Preface

**Section 2 Selected Topics in Applications of Quantum Mechanics 235**

Chapter 9 **Non-Extensive Entropies on Atoms, Molecules and Chemical**

Chapter 10 **Computation of Materials Properties at the Atomic Scale 275**

**Fundamental Physical and Biophysical Models 311**

Chapter 14 **Implications of the "Subquantum Level" in Carcinogenesis and Tumor Progression via Scale Relativity Theory 399** Daniel Timofte, Lucian Eva, Decebal Vasincu, Călin Gh. Buzea,

Chapter 11 **Implications of Quantum Informational Entropy in Some**

Tesloianu, Gabriel Crumpei and Cristina Popa

Chapter 12 **Physical Vacuum is a Special Superfluid Medium 345**

Chapter 13 **Husimi Distribution and the Fisher Information 375**

Sergio Curilef and Flavia Pennini

Maricel Agop and Radu Florin Popa

N. Flores-Gallegos, I. Guillén-Escamilla and J.C. Mixteco-Sánchez

Maricel Agop, Alina Gavriluț, Călin Buzea, Lăcrămioara Ochiuz, Dan

Chapter 8 **The Nuclear Mean Field Theory and Its Application to**

**Nuclear Physics 237** M.R. Pahlavani

**Processes 251**

**VI** Contents

Karlheinz Schwarz

V.I. Sbitnev

Without a doubt, modern science is not meaningful without quantum mechanics. When combined with mathematical physics, the basic laws of microscopic physics produce quan‐ tum mechanics. Despite it being about one century after the discovery of quantum mechan‐ ics, its history and foundations are of great interest to researchers, scientists and students of different branches of science and technology. The philosophy of quantum mechanics, which grows on the foundations, history and basic laws of quantum mechanics, such as uncertain‐ ty and the correspondence principal of quantum mechanics, is a great branch of philosophy. It forms the foundations of quantum mechanics as an open topic for researchers who are studying the history and philosophy of science. Therefore, most authors of quantum me‐ chanics and its applications introduce it via a historical survey of its early success.

The book consists of two sections. The first section is "Selected Topics in Foundations of Quantum Mechanics", which consists of seven chapters. These chapters provide a clear in‐ sight into the foundations of quantum mechanics, through the basic laws.

The title of the first chapter is "Classical or Quantum, What is Reality?" Therefore, as it is clear from the title, the author of this chapter compares classical physics with quantum me‐ chanics. In chapter two, as a basic object of quantum mechanics, photon is compared with the role that is played by signalling information transfer. In chapter three, generalized un‐ certainty is discussed by combining quantum mechanical uncertainty and the path integral method of Feynman. The subject of chapter four is the unification of quantum mechanics with relativistic theory. In chapter 5 and 6, measurement and its validation in experimental computation are studied as a major problem in quantum mechanics. In the last chapter, the lie algebra of QED and its fermionic- Fock space are presented.

The second section, "Selected Topics in Applications of Quantum Mechanics", consists of seven chapters. These are published with the cooperation of international community au‐ thors from different research institutes and universities. The first chapter is dedicated to the application of quantum mechanics, through the mean field method. Thermodynamic entro‐ py and its role in atoms, molecules and matter are presented in chapter two. Chapter three deals with the computation of material properties via atomic structures using quantum me‐ chanics. Quantum information is studied in chapter four. In chapter five, physical vacuum, as a special super fluid, is presented. Husimi distribution and the fisher information are studied in chapter six. The last chapter is dedicated to the application of quantum mechanics in medical cardiography.

The selected topics of the foundations of quantum mechanics are published with the cooper‐ ation of international community authors from different research institutes and universities.

I would like to thank all of them for preparing the chapters in time and the InTech publish‐ ing group, as well as MSc Iva Lipović, for their valuable efforts in the publication of this book.

We hope that this book becomes a rich reference for researchers, as well as students.

**M. R. Pahlavani** Nuclear Physics Department University of Mazandaran Babolsar, Iran **Selected Topics in Foundations of Quantum Mechanics**

I would like to thank all of them for preparing the chapters in time and the InTech publish‐ ing group, as well as MSc Iva Lipović, for their valuable efforts in the publication of this

**M. R. Pahlavani**

Babolsar, Iran

Nuclear Physics Department University of Mazandaran

We hope that this book becomes a rich reference for researchers, as well as students.

book.

VIII Preface

**Provisional chapter**

#### **Classical or Quantum? What is Reality? Classical or Quantum? What is Reality?**

J. J. Sławianowski and V. Kovalchuk J. J. Sławianowski∗ and V. Kovalchuk∗

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59115 10.5772/59115

**1. Introduction**

Discussion of those questions and the struggle with the well-known paradoxes of quantum mechanics like decoherence and measurement problems is as old as the quantum mechanics itself. This was just the reason for the search of formulations of any of both theories in terms of concepts characteristic for the other one. It would be injustice to condemn or disqualify those attempts. They were fruitful, enabled one to solve many concrete problems and did shed some light onto the mutual relationship of both theories. Nevertheless all mysterious features of quantum theory remained mysterious as they were. There is a whole spectrum of views; let us quote the dominant ones in a very simplified form:


There are many papers concerning the first and third possibility. But the second item, i.e., nonlinearity, is not very popular, although it seems to be a good candidate for explaining quantum paradoxes and exorcising the solliptic or any dualistic ideas. Incidentally, from the point of view of the development of physics and other natural sciences this reluctance

©2012 Sławianowski and Kovalchuk, licensee InTech. This is an open access chapter distributed under the

of physicists to nonlinearity in quantum mechanics is rather strange. It contradicts the whole history of physics. Usually one begins, when it is only possible, from linear models, but later on one introduces nonlinear terms to equations. And now we witness the unbelievable development of nonlinear methods, also essentially nonlinear ones, when there is no well-defined linear background term perturbed by a nonlinear correction. It is simply fashionable to formulate and discuss models based on the non-perturbative nonlinearity of geometric origin. Why really to stick to linearity in so fundamental science as quantum physics? Let us also mention that there are objects in the nano-scale like graphenes, fullerenes, and very large molecules the behaviour of which is placed somewhere in the convolution region of the quantum and classical theory. Let us also remind that there are nonlinear computational methods like the Fermi-Dirac procedure in the usual quantum mechanics. It is not excluded that some more fundamental nonlinearity may be formulated and efficiently used.

One of reasons of the reluctance of physicists to nonlinearity in quanta is some non-physical arbitrariness of the nonlinear dynamical models. And really, there were various models introduced "by hand". But recently some essentially nonlinear, non-perturbative models based on deep geometric ideas were suggested. For example, we mean here papers by Doebner and Goldin with co-workers [4, 5], works by Svetlichny [17], and some of our publications [13].

Below we consider two methods of quasi-classical analysis: one based on the limit transition with the Planck constant tending to zero and one based on the analysis of quickly varying wave functions with many modes. They are both connected with the mechanical-optical analogy and both are useful. Namely, the theoretical approach to physics is one based on differential equations, usually partial ones in appropriate variables. But they also contain plenty of so-called physical constants usually experimentally fixed. Let us mention, e.g., ¯*h*, *c*, *G*, *e*, *k*, etc. (respectively, the Planck constant, velocity of light in vacuum, gravitational constant, elementary electric charge, Boltzmann constant, etc.). The system of natural units like ¯*h* = 1, *c* = 1 is not appropriate. First of all because it is incomparable with the limit transition to zero or infinity. The only solution of the problem is to introduce ¯*h*, *c*, etc. as "controlling" parameters and to follow quietly (as possible) the consequence of the mentioned manipulations. Let us mention that there are two aspects of the controlling status of those constants. First, one can think about the anthropic principle and perhaps the divine origin of them. The second aspect may be more material. One can, e.g., think on them as fields and even to formulate the corresponding field or motion equations. Let us mention, incidentally, that putting ¯*h* = 0 we get rid of all paradoxes of quantum mechanics, but obtain the non-realistic world with infinities in electromagnetic radiation theory.

In a sense it might seem strange that quantum mechanics was not formulated or at least suggested some hundred years earlier. Though both geometric and physical optics seemed to be known in the deep of XIX century. And one was also aware that the geometric optics is a short-wave asymptotics of the wave theory. And the analogy between mechanics and geometric optics was also known. Mathematically it was based on the similarity of the Hamilton-Jacobi and eikonal equations. But one interpreted it as a formal similarity between description of single particle trajectories and the geometric theory of optical waves moving along the eikonal rays. The mechanical "waves" were not interpreted as the dynamics of a real wave process even in the sense of the Schrödinger wave picture (as a matter of fact also incorrect, as it turned out later on). It seems that the deciding circumstance here was the absence of the Planck constant, the physical quantity of the dimension of

action. It is necessary to replace the Hamilton-Jacobi action *S t*, *q<sup>i</sup>* by the wave containing factor exp (*i*/¯*h*) *S*(*q*) with the dimensionless argument. It is strange that the ¯*h*-divisor was discovered by Planck in phenomena of electromagnetic radiation, where it is deeply hidden from the direct observation. And only when it was discovered by Planck, the further procedure was open by Planck himself, Bohr, Sommerfeld, de Broglie, Heisenberg, Schrödinger, and many others.

#### **2. Weyl-Wigner-Moyal-Ville description**

2 Quantum Mechanics

and efficiently used.

publications [13].

of physicists to nonlinearity in quantum mechanics is rather strange. It contradicts the whole history of physics. Usually one begins, when it is only possible, from linear models, but later on one introduces nonlinear terms to equations. And now we witness the unbelievable development of nonlinear methods, also essentially nonlinear ones, when there is no well-defined linear background term perturbed by a nonlinear correction. It is simply fashionable to formulate and discuss models based on the non-perturbative nonlinearity of geometric origin. Why really to stick to linearity in so fundamental science as quantum physics? Let us also mention that there are objects in the nano-scale like graphenes, fullerenes, and very large molecules the behaviour of which is placed somewhere in the convolution region of the quantum and classical theory. Let us also remind that there are nonlinear computational methods like the Fermi-Dirac procedure in the usual quantum mechanics. It is not excluded that some more fundamental nonlinearity may be formulated

One of reasons of the reluctance of physicists to nonlinearity in quanta is some non-physical arbitrariness of the nonlinear dynamical models. And really, there were various models introduced "by hand". But recently some essentially nonlinear, non-perturbative models based on deep geometric ideas were suggested. For example, we mean here papers by Doebner and Goldin with co-workers [4, 5], works by Svetlichny [17], and some of our

Below we consider two methods of quasi-classical analysis: one based on the limit transition with the Planck constant tending to zero and one based on the analysis of quickly varying wave functions with many modes. They are both connected with the mechanical-optical analogy and both are useful. Namely, the theoretical approach to physics is one based on differential equations, usually partial ones in appropriate variables. But they also contain plenty of so-called physical constants usually experimentally fixed. Let us mention, e.g., ¯*h*, *c*, *G*, *e*, *k*, etc. (respectively, the Planck constant, velocity of light in vacuum, gravitational constant, elementary electric charge, Boltzmann constant, etc.). The system of natural units like ¯*h* = 1, *c* = 1 is not appropriate. First of all because it is incomparable with the limit transition to zero or infinity. The only solution of the problem is to introduce ¯*h*, *c*, etc. as "controlling" parameters and to follow quietly (as possible) the consequence of the mentioned manipulations. Let us mention that there are two aspects of the controlling status of those constants. First, one can think about the anthropic principle and perhaps the divine origin of them. The second aspect may be more material. One can, e.g., think on them as fields and even to formulate the corresponding field or motion equations. Let us mention, incidentally, that putting ¯*h* = 0 we get rid of all paradoxes of quantum mechanics, but obtain

In a sense it might seem strange that quantum mechanics was not formulated or at least suggested some hundred years earlier. Though both geometric and physical optics seemed to be known in the deep of XIX century. And one was also aware that the geometric optics is a short-wave asymptotics of the wave theory. And the analogy between mechanics and geometric optics was also known. Mathematically it was based on the similarity of the Hamilton-Jacobi and eikonal equations. But one interpreted it as a formal similarity between description of single particle trajectories and the geometric theory of optical waves moving along the eikonal rays. The mechanical "waves" were not interpreted as the dynamics of a real wave process even in the sense of the Schrödinger wave picture (as a matter of fact also incorrect, as it turned out later on). It seems that the deciding circumstance here was the absence of the Planck constant, the physical quantity of the dimension of

the non-realistic world with infinities in electromagnetic radiation theory.

It is neither very easy nor automatic to perform the limit transition ¯*h* → 0 in quantum-mechanical equations. In any case passing to zero with ¯*h* in quantum-mechanical wave functions leads to meaningless results. Rather, one should separately use the explicit *h*¯-dependence of the modulus and phase of the wave function

$$\Psi = \sqrt{D} \exp\left(\frac{i}{\hbar}S\right) = f \exp\left(\frac{i}{\hbar}S\right) \tag{1}$$

and substitute this to the Schrödinger equation (or any other wave equation). And then one should write the system of equations resulting from the comparison of coefficients at the same (but theoretically all possible) powers of ¯*h*. It is important to remember that *D*, *S* are power series of ¯*h*, but the divisor ¯*h* under the exp sign is universally present in equations. The simplest, heuristic way is to use the Weyl-Wigner-Moyal-Ville (WWMV) star product of operators when the phase space is **R**2*<sup>n</sup>* (or any 2*n*-dimensional linear space).

