</sup>3*<sup>x</sup>* (−*ρL*0) <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup>

<sup>2</sup> cos2 *<sup>ϕ</sup>*

completes our calculation of the 'quantum terms' *L*0, *Gi*.

8*m*

3 ∑ *j*=1

<sup>2</sup> , *<sup>ρ</sup>*(2) :<sup>=</sup> *<sup>ρ</sup>* sin<sup>2</sup> *<sup>ϑ</sup>*

*L*�

spinless case [see (51)].

and *L*�

relations (82) become redundant and *<sup>ρ</sup>*(*L*¯ <sup>−</sup> *<sup>L</sup>*0) becomes the Lagrangian density of our problem. Thus, Eqs. (81) and (82) represent a method to construct a Lagrangian.

We assume a functional form *L*0(*χα*, *∂kχα*, *∂k∂lχα*), where *χα* = *ρ*, *ϑ*, *ϕ*. This means *L*<sup>0</sup> does not possess an explicit *x*, *t*-dependence and does not depend on *S* (this would lead to a modification of the continuity equation). We further assume that *L*<sup>0</sup> does not depend on time-derivatives of *χα* (the basic structure of the time-evolution equations should not be affected) and on spatial derivatives higher than second order. These second order derivatives must be taken into account but should not give contributions to the variational equations (a more detailed discussion of the last point has been given in I).

The variation with respect to *S* reproduces the continuity equation which is unimportant for the determination of *L*0, *Gi*. Performing the variation with respect to *ρ*, *ϑ*, *ϕ* and taking the corresponding conditions (79), (74) (73) into account leads to the following differential equations for *L*0, *G*<sup>1</sup> cos *ϕ* − *G*<sup>2</sup> sin *ϕ* and *G*3,

$$-\frac{\partial}{\partial \mathbf{x}\_k} \frac{\partial}{\partial \mathbf{x}\_i} \frac{\partial \rho L\_0}{\partial \frac{\partial^2 \rho}{\partial \mathbf{x}\_k \partial \mathbf{x}\_i}} + \frac{\partial}{\partial \mathbf{x}\_k} \frac{\partial \rho L\_0}{\partial \frac{\partial \rho}{\partial \mathbf{x}\_k}} - \rho \frac{\partial L\_0}{\partial \rho} = 0 \tag{83}$$

$$-\frac{\partial}{\partial \mathbf{x}\_k} \frac{\partial}{\partial \mathbf{x}\_l} \frac{\partial \rho L\_0}{\partial \frac{\partial^2 \theta}{\partial \mathbf{x}\_l \partial \mathbf{x}\_l}} + \frac{\partial}{\partial \mathbf{x}\_k} \frac{\partial \rho L\_0}{\partial \frac{\partial \theta}{\partial \mathbf{x}\_k}} - \frac{\partial \rho L\_0}{\partial \theta} - \frac{\hbar \rho}{2} \left(\mathcal{G}\_1 \cos \varphi - \mathcal{G}\_2 \sin \varphi\right) = 0 \tag{84}$$

$$-\frac{\partial}{\partial \mathbf{x}\_k} \frac{\partial}{\partial \mathbf{x}\_l} \frac{\partial \rho L\_0}{\partial \frac{\partial^2 \rho}{\partial \mathbf{x}\_k \partial \mathbf{x}\_l}} + \frac{\partial}{\partial \mathbf{x}\_k} \frac{\partial \rho L\_0}{\partial \frac{\partial \rho}{\partial \mathbf{x}\_k}} - \frac{\partial \rho L\_0}{\partial \rho} - \frac{\hbar}{2} \rho \mathbf{G}\_3 = 0. \tag{85}$$

The variable *S* does not occur in (83)-(85) in agreement with our assumptions about the form of *L*0. It is easy to see that a proper solution (with vanishing variational contributions from the second order derivatives) of (83)-(85) is given by

$$L\_0 = \frac{\hbar^2}{2m} \left[ \frac{1}{\sqrt{\rho}} \frac{\partial}{\partial \vec{\mathbf{x}}} \frac{\partial}{\partial \vec{\mathbf{x}}} \sqrt{\rho} - \frac{1}{4} \sin^2 \theta \left( \frac{\partial \varphi}{\partial \vec{\mathbf{x}}} \right)^2 - \frac{1}{4} \left( \frac{\partial \theta}{\partial \vec{\mathbf{x}}} \right)^2 \right] \tag{86}$$

$$
\hbar \mathfrak{H}\_1 \cos \varphi - \hbar \mathfrak{H}\_2 \sin \varphi = \frac{\hbar^2}{2m} \left[ \frac{1}{2} \sin 2\theta \left( \frac{\partial \varphi}{\partial \vec{\mathbf{x}}} \right)^2 - \frac{1}{\rho} \frac{\partial}{\partial \vec{\mathbf{x}}} \rho \frac{\partial \theta}{\partial \vec{\mathbf{x}}} \right] \tag{87}
$$

$$\hbar G\_3 = -\frac{\hbar^2}{2m} \frac{1}{\rho} \frac{\partial}{\partial \vec{\mathbf{x}}} (\rho \sin \theta^2 \frac{\partial \rho}{\partial \vec{\mathbf{x}}}). \tag{88}$$

A new adjustable parameter appears on the r.h.s of (86)- (88) which has been identified with *h*¯ 2/2*m*, where ¯*h* is again Planck's constant. This second ¯*h* is related to the quantum-mechanical principle of maximal disorder. It is in the present approach not related in any obvious way to the previous "classical" ¯*h* which denotes the amplitude of a rotation; compare, however, the alternative derivation of spin in section 8.

The solutions for *G*1, *G*<sup>2</sup> may be obtained with the help of the second condition (*Gs* = 0) listed in Eq. (75). The result may be written in the form

$$\begin{split} G\_1 &= \frac{\hbar}{2m} \frac{1}{\rho} \frac{\partial}{\partial \vec{\mathbf{x}}} \rho \left( \frac{1}{2} \sin 2\theta \sin \varphi \frac{\partial \varphi}{\partial \vec{\mathbf{x}}} - \cos \varphi \frac{\partial \theta}{\partial \vec{\mathbf{x}}} \right) \\ G\_2 &= \frac{\hbar}{2m} \frac{1}{\rho} \frac{\partial}{\partial \vec{\mathbf{x}}} \rho \left( \frac{1}{2} \sin 2\theta \cos \varphi \frac{\partial \varphi}{\partial \vec{\mathbf{x}}} + \sin \varphi \frac{\partial \theta}{\partial \vec{\mathbf{x}}} \right). \end{split} \tag{89}$$

20 Will-be-set-by-IN-TECH

relations (82) become redundant and *<sup>ρ</sup>*(*L*¯ <sup>−</sup> *<sup>L</sup>*0) becomes the Lagrangian density of our

We assume a functional form *L*0(*χα*, *∂kχα*, *∂k∂lχα*), where *χα* = *ρ*, *ϑ*, *ϕ*. This means *L*<sup>0</sup> does not possess an explicit *x*, *t*-dependence and does not depend on *S* (this would lead to a modification of the continuity equation). We further assume that *L*<sup>0</sup> does not depend on time-derivatives of *χα* (the basic structure of the time-evolution equations should not be affected) and on spatial derivatives higher than second order. These second order derivatives must be taken into account but should not give contributions to the variational equations (a

The variation with respect to *S* reproduces the continuity equation which is unimportant for the determination of *L*0, *Gi*. Performing the variation with respect to *ρ*, *ϑ*, *ϕ* and taking the corresponding conditions (79), (74) (73) into account leads to the following differential

> *∂ρL*<sup>0</sup> *∂ ∂ρ ∂xk*

*∂ϑ* <sup>−</sup> *<sup>h</sup>*¯ *<sup>ρ</sup>*

− *ρ ∂L*<sup>0</sup>

<sup>−</sup> *∂ρL*<sup>0</sup> *∂ϕ* <sup>−</sup> *<sup>h</sup>*¯ 2

> *∂ϕ ∂x*

*<sup>ρ</sup>* sin *<sup>ϑ</sup>*<sup>2</sup> *∂ϕ ∂x* 

 *∂ϕ ∂x*

<sup>2</sup> − 1 4

*<sup>∂</sup><sup>x</sup>* <sup>−</sup> cos *<sup>ϕ</sup>*

+ sin *ϕ*

*∂x*

*∂ϑ ∂x* 

*∂ϑ ∂x* .

<sup>2</sup> − 1 *ρ ∂ ∂x ρ ∂ϑ ∂x* 

 *∂ϑ ∂x* <sup>2</sup>

*∂ρ* <sup>=</sup> <sup>0</sup> (83)

*ρG*<sup>3</sup> = 0. (85)

. (88)

(86)

(87)

(89)

<sup>2</sup> (*G*<sup>1</sup> cos *<sup>ϕ</sup>* <sup>−</sup> *<sup>G</sup>*<sup>2</sup> sin *<sup>ϕ</sup>*) <sup>=</sup> 0 (84)

problem. Thus, Eqs. (81) and (82) represent a method to construct a Lagrangian.

more detailed discussion of the last point has been given in I).

− *∂ ∂xk*

> + *∂ ∂xk*

*∂ ∂xi*

*∂ρL*<sup>0</sup> *∂ <sup>∂</sup>*2*<sup>ϑ</sup> ∂xk∂xi*

the second order derivatives) of (83)-(85) is given by

 1 √*ρ ∂ ∂x ∂ ∂x*

*hG*¯ <sup>1</sup> cos *<sup>ϕ</sup>* <sup>−</sup> *hG*¯ <sup>2</sup> sin *<sup>ϕ</sup>* <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup>

*<sup>L</sup>*<sup>0</sup> <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup> 2*m*

alternative derivation of spin in section 8.

in Eq. (75). The result may be written in the form

*<sup>G</sup>*<sup>1</sup> <sup>=</sup> *<sup>h</sup>*¯ 2*m* 1 *ρ ∂ ∂x ρ* 1

*<sup>G</sup>*<sup>2</sup> <sup>=</sup> *<sup>h</sup>*¯ 2*m* 1 *ρ ∂ ∂x ρ* 1

− *∂ ∂xk*

*∂ ∂xi*

*∂ρL*<sup>0</sup> *<sup>∂</sup> <sup>∂</sup>*2*<sup>ϕ</sup> ∂xk∂xi*

*∂ρL*<sup>0</sup> *<sup>∂</sup> <sup>∂</sup>*2*<sup>ρ</sup> ∂xk∂xi*

> *∂ρL*<sup>0</sup> *∂ ∂ϑ ∂xk*

> > + *∂ ∂xk*

<sup>√</sup>*<sup>ρ</sup>* <sup>−</sup> <sup>1</sup>

2*m* 1 <sup>2</sup> sin 2*<sup>ϑ</sup>*

A new adjustable parameter appears on the r.h.s of (86)- (88) which has been identified with *h*¯ 2/2*m*, where ¯*h* is again Planck's constant. This second ¯*h* is related to the quantum-mechanical principle of maximal disorder. It is in the present approach not related in any obvious way to the previous "classical" ¯*h* which denotes the amplitude of a rotation; compare, however, the

The solutions for *G*1, *G*<sup>2</sup> may be obtained with the help of the second condition (*Gs* = 0) listed

<sup>2</sup> sin 2*<sup>ϑ</sup>* sin *<sup>ϕ</sup> ∂ϕ*

<sup>2</sup> sin 2*<sup>ϑ</sup>* cos *<sup>ϕ</sup> ∂ϕ*

2*m* 1 *ρ ∂ ∂x* 

*hG*¯ <sup>3</sup> <sup>=</sup> <sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup>

+ *∂ ∂xk*

<sup>−</sup> *∂ρL*<sup>0</sup>

*∂ρL*<sup>0</sup> *∂ ∂ϕ ∂xk*

The variable *S* does not occur in (83)-(85) in agreement with our assumptions about the form of *L*0. It is easy to see that a proper solution (with vanishing variational contributions from

<sup>4</sup> sin2 *<sup>ϑ</sup>*

equations for *L*0, *G*<sup>1</sup> cos *ϕ* − *G*<sup>2</sup> sin *ϕ* and *G*3,

*∂ ∂xi*

− *∂ ∂xk* Eqs. (88) and (89) show that the first condition listed in (75) is also satisfied. The last condition is also fulfilled: *L*<sup>0</sup> can be written as *L*� <sup>0</sup> + Δ*L*0, where

$$L\_0' = -\frac{\hbar^2}{8m} \left[ \sin^2 \theta \left( \frac{\partial \varphi}{\partial \vec{\mathbf{x}}} \right)^2 - \left( \frac{\partial \theta}{\partial \vec{\mathbf{x}}} \right)^2 \right], \quad \Delta L\_0 = \frac{\hbar^2}{2m} \frac{1}{\sqrt{\rho}} \frac{\partial}{\partial \vec{\mathbf{x}}} \frac{\partial}{\partial \vec{\mathbf{x}}} \sqrt{\rho}. \tag{90}$$

and *L*� <sup>0</sup> fulfills (78). We see that *L*� <sup>0</sup> is a quantum-mechanical contribution to the rotational motion while Δ*L*<sup>0</sup> is related to the probability density of the ensemble (as could have been guessed considering the mathematical form of these terms). The last term is the same as in the spinless case [see (51)].

The remaining task is to show that the above solution for *L*<sup>0</sup> does indeed lead to a (appropriately generalized) Fisher functional. This can be done in several ways. The simplest is to use the following result due to Reginatto (1998b):

$$\int \mathrm{d}^3 \mathbf{x} \, (-\rho \mathbf{L}\_0) = \frac{\hbar^2}{8m} \sum\_{j=1}^3 \int \, \mathrm{d}^3 \mathbf{x} \, \sum\_{k=1}^3 \frac{1}{\rho^{(j)}} \left( \frac{\partial \rho^{(j)}}{\partial \mathbf{x}\_k} \right)^2 \,\tag{91}$$

$$\rho^{(1)} := \rho \sin^2 \frac{\theta}{2} \cos^2 \frac{\varphi}{2}, \quad \rho^{(2)} := \rho \sin^2 \frac{\theta}{2} \sin^2 \frac{\varphi}{2}, \quad \rho^{(3)} := \rho \cos^2 \frac{\theta}{2}. \tag{92}$$

The functions *ρ*(*j*) represent the probability that a particle is at space-time point *x*, *t* and *s* points into direction *j*. Inserting (86) the validity of (91) may easily be verified. The r.h.s. of Eq. (91) shows that the averaged value of *L*<sup>0</sup> represents indeed a Fisher functional, which completes our calculation of the 'quantum terms' *L*0, *Gi*.

Summarizing, our assumption, that under certain external conditions four state variables instead of two may be required, led to a nontrivial result, namely the four coupled differential equations (80), (79), (74), (73) with *L*0, *Gi* given by (86), (89), (88). The external condition which stimulates this splitting is given by a gauge field; the most important case is a magnetic field *B* but other possibilities do exist (see below). These four differential equations are equivalent to the much simpler differential equation

$$\left(\frac{\hbar}{\nu}\frac{\partial}{\partial t} + e\phi\right)\hat{\boldsymbol{\psi}} + \frac{1}{2m}\left(\frac{\hbar}{\nu}\frac{\partial}{\partial \vec{X}} - \frac{e}{c}\vec{A}\right)^2 \hat{\boldsymbol{\psi}} + \mu\_B \vec{\sigma} \vec{B} \hat{\boldsymbol{\psi}} = \mathbf{0},\tag{93}$$

which is linear in the complex-valued two-component state variable *ψ*ˆ and is referred to as Pauli equation (the components of the vector *σ* are the three Pauli matrices and *μ<sup>B</sup>* = −*eh*¯ /2*mc*). To see the equivalence one writes, see Takabayasi (1955), Holland (1995),

$$\hat{\psi} = \sqrt{\rho} \,\mathrm{e}^{\frac{\theta}{2}S} \begin{pmatrix} \cos\frac{\theta}{2} \mathrm{e}^{1\frac{\theta}{2}} \\\\ \imath \sin\frac{\theta}{2} \mathrm{e}^{-1\frac{\theta}{2}} \end{pmatrix} \prime \tag{94}$$

and evaluates the real and imaginary parts of the two scalar equations (93). This leads to the four differential equations (80), (79), (74) (73) and completes the present spin theory. In terms of the real-valued functions *ρ*, *S*, *ϑ*, *ϕ* the quantum-mechanical solutions (86), (88), (89) for *L*0, *Gi* look complicated in comparison to the classical solutions *L*<sup>0</sup> = 0, *Gi* = 0. In terms of the variable *ψ*ˆ the situation changes to the contrary: The quantum-mechanical equation becomes simple (linear) and the classical equation, which has been derived by Schiller (1962b), becomes complicated (nonlinear). The simplicity of the

• Chose a free Schrödinger equation with single-valued state function.

• Skip the multi-valued phase. The final state function is again single-valued.

multi-valued phase factor)

terms (potentials) in the differential equation.

 *h*¯ *ı ∂ ∂t* + 1 2*m h*¯ *ı ∂ ∂x*

the above steps) to a multi-valued state function *ψ*¯*multi*.

In order to investigate this possibility, let us rewrite Eq. (96) in the form *p*ˆ0 +

written down. Replace now the derivatives in (97) according to

*M*−<sup>1</sup>

 *h*¯ *ı* <sup>2</sup> *σi ∂ ∂xi σk ∂ ∂xk*

Thus, an alternative free Pauli-equation, besides (96) is given by

 *h*¯ *ı ∂ ∂t* + 1 2*m*

assuming the validity of the condition

This leads to the condition

solution, Eq. (101) takes the form

1 2*m*

• 'Turn on' the interaction by making the state function multi-valued (multiply it with a

A Statistical Derivation of Non-Relativistic Quantum Theory 163

• Shift the multi-valued phase factor to the left of all differential operators, creating new

Let us adapt this method for the derivation of spin (considering spin one-half only). The first and most important step is the identification of the free Pauli equation. An obvious choice is

where *ψ*¯ is a single-valued *two-component* state function; (96) is essentially a duplicate of Schrödinger's equation. We may of course add arbitrary vanishing terms to the expression in brackets. This seems trivial, but some of these terms may vanish *only* if applied to a single-valued *ψ*¯ and may lead to non-vanishing contributions if applied later (in the second of

> *p*ˆ*p*ˆ + *V*

> > *ı ∂ ∂xk*

*<sup>p</sup>*ˆ0 <sup>⇒</sup> *<sup>p</sup>*ˆ0*M*0, *<sup>p</sup>*<sup>ˆ</sup> <sup>⇒</sup> *<sup>p</sup>*ˆ*kMk*, (99)

(*p*ˆ*<sup>i</sup> <sup>p</sup>*ˆ*<sup>k</sup>* <sup>−</sup> *<sup>p</sup>*ˆ*<sup>k</sup> <sup>p</sup>*ˆ*i*) *<sup>ψ</sup>*¯ <sup>=</sup> 0. (100)

<sup>0</sup> *MiMk* = *Eδik* + *Tik*, (101)

*σiσ<sup>k</sup>* = *σ*0*δik* + *ıεiklσl*. (102)

+ *V* 

where *p*ˆ0 is an abbreviation for the first term of (96) and the spatial derivatives are given by

All terms in the bracket in (97) are to be multiplied with a 2*x*2 unit-matrix *E* which has not be

where *M*0, *Mk* are hermitian 2*x*2 matrices with constant coefficients, which should be constructed in such a way that the new equation agrees with (97) for single-valued *ψ*¯, i.e.

where *Tik* is a 2*x*2 matrix with two cartesian indices *i*, *k*, which obeys *Tik* = −*Tki*. A solution of (101) is given by *M*<sup>0</sup> = *σ*0, *Mi* = *σi*, where *σ*0, *σ<sup>i</sup>* are the four Pauli matrices. In terms of this

*<sup>p</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>p</sup>*ˆ*<sup>k</sup>ek*, *<sup>p</sup>*ˆ*<sup>k</sup>* <sup>=</sup> *<sup>h</sup>*¯

<sup>2</sup> + *<sup>V</sup>* 

*ψ*¯ = 0, (96)

*ψ*¯ = 0, (97)

. (98)

*ψ*¯ = 0. (103)

underlying physical principle (principle of maximal disorder) leads to a simple mathematical representation of the final basic equation (if a complex-valued state function is introduced). One may also say that the linearity of the equations is a consequence of this principle of maximal disorder. This is the deeper reason why it has been possible, see Klein (2009), to derive Schrödinger's equation from a set of assumptions including linearity.