Let *q*1,..., *qn*; *p*1,..., *pn* be affine phase space coordinates, *q*1,..., *qn* — the underlying configuration space variables, and (*p*1,..., *pn*) — the induced momentum variables. The operators acting (in principle) in *L*<sup>2</sup> *q*1,..., *qn* are represented by their kernels-functions or rather distributions when we do not insist on remaining within *L*2(*Q*):

$$\left(\mathbf{A}\mathbf{Y}\right)\left(\overline{\boldsymbol{\eta}}\right) = \int A\left[\overline{\boldsymbol{\eta}}, \overline{\boldsymbol{\eta}}'\right] \Psi\left(\overline{\boldsymbol{\eta}}'\right) d\_n \overline{\boldsymbol{\eta}}'.\tag{2}$$

What concerns the adjective "distribution-like" let us stress that such important operators as identity, position, and linear momentum are just represented by distributions:

$$\mathbf{1}\left[\overline{\eta},\overline{\eta}'\right] = \delta\left(\overline{\eta}-\overline{\eta}'\right),\ \mathbf{Q}^a\left[\overline{\eta},\overline{\eta}'\right] = \eta^a\delta\left(\overline{\eta}-\overline{\eta}'\right),\ \mathbf{P}\_a\left[\overline{\eta},\overline{\eta}'\right] = \frac{\hbar}{\mathrm{i}}\frac{\partial}{\partial\overline{\eta}^a}\delta\left(\overline{\eta}-\overline{\eta}'\right).\tag{3}$$

According to the Weyl-Wigner-Moyal-Ville prescription one can represent any function *A q*1,..., *qn*; *p*1,..., *pn* by the kernel

$$A\left[\overline{\eta}, \overline{\eta}'\right] = \int A\left(\frac{1}{2}\left(\overline{\eta} + \overline{\eta}'\right), \overline{p}\right) \exp\left(\frac{i}{\hbar}\left.\overline{p}\cdot\left(\overline{\eta} - \overline{\eta}'\right)\right) \frac{d\_n \overline{p}}{\left(2\pi\hbar\right)^n}\tag{4}$$

and conversely, inverting the Fourier transform:

$$A\left(\overline{q}, \overline{p}\right) = \int \exp\left(-\frac{i}{\hbar}\,\,\overline{p}\cdot\overline{\alpha}\right) A\left[\overline{q} + \frac{\overline{\alpha}}{2}, \overline{q} - \frac{\overline{\alpha}}{2}\right] d\_n \overline{\alpha}.\tag{5}$$

Let us remind that this is a consequence of the Weyl-Wigner-Moyal-Ville star product of the pair of phase-space functions on the affine phase space:

$$\Psi(A\*B)\left(z\right) = 2^{2n} \int \exp\left(\frac{2i}{\hbar}\Gamma\left(z - z\_1, z - z\_2\right)\right) A(z\_1)B(z\_2)d\mu(z\_1)d\mu(z\_2),\tag{6}$$

$$d\mu(z) = d\mu\left(\overline{q}, \overline{p}\right) = \frac{1}{(2\pi\hbar)^n} dq^1 \dots dq^n dp\_1 \dots dp\_n \tag{7}$$

where in the (*q*, *p*)-basis Γ = *O* −*I I O* , *I* is the *n* × *n* identity matrix, and *O* is the *n* × *n* matrix composed of zeros. Obviously, the product *A* ∗ *B* is isomorphic to the product of operators represented by the "matrix" rule:

$$\mathbb{E}\left[AB\right]\left(\overline{q},\overline{q}'\right) = \int A\left[\overline{q},\overline{q}'^{\prime}\right]B\left[\overline{q}^{\prime\prime},\overline{q}'\right]d\_nq^{\prime\prime}.\tag{8}$$

And Hermitian conjugate of operators is represented by the complex conjugate of phase-space functions. The above composition of phase-space functions is non-local and in general the positively definite operators (like, e.g., density operators ρ) are not represented by non-negative phase-space functions.

It is clear from the above formulae that the action of the operator **A** on the configuration space function Ψ is given by

$$\left(\left(\mathbf{A}\mathbf{Y}\right)\left(\overline{q}\right) = \frac{1}{(2\pi\hbar)^{n}} \int \exp\left(\frac{i}{\hbar}\,\overline{p}\cdot\left(\overline{q}-\overline{q}'\right)\right) A\left(\frac{1}{2}\left(\overline{q}+\overline{q}'\right),\overline{p}\right) \Psi\left(\overline{q}'\right) d\_{n}\overline{q}' d\_{n}\overline{p}.\tag{9}$$

Let us remind a few properties of the Weyl-Wigner-Moyal-Ville product of the phase-space functions. So, it is bilinear and associative and preserves the complex conjugation:

$$(\lambda A + \mu B) \ast \mathbb{C} = \lambda A \ast \mathbb{C} + \mu B \ast \mathbb{C}, \quad (A \ast B) \ast \mathbb{C} = A \ast (B \ast \mathbb{C}), \tag{10}$$

$$
\mathbb{C} \* (\lambda A + \mu B) = \lambda \mathbb{C} \* A + \mu \mathbb{C} \* B, \quad \overline{A \* B} = \overline{B} \* \overline{A}, \tag{11}
$$

where **<sup>A</sup>**<sup>+</sup> is represented by *<sup>A</sup>*. Besides 1 ∗ *<sup>A</sup>* = *<sup>A</sup>* ∗ 1, *<sup>A</sup>* ∗ *<sup>A</sup>* �= 0, if *<sup>A</sup>* �= 0 a.e., *A* ∗ *Bdµ* = *ABdµ*, but in general *A* ∗ *B* ∗ *Cdµ* �= *ABCdµ*. Let us notice that

$$\text{Tr}\mathbf{A} = \int A(\overline{\mathbf{z}})d\mu(\overline{\mathbf{z}}), \quad \langle \mathbf{A}, \mathbf{B} \rangle = \text{Tr}\left(\mathbf{A}^+\mathbf{B}\right) = \int \overline{A(\overline{\mathbf{z}})}B(\overline{\mathbf{z}})d\mu(\overline{\mathbf{z}}),\tag{12}$$

and �*C* ∗ *A*, *B*� = �*A*, *C* ∗ *B*� = �*C*, *B* ∗ *A*�. Obviously, the star-product of phase-space functions is non-commutative, just as the operator product. Let us quote some formulae concerning non-commutativity and commutativity:

$$q^a \* p\_b = q^a p\_b + \frac{i\hbar}{2} \delta^a\_{\ b\prime} \ p\_b \* q^a = p\_b q^a - \frac{i\hbar}{2} \delta^a\_{\ b\prime} \tag{13}$$

$$q^a \ast A(q, p) = q^a A(q, p) + \frac{i\hbar}{2} \frac{\partial A}{\partial p\_a}, \; A(q, p) \ast q^a = A(q, p)q^a - \frac{i\hbar}{2} \frac{\partial A}{\partial p\_a}, \tag{14}$$

$$p\_a \* A(q, p) = p\_a A(q, p) - \frac{i\hbar}{2} \frac{\partial A}{\partial q^a}, \; A(q, p) \* p\_a = A(q, p) p\_a + \frac{i\hbar}{2} \frac{\partial A}{\partial q^a}, \tag{15}$$

and obviously for functions depending only on one kind of variables

4 Quantum Mechanics

and conversely, inverting the Fourier transform:

(*A* ∗ *B*)(*z*) = 22*<sup>n</sup>*

where in the (*q*, *p*)-basis Γ =

represented by the "matrix" rule:

by non-negative phase-space functions.

(2*πh*¯)*<sup>n</sup>*

 exp *i h*¯ *p* · *<sup>q</sup>* <sup>−</sup> *<sup>q</sup>*′ *A* 1 2 *<sup>q</sup>* <sup>+</sup> *<sup>q</sup>*′ , *p* Ψ *q*′ *dnq*′

space function Ψ is given by

(**A**Ψ) (*q*) <sup>=</sup> <sup>1</sup>

Tr**A** = 

*A* (*q*, *p*) =

pair of phase-space functions on the affine phase space:

 exp 2*i*

*<sup>d</sup>µ*(*z*) = *<sup>d</sup><sup>µ</sup>* (*q*, *<sup>p</sup>*) <sup>=</sup> <sup>1</sup>

 *O* −*I I O* 

[*AB*] *<sup>q</sup>*, *<sup>q</sup>*′ = *A <sup>q</sup>*, *<sup>q</sup>*′′ *B <sup>q</sup>*′′, *<sup>q</sup>*′ 

 exp − *i <sup>h</sup>*¯ *<sup>p</sup>* · *<sup>α</sup> A q* + *α* 2

Let us remind that this is a consequence of the Weyl-Wigner-Moyal-Ville star product of the

*<sup>h</sup>*¯ <sup>Γ</sup> (*<sup>z</sup>* <sup>−</sup> *<sup>z</sup>*1, *<sup>z</sup>* <sup>−</sup> *<sup>z</sup>*2)

composed of zeros. Obviously, the product *A* ∗ *B* is isomorphic to the product of operators

And Hermitian conjugate of operators is represented by the complex conjugate of phase-space functions. The above composition of phase-space functions is non-local and in general the positively definite operators (like, e.g., density operators ρ) are not represented

It is clear from the above formulae that the action of the operator **A** on the configuration

Let us remind a few properties of the Weyl-Wigner-Moyal-Ville product of the phase-space

(*λA* + *µB*) ∗ *C* = *λA* ∗ *C* + *µB* ∗ *C*, (*A* ∗ *B*) ∗ *C* = *A* ∗ (*B* ∗ *C*), (10) *C* ∗ (*λA* + *µB*) = *λC* ∗ *A* + *µC* ∗ *B*, *A* ∗ *B* = *B* ∗ *A*, (11)

> **A**+**B** =

functions. So, it is bilinear and associative and preserves the complex conjugation:

where **<sup>A</sup>**<sup>+</sup> is represented by *<sup>A</sup>*. Besides 1 ∗ *<sup>A</sup>* = *<sup>A</sup>* ∗ 1, *<sup>A</sup>* ∗ *<sup>A</sup>* �= 0, if *<sup>A</sup>* �= 0 a.e.,

*ABdµ*, but in general *A* ∗ *B* ∗ *Cdµ* �= *ABCdµ*. Let us notice that

*A*(*z*)*dµ*(*z*), �**A**, **B**� = Tr

, *<sup>q</sup>* <sup>−</sup> *<sup>α</sup>* 2 

(2*πh*¯)*<sup>n</sup> dq*<sup>1</sup> ... *dqndp*<sup>1</sup> ... *dpn*, (7)

, *I* is the *n* × *n* identity matrix, and *O* is the *n* × *n* matrix

*dnα*. (5)

*<sup>A</sup>*(*z*1)*B*(*z*2)*dµ*(*z*1)*dµ*(*z*2), (6)

*dnq*′′. (8)

*dn p*. (9)

*A* ∗ *Bdµ* =

*A*(*z*)*B*(*z*)*dµ*(*z*), (12)

$$A(A\*B)(q) = A(q) \* B(q) = A(q)B(q) = (AB)(q),\tag{16}$$

$$A(A\*B)(p) = A(p) \* B(p) = A(p)B(p) = (AB)(p). \tag{17}$$

The star-product is evidently invariant under the action of affine symplectic group, (*U*(*A*, *t*)*f*) ∗ (*U*(*A*, *t*)*g*) = *U*(*A*, *t*)(*f* ∗ *g*), where *t* is a translation vector in **R**2*<sup>n</sup>* and *A* is a linear symplectic transformation, <sup>Γ</sup>*kl <sup>A</sup><sup>k</sup> aA<sup>l</sup> <sup>b</sup>* = <sup>Γ</sup>*ab*, (*U*(*A*, *<sup>t</sup>*)*f*)(*z*) = *<sup>f</sup>*(*Az* + *<sup>t</sup>*).

Quantum states are described by density operators *ρ* which are Hermitian, normalized to unity and positive: �ρ|**A**+**A**� <sup>=</sup> Tr <sup>ρ</sup>**A**+**<sup>A</sup>** ≥ 0, Trρ = 1. To be honest, one can also live without the last normalization condition. When the condition is satisfied, we have Tr ρ<sup>2</sup> ≤ Tr <sup>ρ</sup> <sup>=</sup> 1. Particularly important are pure states described by projectors, <sup>ρ</sup><sup>2</sup> <sup>=</sup> <sup>ρ</sup>, <sup>ρ</sup> <sup>=</sup> <sup>|</sup>Ψ��Ψ|, *ρ <sup>q</sup>*, *<sup>q</sup>*′ = Ψ (*q*) Ψ *q*′ . They are related to the corresponding wave function Ψ as follows:

$$\rho\left(\overline{q}, \overline{q}'\right) = \frac{1}{(2\pi)^n} \int \overline{\Psi}\left(\overline{q} - \frac{\hbar}{2}\overline{\tau}\right) \exp\left(-i\overline{\tau}\cdot\overline{p}\right) \Psi\left(q + \frac{\hbar}{2}\overline{\tau}\right) d\_n\overline{\tau}.\tag{18}$$

In general it takes on negative values, nevertheless it is positive in the quantum-mechanical sense:

$$
\langle \rho | \overline{B} \* B \rangle = \int \rho(z) \left( \overline{B} \* B \right)(z) d\mu(z) > 0 \tag{19}
$$

for all functions *B*. The exceptional Wigner functions are positive in the literal sense and are exponential:

$$E\_{\left(\overline{\xi}, \overline{\pi}\right)}\left(\overline{q}, \overline{p}\right) = \frac{1}{\left(\pi\hbar\right)^{n}} \exp\left(-\frac{1}{\hbar}\left(\left(\overline{q} - \overline{\xi}\right)^{2} + \left(\overline{p} - \overline{\pi}\right)^{2}\right)\right). \tag{20}$$

It is clear that they represent pure states,

$$E\_{\left(\overline{\xi},\overline{\pi}\right)} \* E\_{\left(\overline{\xi},\overline{\pi}\right)} = E\_{\left(\overline{\xi},\overline{\pi}\right)} \qquad \int E\_{\left(\overline{\xi},\overline{\pi}\right)}\left(\overline{\eta},\overline{p}\right) d\_{n}\overline{\eta} \frac{d\_{n}\overline{p}}{(2\pi\hbar)^{n}} = 1. \tag{21}$$

They are coherent states strongly concentrated in the phase space about the point *ξ*, *π* . Therefore, the coarse-grained quantity, so-called Husimi distribution,

$$\widetilde{\rho}\left(\overline{q},\overline{p}\right) = \int E\_{\left(\overline{q},\overline{p}\right)}\left(\overline{\xi},\overline{\pi}\right)\rho\left(\overline{\xi},\overline{\pi}\right)d\_{n}\overline{\xi}\frac{d\_{n}\overline{\pi}}{(2\pi\hbar)^{n}}\tag{22}$$

admits an approximate interpretation of the literally positively-definite probability distribution obtained from the Wigner function *<sup>ρ</sup>*. Indeed, *<sup>E</sup>*(*q*,*p*) is in a sense a pure state Wigner function concentrated at (*q*, *p*) and therefore the above integral is a probability density for the system to be found in the phase-space cell at (*q*, *p*) when it is known to be in a Wigner state *<sup>ρ</sup>*. This interpretation is not bad and certainly *<sup>ρ</sup>* is something that in a sense gives an account of the probability distribution to be found in an ¯*hn*-volume cell about every (*q*, *p*). There is only one drawback of this interpretation. Namely, unlike the true Weyl-Wigner-Moyal-Ville distributions, literally non-positive, the Husimi distributions (22) have non-satisfactory, bad marginal properties, because

$$\int \widetilde{\rho}\left(\overline{q}, \overline{p}\right) d\_n \overline{q} \neq \overline{\widehat{\Psi}}\left(\overline{p}\right) \widehat{\Psi}\left(\overline{p}\right), \qquad \int \widetilde{\rho}\left(\overline{q}, \overline{p}\right) \frac{d\_n \overline{p}}{(2\pi\hbar)^n} \neq \overline{\widehat{\Psi}}\left(\overline{q}\right) \Psi\left(\overline{q}\right). \tag{23}$$

Here, obviously, <sup>Ψ</sup>, <sup>Ψ</sup> are the wave functions underlying *<sup>ρ</sup>*, respectively in the coordinate and momentum representations. For the *ρ* itself the above inequalities become exact equalities.