Besides the Pauli equation we found, as a second important result of our spin calculation, that the following local force is compatible with the statistical constraint:

$$
\vec{F}^L + \vec{F}^I = e\left(\vec{E} + \frac{1}{c}\vec{v} \times \vec{B}\right) - \vec{\mu} \cdot \frac{\partial}{\partial \vec{X}} \vec{B}.\tag{95}
$$

Here, the velocity field ˜ *v*(*x*, *t*) and the magnetic moment field *μ*(*x*, *t*) = −(*e*/*mc*)*s*(*x*, *t*) have been replaced by corresponding particle quantities *v*(*t*) and *μ*(*t*); the dot denotes the inner product between *μ* and *B*. The first force in (95), the Lorentz force, has been derived here from first principles without any additional assumptions. The same cannot be said about the second force which takes this particular form as a consequence of some additional assumptions concerning the form of the 'internal force' *F<sup>R</sup>* [see (71)]. In particular, the field appearing in *F<sup>R</sup>* was arbitrary as well as the proportionality constant (g-factor of the electron) and had to be adjusted by hand. It is well-known that in a relativistic treatment the spin term appears automatically if the potentials are introduced. Interestingly, this unity is not restricted to the relativistic regime. Following Arunsalam (1970) and Gould (1995) we report in the next section an alternative (non-relativistic) derivation of spin, which does not contain any arbitrary fields or constants - but is unable to yield the expression (95) for the macroscopic electromagnetic forces.

In the present treatment spin has been introduced as a property of an ensemble and not of individual particles. Similar views may be found in the literature, see Ohanian (1986). Of course, it is difficult to imagine the properties of an ensemble as being completely independent from the properties of the particles it is made from. The question whether or not a property 'spin' can be ascribed to single particles is a subtle one. Formally, we could assign a probability of being in a state *i* (*i* = 1, 2) to a particle just as we assign a probability for being at a position *<sup>x</sup>* <sup>∈</sup> *<sup>R</sup>*3. But contrary to position, no classical meaning - and no classical measuring device can be associated with the discrete degree of freedom *i*. Experimentally, the measurement of the 'spin of a single electron' is - in contrast to the measurement of its position - a notoriously difficult task. Such experiments, and a number of other interesting questions related to spin, have been discussed by Morrison (2007).

#### **8. Spin as a consequence of a multi-valued phase**

As shown by Arunsalam (1970), Gould (1995), and others, spin in non-relativistic QT may be introduced in exactly the same manner as the electrodynamic potentials. In this section we shall apply a slightly modified version of their method and try to derive spin in an alternative way - which avoids the shortcoming mentioned in the last section.

Arunsalam (1970) and Gould (1995) introduce the potentials by applying the well-known minimal-coupling rule to the free Hamiltonian. In the present treatment this is achieved by making the quantity *S* multi-valued. The latter approach seems intuitively preferable considering the physical meaning of the corresponding classical quantity. Let us first review the essential steps [see Klein (2009) for more details] in the process of creating potentials in the *scalar* Schrödinger equation:

22 Will-be-set-by-IN-TECH

underlying physical principle (principle of maximal disorder) leads to a simple mathematical representation of the final basic equation (if a complex-valued state function is introduced). One may also say that the linearity of the equations is a consequence of this principle of maximal disorder. This is the deeper reason why it has been possible, see Klein (2009), to

Besides the Pauli equation we found, as a second important result of our spin calculation, that

been replaced by corresponding particle quantities *v*(*t*) and *μ*(*t*); the dot denotes the inner

first principles without any additional assumptions. The same cannot be said about the second force which takes this particular form as a consequence of some additional assumptions

*F<sup>R</sup>* was arbitrary as well as the proportionality constant (g-factor of the electron) and had to be adjusted by hand. It is well-known that in a relativistic treatment the spin term appears automatically if the potentials are introduced. Interestingly, this unity is not restricted to the relativistic regime. Following Arunsalam (1970) and Gould (1995) we report in the next section an alternative (non-relativistic) derivation of spin, which does not contain any arbitrary fields or constants - but is unable to yield the expression (95) for the macroscopic electromagnetic

In the present treatment spin has been introduced as a property of an ensemble and not of individual particles. Similar views may be found in the literature, see Ohanian (1986). Of course, it is difficult to imagine the properties of an ensemble as being completely independent from the properties of the particles it is made from. The question whether or not a property 'spin' can be ascribed to single particles is a subtle one. Formally, we could assign a probability of being in a state *i* (*i* = 1, 2) to a particle just as we assign a probability for being at a position *<sup>x</sup>* <sup>∈</sup> *<sup>R</sup>*3. But contrary to position, no classical meaning - and no classical measuring device can be associated with the discrete degree of freedom *i*. Experimentally, the measurement of the 'spin of a single electron' is - in contrast to the measurement of its position - a notoriously difficult task. Such experiments, and a number of other interesting questions related to spin,

As shown by Arunsalam (1970), Gould (1995), and others, spin in non-relativistic QT may be introduced in exactly the same manner as the electrodynamic potentials. In this section we shall apply a slightly modified version of their method and try to derive spin in an alternative

Arunsalam (1970) and Gould (1995) introduce the potentials by applying the well-known minimal-coupling rule to the free Hamiltonian. In the present treatment this is achieved by making the quantity *S* multi-valued. The latter approach seems intuitively preferable considering the physical meaning of the corresponding classical quantity. Let us first review the essential steps [see Klein (2009) for more details] in the process of creating potentials in

<sup>−</sup>*<sup>μ</sup>* · *<sup>∂</sup> ∂x* 

*v*(*x*, *t*) and the magnetic moment field *μ*(*x*, *t*) = −(*e*/*mc*)*s*(*x*, *t*) have

*B*. The first force in (95), the Lorentz force, has been derived here from

*F<sup>R</sup>* [see (71)]. In particular, the field appearing

*B*. (95)

derive Schrödinger's equation from a set of assumptions including linearity.

the following local force is compatible with the statistical constraint:

*F<sup>I</sup>* = *e E* + 1 *c <sup>v</sup>* <sup>×</sup> *B* 

 *F<sup>L</sup>* +

concerning the form of the 'internal force'

have been discussed by Morrison (2007).

the *scalar* Schrödinger equation:

**8. Spin as a consequence of a multi-valued phase**

way - which avoids the shortcoming mentioned in the last section.

Here, the velocity field ˜

product between *μ* and

in

forces.


Let us adapt this method for the derivation of spin (considering spin one-half only). The first and most important step is the identification of the free Pauli equation. An obvious choice is

$$
\left[\frac{\hbar}{\nu}\frac{\partial}{\partial t} + \frac{1}{2m}(\frac{\hbar}{\nu}\frac{\partial}{\partial \vec{x}})^2 + V\right]\psi = 0,\tag{96}
$$

where *ψ*¯ is a single-valued *two-component* state function; (96) is essentially a duplicate of Schrödinger's equation. We may of course add arbitrary vanishing terms to the expression in brackets. This seems trivial, but some of these terms may vanish *only* if applied to a single-valued *ψ*¯ and may lead to non-vanishing contributions if applied later (in the second of the above steps) to a multi-valued state function *ψ*¯*multi*.

In order to investigate this possibility, let us rewrite Eq. (96) in the form

$$
\left[\not p\_0 + \frac{1}{2m}\vec{\not p}\vec{\not p} + V\right]\bar{\psi} = 0,\tag{97}
$$

where *p*ˆ0 is an abbreviation for the first term of (96) and the spatial derivatives are given by

$$
\vec{\not p} = \not p\_k \vec{e}\_{k\prime} \qquad \qquad \not p\_k = \frac{\hbar}{n} \frac{\partial}{\partial \mathbf{x}\_k} . \tag{98}
$$

All terms in the bracket in (97) are to be multiplied with a 2*x*2 unit-matrix *E* which has not be written down. Replace now the derivatives in (97) according to

$$
\not p\_0 \Rightarrow \not p\_0 M\_{0\prime} \qquad \qquad \overline{\not p} \Rightarrow \overline{\not p}\_k M\_{k\prime} \tag{99}
$$

where *M*0, *Mk* are hermitian 2*x*2 matrices with constant coefficients, which should be constructed in such a way that the new equation agrees with (97) for single-valued *ψ*¯, i.e. assuming the validity of the condition

$$\left(\left(\not p\_i\not p\_k - \not p\_k\not p\_i\right)\not \psi = 0.\tag{100}$$

This leads to the condition

$$M\_0^{-1} M\_{\bar{i}} M\_{\bar{k}} = E \delta\_{i\bar{k}} + T\_{i\bar{k}\prime} \tag{101}$$

where *Tik* is a 2*x*2 matrix with two cartesian indices *i*, *k*, which obeys *Tik* = −*Tki*. A solution of (101) is given by *M*<sup>0</sup> = *σ*0, *Mi* = *σi*, where *σ*0, *σ<sup>i</sup>* are the four Pauli matrices. In terms of this solution, Eq. (101) takes the form

$$
\sigma\_{l}\sigma\_{k} = \sigma\_{0}\delta\_{lk} + \imath\varepsilon\_{ikl}\sigma\_{l}.\tag{102}
$$

Thus, an alternative free Pauli-equation, besides (96) is given by

$$
\left[\frac{\hbar}{\imath}\frac{\partial}{\partial t} + \frac{1}{2m} \left(\frac{\hbar}{\imath}\right)^2 \sigma\_i \frac{\partial}{\partial \mathbf{x}\_i} \sigma\_k \frac{\partial}{\partial \mathbf{x}\_k} + V\right] \bar{\Psi} = 0. \tag{103}
$$

The index *I* = 1, ...*N* is used to distinguish particles, while indices *i*, *k*, .. are used here to distinguish the 3*N* coordinates *q*1, ...*qn*. No new symbol has been introduced in (105) to distinguish the masses *mI* and *mi* since there is no danger of confusion in anyone of the formulas below. However, the indices of masses will be frequently written in the form *m*(*i*) in order to avoid ambiguities with regard to the summation convention. The symbol *Q* in arguments denotes dependence on all *q*1, ...*qn*. In order to generalize the results of section 8 a notation *xI*,*k*, *xI*, and *mI* (with *I* = 1, ..., *N* and *k* = 1, 2, 3) for coordinates, positions, and

A Statistical Derivation of Non-Relativistic Quantum Theory 165

The basic relations of section 2, generalized in an obvious way to *N* particles, take the form

*qk* <sup>=</sup> <sup>1</sup> *m*(*k*)

Here, *S* is a single-valued variable; the multi-valuedness will be added later, following the

The following calculations may be performed in complete analogy to the corresponding steps of section 2. For the present *N*−dimensional problem, the vanishing of the surface integrals, occurring in the course of various partial integrations, requires that *ρ* vanishes exponentially in arbitrary directions of the configuration space. The final conclusion to be drawn from

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>V</sup>* <sup>=</sup> *<sup>L</sup>*0,

The remaining problem is the determination of the unknown function *L*0, whose form is

*L*<sup>0</sup> can be determined using again the principle of minimal Fisher information, see I for details.

where *ES* = 0, *E<sup>ρ</sup>* = 0 are shorthand notations for the two basic equations (110) and (106). As before, Eqs. (111), (112) represent a method to construct a Lagrangian. After determination of *L*<sup>0</sup> the three relations listed in (111), (112) become redundant and (112) become the

The following calculation can be performed in complete analogy to the case *N* = 1 reported in section 5. All relations remain valid if the upper summation limit 3 is replaced by 3*N*. This

> *∂ρ ∂qj*

*∂ρ ∂qj* + 1 *m*(*j*)

1 *m*(*j*) *ρ*(*Q*, *t*) *m*(*k*)

*∂S*(*Q*, *t*) *∂qk*

<sup>d</sup>*<sup>Q</sup> ∂ρ ∂qk*

d*Q ρ* (*L* − *L*0) = 0 (111)

*Ea* = 0, *a* = *S*, *ρ*, (112)

*∂*2*ρ ∂qj∂qj* = 0 (106)

*L*<sup>0</sup> = 0, (110)

. (113)

*pk* (107)

*pk* = *Fk*(*Q*, *t*) (108)

d*Q ρ*(*Q*, *t*) *qk* (109)

*∂ ∂qk*

d d*t*

d d*t*

*qk* = 

*∂ρ*(*Q*, *t*) *<sup>∂</sup><sup>t</sup>* <sup>+</sup>

masses will be more convenient.

method of section 8.

Eqs. (106)- (109) takes the form

*n* ∑ *j*=1

1 2*m*(*j*)

constrained by the condition defined in Eq. (110).

fundamental equations of the *N*−particle theory.

is also true for the final result, which takes the form

*<sup>L</sup>*<sup>0</sup> <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup> 4*ρ*  − 1 2*ρ*

 *∂S ∂qj*

Its implementation in the present framework takes the form

*δ* d*t* 

<sup>2</sup> + *∂S*

The quantity in the bracket is the generalized Hamiltonian constructed by Arunsalam (1970) and Gould (1995). In the present approach gauge fields are introduced by means of a multi-valued phase. This leads to the same formal consequences as the minimal coupling rule but allows us to conclude that the second free Pauli equation (103) is *more appropriate* than the first, Eq. (96), because it is more general with regard to the consequences of multi-valuedness. This greater generality is due to the presence of the second term on the r.h.s. of (102).

The second step is to turn on the multi-valuedness in Eq. (103), *<sup>ψ</sup>*¯ <sup>⇒</sup> *<sup>ψ</sup>*¯*multi*, by multiplying *ψ*¯ with a multi-valued two-by-two matrix. This matrix must be chosen in such a way that the remaining steps listed above lead to Pauli's equation (93) in presence of an gauge field. Since in our case the final result (93) is known, this matrix may be found by performing the inverse process, i.e. performing a singular gauge transformation *ψ*ˆ = Γ*ψ*¯*multi* of Pauli's equation (93) from *ψ*ˆ to *ψ*¯*multi*, which *removes* all electrodynamic terms from (93) and creates Eq. (103). The final result for the matrix Γ is given by

$$\Gamma = E \exp\left\{ \imath \frac{e}{\hbar c} \int^{\bf x,t} \left[ \mathrm{d} \mathbf{x}\_k^{\prime} A\_k(\mathbf{x}^{\prime}, t^{\prime}) - c \mathrm{d} \mathbf{t}^{\prime} \phi(\mathbf{x}^{\prime}, t^{\prime}) \right] \right\}, \tag{104}$$

and agrees, apart from the unit matrix *E*, with the multi-valued factor introduced previously [see (17) and (52)] leading to the electrodynamic potentials. The inverse transition from (103) to (93), i.e. the creation of the potentials and the Zeeman term, can be performed by using the inverse of (104).

The Hamiltonian (103) derived by Arunsalam (1970) and Gould (1995) shows that spin can be described by means of the same abelian gauge theory that leads to the standard quantum mechanical gauge coupling terms; no new adjustable fields or parameters appear. The only requirement is that the appropriate free Pauli equation (103) is chosen as starting point. The theory of Dartora & Cabrera (2008), on the other hand, started from the alternative (from the present point of view inappropriate) free Pauli equation (96) and leads to the conclusion that spin must be described by a non-abelian gauge theory.

As far as our derivation of non-relativistic QT is concerned we have now two alternative, and in a sense complementary, possibilities to introduce spin. The essential step in the second (Arunsalam-Gould) method is the transition from (96) to the equivalent free Pauli equation (103). This step is a remarkable short-cut for the complicated calculations, performed in the last section, leading to the various terms required by the principle of minimal Fisher information. The Arunsalam-Gould method is unable to provide the shape (95) of the corresponding macroscopic forces but is very powerful insofar as no adjustable quantities are required. It will be used in the next section to perform the transition to an arbitrary number of particles.

#### **9. Transition to N particles as final step to non-relativistic quantum theory**

In this section the present derivation of non-relativistic QT is completed by deriving Schrödinger's equation for an arbitrary number *N* of particles or, more precisely, for statistical ensembles of identically prepared experimental arrangements involving *N* particles.

In order to generalize the results of sections 2 and 5, a convenient set of *n* = 3*N* coordinates *q*1, ...*qn* and masses *m*1, ...*mn* is defined by

$$\begin{aligned} \left(q\_1, q\_2, q\_3, \dots, q\_{n-2}, q\_{n-1}, q\_n\right) &= \left(\mathbf{x}\_1, y\_1, z\_1, \dots, \mathbf{x}\_{N\_I} y\_{N\_I} z\_N\right), \\ \left(m\_1, m\_2, m\_3, \dots, m\_{n-2}, m\_{n-1}, m\_n\right) &= \left(m\_1, m\_{1'}, m\_{1'}, \dots, m\_{N\_I}, m\_{N\_I}, m\_N\right). \end{aligned} \tag{105}$$

24 Will-be-set-by-IN-TECH

The quantity in the bracket is the generalized Hamiltonian constructed by Arunsalam (1970) and Gould (1995). In the present approach gauge fields are introduced by means of a multi-valued phase. This leads to the same formal consequences as the minimal coupling rule but allows us to conclude that the second free Pauli equation (103) is *more appropriate* than the first, Eq. (96), because it is more general with regard to the consequences of multi-valuedness.

The second step is to turn on the multi-valuedness in Eq. (103), *<sup>ψ</sup>*¯ <sup>⇒</sup> *<sup>ψ</sup>*¯*multi*, by multiplying *ψ*¯ with a multi-valued two-by-two matrix. This matrix must be chosen in such a way that the remaining steps listed above lead to Pauli's equation (93) in presence of an gauge field. Since in our case the final result (93) is known, this matrix may be found by performing the inverse process, i.e. performing a singular gauge transformation *ψ*ˆ = Γ*ψ*¯*multi* of Pauli's equation (93) from *ψ*ˆ to *ψ*¯*multi*, which *removes* all electrodynamic terms from (93) and creates Eq. (103). The

and agrees, apart from the unit matrix *E*, with the multi-valued factor introduced previously [see (17) and (52)] leading to the electrodynamic potentials. The inverse transition from (103) to (93), i.e. the creation of the potentials and the Zeeman term, can be performed by using the

The Hamiltonian (103) derived by Arunsalam (1970) and Gould (1995) shows that spin can be described by means of the same abelian gauge theory that leads to the standard quantum mechanical gauge coupling terms; no new adjustable fields or parameters appear. The only requirement is that the appropriate free Pauli equation (103) is chosen as starting point. The theory of Dartora & Cabrera (2008), on the other hand, started from the alternative (from the present point of view inappropriate) free Pauli equation (96) and leads to the conclusion that

As far as our derivation of non-relativistic QT is concerned we have now two alternative, and in a sense complementary, possibilities to introduce spin. The essential step in the second (Arunsalam-Gould) method is the transition from (96) to the equivalent free Pauli equation (103). This step is a remarkable short-cut for the complicated calculations, performed in the last section, leading to the various terms required by the principle of minimal Fisher information. The Arunsalam-Gould method is unable to provide the shape (95) of the corresponding macroscopic forces but is very powerful insofar as no adjustable quantities are required. It will be used in the next section to perform the transition to an arbitrary number

**9. Transition to N particles as final step to non-relativistic quantum theory**

ensembles of identically prepared experimental arrangements involving *N* particles.

In this section the present derivation of non-relativistic QT is completed by deriving Schrödinger's equation for an arbitrary number *N* of particles or, more precisely, for statistical

In order to generalize the results of sections 2 and 5, a convenient set of *n* = 3*N* coordinates

(*q*1, *<sup>q</sup>*2, *<sup>q</sup>*3, ...*qn*−2, *qn*−1, *qn*) = (*x*1, *<sup>y</sup>*1, *<sup>z</sup>*1, ..., *xN*, *yN*, *zN*),

(*m*1, *<sup>m</sup>*2, *<sup>m</sup>*3, ...*mn*−2, *mn*−1, *mn*) <sup>=</sup> (*m*1, *<sup>m</sup>*1, *<sup>m</sup>*1, ..., *mN*, *mN*, *mN*). (105)

, (104)

This greater generality is due to the presence of the second term on the r.h.s. of (102).

final result for the matrix Γ is given by

inverse of (104).

of particles.

Γ = *E* exp

spin must be described by a non-abelian gauge theory.

*q*1, ...*qn* and masses *m*1, ...*mn* is defined by

 *ı e hc*¯  *x*,*t* d*x*� *kAk*(*x*� , *t* � ) − *c*d*t* � *φ*(*x*� , *t* � )  The index *I* = 1, ...*N* is used to distinguish particles, while indices *i*, *k*, .. are used here to distinguish the 3*N* coordinates *q*1, ...*qn*. No new symbol has been introduced in (105) to distinguish the masses *mI* and *mi* since there is no danger of confusion in anyone of the formulas below. However, the indices of masses will be frequently written in the form *m*(*i*) in order to avoid ambiguities with regard to the summation convention. The symbol *Q* in arguments denotes dependence on all *q*1, ...*qn*. In order to generalize the results of section 8 a notation *xI*,*k*, *xI*, and *mI* (with *I* = 1, ..., *N* and *k* = 1, 2, 3) for coordinates, positions, and masses will be more convenient.

The basic relations of section 2, generalized in an obvious way to *N* particles, take the form

$$\frac{\partial \rho(Q,t)}{\partial t} + \frac{\partial}{\partial q\_k} \frac{\rho(Q,t)}{m\_{(k)}} \frac{\partial S(Q,t)}{\partial q\_k} = 0 \tag{106}$$

$$\frac{\text{d}}{\text{d}t}\overline{q\_k} = \frac{1}{m\_{(k)}}\overline{p\_k} \tag{107}$$

$$\frac{\mathbf{d}}{\mathbf{d}t}\overline{p\_k} = \overline{F\_k(\mathbf{Q}\_\prime t)}\tag{108}$$

$$\overline{q\_k} = \int \mathbf{d}Q \,\rho(Q\_\prime \, t) \, q\_k \tag{109}$$

Here, *S* is a single-valued variable; the multi-valuedness will be added later, following the method of section 8.