Let us now discuss the problem of the WKB approximation from the point of view of the above remarks. It is clear that from the point of view of the above statements, in the lowest-order approximation of *D*, *S* in ¯*h*, we have the following ¯*h*-independent interpretation of the *D*, *S*-functions in terms of the expectation values:

$$
\langle \Psi | \mathbf{Q}^{\dot{i}} | \Psi \rangle = \int D \left( q^1, \dots, q^n \right) q^i dq^1 \dots dq^n,\tag{24}
$$

$$
\langle \Psi | \mathbf{P}\_i | \Psi \rangle = \int D \left( q^1, \dots, q^n \right) \partial\_i \mathcal{S} \left( q^1, \dots, q^n \right) dq^1 \dots dq^n. \tag{25}
$$

It is clear that the Planck constant ¯*h* is absent in those expressions, so really the functions *D*, *S* are ¯*h*-independent up to higher orders. In any case it is so at places distant from the turning points. Let us consider the *<sup>n</sup>*-dimensional submanifold given by equations *pi* = *<sup>∂</sup>S*/*∂q<sup>i</sup>* , *<sup>i</sup>* = 1, . . . , *<sup>n</sup>*, i.e., *Fi* = *pi* − *∂S*/*∂q<sup>i</sup>* = 0. This submanifold, <sup>m</sup>*S*, is a special case of what is called Lagrangian manifold, because the Poisson brackets of the left-hand sides of its equations vanish; moreover, they vanish after the restriction to m*S*,

$$\left\{ F\_{\dot{i}}, F\_{\dot{j}} \right\} = \frac{\partial F\_{\dot{i}}}{\partial q^a} \frac{\partial F\_{\dot{j}}}{\partial p\_a} - \frac{\partial F\_{\dot{i}}}{\partial p\_a} \frac{\partial F\_{\dot{j}}}{\partial q^a} = \left( \dot{\theta}\_{\dot{i}\dot{j}}^2 - \dot{\theta}\_{\dot{j}\dot{i}}^2 \right) S = 0. \tag{26}$$

Let us take the singular probability distribution concentrated on m*S*:

$$\rho\_{\rm cl}[D, S] = \lim\_{\hbar \to 0} \rho[D, S] = D\left(q^1, \dots, q^n\right) \delta\left(p\_1 - \frac{\partial S}{\partial q^1}\right) \dots \delta\left(p\_n - \frac{\partial S}{\partial q^n}\right)$$

$$= \left|\Psi\left(q^1, \dots, q^n\right)\right|^2 \delta\left(p\_1 - \frac{\partial S}{\partial q^1}\right) \dots \delta\left(p\_n - \frac{\partial S}{\partial q^n}\right) . \tag{27}$$

Obviously, it is different from *ρ*(*q*, *p*), nevertheless the expectation values of *q<sup>i</sup>* , *pj* and their linear combinations on *<sup>ρ</sup>*cl are just the same as those on *<sup>ρ</sup>*,

6 Quantum Mechanics

They are coherent states strongly concentrated in the phase space about the point

admits an approximate interpretation of the literally positively-definite probability distribution obtained from the Wigner function *<sup>ρ</sup>*. Indeed, *<sup>E</sup>*(*q*,*p*) is in a sense a pure state Wigner function concentrated at (*q*, *p*) and therefore the above integral is a probability density for the system to be found in the phase-space cell at (*q*, *p*) when it is known to be in a Wigner state *<sup>ρ</sup>*. This interpretation is not bad and certainly *<sup>ρ</sup>* is something that in a sense gives an account of the probability distribution to be found in an ¯*hn*-volume cell about every (*q*, *p*). There is only one drawback of this interpretation. Namely, unlike the true Weyl-Wigner-Moyal-Ville distributions, literally non-positive, the Husimi distributions (22)

Here, obviously, <sup>Ψ</sup>, <sup>Ψ</sup> are the wave functions underlying *<sup>ρ</sup>*, respectively in the coordinate and momentum representations. For the *ρ* itself the above inequalities become exact equalities. Let us now discuss the problem of the WKB approximation from the point of view of the above remarks. It is clear that from the point of view of the above statements, in the lowest-order approximation of *D*, *S* in ¯*h*, we have the following ¯*h*-independent interpretation

> *qi*

 *∂iS* 

It is clear that the Planck constant ¯*h* is absent in those expressions, so really the functions *D*, *S* are ¯*h*-independent up to higher orders. In any case it is so at places distant from the turning points. Let us consider the *<sup>n</sup>*-dimensional submanifold given by equations *pi* = *<sup>∂</sup>S*/*∂q<sup>i</sup>*

is called Lagrangian manifold, because the Poisson brackets of the left-hand sides of its

*∂Fj <sup>∂</sup>q<sup>a</sup>* <sup>=</sup> *∂*2 *ij* <sup>−</sup> *<sup>∂</sup>*<sup>2</sup> *ji* 

<sup>−</sup> *<sup>∂</sup>Fi ∂pa*

*q*1,..., *q<sup>n</sup>*

*q*1,..., *q<sup>n</sup>*

*<sup>ρ</sup>*(*q*, *<sup>p</sup>*) *dn <sup>p</sup>*

*q*1,..., *q<sup>n</sup>*

= 0. This submanifold, <sup>m</sup>*S*, is a special case of what

*dn<sup>ξ</sup> dn<sup>π</sup>*

Therefore, the coarse-grained quantity, so-called Husimi distribution,

*E*(*q*,*p*) *ξ*, *π ρ ξ*, *π* 

*<sup>ρ</sup>*(*q*, *<sup>p</sup>*) <sup>=</sup>

have non-satisfactory, bad marginal properties, because

*<sup>ρ</sup>*(*q*, *<sup>p</sup>*) *dnq* �<sup>=</sup> <sup>Ψ</sup> (*p*) <sup>Ψ</sup> (*p*),

of the *D*, *S*-functions in terms of the expectation values:


> *∂S*/*∂q<sup>i</sup>*

equations vanish; moreover, they vanish after the restriction to m*S*,

Let us take the singular probability distribution concentrated on m*S*:

*∂Fj ∂pa*

 *D* 

�Ψ|**P***i*|Ψ� =

 *Fi*, *Fj* <sup>=</sup> *<sup>∂</sup>Fi ∂q<sup>a</sup>*

�Ψ|**Q***i*

*<sup>i</sup>* = 1, . . . , *<sup>n</sup>*, i.e., *Fi* = *pi* −

*ξ*, *π* .

(2*πh*¯)*<sup>n</sup>* (22)

(2*πh*¯)*<sup>n</sup>* �<sup>=</sup> <sup>Ψ</sup> (*q*) <sup>Ψ</sup> (*q*). (23)

*dq*<sup>1</sup> ... *dqn*, (24)

*dq*<sup>1</sup> ... *dqn*. (25)

*S* = 0. (26)

,

$$
\begin{split}
\langle\Psi|a\_{i}\mathbf{q}^{i}+\beta^{i}\mathbf{p}\_{i}|\Psi\rangle&=\int\left(a\_{i}q^{i}+\beta^{i}p\_{i}\right)\rho[D,S]d\_{n}\overline{q}\frac{d\_{n}\overline{p}}{(2\pi\hbar)^{n}}\\&=\int\left(a\_{i}q^{i}+\beta^{i}p\_{i}\right)\rho\_{\text{cl}}[D,S]d\_{n}\overline{q}\frac{d\_{n}\overline{p}}{(2\pi\hbar)^{n}}.\end{split}\tag{28}$$

Let us mention that all limit transitions here, in particular the one between *ρ*[*D*, *S*] and *<sup>ρ</sup>*cl[*D*, *<sup>S</sup>*] are meant in the distribution theory sense.

It is important that the Weyl-Wigner-Moyal-Ville product may be expanded as a power series in ¯*h* and that the functional coefficients are interpretable in terms of the symplectic geometry of the classical phase space. The first two terms of the expansion are given by

$$A \ast B \simeq AB + \frac{i\hbar}{2} \{A, B\} + \dots \,\tag{29}$$

the next terms are given by the multiple Poisson brackets. In any case, the Weyl-Wigner-Moyal-Ville product and the corresponding quantum Poisson bracket are given in the limit ¯*h* → 0 by the following ¯*h*-independent expressions:

$$\lim\_{\hbar \to 0} A \ast B = AB\_{\prime} \tag{30}$$

$$\lim\_{\hbar \to 0} \{A, B\}\_{\text{QFB}} = \lim\_{\hbar \to 0} \frac{1}{i\hbar} \left(A \ast B - B \ast A\right) = \{A, B\}.\tag{31}$$

Let us mention that these formulae have interesting features and interpretation. Namely, the eigenequation for the wave function implies the following eigenequation for the corresponding density operator: **A**ρ = *a*ρ, i.e., in terms of the Weyl-Wigner-Moyal-Ville approach *A* ∗ *ρ* = *aρ*. But this implies *ρ* ∗ *A* = *aρ* if *A* is real, i.e., **A** is hermitian, and therefore

$$[\mathbf{A}, \rho]\_{\rm QFB} = \frac{1}{i\hbar}[\mathbf{A}, \rho] = 0,\qquad \text{thus,}\qquad \frac{1}{i\hbar} \left( A \ast \rho - \rho \ast A \right) = 0.\tag{32}$$

On the quantum level this equation is a direct consequence of the eigenequation *A* ∗ *ρ* = *aρ*. But these equations have quite different qualitative interpretation in physical terms. Namely, **A**ρ = *a*ρ has a purely informational content. It tells us that on the state ρ, or *ρ* in the Weyl-Wigner-Moyal-Ville language, the physical quantity **A** (*A* in the Weyl-Wigner-Moyal-Ville terms) takes spread-freely the value *a*. This is the purely informational property. But the Poisson bracket property, mathematically following from it, has a qualitatively different interpretation, namely, such a *ρ* is invariant under the one-parameter group of unitary transformations generated by **A** (*A*),

$$\exp\left(\frac{i}{\hbar}\mathbf{A}\tau\right)\rho\exp\left(-\frac{i}{\hbar}\mathbf{A}\tau\right) = \rho.\tag{33}$$

This is a symmetry property. Therefore, on the quantum level information implies symmetry. But in classical physics Poisson bracket and the pointwise product of functions are algebraically independent. Therefore, information and symmetry of statistical states become logically independent. This implies that in the classical limit Schrödinger equation or the corresponding eigenequation for the density operator must be in the lowest order of approximation replaced by the pair of equations for the phase and modulus of the wave function. Therefore, substituting (1) to (9) and taking the limit ¯*h* → 0 we obtain:

$$(\mathbf{A}\Psi)(q) \approx A\left(q^i, \frac{\partial S}{\partial q^i}\right) \Psi(q) + \frac{\hbar}{i} \left(\boldsymbol{\varepsilon}\_{\boldsymbol{v}} f\right) \exp\left(\frac{i}{\hbar} S(q)\right);\tag{34}$$

higher order terms in ¯*h* are omitted. The symbol *£v* denotes the Lie derivative of *f* with respect to the vector field *v*[*A*, *S*] which equals

$$v^i = \frac{\partial A}{\partial p\_i} \left( q^j \prime \frac{\partial S}{\partial q^j} \right) . \tag{35}$$

In spite of the use of analytical symbols, *v<sup>i</sup>* is a well-defined vector field tangent to the manifold <sup>m</sup>*<sup>S</sup>* given by equations *pj* = *<sup>∂</sup>S*/*∂q<sup>j</sup>* , *j* = 1, . . . , *n*. It is obtained from the Hamiltonian vector field generated by the function *A*,

$$X[A] = \frac{\partial A}{\partial p\_i} \frac{\partial}{\partial q^i} - \frac{\partial A}{\partial q^i} \frac{\partial}{\partial p\_i}.\tag{36}$$

This vector field is tangent to m*S*, so we restrict it to some vector field on this manifold and project it to the configuration space *Q*, i.e., to the manifold of *qa*-variables. It is clear that *f* geometrically is not a scalar field, but the scalar *W*-density of weight 1/2. Therefore,

$$
\mathcal{E}\_{\upsilon}f = \upsilon^a \frac{\partial f}{\partial q^a} + \frac{1}{2} \frac{\partial \upsilon^a}{\partial q^a} f. \tag{37}
$$

*D* is a scalar density of weight one, thus,

8 Quantum Mechanics

On the quantum level this equation is a direct consequence of the eigenequation *A* ∗ *ρ* = *aρ*. But these equations have quite different qualitative interpretation in physical terms. Namely, **A**ρ = *a*ρ has a purely informational content. It tells us that on the state ρ, or *ρ* in the Weyl-Wigner-Moyal-Ville language, the physical quantity **A** (*A* in the Weyl-Wigner-Moyal-Ville terms) takes spread-freely the value *a*. This is the purely informational property. But the Poisson bracket property, mathematically following from it, has a qualitatively different interpretation, namely, such a *ρ* is invariant under the

This is a symmetry property. Therefore, on the quantum level information implies symmetry. But in classical physics Poisson bracket and the pointwise product of functions are algebraically independent. Therefore, information and symmetry of statistical states become logically independent. This implies that in the classical limit Schrödinger equation or the corresponding eigenequation for the density operator must be in the lowest order of approximation replaced by the pair of equations for the phase and modulus of the wave

<sup>Ψ</sup>(*q*) + *<sup>h</sup>*¯

higher order terms in ¯*h* are omitted. The symbol *£v* denotes the Lie derivative of *f* with

In spite of the use of analytical symbols, *v<sup>i</sup>* is a well-defined vector field tangent to the

*∂ <sup>∂</sup>q<sup>i</sup>* <sup>−</sup> *<sup>∂</sup><sup>A</sup> ∂qi*

This vector field is tangent to m*S*, so we restrict it to some vector field on this manifold and project it to the configuration space *Q*, i.e., to the manifold of *qa*-variables. It is clear that *f*

> *<sup>∂</sup>q<sup>a</sup>* <sup>+</sup> 1 2 *∂v<sup>a</sup>*

*<sup>i</sup>* (*£v <sup>f</sup>*) exp

*∂ ∂pi*  *i h*¯ *S*(*q*) 

function. Therefore, substituting (1) to (9) and taking the limit ¯*h* → 0 we obtain:

*<sup>v</sup><sup>i</sup>* <sup>=</sup> *<sup>∂</sup><sup>A</sup> ∂pi qj* , *∂S ∂qj* 

*<sup>X</sup>*[*A*] = *<sup>∂</sup><sup>A</sup>*

*∂pi*

geometrically is not a scalar field, but the scalar *W*-density of weight 1/2. Therefore,

*£v <sup>f</sup>* <sup>=</sup> *<sup>v</sup><sup>a</sup> <sup>∂</sup> <sup>f</sup>*

 *qi* , *∂S ∂qi*  = ρ. (33)

. (35)

. (36)

*<sup>∂</sup>q<sup>a</sup> <sup>f</sup>* . (37)

, *j* = 1, . . . , *n*. It is obtained from the

; (34)

one-parameter group of unitary transformations generated by **A** (*A*),

exp *i h*¯ **A***τ* ρ exp − *i h*¯ **A***τ* 

(**A**Ψ)(*q*) ≈ *A*

respect to the vector field *v*[*A*, *S*] which equals

manifold <sup>m</sup>*<sup>S</sup>* given by equations *pj* = *<sup>∂</sup>S*/*∂q<sup>j</sup>*

Hamiltonian vector field generated by the function *A*,

$$
\varepsilon \varepsilon\_{\upsilon} D = v^{a} \frac{\partial D}{\partial q^{a}} + \frac{\partial v^{a}}{\partial q^{a}} D = \frac{\partial}{\partial q^{a}} \left( D v^{a} \right) . \tag{38}
$$

If we consider the Schrödinger equation

$$i\hbar\frac{\partial\Psi}{\partial t} = \mathbf{H}\Psi,\tag{39}$$

then in the quasiclassical limit we obtain the following system of equations:

$$\frac{\partial S}{\partial t} + H\left(q\_{\prime}\frac{\partial S}{\partial q^{\prime}}, t\right) = 0, \qquad \frac{\partial D}{\partial t} + \frac{\partial}{\partial q^{a}}\left(D\frac{\partial H}{\partial p\_{a}}\left(q\_{\prime}\frac{\partial S}{\partial q}\right)\right) = 0. \tag{40}$$