The following calculations may be performed in complete analogy to the corresponding steps of section 2. For the present *N*−dimensional problem, the vanishing of the surface integrals, occurring in the course of various partial integrations, requires that *ρ* vanishes exponentially in arbitrary directions of the configuration space. The final conclusion to be drawn from Eqs. (106)- (109) takes the form

$$\sum\_{j=1}^{n} \frac{1}{2m\_{(j)}} \left(\frac{\partial \mathcal{S}}{\partial q\_{j}}\right)^{2} + \frac{\partial \mathcal{S}}{\partial t} + V = L\_{0}, \qquad \int \mathrm{d}\mathcal{Q} \, \frac{\partial \rho}{\partial q\_{k}} L\_{0} = 0,\tag{110}$$

The remaining problem is the determination of the unknown function *L*0, whose form is constrained by the condition defined in Eq. (110).

*L*<sup>0</sup> can be determined using again the principle of minimal Fisher information, see I for details. Its implementation in the present framework takes the form

$$\delta \int \mathrm{d}t \int \mathrm{d}Q \,\rho \,(L - L\_0) = 0\tag{111}$$

$$E\_{\mathfrak{a}} = 0, \quad \mathfrak{a} = \mathcal{S}\_{\prime} \mathfrak{a}, \tag{112}$$

where *ES* = 0, *E<sup>ρ</sup>* = 0 are shorthand notations for the two basic equations (110) and (106). As before, Eqs. (111), (112) represent a method to construct a Lagrangian. After determination of *L*<sup>0</sup> the three relations listed in (111), (112) become redundant and (112) become the fundamental equations of the *N*−particle theory.

The following calculation can be performed in complete analogy to the case *N* = 1 reported in section 5. All relations remain valid if the upper summation limit 3 is replaced by 3*N*. This is also true for the final result, which takes the form

$$L\_0 = \frac{\hbar^2}{4\rho} \left[ -\frac{1}{2\rho} \frac{1}{m\_{(j)}} \frac{\partial \rho}{\partial q\_j} \frac{\partial \rho}{\partial q\_j} + \frac{1}{m\_{(j)}} \frac{\partial^2 \rho}{\partial q\_j \partial q\_j} \right]. \tag{113}$$

**10. The classical limit of quantum theory is a statistical theory**

*∂ρ <sup>∂</sup><sup>t</sup>* <sup>+</sup>

*∂S <sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>e</sup><sup>φ</sup>* <sup>+</sup>

*∂ ∂xk ρ m*

1 <sup>2</sup>*<sup>m</sup>* ∑ *k*

equations

aspects described by *ρ*.

measurements.

form *<sup>p</sup>*˜2(*x*, *<sup>t</sup>*)

The classical limit of Schrödinger's equation plays an important role for two topics discussed in the next section, namely the interpretation of QT and the particular significance of potentials in QT; to study these questions it is sufficient to consider a single-particle ensemble described by a single state function. This 'classical limit theory' is given by the two differential

A Statistical Derivation of Non-Relativistic Quantum Theory 167

 *∂S ∂xk*

 *∂S ∂xk*

which are obtained from Eqs. (50) and (51) by performing the limit ¯*h* → 0. The quantum mechanical theory (50) and (51) and the classical theory (118) and (119) show fundamentally the same mathematical structure; both are initial value problems for the variables *S* and *ρ* obeying two partial differential equations. The difference is the absence of the last term on the l.h.s. of (51) in the corresponding classical equation (119). This leads to a *decoupling* of *S* and *ρ* in (119); the identity of the classical object described by *S* is no longer affected by statistical

The field theory (118), (119) for the two 'not decoupled' fields *S* and *ρ* is obviously very different from classical mechanics which is formulated in terms of trajectories. The fact that one of these equations, namely (119), agrees with the Hamilton-Jacobi equation, does not change the situation since the presence of the continuity equation (118) cannot be neglected. On top of that, even if it could be neglected, Eq. (119) would still be totally different from classical mechanics: In order to construct particle trajectories from the partial differential equation (119) for the field *S*(*x*, *t*), a number of clearly defined mathematical manipulations, which are part of the classical theory of canonical transformations, see Greiner (1989), must be performed. The crucial point is that the latter theory is *not* part of QT and cannot be added 'by hand' in the limit ¯*h* → 0. Thus, (118), (119) is, like QT, an *indeterministic* theory predicting not values of single event observables but only probabilities, which must be verified by ensemble

Given that we found a solution *S*(*x*, *t*), *ρ*(*x*, *t*) of (118), (119) for given initial values, we may ask which experimental predictions can be made with the help of these quantities. Using the fields *p*˜(*x*, *t*), *E*˜(*x*, *t*) defined by Eqs. (19), (18), the Hamilton-Jacobi equation (119) takes the

The l.h.s. of (120) depends on the field *p*˜ in the same way as a classical particle Hamiltonian on the (gauge-invariant) kinetic momentum *p*. We conclude that the field *p*˜(*x*, *t*) describes a mapping from space-time points to particle momenta: If a particle (in an external electromagnetic field) is found at time *t* at the point *x*, then its kinetic momentum is given by *p*˜(*x*, *t*). This is not a deterministic prediction since we can not predict if a single particle will be or will not be at time *t* at point *x*; the present theory gives only a probability *ρ*(*x*, *t*) for such an event. Combining our findings about *p*˜(*x*, *t*) and *x* we conclude that the experimental prediction which can be made with the help of *S*(*x*, *t*), *ρ*(*x*, *t*) is given by the following phase

+ *V*(*x*, *t*) = *E*˜(*x*, *t*), (120)

2*m*

− *e c Ak* 

− *e c Ak* <sup>2</sup>

= 0, (118)

+ *V* = 0, (119)

If a complex-valued variable *ψ*, defined as in (52), is introduced, the two basic relations *Ea* = 0 may be condensed into the single differential equation,

$$
\left[\frac{\hbar}{\imath}\frac{\partial}{\partial t} + \sum\_{I=1}^{N} \frac{1}{2m\_{(I)}} \left(\frac{\hbar}{\imath}\frac{\partial}{\partial \mathbf{x}\_{I,k}}\right) \left(\frac{\hbar}{\imath}\frac{\partial}{\partial \mathbf{x}\_{I,k}}\right) + V\right] \psi = 0,\tag{114}
$$

which is referred to as *N*−particle Schrödinger equation, rewritten here in the more familiar form using particle indices. As is well-known, only approximate solutions of this partial differential equation of order 3*N* + 1 exist for realistic systems. The inaccessible complexity of quantum-mechanical solutions for large *N* is not reflected in the abstract Hilbert space structure (which is sometimes believed to characterize the whole of QT) but plays probably a decisive role for a proper description of the mysterious relation between QT and the macroscopic world.

Let us now generalize the Arunsalam-Gould method, discussed in section 8, to an arbitrary number of particles. We assume, that the considered *N*−particle statistical ensemble responds in 2*<sup>N</sup>* ways to the external electromagnetic field. This means we restrict ourselves again, like in section 6, 7 to spin one-half. Then, the state function may be written as *ψ*(*x*1,*s*1; ....*xI*,*sI*; ...*xN*,*sN*) where *sI* = 1, 2. In the first of the steps listed at the beginning of section 8, a differential equation, which is equivalent to Eq. (114) for single-valued *ψ* but may give non-vanishing contributions for multi-valued *ψ*, has to be constructed. The proper generalization of Eq. (103) to arbitrary *N* takes the form

$$
\left[\frac{\hbar}{\hbar}\frac{\partial}{\partial t} + \sum\_{I=1}^{N} \frac{1}{2m\_{(I)}} \left(\frac{\hbar}{\hbar}\sigma\_{k}^{(I)}\frac{\partial}{\partial \mathbf{x}\_{I,k}}\right) \left(\frac{\hbar}{\hbar}\sigma\_{l}^{(I)}\frac{\partial}{\partial \mathbf{x}\_{I,l}}\right) + V\right] \psi = 0,\tag{115}
$$

where the Pauli matrices *<sup>σ</sup>*(*I*) *<sup>k</sup>* operate by definition only on the two-dimensional subspace spanned by the variable *sI*. In the second step we perform the replacement

$$\Psi \Rightarrow \exp\left\{ -\frac{1}{\hbar} \sum\_{I=1}^{N} \frac{e\_I}{c} \sum\_{k=1}^{3} \int^{\vec{\mathbf{x}}\_{I,t}} \left[ \mathbf{d} \mathbf{x}\_{I,k}' A\_k(\mathbf{x}\_{I\prime}' t\prime) - c \mathbf{d} t\prime \phi(\mathbf{x}\_{I\prime}' t\prime) \right] \right\} \psi,\tag{116}$$

using a multi-valued phase factor, which is an obvious generalization of Eq. (104). The remaining steps, in the listing of section 8, lead in a straightforward way to the final result

$$\begin{aligned} \left[\frac{\hbar}{l}\frac{\partial}{\partial t} + \sum\_{I=1}^{N} \varepsilon\_I \Phi(\mathbf{x}\_I, t) + \sum\_{I=1}^{N} \sum\_{k=1}^{3} \frac{1}{2m\_{(I)}} \left(\frac{\hbar}{l} \frac{\partial}{\partial \mathbf{x}\_{I,k}} - \frac{e\_I}{c} A\_k(\mathbf{x}\_{I,t})\right)^2 \\ + \sum\_{I=1}^{N} \mu\_B^{(I)} \sigma\_k^{(I)} B\_k(\mathbf{x}\_I, t) + V(\mathbf{x}\_1, \dots, \mathbf{x}\_N, t) \right] \psi &= 0 \end{aligned} \tag{117}$$

where *<sup>μ</sup>*(*I*) *<sup>B</sup>* <sup>=</sup> <sup>−</sup>*he*¯ *<sup>I</sup>*/2*mI <sup>c</sup>* and *B* = rot*A* . The mechanical potential *V*(*x*1, ..., *xN*, *t*) describes a general many-body force but contains, of course, the usual sum of two-body potentials as a special case. Eq. (117) is the *N*−body version of Pauli's equation and completes - in the sense discussed at the very beginning of this paper - the present derivation of non-relativistic QT.

26 Will-be-set-by-IN-TECH

If a complex-valued variable *ψ*, defined as in (52), is introduced, the two basic relations *Ea* = 0

which is referred to as *N*−particle Schrödinger equation, rewritten here in the more familiar form using particle indices. As is well-known, only approximate solutions of this partial differential equation of order 3*N* + 1 exist for realistic systems. The inaccessible complexity of quantum-mechanical solutions for large *N* is not reflected in the abstract Hilbert space structure (which is sometimes believed to characterize the whole of QT) but plays probably a decisive role for a proper description of the mysterious relation between QT and the

Let us now generalize the Arunsalam-Gould method, discussed in section 8, to an arbitrary number of particles. We assume, that the considered *N*−particle statistical ensemble responds in 2*<sup>N</sup>* ways to the external electromagnetic field. This means we restrict ourselves again, like in section 6, 7 to spin one-half. Then, the state function may be written as *ψ*(*x*1,*s*1; ....*xI*,*sI*; ...*xN*,*sN*) where *sI* = 1, 2. In the first of the steps listed at the beginning of section 8, a differential equation, which is equivalent to Eq. (114) for single-valued *ψ* but may give non-vanishing contributions for multi-valued *ψ*, has to be constructed. The proper

> *∂ ∂xI*,*<sup>k</sup>*

 *h*¯ *ı <sup>σ</sup>*(*I*) *l*

*∂ ∂xI*,*<sup>l</sup>*

*<sup>k</sup>* operate by definition only on the two-dimensional subspace

 + *V* 

− *eI c*

 *ψ* = 0

*B* = rot*A* . The mechanical potential *V*(*x*1, ..., *xN*, *t*) describes a

*Ak*(*xI*, *t*)

<sup>2</sup>

 *h*¯ *ı ∂ ∂xI*,*<sup>k</sup>*  + *V* 

*ψ* = 0, (114)

*ψ* = 0, (115)

*ψ*, (116)

, (117)

may be condensed into the single differential equation,

generalization of Eq. (103) to arbitrary *N* takes the form

1 2*m*(*I*)

*N* ∑ *I*=1

*eI c*

*eIφ*(*xI*, *t*) +

*<sup>μ</sup>*(*I*) *<sup>B</sup> <sup>σ</sup>*(*I*)  *h*¯ *ı <sup>σ</sup>*(*I*) *k*

spanned by the variable *sI*. In the second step we perform the replacement

3 ∑ *k*=1 *xI*,*<sup>t</sup>*

*N* ∑ *I*=1

3 ∑ *k*=1

*<sup>k</sup> Bk*(*xI*, *t*) + *V*(*x*1, ..., *xN*, *t*)

general many-body force but contains, of course, the usual sum of two-body potentials as a special case. Eq. (117) is the *N*−body version of Pauli's equation and completes - in the sense discussed at the very beginning of this paper - the present derivation of non-relativistic QT.

d*x*�

using a multi-valued phase factor, which is an obvious generalization of Eq. (104). The remaining steps, in the listing of section 8, lead in a straightforward way to the final result

> 1 2*m*(*I*)

*I*,*kAk*(*x*� *I*, *t* � ) − *c*d*t* � *φ*(*x*� *I*, *t* � ) 

> *h*¯ *ı ∂ ∂xI*,*<sup>k</sup>*

 *h*¯ *ı ∂ ∂t* + *N* ∑ *I*=1

*ψ* ⇒ exp

 *h*¯ *ı ∂ ∂t* + *N* ∑ *I*=1

where *<sup>μ</sup>*(*I*)

 − *ı h*¯

+ *N* ∑ *I*=1

*<sup>B</sup>* <sup>=</sup> <sup>−</sup>*he*¯ *<sup>I</sup>*/2*mI <sup>c</sup>* and

where the Pauli matrices *<sup>σ</sup>*(*I*)

1 2*m*(*I*)  *h*¯ *ı ∂ ∂xI*,*<sup>k</sup>*

 *h*¯ *ı ∂ ∂t* + *N* ∑ *I*=1

macroscopic world.

#### **10. The classical limit of quantum theory is a statistical theory**

The classical limit of Schrödinger's equation plays an important role for two topics discussed in the next section, namely the interpretation of QT and the particular significance of potentials in QT; to study these questions it is sufficient to consider a single-particle ensemble described by a single state function. This 'classical limit theory' is given by the two differential equations

$$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_k} \frac{\rho}{m} \left( \frac{\partial \mathbf{S}}{\partial \mathbf{x}\_k} - \frac{e}{c} A\_k \right) = \mathbf{0},\tag{118}$$

$$\frac{\partial \mathcal{S}}{\partial t} + e\phi + \frac{1}{2m} \sum\_{k} \left( \frac{\partial \mathcal{S}}{\partial \mathbf{x}\_k} - \frac{e}{c} A\_k \right)^2 + V = 0,\tag{119}$$

which are obtained from Eqs. (50) and (51) by performing the limit ¯*h* → 0. The quantum mechanical theory (50) and (51) and the classical theory (118) and (119) show fundamentally the same mathematical structure; both are initial value problems for the variables *S* and *ρ* obeying two partial differential equations. The difference is the absence of the last term on the l.h.s. of (51) in the corresponding classical equation (119). This leads to a *decoupling* of *S* and *ρ* in (119); the identity of the classical object described by *S* is no longer affected by statistical aspects described by *ρ*.

The field theory (118), (119) for the two 'not decoupled' fields *S* and *ρ* is obviously very different from classical mechanics which is formulated in terms of trajectories. The fact that one of these equations, namely (119), agrees with the Hamilton-Jacobi equation, does not change the situation since the presence of the continuity equation (118) cannot be neglected. On top of that, even if it could be neglected, Eq. (119) would still be totally different from classical mechanics: In order to construct particle trajectories from the partial differential equation (119) for the field *S*(*x*, *t*), a number of clearly defined mathematical manipulations, which are part of the classical theory of canonical transformations, see Greiner (1989), must be performed. The crucial point is that the latter theory is *not* part of QT and cannot be added 'by hand' in the limit ¯*h* → 0. Thus, (118), (119) is, like QT, an *indeterministic* theory predicting not values of single event observables but only probabilities, which must be verified by ensemble measurements.

Given that we found a solution *S*(*x*, *t*), *ρ*(*x*, *t*) of (118), (119) for given initial values, we may ask which experimental predictions can be made with the help of these quantities. Using the fields *p*˜(*x*, *t*), *E*˜(*x*, *t*) defined by Eqs. (19), (18), the Hamilton-Jacobi equation (119) takes the form *<sup>p</sup>*˜2(*x*, *<sup>t</sup>*)

$$\frac{\tilde{p}^2(\mathbf{x},t)}{2m} + V(\mathbf{x},t) = \mathcal{E}(\mathbf{x},t),\tag{120}$$

The l.h.s. of (120) depends on the field *p*˜ in the same way as a classical particle Hamiltonian on the (gauge-invariant) kinetic momentum *p*. We conclude that the field *p*˜(*x*, *t*) describes a mapping from space-time points to particle momenta: If a particle (in an external electromagnetic field) is found at time *t* at the point *x*, then its kinetic momentum is given by *p*˜(*x*, *t*). This is not a deterministic prediction since we can not predict if a single particle will be or will not be at time *t* at point *x*; the present theory gives only a probability *ρ*(*x*, *t*) for such an event. Combining our findings about *p*˜(*x*, *t*) and *x* we conclude that the experimental prediction which can be made with the help of *S*(*x*, *t*), *ρ*(*x*, *t*) is given by the following phase

meaningful physical observables4. On the other hand, it cannot be expected that the rules (1) hold for arbitrary functions of *x*, *p*; each case has to be investigated separately. Thus, the

A Statistical Derivation of Non-Relativistic Quantum Theory 169

The fundamental Ehrenfest-like relations of the present theory establish [like the formal rules (1)] a *correspondence* between particle mechanics and QT. Today, philosophical questions concerning, in particular, the 'reality' of particles play an important role in the thinking of some physicists. So: 'What is this theory about.. ?' While the present author is no expert in this field, the concept of *indeterminism*, as advocated by the philosopher Popper (1982), seems

The present method to introduce gauge fields by means of a multi-valued dynamic variable ('phase function') has been invented many years ago but leads, in the context of the present statistical theory, nevertheless to several new results. In particular, it has been shown in section 3, that only the Lorentz force can exist as fundamental macroscopic force if the statistical assumptions of section 2 are valid. It is the only force (in the absence of spin effects, see the remarks below) that can be incorporated in a 'standard' differential equation for the dynamical variables *ρ*, *S*. The corresponding terms in the statistical field equations, *representing* the Lorentz force, are given by the familiar gauge (minimal) coupling terms containing the potentials. The important fact that all forces in nature follow this 'principle of minimal coupling' is commonly explained as a consequence of local gauge invariance. The

Let us use the following symbolic notation to represent the relation between the local force

*E* + *e c <sup>v</sup>* <sup>×</sup>

inverse is not true. Roughly speaking, the local fields are 'derivatives' of the potentials - and the potentials are 'integrals' of the local field; this mathematical relation reflects the physical

as well as their non-uniqueness. It might seem that the logical chain displayed in (122) is already realized in the classical treatment of a particle-field system, where potentials have to be introduced in order to construct a Lagrangian, see e.g. Landau & Lifshitz (1967). However, in this case, the form of the local force is not derived but postulated. The present treatment 'explains' the form of the Lagrangian - as a consequence of the basic assumptions listed in

The generalization of the present theory to spin, reported in sections 6 and 7, leads to a

*<sup>B</sup>* <sup>→</sup> *<sup>μ</sup>* · *<sup>∂</sup>*

r.h.s. The points discussed after Eq. (122) apply here as well [As a matter of fact we consider

*B* as a unique physical quantity; it would not be unique if it would be defined in terms of the tensor on the r.h.s. of (123)]. We see here a certain analogy between gauge and spin interaction terms. Unfortunately, the derivation of the spin force on the r.h.s. of (123) requires - in contrast

<sup>4</sup> As indicated by preliminary calculations of the angular momentum relation corresponding to Eq. (8)

*∂x* 

*B*, on the l.h.s. of (123), plays the role of a 'potential' for the local force on the

*μ*

to the Lorentz force - additional assumptions (see the remarks in sections 7, 8).

*B* are uniquely defined in terms of the potentials *φ* and *A* [see (23)] while the

*B*. (122)

*B*. (123)

*E* and *B*,

<sup>Φ</sup>, *<sup>A</sup>* <sup>⇒</sup> *<sup>e</sup>*

role of the potentials *φ* and *A* as statistical representatives of the the local fields

breakdown of (1), as expressed by Groenewold's theorem, is no surprise.

to provide an appropriate philosophical basis for the present work.

present treatment offers an alternative explanation.

correspondence similar to Eq. (122), namely

The fields

section 2.