This is the system composed of the Hamilton-Jacobi equation for *S* and the continuity equation for *D*. The second equation is dependent on the solution of the first one. Geometrically it may be written in the form

$$\frac{\partial D}{\partial t} + \mathcal{L}\_{v[H,S]}D = 0. \tag{41}$$

Let us take a system of *n* functions *Ai* on the phase space with pairwise vanishing Weyl-Wigner-Moyal-Ville commutators,

$$A\_{\dot{l}} \* A\_{\dot{j}} - A\_{\dot{j}} \* A\_{\dot{l}} = 0. \tag{42}$$

Consider the family of eigenequations for the Weyl-Wigner-Moyal-Ville density function *ρ*:

$$A\_i \* \rho = a\_i \rho.\tag{43}$$

Obviously, they imply that (1/*ih*¯)(*Ai* ∗ *<sup>ρ</sup>* − *<sup>ρ</sup>* ∗ *Ai*) = 0. In the classical limit this system becomes

$$A\_i \rho = a\_i \rho\_\prime \qquad \{A\_{i\prime} \rho\} = 0. \tag{44}$$

The quantum compatibility condition (42) for (43) implies that in the classical limit the corresponding condition for (44), i.e., {*Ai*, *Aj*} = 0, also holds. The corresponding solution for (44) may be given as:

$$
\rho(q, p) = \delta\left(A\_1(q, p) - a\_1\right) \dots \delta\left(A\_n(q, p) - a\_n\right). \tag{45}
$$

To be more precise, this holds when *A*1,..., *An* is a system of functionally independent analytic functions. This distribution is concentrated on the Lagrangian manifold <sup>m</sup>(*A*,*a*) given by equations: *Ai*(*q*, *<sup>p</sup>*) = *ai*, *<sup>i</sup>* = 1, . . . , *<sup>n</sup>*. Solving them with respect to *pi* we obtain the transformed equations in the potential form: *pj* = *<sup>∂</sup>S*(*q*, *<sup>a</sup>*)/*∂q<sup>j</sup>* . Short calculation shows that *ρ* may be written as follows:

$$\rho(q, p) = \left| \det \left[ \frac{\partial^2 S}{\partial q^i \partial a^j} \right] \right| \delta \left( p\_1 - \frac{\partial S(q, a)}{\partial q^1} \right) \dots \delta \left( p\_n - \frac{\partial S(q, a)}{\partial q^n} \right) . \tag{46}$$

The quantity det *∂*2*S*/*∂q<sup>i</sup> ∂aj* is known as the Van Vleck determinant [20]. The corresponding quasiclassical wave function is given by:

$$\Psi(q, a) = \sqrt{\det\left[\frac{\partial^2 S}{\partial q^i \partial a^j}\right]} \exp\left(\frac{i}{\hbar} S(q, a)\right). \tag{47}$$

This expression is convenient when one of the functions *A*1,..., *An* is physically interpretable as a Hamiltonian *H*. Or when Hamiltonian is a simple function of other quantities *Ai* constants of motion, *<sup>H</sup>* = *<sup>E</sup>* (*A*1(*q*, *<sup>p</sup>*),..., *An*(*q*, *<sup>p</sup>*)). Then the function

$$\Psi(q, t; a) = \sqrt{\det\left[\frac{\partial^2 S}{\partial q^i \partial a^j}\right]} \exp\left(\frac{i}{\hbar} S(q, a) - E\left(a\_1, \dots, a\_{n\prime}t\right)\right). \tag{48}$$

is an approximate quasiclassical solution of the Schrödinger equation (39) with the continuous spectrum of *A*1,..., *An*. And here some additional remarks are necessary. The first one is that (47), (48) are valid only far from the turning points. So, they are valid only in the non-compact spaces **R***n*, **R**2*n*, when there is no quantization of *Aj* at all, or one must modify them so as to admit compact configuration spaces. But then the above version of the Weyl-Wigner-Moyal-Ville formalism does not work and must be replaced by something else. Some way to remain within the framework is to unify the solutions (48) with the quantization of *A*1,..., *An* by the Bohr-Sommerfeld quantum conditions. Roughly speaking, the idea is then that only such submanifolds <sup>m</sup>(*a*1,...,*an*) are admitted that the periods of *<sup>ω</sup>* <sup>=</sup> *pidq<sup>i</sup>* on <sup>m</sup>(*a*1,...,*an*) are integer multiples of the Planck constant. This condition gives rise to the "quantization" of *A*1,..., *An*. And this is what one really does in the Old Quantum Theory. But in general some difficulties appear on the level of wave functions (47), (48), namely one has to use some Maslov modifications and use the Airy special functions.

Nevertheless, the very heart of idea survives: quasiclassical pure quantum states are represented by probability distributions concentrated on *n*-dimensional submanifolds of the phase space; let us repeat that *n* is the number of degrees of freedom. At least locally the expressions (47), (48) are qualitatively correct. This is very interesting from the geometrical point of view. Namely, in spite of using analytic expressions, the Van Vleck determinant is a well-defined, coordinate-independent scalar density of weight two both in the configuration space *Q* (*qa*-variables) and in the **R***n*-space of the values *a*1,..., *a<sup>n</sup>* of constants of motion *A*1,..., *An*. And its square root is a well-defined scalar *W*-density of weight one. By its very geometric interpretation, this quantity is a priori the best candidate for the quasiclassical probability distribution of the wave functions (47), (48). Obviously, the care must be taken

concerning the mentioned problems, in particular the behaviour at the classical turning points.

In any case, expressions (47), (48) are almost true (in the quasiclassical sense), when the variables *q<sup>i</sup>* are taken modulo 2*π*, i.e., when the configuration space is topologically a torus, and when there are no turning points at all. Then the Bohr-Sommerfeld quantum conditions work literally (in approximation) and there is no need to introduce the Airy functions. And Van Vleck determinant is a good approximation to the quantum density function.

**Summary of Section 2:** The main message following from the above study is that the classical limit transition, when correctly carried out, indicates that it is not points of the classical phase space, but rather *n*-parameter Lagrangian submanifolds in the phase space that is to correspond to the quantum pure states (*n* is the number of degrees of freedom). Or more precisely, it is probability distributions on those manifolds that are to describe the pure states. When the time variable is taken into account, then it turns out that the pure states evolutions are what J. L. Synge used to call the coherent *n*-parameter families of classical trajectories. It is in a sense a surprising result that both on the level of wave functions phases and on the level of probability distributions, the corresponding quantities may be a priori guessed on the basis of the classical Hamilton-Jacobi theory and the Van Vleck determinant following from it.

#### **3. Symplectic and contact interpretation**

10 Quantum Mechanics

*ρ* may be written as follows:

The quantity det

*ρ*(*q*, *p*) =

 *∂*2*S*/*∂q<sup>i</sup>*

Ψ(*q*, *t*; *a*) =

 det

by equations: *Ai*(*q*, *<sup>p</sup>*) = *ai*, *<sup>i</sup>* = 1, . . . , *<sup>n</sup>*. Solving them with respect to *pi* we obtain the

*<sup>p</sup>*<sup>1</sup> <sup>−</sup> *<sup>∂</sup>S*(*q*, *<sup>a</sup>*) *∂q*<sup>1</sup>

 *∂*2*S ∂qi∂a<sup>j</sup>*

This expression is convenient when one of the functions *A*1,..., *An* is physically interpretable as a Hamiltonian *H*. Or when Hamiltonian is a simple function of other quantities *Ai* —

is an approximate quasiclassical solution of the Schrödinger equation (39) with the continuous spectrum of *A*1,..., *An*. And here some additional remarks are necessary. The first one is that (47), (48) are valid only far from the turning points. So, they are valid only in the non-compact spaces **R***n*, **R**2*n*, when there is no quantization of *Aj* at all, or one must modify them so as to admit compact configuration spaces. But then the above version of the Weyl-Wigner-Moyal-Ville formalism does not work and must be replaced by something else. Some way to remain within the framework is to unify the solutions (48) with the quantization of *A*1,..., *An* by the Bohr-Sommerfeld quantum conditions. Roughly speaking, the idea is then that only such submanifolds <sup>m</sup>(*a*1,...,*an*) are admitted that the periods of *<sup>ω</sup>* <sup>=</sup> *pidq<sup>i</sup>* on <sup>m</sup>(*a*1,...,*an*) are integer multiples of the Planck constant. This condition gives rise to the "quantization" of *A*1,..., *An*. And this is what one really does in the Old Quantum Theory. But in general some difficulties appear on the level of wave functions (47), (48), namely one

Nevertheless, the very heart of idea survives: quasiclassical pure quantum states are represented by probability distributions concentrated on *n*-dimensional submanifolds of the phase space; let us repeat that *n* is the number of degrees of freedom. At least locally the expressions (47), (48) are qualitatively correct. This is very interesting from the geometrical point of view. Namely, in spite of using analytic expressions, the Van Vleck determinant is a well-defined, coordinate-independent scalar density of weight two both in the configuration space *Q* (*qa*-variables) and in the **R***n*-space of the values *a*1,..., *a<sup>n</sup>* of constants of motion *A*1,..., *An*. And its square root is a well-defined scalar *W*-density of weight one. By its very geometric interpretation, this quantity is a priori the best candidate for the quasiclassical probability distribution of the wave functions (47), (48). Obviously, the care must be taken

 exp *i h*¯ *S*(*q*, *a*) 

 ... *δ* 

. Short calculation shows that

. (47)

. (48)

. (46)

*pn* <sup>−</sup> *<sup>∂</sup>S*(*q*, *<sup>a</sup>*) *∂q<sup>n</sup>*

is known as the Van Vleck determinant [20]. The

*<sup>S</sup>*(*q*, *<sup>a</sup>*) − *<sup>E</sup>* (*a*1,..., *an*, *<sup>t</sup>*)

transformed equations in the potential form: *pj* = *<sup>∂</sup>S*(*q*, *<sup>a</sup>*)/*∂q<sup>j</sup>*

 *∂*2*S ∂qi∂a<sup>j</sup>*

*∂aj* 

Ψ(*q*, *a*) =

 det

corresponding quasiclassical wave function is given by:

 *δ* 

 det

constants of motion, *<sup>H</sup>* = *<sup>E</sup>* (*A*1(*q*, *<sup>p</sup>*),..., *An*(*q*, *<sup>p</sup>*)). Then the function

 *∂*2*S ∂qi∂a<sup>j</sup>*

has to use some Maslov modifications and use the Airy special functions.

 exp *i h*¯

> Our arguments above were based on the assumed affine geometry of the phase space. However, it is clear that this fact is not very important. It did not influence our views. Affine geometry and the Weyl-Wigner-Moyal-Ville procedure were merely the auxiliary tools of our analysis. Nevertheless, it is convenient to comment our results in general symplectic terms.

> Let (*P*, *γ*) be a symplectic manifold, i.e., a differential manifold *P* endowed with the differential two-form *γ* satisfying the following conditions: it is closed and non-degenerate. In coordinates *ξ<sup>a</sup>* this means that

$$
\gamma = \frac{1}{2} \gamma\_{ab} d\mathfrak{f}^a \wedge d\mathfrak{f}^b \, , \tag{49}
$$

where *<sup>γ</sup>ab*,*<sup>c</sup>* + *<sup>γ</sup>bc*,*<sup>a</sup>* + *<sup>γ</sup>ca*,*<sup>b</sup>* = 0, det [*γab*] �= 0 all over *<sup>P</sup>*. The comma symbol denotes the partial derivative. Therefore, dim *P* = 2*n*, *n* being natural. As *dγ* = 0, then locally *γ* = *dω*. Not always, but in majority of applications *P* is a cotangent bundle over some *n*-dimensional configuration space,

$$P = T^\*Q = \bigcup\_{q \in Q} T\_q^\*Q\_\prime \tag{50}$$

where *TqQ*, *<sup>T</sup>*<sup>∗</sup> *<sup>q</sup> <sup>Q</sup>* denote as usual the tangent space at *<sup>q</sup>* <sup>∈</sup> *<sup>Q</sup>* and its dual — the cotangent space. If *q<sup>i</sup>* , *i* = 1, . . . , *n*, are coordinates in an open domain of *Q*, then the induced coordinates in *<sup>T</sup>*∗*<sup>Q</sup>* are denoted by *qi* , *pi* , where *pi* are components of the canonical momentum attached of *q* ∈ *Q*. This structure gives rise to the Cartan one-form *ω* given locally by *<sup>ω</sup>* = *pidq<sup>i</sup>* ; the coordinate-free definition is easily possible but we do not quote it here. In any case the symplectic form in *<sup>T</sup>*∗*<sup>Q</sup>* is given by *<sup>γ</sup>* <sup>=</sup> *<sup>d</sup><sup>ω</sup>* <sup>=</sup> *dpi* <sup>∧</sup> *dq<sup>i</sup>* . Being non-degenerate, *<sup>γ</sup>* does possess the inverse form *<sup>γ</sup>* with coordinates *<sup>γ</sup>ab* such that *<sup>γ</sup>acγcb* = *<sup>δ</sup><sup>a</sup> <sup>b</sup>*. This gives rise to the Poisson bracket construction

$$\{F, G\} = \gamma^{ab} \frac{\partial F}{\partial \xi^a} \frac{\partial G}{\partial \xi^b} \,, \tag{51}$$

in the induced coordinates *qi* , *pi* :

$$\{F, G\} = \frac{\partial F}{\partial q^a} \frac{\partial G}{\partial p\_a} - \frac{\partial F}{\partial p\_a} \frac{\partial G}{\partial q^a}. \tag{52}$$

Canonical transformations preserve the two-form *γ*, *ϕ*∗*γ* = *γ*, and infinitesimal ones, i.e., canonical vector fields *<sup>X</sup>* satisfy *£X<sup>γ</sup>* = 0. Of course, the identity *£X<sup>γ</sup>* = (*X*⌋*dγ*) + *<sup>d</sup>*(*X*⌋*γ*) implies that because of *dγ* = 0,

$$(d\left(X\left[\gamma\right)\right)\_{ab} = \left(X^c \gamma\_{ca}\right)\_{,b} - \left(X^c \gamma\_{cb}\right)\_{,a} = 0,\tag{53}$$

therefore, at least locally the vector field *<sup>X</sup>* is Hamiltonian (*X*⌋*γ*)*<sup>a</sup>* <sup>=</sup> *<sup>X</sup>cγca* <sup>=</sup> <sup>−</sup>*∂F*/*∂ξa*. It is denoted by *XF* <sup>=</sup> <sup>−</sup>*dF* and called the Hamiltonian vector field generated by the local Hamiltonian *F*. If *F* is globally one-valued, we say that *XF* is a Hamiltonian field generated by *F*. Therefore, unlike the symmetry group of the symmetric metric tensor on a manifold *M*, which is a finite-dimensional Lie group of dimension at most *n*(*n* + 1)/2, the group of symplectomorphisms, i.e., one of canonical transformations is always infinite-dimensional, labelled by arbitrary sufficiently smooth functions on *P*.