The term linear in

*E* and

and the terms representing its action in a statistical context:

space probability density:

$$w(\mathbf{x}, p, t) = \rho(\mathbf{x}, t)\delta^{(3)}(p - \frac{\partial \bar{\mathcal{S}}(\mathbf{x}, t)}{\partial \mathbf{x}}).\tag{121}$$

Eq. (121) confirms our claim that the classical limit theory is a statistical theory. The one-dimensional version of (121) has been obtained before by means of a slightly different method in I. The deterministic element [realized by the delta-function shaped probability in (121)] contained in the classical statistical theory (118), (119) is *absent* in QT, see I.

Eqs. (118), (119) constitute the mathematically well-defined limit ¯*h* → 0 of Schrödinger's equation. Insofar as there is general agreement with regard to two points, namely that (i) 'non-classicality' (whatever this may mean precisely) is expressed by a nonzero ¯*h*, and that (ii) Schrödinger's equation is the most important relation of quantum theory, one would also expect general agreement with regard to a further point, namely that Eqs. (118), (119) present essentially (for a three-dimensional configuration space) the *classical limit* of quantum mechanics. But this is, strangely enough, not the case. With a few exceptions, see Van Vleck (1928), Schiller (1962a), Ballentine (1994), Shirai (1998), Klein (2009), most works (too many to be quoted) take it for granted that the classical limit of quantum theory is classical mechanics. The objective of papers like Rowe (1991), Werner & Wolff (1995), Landau (1996), Allori & Zanghi (2009) devoted to "..the classical limit of quantum mechanics.." is very often not the problem: "*what is* the classical limit of quantum mechanics ?" but rather: "*how to bridge the gap* between quantum mechanics and classical mechanics ?". Thus, the fact that classical mechanics is the classical limit of quantum mechanics is considered as *evident* and any facts not compatible with it - like Eqs. (118), (119) - are denied.

What, then, is the reason for this widespread denial of reality ? One of the main reasons is the principle of reductionism which still rules the thinking of most physicists today. The reductionistic ideal is a hierarchy of physical theories; better theories have an enlarged domain of validity and contain 'inferior' theories as special cases. This principle which has been extremely successful in the past *dictates* that classical mechanics is a special case of quantum theory. Successful as this idea might have been during a long period of time it is not necessarily universally true; quantum mechanics and classical mechanics describe different domains of reality, both may be true in their own domains of validity. Many phenomena in nature indicate that the principle of reductionism (alone) is insufficient to describe reality, see Laughlin & Pines (2000). Releasing ourselves from the metaphysical principle of reductionism, we accept that the classical limit of quantum mechanics for a three-dimensional configuration space is the statistical theory defined by Eqs. (118), (119). It is clear that this theory is not realized in nature (with the same physical meaning of the variables) because ¯*h* is different from zero. But this is a different question and does not affect the conclusion.

#### **11. Extended discussion**

In this paper it has been shown, continuing the work of I, that the basic differential equation of non-relativistic QT may be derived from a number of clearly defined assumptions of a statistical nature. Although this does not exclude the possibility of other derivations, we consider this success as a strong argument in favor of the statistical interpretation of QT. This result explains also, at least partly, the success of the canonical quantization rules (1). Strictly speaking, these rules have only be derived for a particular (though very important) special case, the Hamiltonian. However, one can expect that (1) can be verified for all 28 Will-be-set-by-IN-TECH

Eq. (121) confirms our claim that the classical limit theory is a statistical theory. The one-dimensional version of (121) has been obtained before by means of a slightly different method in I. The deterministic element [realized by the delta-function shaped probability

Eqs. (118), (119) constitute the mathematically well-defined limit ¯*h* → 0 of Schrödinger's equation. Insofar as there is general agreement with regard to two points, namely that (i) 'non-classicality' (whatever this may mean precisely) is expressed by a nonzero ¯*h*, and that (ii) Schrödinger's equation is the most important relation of quantum theory, one would also expect general agreement with regard to a further point, namely that Eqs. (118), (119) present essentially (for a three-dimensional configuration space) the *classical limit* of quantum mechanics. But this is, strangely enough, not the case. With a few exceptions, see Van Vleck (1928), Schiller (1962a), Ballentine (1994), Shirai (1998), Klein (2009), most works (too many to be quoted) take it for granted that the classical limit of quantum theory is classical mechanics. The objective of papers like Rowe (1991), Werner & Wolff (1995), Landau (1996), Allori & Zanghi (2009) devoted to "..the classical limit of quantum mechanics.." is very often not the problem: "*what is* the classical limit of quantum mechanics ?" but rather: "*how to bridge the gap* between quantum mechanics and classical mechanics ?". Thus, the fact that classical mechanics is the classical limit of quantum mechanics is considered as *evident* and any facts

What, then, is the reason for this widespread denial of reality ? One of the main reasons is the principle of reductionism which still rules the thinking of most physicists today. The reductionistic ideal is a hierarchy of physical theories; better theories have an enlarged domain of validity and contain 'inferior' theories as special cases. This principle which has been extremely successful in the past *dictates* that classical mechanics is a special case of quantum theory. Successful as this idea might have been during a long period of time it is not necessarily universally true; quantum mechanics and classical mechanics describe different domains of reality, both may be true in their own domains of validity. Many phenomena in nature indicate that the principle of reductionism (alone) is insufficient to describe reality, see Laughlin & Pines (2000). Releasing ourselves from the metaphysical principle of reductionism, we accept that the classical limit of quantum mechanics for a three-dimensional configuration space is the statistical theory defined by Eqs. (118), (119). It is clear that this theory is not realized in nature (with the same physical meaning of the variables) because ¯*h* is different from zero. But this is a different question and does not affect

In this paper it has been shown, continuing the work of I, that the basic differential equation of non-relativistic QT may be derived from a number of clearly defined assumptions of a statistical nature. Although this does not exclude the possibility of other derivations, we consider this success as a strong argument in favor of the statistical interpretation of QT. This result explains also, at least partly, the success of the canonical quantization rules (1). Strictly speaking, these rules have only be derived for a particular (though very important) special case, the Hamiltonian. However, one can expect that (1) can be verified for all

(*<sup>p</sup>* <sup>−</sup> *<sup>∂</sup>S*˜(*x*, *<sup>t</sup>*)

*<sup>∂</sup><sup>x</sup>* ). (121)

*w*(*x*, *p*, *t*) = *ρ*(*x*, *t*)*δ*(3)

in (121)] contained in the classical statistical theory (118), (119) is *absent* in QT, see I.

not compatible with it - like Eqs. (118), (119) - are denied.

space probability density:

the conclusion.

**11. Extended discussion**

meaningful physical observables4. On the other hand, it cannot be expected that the rules (1) hold for arbitrary functions of *x*, *p*; each case has to be investigated separately. Thus, the breakdown of (1), as expressed by Groenewold's theorem, is no surprise.

The fundamental Ehrenfest-like relations of the present theory establish [like the formal rules (1)] a *correspondence* between particle mechanics and QT. Today, philosophical questions concerning, in particular, the 'reality' of particles play an important role in the thinking of some physicists. So: 'What is this theory about.. ?' While the present author is no expert in this field, the concept of *indeterminism*, as advocated by the philosopher Popper (1982), seems to provide an appropriate philosophical basis for the present work.

The present method to introduce gauge fields by means of a multi-valued dynamic variable ('phase function') has been invented many years ago but leads, in the context of the present statistical theory, nevertheless to several new results. In particular, it has been shown in section 3, that only the Lorentz force can exist as fundamental macroscopic force if the statistical assumptions of section 2 are valid. It is the only force (in the absence of spin effects, see the remarks below) that can be incorporated in a 'standard' differential equation for the dynamical variables *ρ*, *S*. The corresponding terms in the statistical field equations, *representing* the Lorentz force, are given by the familiar gauge (minimal) coupling terms containing the potentials. The important fact that all forces in nature follow this 'principle of minimal coupling' is commonly explained as a consequence of local gauge invariance. The present treatment offers an alternative explanation.

Let us use the following symbolic notation to represent the relation between the local force and the terms representing its action in a statistical context:

$$
\Phi \ll \vec{A} \Rightarrow \varepsilon \vec{E} + \frac{\varepsilon}{c} \vec{v} \times \vec{B}.\tag{122}
$$

The fields *E* and *B* are uniquely defined in terms of the potentials *φ* and *A* [see (23)] while the inverse is not true. Roughly speaking, the local fields are 'derivatives' of the potentials - and the potentials are 'integrals' of the local field; this mathematical relation reflects the physical role of the potentials *φ* and *A* as statistical representatives of the the local fields *E* and *B*, as well as their non-uniqueness. It might seem that the logical chain displayed in (122) is already realized in the classical treatment of a particle-field system, where potentials have to be introduced in order to construct a Lagrangian, see e.g. Landau & Lifshitz (1967). However, in this case, the form of the local force is not derived but postulated. The present treatment 'explains' the form of the Lagrangian - as a consequence of the basic assumptions listed in section 2.

The generalization of the present theory to spin, reported in sections 6 and 7, leads to a correspondence similar to Eq. (122), namely

$$
\vec{\mu}\vec{\mu}\vec{B} \to \vec{\mu} \cdot \frac{\partial}{\partial \vec{\lambda}}\vec{B}.\tag{123}
$$

The term linear in *B*, on the l.h.s. of (123), plays the role of a 'potential' for the local force on the r.h.s. The points discussed after Eq. (122) apply here as well [As a matter of fact we consider *B* as a unique physical quantity; it would not be unique if it would be defined in terms of the tensor on the r.h.s. of (123)]. We see here a certain analogy between gauge and spin interaction terms. Unfortunately, the derivation of the spin force on the r.h.s. of (123) requires - in contrast to the Lorentz force - additional assumptions (see the remarks in sections 7, 8).

<sup>4</sup> As indicated by preliminary calculations of the angular momentum relation corresponding to Eq. (8)

equations (118), (119) with those of the quantum mechanical, linear equations, (50), (51), in order to find out which 'typical quantum-mechanical features' are already given by statistical (nonlocal) correlations of the classical limit theory and which features are really

A Statistical Derivation of Non-Relativistic Quantum Theory 171

In the present paper it has been shown that the method reported in I, for the derivation of Schrödingers's equation, can be generalized in such a way that essentially all aspects of non-relativistic QT are taken into account. The success of this derivation from statistical origins is interpreted as an argument in favor of the SI. The treatment of gauge fields and spin in the present statistical framework led to several remarkable new insights. We understand now why potentials (and not local fields) occur in the field equations of QT. The non-uniqueness of the potentials and the related concept of gauge invariance is not a mystery any more. Spin is derived as a kind of two-valuedness of a statistical ensemble. The local forces associated with the gauge potentials, the Lorentz force and the force experienced by a particle with magnetic moment, can also be derived. Apart from some open questions in the area of non-relativistic physics, a major problem for future research is a relativistic

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Ali, A. H. (2009). The ensemble quantum state of a single particle, *Int. J. Theor. Phys.*

Allori, V. & Zanghi, N. (2009). On the classical limit of quantum mechanics, *Foundations of*

Arunsalam, V. (1970). Hamiltonians and wave equations for particles of spin 0 and spin <sup>1</sup>

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Ballentine, L. E. (1994). Inadequacy of Ehrenfest's theorem to characterize the classical regime,

Belinfante, F. J. (1975). *Measurements and Time Reversal in Objective Quantum Theory*, Pergamon

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Dirac, P. A. M. (1931). Quantised singularities in the electromagnetic field, *Proc. R. Soc. London,*

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2

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quantum-theoretical in nature - related to the nonzero value of ¯*h*.