An important problem is a classification of submanifolds in a symplectic manifold. This is completely new in comparison to submanifolds in positively definite Riemann spaces. So, let *M* ⊂ *P* be a (2*n* − *m*)-dimensional submanifold ("constraints") in a symplectic manifold (*P*, *γ*), e.g., given by equations

$$F\_a(\xi) = F\_a(q, p) = 0, \qquad a = 1, \ldots, m. \tag{54}$$

The system of those functions is functionally independent, at least in some neighbourhood of *M*. Sometimes it is also convenient to take the foliation by submanifolds *Ma*

$$F\_{\mathbf{a}}(q, p) = c\_{\mathbf{a}} \tag{55}$$

where *ca* are constants. At every *p* ∈ *M* there is a tangent space *TpM* and its symplectic

orthogonal (dual) space *TpM*<sup>⊥</sup> which consists of vectors *<sup>γ</sup>p*-"orthogonal" to *TpM*:

12 Quantum Mechanics

*<sup>γ</sup>acγcb* = *<sup>δ</sup><sup>a</sup>*

locally by *<sup>ω</sup>* = *pidq<sup>i</sup>*

in the induced coordinates

implies that because of *dγ* = 0,

It is denoted by *XF* = −*dF*

(*P*, *γ*), e.g., given by equations

momentum attached of *q* ∈ *Q*. This structure gives rise to the Cartan one-form *ω* given

it here. In any case the symplectic form in *<sup>T</sup>*∗*<sup>Q</sup>* is given by *<sup>γ</sup>* <sup>=</sup> *<sup>d</sup><sup>ω</sup>* <sup>=</sup> *dpi* <sup>∧</sup> *dq<sup>i</sup>*

Being non-degenerate, *<sup>γ</sup>* does possess the inverse form *<sup>γ</sup>* with coordinates *<sup>γ</sup>ab* such that

*∂ξa*

<sup>−</sup> *<sup>∂</sup><sup>F</sup> ∂pa*

*<sup>γ</sup>ca*),*<sup>b</sup>* <sup>−</sup> (*X<sup>c</sup>*

*∂G*

and called the Hamiltonian vector field generated by the local

*Fa*(*ξ*) = *Fa*(*q*, *p*) = 0, *a* = 1, . . . , *m*. (54)

*Fa*(*q*, *p*) = *ca*, (55)

*∂G*

*∂ξ<sup>b</sup>* , (51)

*<sup>∂</sup>q<sup>a</sup>* . (52)

*<sup>γ</sup>cb*),*<sup>a</sup>* <sup>=</sup> 0, (53)

{*F*, *<sup>G</sup>*} <sup>=</sup> *<sup>γ</sup>ab <sup>∂</sup><sup>F</sup>*

*∂q<sup>a</sup>*

*∂G ∂pa*

Canonical transformations preserve the two-form *γ*, *ϕ*∗*γ* = *γ*, and infinitesimal ones, i.e., canonical vector fields *<sup>X</sup>* satisfy *£X<sup>γ</sup>* = 0. Of course, the identity *£X<sup>γ</sup>* = (*X*⌋*dγ*) + *<sup>d</sup>*(*X*⌋*γ*)

therefore, at least locally the vector field *<sup>X</sup>* is Hamiltonian (*X*⌋*γ*)*<sup>a</sup>* <sup>=</sup> *<sup>X</sup>cγca* <sup>=</sup> <sup>−</sup>*∂F*/*∂ξa*.

Hamiltonian *F*. If *F* is globally one-valued, we say that *XF* is a Hamiltonian field generated by *F*. Therefore, unlike the symmetry group of the symmetric metric tensor on a manifold *M*, which is a finite-dimensional Lie group of dimension at most *n*(*n* + 1)/2, the group of symplectomorphisms, i.e., one of canonical transformations is always infinite-dimensional,

An important problem is a classification of submanifolds in a symplectic manifold. This is completely new in comparison to submanifolds in positively definite Riemann spaces. So, let *M* ⊂ *P* be a (2*n* − *m*)-dimensional submanifold ("constraints") in a symplectic manifold

The system of those functions is functionally independent, at least in some neighbourhood

where *ca* are constants. At every *p* ∈ *M* there is a tangent space *TpM* and its symplectic

of *M*. Sometimes it is also convenient to take the foliation by submanifolds *Ma*

*<sup>b</sup>*. This gives rise to the Poisson bracket construction

{*F*, *<sup>G</sup>*} <sup>=</sup> *<sup>∂</sup><sup>F</sup>*

(*<sup>d</sup>* (*X*⌋*γ*))*ab* <sup>=</sup> (*X<sup>c</sup>*

 *qi* , *pi* :

labelled by arbitrary sufficiently smooth functions on *P*.

; the coordinate-free definition is easily possible but we do not quote

.

$$T\_p M^\perp = \left\{ \upsilon \in T\_p P : \gamma(p)\_{ab} \upsilon^b X^a \quad \text{if} \quad X \in T\_p M \right\},\tag{56}$$

or in more sophisticated terms: �*v*⌋*γp*, ·�|*TpM* = 0. It is a peculiarity of symplectic geometry that *TpM*<sup>⊥</sup> need not be complementary to *TpM*. The following class index was introduced to describe this.

If dim *Kp*(*M*) = dim *TpM* <sup>∩</sup> *TpM*<sup>⊥</sup> = *k*, then we put Cl*pM* = (*k*, *m* − *k*). We are interested mainly in situation when this does not depend on *p*, that is, incidentally, a typical situation. Then we write simply Cl *M* = (*k*, *m* − *k*). If *k* = *m*, then we write simply Cl *M* = I and say that *M* is co-isotropic. This means that the subspaces *γ*-orthogonal to *M* are tangent to *M*. If *k* = 0, then Cl *M* = II, and the subspace *γ*-orthogonal to *M* are at the same time transversal (complementary) to *M*. Cl *M* = I implies that the functions *Fa* in (54) or (55) satisfy respectively {*Fa*, *Fb*} |*<sup>M</sup>* = 0 or {*Fa*, *Fb*} = 0. Similarly, Cl *<sup>M</sup>* = II implies that det [{*Fa*, *Fb*}] �<sup>=</sup> 0, at least in a neighbourhood of *<sup>M</sup>*. If *TpM* <sup>⊂</sup> *TpM*⊥, then we say that *<sup>M</sup>* is isotropic. Then for any pair of tangent vectors at any *<sup>p</sup>* ∈ *<sup>M</sup>* we have: *<sup>γ</sup>*(*p*)*abuavb* = 0, when *<sup>u</sup>*, *<sup>v</sup>* <sup>∈</sup> *TpM*. If *<sup>M</sup>* is isotropic, then dim *<sup>M</sup>* <sup>≤</sup> *<sup>n</sup>*. If dim *<sup>M</sup>* <sup>=</sup> *<sup>n</sup>*, i.e., if *TpM* <sup>=</sup> *TpM*⊥, we say that *M* is Lagrangian. It is described by the system of equations *Fa* = 0, *a* = 1, . . . , *n*, {*Fa*, *Fb*} |*<sup>M</sup>* = 0, or, if we deal with a foliation by Lagrangian manifolds, i.e., with a polarization, then *Fa* = *ca*, {*Fa*, *Fb*} = 0. Equations for the Lagrangian submanifold may be solved in the following way with respect to canonical momenta:

$$p\_i = \frac{\partial S}{\partial q^i}, \qquad i = 1, \dots, n,\tag{57}$$

when it is transversal to the fibres of constant *qa*, *a* = 1, . . . , *n*. Let us denote the corresponding manifold by m*S*. The Hamilton-Jacobi equation

$$
\Omega\left(\ldots, q^{\mu}, \ldots; \ldots, \frac{\partial \mathcal{S}}{\partial q^{\mu}}, \ldots\right) = 0\tag{58}
$$

means that m*<sup>S</sup>* belongs to the zero-valued surface of Ω. We have used here the Greek symbols *µ* to indicate that the time variable may be included into coordinates. For example, in non-relativistic mechanics:

$$\frac{\partial S}{\partial t} + H\left(t, \dots, q^i, \dots; \dots; \frac{\partial S}{\partial q^i}, \dots\right) = 0. \tag{59}$$

The integrability condition for the system of Hamilton-Jacobi equations with functions Ω*ν*, *ν* = 0, 1, . . . , *n*, is given by the equation Ω*µ*, Ω*<sup>ν</sup>* = 0, i.e., the manifold Ω*<sup>µ</sup>* = 0, *µ* = 0, 1, . . . , *n*, has the class I, i.e., is co-isotropic.

One can show that on every regular submanifold *M* the assignment *M* ∋ *p* �→ *Kp*(*M*) = *TpM*<sup>⊥</sup> <sup>∩</sup> *TpM* is an integrable distribution, therefore, the quotient manifold *<sup>P</sup>*′ (*M*) = *<sup>M</sup>*/*K*(*M*) carries the canonical symplectic structure *<sup>γ</sup>*′ such that *<sup>γ</sup>*�*<sup>M</sup>* = *<sup>π</sup>*∗*γ*′ , where *<sup>π</sup>* : *<sup>M</sup>* → *<sup>P</sup>*′ (*M*) is the natural projection. Obviously, *K*(*M*) denotes the system of leaves of the distribution. It is clear that dim *<sup>P</sup>*′ (*M*) = 2 (*n* − (*m* + *k*)/2).

Lagrange manifolds, i.e., isotropic ones of dimension *n*, are placed, as seen from the formula, only on co-isotropic, i.e., first class submanifolds. And if they are transversal to the configuration *Q*-fibres (*X*-fibres), then using the formula (57) we obtain the Hamilton-Jacobi equations (58), (59) or simply

$$A\left(\ldots, q^i, \ldots; \ldots; \frac{\partial S}{\partial q^i}, \ldots\right) = a \tag{60}$$

for the potential *<sup>S</sup>*. Every <sup>m</sup>*<sup>S</sup>* ⊂ *<sup>M</sup>* is composed of the foliation of singular fibres *<sup>K</sup>*(*M*). Singular fibres, first of all one-dimensional ones, i.e., integral curves of the Hamiltonian vector fields *XF*, *X*<sup>Ω</sup> are classical trajectories. Those of which m*<sup>S</sup>* are composed were called by Synge coherent families with the potential *S* [18, 19]. As seen, they correspond to quasiclassical wave functions. And classically, being dependent on *n* parameters, they correspond to the complete integrals of Hamilton-Jacobi equation (59). Let us summarize our symplectic interpretation of them.

In quantum mechanics the eigenstates of the physical quantity represented by the Hermitian operator **<sup>A</sup>** are given by density operators <sup>ρ</sup> satisfying the operator eigenequation <sup>ρ</sup>. Let us stress that in general this is the equation both on ρ and *a*. Taking its Hermitian conjugate we obtain ρ**A** = *a*ρ. One can write these equations as (**A** − *a*Id) ρ = 0, ρ (**A** − *a*Id) = 0. This is the afore-mentioned informative aspect of the eigenequation. But just as it was within the Weyl-Wigner-Moyal-Ville framework, this information context implies the formal consequence, but qualitatively a completely different symmetry property, namely the invariance of *ρ* under the unitary group generated by **A**: (1/*ih*¯)[**A**, ρ]=(1/*ih*¯)(**A**ρ − ρ**A**) = 0. Therefore, in the classical limit one must assume that the quasiclassical *ρ* satisfies a pair of mathematically independent, but physically interpretable just as above, conditions: *Aρ* = *aρ* — information, {*A*, *ρ*} = 0 — symmetry.

Let us introduce the set of operators: *<sup>E</sup>*<sup>ρ</sup> := {**<sup>F</sup>** ∈ *<sup>B</sup>*(*H*) : **<sup>F</sup>**<sup>ρ</sup> = <sup>0</sup>}. In principle *<sup>B</sup>*(*H*) denotes the set of bounded operators acting in the Hilbert space *H*. Although, to be honest, one can weaken this assumption. It is also clear that similarly as in classical statistics, the following holds in quanta:

$$-S\left(\rho\_1\right) = \text{Tr}\left(\rho\_1 \ln \rho\_1\right) \le \text{Tr}\left(\rho\_2 \ln \rho\_2\right) = -S\left(\rho\_2\right),\tag{61}$$

when *<sup>E</sup>*ρ<sup>1</sup> ⊂ *<sup>E</sup>*ρ<sup>2</sup> . In other words, the larger *<sup>E</sup>*ρ, the greater informational content of <sup>ρ</sup>. Of course, we mean here the quantum concept of the Shannon entropy and the mathematical sense of Tr(<sup>ρ</sup> ln <sup>ρ</sup>). Quantum pure states are defined in such a way that *<sup>E</sup>*<sup>ρ</sup> is a maximal ideal. It answers uniquely the maximal number of experimental questions. There exists then the one-dimensional linear subspace *<sup>V</sup>* ⊂ *<sup>H</sup>* such that *<sup>E</sup>*<sup>ρ</sup> consists of operators which vanish on *V*,

(*M*) =

, where

= *a* (60)

14 Quantum Mechanics

*<sup>π</sup>* : *<sup>M</sup>* → *<sup>P</sup>*′

of the distribution. It is clear that dim *<sup>P</sup>*′

our symplectic interpretation of them.

— information, {*A*, *ρ*} = 0 — symmetry.

holds in quanta:

*A* ..., *q<sup>i</sup>*

equations (58), (59) or simply

One can show that on every regular submanifold *M* the assignment *M* ∋ *p* �→ *Kp*(*M*) =

Lagrange manifolds, i.e., isotropic ones of dimension *n*, are placed, as seen from the formula, only on co-isotropic, i.e., first class submanifolds. And if they are transversal to the configuration *Q*-fibres (*X*-fibres), then using the formula (57) we obtain the Hamilton-Jacobi

,...;...,

for the potential *<sup>S</sup>*. Every <sup>m</sup>*<sup>S</sup>* ⊂ *<sup>M</sup>* is composed of the foliation of singular fibres *<sup>K</sup>*(*M*). Singular fibres, first of all one-dimensional ones, i.e., integral curves of the Hamiltonian vector fields *XF*, *X*<sup>Ω</sup> are classical trajectories. Those of which m*<sup>S</sup>* are composed were called by Synge coherent families with the potential *S* [18, 19]. As seen, they correspond to quasiclassical wave functions. And classically, being dependent on *n* parameters, they correspond to the complete integrals of Hamilton-Jacobi equation (59). Let us summarize

In quantum mechanics the eigenstates of the physical quantity represented by the Hermitian operator **<sup>A</sup>** are given by density operators <sup>ρ</sup> satisfying the operator eigenequation <sup>ρ</sup>. Let us stress that in general this is the equation both on ρ and *a*. Taking its Hermitian conjugate we obtain ρ**A** = *a*ρ. One can write these equations as (**A** − *a*Id) ρ = 0, ρ (**A** − *a*Id) = 0. This is the afore-mentioned informative aspect of the eigenequation. But just as it was within the Weyl-Wigner-Moyal-Ville framework, this information context implies the formal consequence, but qualitatively a completely different symmetry property, namely the invariance of *ρ* under the unitary group generated by **A**: (1/*ih*¯)[**A**, ρ]=(1/*ih*¯)(**A**ρ − ρ**A**) = 0. Therefore, in the classical limit one must assume that the quasiclassical *ρ* satisfies a pair of mathematically independent, but physically interpretable just as above, conditions: *Aρ* = *aρ*

Let us introduce the set of operators: *<sup>E</sup>*<sup>ρ</sup> := {**<sup>F</sup>** ∈ *<sup>B</sup>*(*H*) : **<sup>F</sup>**<sup>ρ</sup> = <sup>0</sup>}. In principle *<sup>B</sup>*(*H*) denotes the set of bounded operators acting in the Hilbert space *H*. Although, to be honest, one can weaken this assumption. It is also clear that similarly as in classical statistics, the following

when *<sup>E</sup>*ρ<sup>1</sup> ⊂ *<sup>E</sup>*ρ<sup>2</sup> . In other words, the larger *<sup>E</sup>*ρ, the greater informational content of <sup>ρ</sup>. Of course, we mean here the quantum concept of the Shannon entropy and the mathematical sense of Tr(<sup>ρ</sup> ln <sup>ρ</sup>). Quantum pure states are defined in such a way that *<sup>E</sup>*<sup>ρ</sup> is a maximal ideal. It answers uniquely the maximal number of experimental questions. There exists then the one-dimensional linear subspace *<sup>V</sup>* ⊂ *<sup>H</sup>* such that *<sup>E</sup>*<sup>ρ</sup> consists of operators which vanish

− *<sup>S</sup>* (ρ1) = Tr(ρ<sup>1</sup> ln <sup>ρ</sup>1) ≤ Tr(ρ<sup>2</sup> ln <sup>ρ</sup>2) = −*<sup>S</sup>* (ρ2), (61)

(*M*) is the natural projection. Obviously, *K*(*M*) denotes the system of leaves

(*M*) = 2 (*n* − (*m* + *k*)/2).

*∂S ∂qi* ,... 

*TpM*<sup>⊥</sup> <sup>∩</sup> *TpM* is an integrable distribution, therefore, the quotient manifold *<sup>P</sup>*′

*<sup>M</sup>*/*K*(*M*) carries the canonical symplectic structure *<sup>γ</sup>*′ such that *<sup>γ</sup>*�*<sup>M</sup>* = *<sup>π</sup>*∗*γ*′

$$E\_{\rho} = \left\{ \mathbf{F} \in \mathcal{B}(H) : \mathbf{F}|V = 0 \right\}. \tag{62}$$

This means that the subspace *V* ⊂ *H* given by *V* = **<sup>F</sup>**∈*E*<sup>ρ</sup> Ker**<sup>F</sup>** satisfies conditions: <sup>ρ</sup>(*H*) = *<sup>V</sup>*, <sup>ρ</sup>|*<sup>V</sup>* = Id*V*, ρρ = <sup>ρ</sup>, therefore, *<sup>ρ</sup>* is a projector of *<sup>H</sup>* onto *<sup>V</sup>*. The entropy (information) takes on *<sup>ρ</sup>* the minimal (maximal) value, Tr(<sup>ρ</sup> ln <sup>ρ</sup>) <sup>=</sup> 0. When *<sup>P</sup>* <sup>=</sup> *<sup>T</sup>*∗*Q*, then the formulae (47), (48) may be literally applied together with the Bohr-Sommerfeld quantum rules: *ω* = *pidq<sup>i</sup>* = *nh* on any closed curve on <sup>m</sup>*S*. This defines the quantized values of *<sup>a</sup><sup>i</sup>* in terms of integers and Planck constant.

Expression for the Van Vleck determinant is correct independently on the additional phase space structures like the affine one. Just because of the structure of this determinant. This is seen from the density formula:

$$\mathcal{V} = \det \left[ \frac{\partial \mathcal{S}}{\partial q^i \partial a^j} \right] dq^1 \wedge \dots \wedge dq^n \otimes da^1 \wedge \dots \wedge da^n. \tag{63}$$

Moreover, it turns out that this determinant is much more general and even the cotangent bundle structure is not necessary for it. Namely, let us assume a pair of polarizations, i.e., a pair of complementary foliations of a general phase space (*P*, *γ*) by Lagrangian manifolds. Let us observe that *<sup>P</sup>* need not be identical with *<sup>T</sup>*∗*<sup>Q</sup>* and everything we assume is just a pair of foliations. Lagrangian submanifolds of any foliation have a local affine structure. Introducing coordinates *q<sup>i</sup>* , *a<sup>i</sup>* , we can formally describe them in terms of equations: *pi* = *∂S*(*q*, *a*)/*∂q<sup>i</sup>* , but *<sup>S</sup>* is non-unique up to the gauging: *<sup>S</sup>* �→ *<sup>S</sup>* <sup>+</sup> *<sup>ϕ</sup>* ◦ pr1 <sup>+</sup> <sup>Ψ</sup> ◦ pr2, where pr1, pr2 are projections from *<sup>P</sup>* to the *<sup>Q</sup>*, **<sup>R</sup>***n*-manifolds. But this gauging does not influence the value of the Van Vleck determinant.

It is interesting to see what follows when we consider Hamiltonian and quantum dynamics in a homogeneous formulation of Hamiltonian/quantum dynamics. So, let us consider the motion of a particle in an (*n* + 1)-dimensional space-time manifold *X* and take a complete integral *S xµ*, *a<sup>i</sup>* depending on *n* arbitrary constants. Then instead of the above quantities we obtain the following vector-density object:

$$\mathcal{V} = D^{\mu}dx^{0} \wedge dx^{1} \wedge \dots \wedge \mu \wedge \dots \wedge dx^{n} \otimes da^{1} \wedge \dots \wedge da^{n},\tag{64}$$

where *<sup>D</sup><sup>µ</sup>* is a minor of the matrix *∂*2*S*/ *∂xµ∂a<sup>i</sup>* obtained by removing the *<sup>µ</sup>*-th column. The symbol *µ* in the exterior product means that *dx<sup>µ</sup>* is omitted. It is clear that the above expressions imply that

$$\frac{\partial j^{\mu}}{\partial x^{\mu}} = 0,\tag{65}$$

where *j <sup>µ</sup>* = (−1)*µDµ*. This formula is geometrically correct, because *j <sup>µ</sup>* is a contravariant vector density of weight one. Therefore, the left-hand side of (65) is well defined in spite of using the usual partial differentiation. One can easily check that it follows from (58). In particular, if <sup>Ω</sup> in (58) equals the non-relativistic <sup>Ω</sup> = *<sup>p</sup>*<sup>0</sup> + *<sup>H</sup> x*0, *x<sup>i</sup>* ; *pi* = −*E* + *H t*, *q<sup>i</sup>* , *pi* (*E* denotes the energy variable), then *j <sup>µ</sup>* in (64), (65) equals the formerly written non-relativistic four-current

$$(j^{\mu}) = \left( \det \left[ \frac{\partial^2 S}{\partial q^i \partial a^j} \right], \det \left[ \frac{\partial^2 S}{\partial q^i \partial a^j} \right] \frac{\partial H}{\partial p\_i} \left( q, \frac{\partial S}{\partial q} \right) \right). \tag{66}$$

This *j <sup>µ</sup>* satisfies the continuity equation (65) in virtue of (40). For the general relativistically written Ω *xµ*, *p<sup>µ</sup>* , one obtains the four-current density, e.g., for the quasiclassical Klein-Gordon equation. The current (66) corresponds to some choice of the complete integral of Hamilton-Jacobi equations.

Let us mention that for the system of Hamilton-Jacobi equations

$$
\Omega\_{\wedge} \left( \mathbf{x}^{\mu}, \frac{\partial \mathcal{S}}{\partial \mathbf{x}^{\mu}} \right) = \mathbf{0} \tag{67}
$$

we obtain generalized continuity equations. However, there is no place to stop at this topic here. In any case *j <sup>µ</sup>* is also built of the complete integral of (67).

It is difficult not to be astonished by the fact that the above structures were not discovered some hundred years earlier. They are based on the purely classical and deeply geometric concepts. As mentioned, this may be explained only by the fact that the Planck constant was not known then. To be more precise, it was hidden deeply in the thickest of radiation theory and its thermodynamics.

Let us mention some additional facts. We said that the Van Vleck symbol may be assigned to any complementary pair of polarizations *Q* × **R***<sup>n</sup>* ∋ (*q*, *a*) → *V*(*q*, *a*). It may be interpreted in a statistical way due to its structure of the double scalar density. Indeed, the quantity V(*q*, *<sup>a</sup>*) = det *∂*2*S*/*∂q<sup>i</sup> ∂aj* may be dualistically interpreted as the density of probability both in *Q* and in **R***n*. If *A* ⊂ *Q*, *B* ⊂ **R***n*, then

$$P(A,B) = \int\_{A \times B} \mathcal{V}(q,a) dq^1 \dots dq^n da^1 \dots da^n \tag{68}$$

may be interpreted as the quasiclassical probability that the system with values of integrals of motion in *B* ⊂ **R***<sup>n</sup>* will be found in the region *A* ⊂ *Q* of the configuration space. And conversely, it is equal to the probability that the system placed in *A* ⊂ *Q* will show the values of integration constants in *B* ⊂ **R***n*. To be honest, in general they are non-normalized to unity relative probabilities.

When performing pull-backs of probability densities on *Q* to m*S*, we obtain some probability distributions on the Lagrangian manifold. Therefore, quasiclassical pure quantum states may be interpreted as probability distributions concentrated on submanifolds m*S*. So, their supports are *<sup>n</sup>*-dimensional and distinguished by the fact that *<sup>γ</sup>*�m*<sup>S</sup>* = 0. Quasiclassical

mixed states are usually smeared out as 2*n*-dimensional probability distributions on *P* = *<sup>T</sup>*∗*Q*.

Let us quote yet some another quasiclassical structures. To do this we begin with the linear symplectic spaces. Let *D*(*P*) denote the set of all linear Lagrangian subspaces of *P*. Let *M* ⊂ *P* be some co-isotropic linear subspace of *P* and *D*(*M*) ⊂ *D*(*P*) denote the set Lagrangian subspaces contained in *M*. One can show that any m ⊂ *D*(*P*) intersects *M* along some at least (*<sup>n</sup>* − *<sup>m</sup>*)-dimensional isotropic subspace. But the singular fibre of *<sup>M</sup>*, i.e., *<sup>M</sup>*<sup>⊥</sup> ⊂ *<sup>M</sup>* is *m*-dimensional. Therefore, the subspace

$$E\_M(\mathfrak{m}) := \mathfrak{m} \cap M + M^\perp \tag{69}$$

is also Lagrangian and contained in *M*. Therefore, without any additional structure *M* gives rise to the mapping *EM* : *<sup>D</sup>*(*P*) → *<sup>D</sup>*(*M*) with the following properties:

1. *EM* is a retraction onto the subset *<sup>D</sup>*(*M*), moreover, it is a projection:

$$E\_M|D(M) = \text{id}\_{D(M)'} \qquad E\_M \circ E\_M = E\_M. \tag{70}$$

2. *<sup>M</sup>*, *<sup>N</sup>* are co-isotropic and compatible, i.e., *<sup>M</sup>* ∩ *<sup>N</sup>* is also co-isotropic, then *EM*, *EN* commute and

$$E\_M \circ E\_N = E\_N \circ E\_M = E\_{M \cap N}.\tag{71}$$


$$E\_{f(M)} = F \circ E\_M \circ F^{-1} \, \prime \tag{72}$$

where *F* : *D*(*P*) → *D*(*P*) is induced by *f* .

16 Quantum Mechanics

−*E* + *H*

This *j*

written Ω

here. In any case *j*

V(*q*, *a*) = det

and its thermodynamics.

 *∂*2*S*/*∂q<sup>i</sup>*

relative probabilities.

*∂aj* 

*P*(*A*, *B*) =

 *A*×*B*

in *Q* and in **R***n*. If *A* ⊂ *Q*, *B* ⊂ **R***n*, then

 *t*, *q<sup>i</sup>* , *pi* 

written non-relativistic four-current

*xµ*, *p<sup>µ</sup>* 

of Hamilton-Jacobi equations.

(*j <sup>µ</sup>*) = det

spite of using the usual partial differentiation. One can easily check that it follows from

 *∂*2*S ∂qi∂a<sup>j</sup>*

*<sup>µ</sup>* satisfies the continuity equation (65) in virtue of (40). For the general relativistically

Klein-Gordon equation. The current (66) corresponds to some choice of the complete integral

we obtain generalized continuity equations. However, there is no place to stop at this topic

It is difficult not to be astonished by the fact that the above structures were not discovered some hundred years earlier. They are based on the purely classical and deeply geometric concepts. As mentioned, this may be explained only by the fact that the Planck constant was not known then. To be more precise, it was hidden deeply in the thickest of radiation theory

Let us mention some additional facts. We said that the Van Vleck symbol may be assigned to any complementary pair of polarizations *Q* × **R***<sup>n</sup>* ∋ (*q*, *a*) → *V*(*q*, *a*). It may be interpreted in a statistical way due to its structure of the double scalar density. Indeed, the quantity

may be interpreted as the quasiclassical probability that the system with values of integrals of motion in *B* ⊂ **R***<sup>n</sup>* will be found in the region *A* ⊂ *Q* of the configuration space. And conversely, it is equal to the probability that the system placed in *A* ⊂ *Q* will show the values of integration constants in *B* ⊂ **R***n*. To be honest, in general they are non-normalized to unity

When performing pull-backs of probability densities on *Q* to m*S*, we obtain some probability distributions on the Lagrangian manifold. Therefore, quasiclassical pure quantum states may be interpreted as probability distributions concentrated on submanifolds m*S*. So, their supports are *<sup>n</sup>*-dimensional and distinguished by the fact that *<sup>γ</sup>*�m*<sup>S</sup>* = 0. Quasiclassical

may be dualistically interpreted as the density of probability both

V(*q*, *a*)*dq*<sup>1</sup> ... *dqnda*<sup>1</sup> ... *da<sup>n</sup>* (68)

 *∂H ∂pi q*, *∂S ∂q* 

, one obtains the four-current density, e.g., for the quasiclassical

 *x*0, *x<sup>i</sup>* ; *pi* =

. (66)

*<sup>µ</sup>* in (64), (65) equals the formerly

= 0 (67)

(58). In particular, if <sup>Ω</sup> in (58) equals the non-relativistic <sup>Ω</sup> = *<sup>p</sup>*<sup>0</sup> + *<sup>H</sup>*

(*E* denotes the energy variable), then *j*

 *∂*2*S ∂qi∂a<sup>j</sup>*

Let us mention that for the system of Hamilton-Jacobi equations

Ω∧ *<sup>x</sup>µ*, *<sup>∂</sup><sup>S</sup> ∂x<sup>µ</sup>* 

*<sup>µ</sup>* is also built of the complete integral of (67).

 , det

> This is interesting and easily interpretable in terms of quasiclassical wave functions. The relationship with the corresponding quantum relations is also readable. Let us illustrate this with the following simple example.