**12. Summary**

**13. References**

generalization of the present theory.

*Phys. Rev.* 115(3): 485.

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*Physical Review* A 50: 2854–2859.

Our notation for potentials *φ*, *A* , fields *E*, *B*, and parameters *e*, *c* suggests that these quantities are electrodynamical in nature. However, this is not necessarily true. By definition, the fields *E*, *B* obey four equations (the homogeneous Maxwell equations), which means that additional conditions are required in order to determine these six fields. The most familiar possibility is, of course, the second pair of Maxwell's equations. A second possible realization for the fields *E*, *B* is given by the *inertial* forces acting on a mass *m* in an arbitrarily accelerated reference frame, see Hughes (1992). The inertial gauge field may also lead to a spin response of the ensemble; such experiments have been proposed by Mashhoon & Kaiser (2006). It is remarkable that the present theory establishes a (admittedly somewhat vague) link between the two extremely separated physical fields of inertia and QT.

It is generally assumed that the electrodynamic potentials have a particular significance in QT which they do not have in classical physics. Let us analyze this statement in detail. The first part of the statement, concerning the significance of the potentials, is of course true. The second part, asserting that in classical physics all external influences *can* be described solely in terms of field strengths, is wrong. More precisely, it is true for classical mechanics but not for classical physics in general. A counterexample - a theory belonging to classical physics but with potentials playing an indispensable role - is provided by the classical limit (118),(119) of Schrödinger's equation. In this field theory the potentials play an indispensable role because (in contrast to particle theories, like the canonical equations) no further derivatives of the Hamiltonian, which could restore the fields, are to be performed. This means that the significance of the potentials is not restricted to quantum theory but rather holds for the whole class of *statistical* theories discussed above, which contains both quantum theory and its classical limit theory as special cases. This result is in agreement with the statistical interpretation of potentials proposed in section 3.

The precise characterization of the role of the potentials is of particular importance for the interpretation of the Aharonov-Bohm effect. The 'typical quantum-mechanical features' observed in these phase shift experiments should be identified by comparing the quantum mechanical results not with classical mechanics but with the predictions of the classical statistical theory (118), (119). The predictions of two statistical theories, both of which use potentials to describe the influence of the external field, have to be compared.

The limiting behavior of Schrödinger's equation as ¯*h* → 0, discussed in section 10, is very important for the proper interpretation of QT. The erroneous belief (wish) that this limit can (must) be identified with classical mechanics is closely related to the erroneous belief that QT is able to describe the dynamics of individual particles. In this respect QT is obviously an *incomplete* theory, as has been pointed out many times before, during the last eighty years, see e.g. Einstein (1949), Margenau (1935) , Ballentine (1970), Held (2008). Unfortunately, this erroneous opinion is historically grown and firmly established in our thinking as shown by the ubiquitous use of phrases like 'the wave function of the electron'. But it is clear that an erroneous identification of the domain of validity of a physical theory will automatically create all kinds of mysteries and unsolvable problems - and this is exactly what happens. Above, we have identified one of the (more subtle) problems of this kind, concerning the role of potentials in QT, but many more could be found. Generalizing the above argumentation concerning potentials, we claim that characteristic features of QT cannot be identified by comparison with classical mechanics. Instead, quantum theory should be compared with its classical limit, which is in the present 3*D*-case given by (118), (119) - we note in this context that several 'typical' quantum phenomena have been explained by Kirkpatrick (2003) in terms of classical probability theory. One has to compare the solutions of the classical, nonlinear equations (118), (119) with those of the quantum mechanical, linear equations, (50), (51), in order to find out which 'typical quantum-mechanical features' are already given by statistical (nonlocal) correlations of the classical limit theory and which features are really quantum-theoretical in nature - related to the nonzero value of ¯*h*.

#### **12. Summary**

30 Will-be-set-by-IN-TECH

are electrodynamical in nature. However, this is not necessarily true. By definition, the fields

reference frame, see Hughes (1992). The inertial gauge field may also lead to a spin response of the ensemble; such experiments have been proposed by Mashhoon & Kaiser (2006). It is remarkable that the present theory establishes a (admittedly somewhat vague) link between

It is generally assumed that the electrodynamic potentials have a particular significance in QT which they do not have in classical physics. Let us analyze this statement in detail. The first part of the statement, concerning the significance of the potentials, is of course true. The second part, asserting that in classical physics all external influences *can* be described solely in terms of field strengths, is wrong. More precisely, it is true for classical mechanics but not for classical physics in general. A counterexample - a theory belonging to classical physics but with potentials playing an indispensable role - is provided by the classical limit (118),(119) of Schrödinger's equation. In this field theory the potentials play an indispensable role because (in contrast to particle theories, like the canonical equations) no further derivatives of the Hamiltonian, which could restore the fields, are to be performed. This means that the significance of the potentials is not restricted to quantum theory but rather holds for the whole class of *statistical* theories discussed above, which contains both quantum theory and its classical limit theory as special cases. This result is in agreement with the statistical

The precise characterization of the role of the potentials is of particular importance for the interpretation of the Aharonov-Bohm effect. The 'typical quantum-mechanical features' observed in these phase shift experiments should be identified by comparing the quantum mechanical results not with classical mechanics but with the predictions of the classical statistical theory (118), (119). The predictions of two statistical theories, both of which use

The limiting behavior of Schrödinger's equation as ¯*h* → 0, discussed in section 10, is very important for the proper interpretation of QT. The erroneous belief (wish) that this limit can (must) be identified with classical mechanics is closely related to the erroneous belief that QT is able to describe the dynamics of individual particles. In this respect QT is obviously an *incomplete* theory, as has been pointed out many times before, during the last eighty years, see e.g. Einstein (1949), Margenau (1935) , Ballentine (1970), Held (2008). Unfortunately, this erroneous opinion is historically grown and firmly established in our thinking as shown by the ubiquitous use of phrases like 'the wave function of the electron'. But it is clear that an erroneous identification of the domain of validity of a physical theory will automatically create all kinds of mysteries and unsolvable problems - and this is exactly what happens. Above, we have identified one of the (more subtle) problems of this kind, concerning the role of potentials in QT, but many more could be found. Generalizing the above argumentation concerning potentials, we claim that characteristic features of QT cannot be identified by comparison with classical mechanics. Instead, quantum theory should be compared with its classical limit, which is in the present 3*D*-case given by (118), (119) - we note in this context that several 'typical' quantum phenomena have been explained by Kirkpatrick (2003) in terms of classical probability theory. One has to compare the solutions of the classical, nonlinear

potentials to describe the influence of the external field, have to be compared.

*B* obey four equations (the homogeneous Maxwell equations), which means that additional conditions are required in order to determine these six fields. The most familiar possibility is, of course, the second pair of Maxwell's equations. A second possible realization for the

*B* is given by the *inertial* forces acting on a mass *m* in an arbitrarily accelerated

*B*, and parameters *e*, *c* suggests that these quantities

*E*,

the two extremely separated physical fields of inertia and QT.

interpretation of potentials proposed in section 3.

Our notation for potentials *φ*, *A* , fields

 *E*,

fields *E*,

> In the present paper it has been shown that the method reported in I, for the derivation of Schrödingers's equation, can be generalized in such a way that essentially all aspects of non-relativistic QT are taken into account. The success of this derivation from statistical origins is interpreted as an argument in favor of the SI. The treatment of gauge fields and spin in the present statistical framework led to several remarkable new insights. We understand now why potentials (and not local fields) occur in the field equations of QT. The non-uniqueness of the potentials and the related concept of gauge invariance is not a mystery any more. Spin is derived as a kind of two-valuedness of a statistical ensemble. The local forces associated with the gauge potentials, the Lorentz force and the force experienced by a particle with magnetic moment, can also be derived. Apart from some open questions in the area of non-relativistic physics, a major problem for future research is a relativistic generalization of the present theory.

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**7** 

Carsten Held

*Germany* 

**The Quantum Completeness Problem** 

Quantum mechanics (QM) is complete in a precise sense. It cannot be supplemented by more informative descriptions of physical systems given certain reasonable assumptions; this is what the no-hidden-variables proofs show. Theorems of Kochen-Specker-type (KS theorems) crucially employ an assumption of context-independence of the observables considered while those of Bell-type theorems use an assumption of locality. Since locality amounts to context-independence of local observables the former theorems can be considered as more general than the latter. What exactly do these theorems show in terms of

The standard answer to these questions is indeed older than the theorems themselves. It was given by the inventors of QM, notably von Neumann (who himself devised a no-hiddenvariables argument) and Dirac. It says that the mathematical entity representing the maximal QM information we can have about a system S represents all the physical properties that S has. This is an informative physical interpretation of the theorems. Since it embodies the idea that the QM information about S is the complete representation of its

In this paper, I will try to show that the completeness assumption is not the correct interpretation of the KS theorems for an ultimately simple reason: it cannot be squared with QM itself. The argument proceeds as follows. Initially, I explicate four properties of QM – properties not mentioned but represented in standard axiomatisations – that will drive the argument, and moreover cast the completeness assumption into a precise form: COMP (basically a weakened version of the well-known eigenstate-eigenvalue link) (sec.2). Then I consider the central equation of the QM statistical algorithm: the trace formula. From the QM properties I conclude that the formula must be explicated in one of two ways. Neither option, however, can be harmonised with both QM and a general probability principle (sec.3). I discuss whether the argument is just an involved form of the measurement problem or else unduly neglects the notion of QM measurement – both with a negative result (sec.4). I consider a fundamental objection that indeed circumvents the argument and show that it violates one of the four QM properties (sec.5). Finally, I briefly consider what the KS theorems tell us about QM given that the argument against COMP is correct (sec.6).

Initially, we must identify the features of QM needed for the argument. The theory can be introduced in various ways, hence comes in different versions but the four properties we

physics? In which sense do they prove QM to be complete?

physical state it may be called the *completeness assumption*.

**1. Introduction** 

**2. Preliminaries** 

**2.1 Properties of quantum mechanics** 

*Philosophisches Seminar, Universität Erfurt, Erfurt* 