> We consider an affine (or linear) phase space with affine coordinates *qi* , *pi* . Then we have *<sup>ω</sup>* = *pidq<sup>i</sup>* , *<sup>γ</sup>* = *<sup>d</sup><sup>ω</sup>* = *dpi* ∧ *dq<sup>i</sup>* . Let us take as *M* the manifold of states on which the momentum variable *p*<sup>1</sup> takes on a fixed values *b*,

$$M = \{ p \in P : p\_1(p) = b \}, \qquad D(P) \ni \mathfrak{m} = \left\{ p \in P : q^i(p) = a^i \right\},\tag{73}$$

where *<sup>i</sup>* = 1, . . . , *<sup>n</sup>*, therefore, *<sup>M</sup>* is a manifold with fixed values of *<sup>p</sup>*<sup>1</sup> equal to *<sup>b</sup>*, and <sup>m</sup> is the Lagrangian space (the carrier of a quasiclassical state) with fixed positions *a<sup>i</sup>* . One can easily show that

$$E\_M(\mathfrak{m}) = \left\{ p \in P : p\_1(p) = b, q^2(p) = a^2, \dots, q^n(p) = a^n \right\}.\tag{74}$$

Therefore, if we fix the value of *p*<sup>1</sup> with the help of *EM*, we result in a complete indeterminacy of *q*1. But this is just the classical uncertainty principle. Simply on the classical, or rather semi-classical level, it is not a point in the phase space, but rather Legendre submanifold, or to be more precise, a probability distribution on it, that is a counterpart of the quantum wave state.

*P* was assumed here to be a linear space endowed with a symplectic structure. But it turns out that the above prescription may be globalized to the general symplectic manifold. Roughly speaking, this follows from its "flatness" which makes it similar to a linear symplectic space in finite domains due to the existence of Darboux coordinates which enable one to write *<sup>γ</sup>* = *dpi* ∧ *dq<sup>i</sup>* . This holds in every symplectic manifold, not necessarily cotangent bundle, locally, but in finite domains. Indeed, let *M* ⊂ *P* be a co-isotropic submanifold, *K*(*M*) its singular foliation, and m ⊂ *P* — Lagrangian submanifold. The manifolds *M* and m need not to intersect; in such situation we say that *EM*(m) = ∅. In this way the empty set ∅ is joined to *<sup>D</sup>*(*P*). It corresponds to the vanishing wave function. We put also *EM*(∅) = ∅. Similarly we do when m, *M* intersect in a non-clean way, i.e., otherwise than linear subspaces. Let us assume the generic case, when m, *M* intersect in a regular way, i.e., when *Tp*m ∩ *TpM* = *Tp*(m ∩ *M*) for any *p* ∈ m ∩ *M*. To be more precise, we assume that the subset of points *p* at which this is satisfied is a Lagrangian submanifold. Obviously, m ∩ *M* is an isotropic submanifold. If the intersection m ∩ *M* is regular at its every point *p*, then *EM*(m) is defined as the maximal extension of <sup>m</sup> ∩ *<sup>M</sup>* by the singular foliation *<sup>K</sup>*(*M*), *EM*(m) = *<sup>π</sup>*−<sup>1</sup> (<sup>m</sup> <sup>∩</sup> *<sup>M</sup>*). It is evidently Lagrangian and *EM* : *<sup>D</sup>*(*P*) <sup>→</sup> *<sup>D</sup>*(*M*) satisfies the above properties (70)–(72). One should mention only that (72) is satisfied by every canonical mapping, i.e., every diffeomorphism preserving the two-form *γ*. The finite-dimensional symplectic group is replaced by the infinite-dimensional group "parameterized" by arbitrary functions.

It is also interesting to know that there are classical counterparts of superpositions and scalar products. We have seen that pure states are represented in a sense by probability distributions concentrated on Lagrange manifolds m*S*. But the function *S* itself, i.e., the phase of wave functions, is not contained in the corresponding analogy. One should adjoint an additional dimension, action, and consider locally the manifolds *Q* × **R**, *P* × **R**, or rather *Q* × SU(1), *P* × SU(1). To be mathematically more honest, one should use the principal fibre bundles with the bases *<sup>Q</sup>*, *<sup>P</sup>* and the structure group **<sup>R</sup>**additive or SU(1)multiplicative. The corresponding geometry of the contact fibre bundle *C* over *P* is locally given by

$$
\Omega = p\_i dq^i - dz. \tag{75}
$$

Take the set of Legendre submanifolds corresponding to the complete integral {*Sa* : *a* ∈ *A*} of the Hamilton-Jacobi equation (40), (58), (59). These solutions may be represented by their diagrams (locally) in *Q* × **R** or *Q* × SU(1):

18 Quantum Mechanics

show that

state.

*<sup>γ</sup>* = *dpi* ∧ *dq<sup>i</sup>*

functions.

*EM*(m) =

*<sup>p</sup>* ∈ *<sup>P</sup>* : *<sup>p</sup>*1(*p*) = *<sup>b</sup>*, *<sup>q</sup>*2(*p*) = *<sup>a</sup>*2,..., *<sup>q</sup>n*(*p*) = *<sup>a</sup><sup>n</sup>*

Therefore, if we fix the value of *p*<sup>1</sup> with the help of *EM*, we result in a complete indeterminacy of *q*1. But this is just the classical uncertainty principle. Simply on the classical, or rather semi-classical level, it is not a point in the phase space, but rather Legendre submanifold, or to be more precise, a probability distribution on it, that is a counterpart of the quantum wave

*P* was assumed here to be a linear space endowed with a symplectic structure. But it turns out that the above prescription may be globalized to the general symplectic manifold. Roughly speaking, this follows from its "flatness" which makes it similar to a linear symplectic space in finite domains due to the existence of Darboux coordinates which enable one to write

locally, but in finite domains. Indeed, let *M* ⊂ *P* be a co-isotropic submanifold, *K*(*M*) its singular foliation, and m ⊂ *P* — Lagrangian submanifold. The manifolds *M* and m need not to intersect; in such situation we say that *EM*(m) = ∅. In this way the empty set ∅ is joined to *<sup>D</sup>*(*P*). It corresponds to the vanishing wave function. We put also *EM*(∅) = ∅. Similarly we do when m, *M* intersect in a non-clean way, i.e., otherwise than linear subspaces. Let us assume the generic case, when m, *M* intersect in a regular way, i.e., when *Tp*m ∩ *TpM* = *Tp*(m ∩ *M*) for any *p* ∈ m ∩ *M*. To be more precise, we assume that the subset of points *p* at which this is satisfied is a Lagrangian submanifold. Obviously, m ∩ *M* is an isotropic submanifold. If the intersection m ∩ *M* is regular at its every point *p*, then *EM*(m) is defined as the maximal extension of <sup>m</sup> ∩ *<sup>M</sup>* by the singular foliation *<sup>K</sup>*(*M*), *EM*(m) = *<sup>π</sup>*−<sup>1</sup> (<sup>m</sup> <sup>∩</sup> *<sup>M</sup>*). It is evidently Lagrangian and *EM* : *<sup>D</sup>*(*P*) <sup>→</sup> *<sup>D</sup>*(*M*) satisfies the above properties (70)–(72). One should mention only that (72) is satisfied by every canonical mapping, i.e., every diffeomorphism preserving the two-form *γ*. The finite-dimensional symplectic group is replaced by the infinite-dimensional group "parameterized" by arbitrary

It is also interesting to know that there are classical counterparts of superpositions and scalar products. We have seen that pure states are represented in a sense by probability distributions concentrated on Lagrange manifolds m*S*. But the function *S* itself, i.e., the phase of wave functions, is not contained in the corresponding analogy. One should adjoint an additional dimension, action, and consider locally the manifolds *Q* × **R**, *P* × **R**, or rather *Q* × SU(1), *P* × SU(1). To be mathematically more honest, one should use the principal fibre bundles with the bases *<sup>Q</sup>*, *<sup>P</sup>* and the structure group **<sup>R</sup>**additive or SU(1)multiplicative. The corresponding

Take the set of Legendre submanifolds corresponding to the complete integral {*Sa* : *a* ∈ *A*} of the Hamilton-Jacobi equation (40), (58), (59). These solutions may be represented by their

<sup>Ω</sup> = *pidq<sup>i</sup>* − *dz*. (75)

geometry of the contact fibre bundle *C* over *P* is locally given by

. This holds in every symplectic manifold, not necessarily cotangent bundle,

. (74)

$$\text{Graph } \mathbb{S}\_{\mathfrak{a}} = \left\{ (\mathfrak{q}, \mathbb{S}\_{\mathfrak{a}}(\mathfrak{q})) : \mathfrak{q} \in \mathbb{Q} \right\}.\tag{76}$$

The independence of the Hamilton-Jacobi equation on the algebraic presence of the variable *S* implies that for any value of *a* the function *Sa* + *t*(*a*) with *t*(*a*) ∈ **R** is a solution too. The Hamilton-Jacobi equation imposes only conditions on the tangent elements of functions, therefore, the envelope of diagrams {(*q*, *Sa* + *t*(*a*)) : *q* ∈ *Q*} denoted by

$$\text{Env}\_{a \in A} \left\{ (q, \mathcal{S}\_a(q) + t(a)) : q \in \mathcal{Q} \right\} \tag{77}$$

also represents some solution of (40), (58), (59). The arbitrariness of these solutions corresponds exactly to the arbitrariness of functions *t* : *A* → **R**. Let us repeat that (77) is a diagram of the set of values {(*q*, *S*(*q*)) : *q* ∈ *Q*}, where the function *S* is obtained from the family of *Sa*-s and *t* in the following way:

i We start from equations:

$$\frac{\partial}{\partial a^i} \left( S\_a(q) + t(a) \right) = 0 \tag{78}$$

and solve them, at least in principle, with respect to *a*. One obtains (also in principle) some *q*-dependent solution, *a*(*q*).

ii This solution is substituted to *Sa*(*q*) + *t*(*a*) and one obtains the expression denoted by the Stat-symbol,

$$S(q) = S\_{a(q)}(q) + t(a(q)) = \text{Stat}\_{a \in A} \left( S\_a(q) + t(a) \right). \tag{79}$$

This follows from the theory of the Hamilton-Jacobi equation. But the same may be shown from "deriving" the continuous superpositions of wave functions satisfying the Schrödinger equation, by performing the WKB-limit transition ¯*h* → 0 in the following expression:

$$\int \sqrt{w(a)} \exp\left(\frac{i}{\hbar}t(a)\right) \sqrt{D(a)} \exp\left(\frac{i}{\hbar}S(q,a)\right) d\_{\text{n}}a. \tag{80}$$

In the WKB-limit ¯*h* → 0 one obtains just (79) as the limit condition. Let us mention that for any function *S* on a differentiable manifold *A* the symbol Stat*f* denotes the value of *S* at the stationary point *a* ∈ *A*, where *dSa* = 0. If there are many stationary points, then *S* in (79) is multivalued. This means that the quasiclassical superposition consists of several waves with various values of phases. We omit the relatively complicated quasiclassical behaviour of moduli, just for simplicity. Similarly, for the pair of wave functions <sup>Ψ</sup><sup>1</sup> <sup>=</sup> <sup>√</sup>*D*<sup>1</sup> exp (*i*/¯*h*) *<sup>S</sup>*1, <sup>Ψ</sup><sup>2</sup> <sup>=</sup> <sup>√</sup>*D*<sup>2</sup> exp (*i*/¯*h*) *<sup>S</sup>*<sup>2</sup> we can investigate the quasiclassical behaviour of the scalar product:

$$
\langle \Psi\_1 | \Psi\_2 \rangle = \int \overline{\Psi}\_1(q) \Psi\_2(q) d\_{\mathfrak{n}} q = \int \sqrt{D\_1 D\_2} \exp\left(\frac{i}{\hbar} \left(S\_1 - S\_2\right)\right) d\_{\mathfrak{n}} q. \tag{81}
$$

Denoting �Ψ1|Ψ2� <sup>=</sup> <sup>√</sup>*<sup>D</sup>* exp (*i*/¯*h*) *<sup>S</sup>* and applying the method of stationary phase we again obtain *<sup>S</sup>* = Stat(*S*<sup>2</sup> − *<sup>S</sup>*1) = Stat*q*∈*<sup>Q</sup>* (*S*2(*q*) − *<sup>S</sup>*1(*q*)), where just as previously, Stat*q*∈*<sup>Q</sup> <sup>ϕ</sup>*(*q*) denotes the value of *ϕ* at the point *q* ∈ *Q*, where the differential of *ϕ* vanishes:

$$d\varphi\_q = 0.\tag{82}$$

The situation is simple when (82) has exactly one solution. If there are a few *q*1,..., *qk*, then the scalar product is a superposition of ones with the corresponding phases

$$
\lambda\_1 \exp\left(\frac{i}{\hbar}\varphi\_1\right) + \dots + \lambda\_k \exp\left(\frac{i}{\hbar}\varphi\_k\right) \tag{83}
$$

where *λ*-s are obtained from the quasiclassical limits of *D*. In any case there is a multivalued phase *<sup>ϕ</sup>*1,..., *<sup>ϕ</sup>k*. If (82) has no solutions, then we say that <sup>Ψ</sup>1, <sup>Ψ</sup><sup>2</sup> are quasiclassically orthogonal.

It would be nice to express those concepts in terms of the contact geometry (75). Let us remind that the Lagrange submanifolds <sup>m</sup>*<sup>S</sup>* ⊂ *<sup>P</sup>* represent rather the density operators of quasiclassical pure states than their wave functions. The latter ones are represented by horizontal lifts of Lagrange submanifolds, i.e., by the maximal, thus *n*-dimensional horizontal submanifolds in *C*, i.e., such ones M that Ω�M = 0. In particular, they are given as <sup>M</sup>*<sup>S</sup>* = hor <sup>m</sup>*S*, where

$$\mathfrak{M}\_{\mathbb{S}} := \left\{ \left( d\mathbb{S}\_{q\prime} \mathcal{S}(q) \right) : q \in \mathbb{Q} \right\}.\tag{84}$$

But they need not be so; another extreme example is the horizontal lift of *<sup>T</sup>*<sup>∗</sup> *<sup>q</sup> Q*,

$$\mathfrak{M}\_{\emptyset} := \left( T\_{\emptyset}^{\*} \mathbb{Q}, 0 \right) = \left\{ (p, 0) : p \in T\_{\emptyset}^{\*} \mathbb{Q} \right\}. \tag{85}$$

Intermediate examples between (84), (85) are labelled by pairs (*M*, *S*) where *M* ⊂ *Q* is a submanifold of *Q* and *S* : *M* → **R** is a real-valued function on *M*. The corresponding Legendre submanifold in *<sup>T</sup>*∗*<sup>Q</sup>* is given by

$$\mathfrak{M}\_{(M,S)} := \left\{ (p\_\prime S(\pi(p))) : \pi(p) \in M, \ p|T\_{\pi(p)}M = dS\_{\pi(p)} \right\},\tag{86}$$

*<sup>π</sup>* : *<sup>T</sup>*∗*<sup>Q</sup>* → *<sup>Q</sup>* is the natural projection of the co-tangent bundle onto its base.

Let us now translate the above formulae into the language of contact geometry. Some similarities to the rigorous quantum expressions will be obvious. The vertical fibre bundle action of the structural group **R** or SU(1) when operating on the elements M of the set of Legendre manifolds H(*C*) will be denoted as follows: [*t*]M := {*gt*(*z*) : *z* ∈ M}, where *gt* is the action of the group element in *C*. If *T* is a countable subset of the group elements, then we put:

$$T\mathfrak{M} := \bigcup\_{t \in G} [t]\mathfrak{M} = \bigcup\_{t \in G} \left\{ \mathfrak{g}\_t(z) : z \in \mathfrak{M} \right\}.\tag{87}$$

In the case of empty sets (which correspond formally to zero), we have ∅M = ∅, *T*∅ = ∅.

Now let us express the quasiclassical "superposition" and phases of the "scalar products" in terms of the contact geometry. Let *M* ⊂ *C* be a submanifold. Its characteristic subset Σ(*M*) is defined as the set of all points *z* ∈ *M* at which Ω*z*|*TzM* = 0. In practical applications we often deal with the situation that Σ(*M*), which is always horizontal, is at the same time *n*-dimensional, therefore, it is a Legendre submanifold, i.e., an element of H(*C*). Let us assume that {M*<sup>a</sup>* : *a* ∈ *A*} is a family of elements of H(*C*), i.e., a family of Legendre submanifolds. We say that its superposition is the maximal element of H(*C*) which is contained in the determinant set of *<sup>a</sup>*∈*<sup>A</sup>* <sup>M</sup>*a*. We denote it by <sup>M</sup> <sup>=</sup> *Ea*∈*A*M*a*. Without going into details we show with the help of examples below that this superposition is in fact, from the point of view of *Q* × **R**, the envelope or "generalized envelope" of the family of surfaces:

i Let us again consider the contact manifold *<sup>C</sup>* = *<sup>T</sup>*∗*<sup>Q</sup>* × **<sup>R</sup>** with the natural contact form *pidq<sup>i</sup>* − *dz*. We take a manifold *<sup>A</sup>* parameterizing functions *Sa*(*q*) = *<sup>S</sup>*(*q*, *<sup>a</sup>*) and the coefficients function *f* : *A* → **R**. *S* gives rise to the following family of Legendre manifolds:

$$\mathfrak{M}\_{\mathfrak{q}} := \mathfrak{M}\_{\mathbb{S}(\cdot,\mathfrak{a})} = \left\{ \left( d\mathbb{S}\left(\cdot,a\right)\_{\mathfrak{q}}, \mathbb{S}(\mathfrak{q},a) \right) : \mathfrak{q} \in \mathbb{Q} \right\}.\tag{88}$$

If it happens (it need not be so) that

20 Quantum Mechanics

orthogonal.

as <sup>M</sup>*<sup>S</sup>* = hor <sup>m</sup>*S*, where

Denoting �Ψ1|Ψ2� <sup>=</sup> <sup>√</sup>*<sup>D</sup>* exp (*i*/¯*h*) *<sup>S</sup>* and applying the method of stationary phase we again obtain *<sup>S</sup>* = Stat(*S*<sup>2</sup> − *<sup>S</sup>*1) = Stat*q*∈*<sup>Q</sup>* (*S*2(*q*) − *<sup>S</sup>*1(*q*)), where just as previously, Stat*q*∈*<sup>Q</sup> <sup>ϕ</sup>*(*q*)

The situation is simple when (82) has exactly one solution. If there are a few *q*1,..., *qk*, then

where *λ*-s are obtained from the quasiclassical limits of *D*. In any case there is a multivalued phase *<sup>ϕ</sup>*1,..., *<sup>ϕ</sup>k*. If (82) has no solutions, then we say that <sup>Ψ</sup>1, <sup>Ψ</sup><sup>2</sup> are quasiclassically

It would be nice to express those concepts in terms of the contact geometry (75). Let us remind that the Lagrange submanifolds <sup>m</sup>*<sup>S</sup>* ⊂ *<sup>P</sup>* represent rather the density operators of quasiclassical pure states than their wave functions. The latter ones are represented by horizontal lifts of Lagrange submanifolds, i.e., by the maximal, thus *n*-dimensional horizontal submanifolds in *C*, i.e., such ones M that Ω�M = 0. In particular, they are given

> : *q* ∈ *Q*

Intermediate examples between (84), (85) are labelled by pairs (*M*, *S*) where *M* ⊂ *Q* is a submanifold of *Q* and *S* : *M* → **R** is a real-valued function on *M*. The corresponding

Let us now translate the above formulae into the language of contact geometry. Some similarities to the rigorous quantum expressions will be obvious. The vertical fibre bundle action of the structural group **R** or SU(1) when operating on the elements M of the set of Legendre manifolds H(*C*) will be denoted as follows: [*t*]M := {*gt*(*z*) : *z* ∈ M}, where *gt* is

(*p*, 0) : *<sup>p</sup>* <sup>∈</sup> *<sup>T</sup>*<sup>∗</sup>

(*p*, *<sup>S</sup>*(*π*(*p*))) : *<sup>π</sup>*(*p*) <sup>∈</sup> *<sup>M</sup>*, *<sup>p</sup>*|*Tπ*(*p*)*<sup>M</sup>* <sup>=</sup> *dSπ*(*p*)

*<sup>q</sup> Q* 

<sup>M</sup>*<sup>S</sup>* := *dSq*, *<sup>S</sup>*(*q*)

But they need not be so; another extreme example is the horizontal lift of *<sup>T</sup>*<sup>∗</sup>

*<sup>π</sup>* : *<sup>T</sup>*∗*<sup>Q</sup>* → *<sup>Q</sup>* is the natural projection of the co-tangent bundle onto its base.

M*<sup>q</sup>* := *T*∗ *<sup>q</sup> Q*, 0 = 

Legendre submanifold in *<sup>T</sup>*∗*<sup>Q</sup>* is given by

<sup>M</sup>(*M*,*S*) :<sup>=</sup>

+ ... + *<sup>λ</sup><sup>k</sup>* exp

 *i h*¯ *ϕk* 

*dϕ<sup>q</sup>* = 0. (82)

, (83)

. (84)

*<sup>q</sup> Q*,

. (85)

, (86)

denotes the value of *ϕ* at the point *q* ∈ *Q*, where the differential of *ϕ* vanishes:

the scalar product is a superposition of ones with the corresponding phases

 *i h*¯ *ϕ*1 

*λ*<sup>1</sup> exp

$$E\_{a \in A}[f(a)]\mathfrak{M}\_a = \mathfrak{M}\_S = \left\{ \left( dS\_{q\prime} S(q) \right) : q \in Q \right\},\tag{89}$$

then we obtain *<sup>S</sup>*(*q*) = Stat*a*∈*<sup>A</sup>* (*S*(*q*, *<sup>a</sup>*) + *<sup>f</sup>*(*a*)). And this means that the manifold *<sup>ξ</sup><sup>S</sup>* := {(*q*, *S*(*q*)) : *q* ∈ *Q*} ⊂ *Q* × **R** is really the literal envelope of the family of submanifolds *<sup>ξ</sup><sup>a</sup>* :<sup>=</sup> *<sup>ξ</sup>S*(·,*a*) <sup>=</sup> {(*q*, *<sup>S</sup>*(*q*, *<sup>a</sup>*)) : *<sup>q</sup>* <sup>∈</sup> *<sup>Q</sup>*} <sup>⊂</sup> *<sup>Q</sup>* <sup>×</sup> **<sup>R</sup>**.


$$\hat{S}(p) = \text{Stat}\_{\mathbf{x} \in V} \left( S(\mathbf{x}) - \langle p, \mathbf{x} \rangle \right), \ S(\mathbf{x}) = \text{Stat}\_{p \in V^\*} \left( \hat{S}(p) + \langle p, \mathbf{x} \rangle \right). \tag{90}$$

When we take into account that the analogy between phases of <sup>Ψ</sup>(*x*) and <sup>Ψ</sup>(*p*) is seriously accepted, we see immediately the obvious quasiclassical relationship between Ψ(*x*) and <sup>Ψ</sup>(*p*). It is based on the concept of generalized envelope.

Let us also notice that all those concepts are invariant with respect to the special contact transformations in *C*. First of all, let us remind that *u* : *C* → *C* is a special contact transformation when it preserves <sup>Ω</sup>, *<sup>u</sup>*∗<sup>Ω</sup> = <sup>Ω</sup>. If *<sup>u</sup>* is such and *<sup>U</sup>* : H(*C*) → H(*C*) is the corresponding transformation of H(*C*), then *UEa*∈*A*[*ta*]M*<sup>a</sup>* = *Ea*∈*A*[*ta*]*U*M*a*.

Now let us begin with the concept of the vertical distance, i.e., scalar product of Legendre submanifolds. Let us take a pair of Legendre manifolds <sup>M</sup>1,M<sup>2</sup> ∈ H(*C*) with the property that their Lagrange projections <sup>m</sup>1, <sup>m</sup><sup>2</sup> and also <sup>m</sup><sup>1</sup> ∩ <sup>m</sup><sup>2</sup> are connected and simply-connected. Then there exists exactly one number *t* ∈ **R** (or exp(*it*) ∈ SU(1)) of the property that (*gt*M1) ∩ <sup>M</sup><sup>2</sup> �= ∅. This number *<sup>t</sup>* or better its unitary exponent exp(*it*) is the scalar product of <sup>M</sup><sup>1</sup> and <sup>M</sup>2. To be more precise, the classical scalar product is exp(*it*), where *<sup>t</sup>* is its phase. More generally, we say that the vertical distance, or the Huygens scalar product [M1|M2] of the pair of Legendre submanifolds <sup>M</sup>1, <sup>M</sup><sup>2</sup> is a subset of **<sup>R</sup>** (or exponentially of SU(1)) such that if *<sup>t</sup>* ∈ [M1|M2], then <sup>M</sup><sup>2</sup> ∩ *gt* (M1) �= ∅. If [M1|M2] is empty, then we say that <sup>M</sup>1, <sup>M</sup><sup>2</sup> are orthogonal. Their Lagrange projections <sup>m</sup>1, <sup>m</sup><sup>2</sup> onto *<sup>P</sup>* <sup>=</sup> *<sup>T</sup>*∗*<sup>Q</sup>* are then disjoint. It is clear that any mapping *U* : H(*C*) → H(*C*) generated by a special contact transformation *u* : *C* → *C* is then "unitary" in the sense of the scalar product [·, ·], i.e., [*U*M1|*U*M2] = [M1|M2] for any pair of Legendre submanifolds M1, M2.

Let *<sup>M</sup>* <sup>⊂</sup> *<sup>P</sup>* <sup>=</sup> *<sup>T</sup>*∗*<sup>Q</sup>* be any co-isotropic submanifold and <sup>H</sup>*M*(*C*) ⊂ H(*C*) denote the set of Legendre submanifolds with Lagrange projections to *<sup>P</sup>* = *<sup>T</sup>*∗*<sup>Q</sup>* placed within *<sup>M</sup>*. Then the operations *EM* on Lagrangian submanifolds introduced above may be canonically lifted to the operations <sup>Π</sup>*<sup>M</sup>* acting on the horizontal lifts of <sup>m</sup> <sup>⊂</sup> *<sup>D</sup>*(*P*) = *<sup>D</sup>* (*T*∗*Q*). Namely, for any co-isotropic *<sup>M</sup>* there exists the canonical mapping <sup>Π</sup>*<sup>M</sup>* : H(*C*) → H*M*(*C*) with the property: <sup>Π</sup> ◦ <sup>Π</sup>*<sup>M</sup>* = <sup>Λ</sup>*<sup>M</sup>* ◦ <sup>Π</sup>, (Π*M*M) ∩ <sup>M</sup> = *<sup>π</sup>*−1(*M*) ∩ <sup>M</sup>, where <sup>Π</sup> : H(*C*) → *<sup>D</sup>*(*P*) = *<sup>D</sup>* (*T*∗*Q*) is the natural projection induced by the fibre bundle projection *<sup>π</sup>* : *<sup>C</sup>* → *<sup>P</sup>*. In fact, <sup>Π</sup>*M*<sup>M</sup> is the horizontal lift of <sup>Λ</sup>*M*<sup>m</sup> which contains <sup>M</sup> ∩ *<sup>π</sup>*−1(*M*) . If m intersects *M* in a regular way, then <sup>Π</sup>*M*<sup>M</sup> is a maximal extension of the intersection <sup>M</sup> ∩ *<sup>π</sup>*−1(*M*) along the fibres of the Ω-horizontal lift *K*Ω(*M*) = lift *K*(*M*). If m ∩ *M* = ∅ or if it is not regular, then it is assumed that <sup>Π</sup>*M*<sup>M</sup> = ∅.

Let us repeat again that Π*<sup>M</sup>* satisfy the properties (69)–(72) modified by the admitted empty-set values: <sup>Π</sup>*M*|H*M*(*C*) = IdH*<sup>M</sup>* (*C*), <sup>Π</sup>*<sup>M</sup>* ◦ <sup>Π</sup>*<sup>M</sup>* <sup>=</sup> <sup>Π</sup>*M*, <sup>Π</sup>*<sup>M</sup>* ◦ <sup>Π</sup>*<sup>N</sup>* <sup>=</sup> <sup>Π</sup>*<sup>N</sup>* ◦ <sup>Π</sup>*<sup>M</sup>* <sup>=</sup> <sup>Π</sup>*M*∩*<sup>N</sup>* if Cl *<sup>M</sup>* ∩ *<sup>N</sup>* = I. If <sup>Π</sup>*<sup>M</sup>* ◦ <sup>Π</sup>*<sup>N</sup>* = <sup>Π</sup>*<sup>N</sup>* ◦ <sup>Π</sup>*M*, then *<sup>M</sup>* ∩ *<sup>N</sup>*-compatible and the both sides equal <sup>Π</sup>*M*∩*N*. For any special contact transformation <sup>Π</sup>*f M* <sup>=</sup> *<sup>F</sup>* ◦ <sup>Π</sup>*<sup>M</sup>* ◦ *<sup>F</sup>*<sup>−</sup>1, where *<sup>F</sup>* is a transformation of H(*C*) induced by *f* .

It is clear that every special contact transformation *u*, i.e., diffeomorphisms of *C* preserving Ω, projects to *P* onto canonical transformation *u* preserving *γ*, *π* ◦ *u* = *u* ◦ *π*. Let M*<sup>q</sup>* : *q* ∈ *Q* be a system of Legendre submanifolds of *C* such that *<sup>q</sup>*∈*<sup>Q</sup>* M*<sup>q</sup>* is an image of a cross-section of *C* over *P*, such that the projections to *P*, m*<sup>q</sup>* form a polarization, i.e., a family of mutually disjoint Lagrange submanifolds of *P*. Then *u* acts on M*<sup>q</sup>* in such a way that

22 Quantum Mechanics

*<sup>S</sup>*(*p*) = Stat*x*∈*<sup>V</sup>* (*S*(*x*) − �*p*, *<sup>x</sup>*�), *<sup>S</sup>*(*x*) = Stat*p*∈*V*<sup>∗</sup>

the corresponding transformation of H(*C*), then *UEa*∈*A*[*ta*]M*<sup>a</sup>* = *Ea*∈*A*[*ta*]*U*M*a*.

<sup>Ψ</sup>(*p*). It is based on the concept of generalized envelope.

pair of Legendre submanifolds M1, M2.

<sup>Π</sup> ◦ <sup>Π</sup>*<sup>M</sup>* = <sup>Λ</sup>*<sup>M</sup>* ◦ <sup>Π</sup>, (Π*M*M) ∩ <sup>M</sup> =

transformation of H(*C*) induced by *f* .

that <sup>Π</sup>*M*<sup>M</sup> = ∅.

horizontal lift of <sup>Λ</sup>*M*<sup>m</sup> which contains <sup>M</sup> ∩

then <sup>Π</sup>*M*<sup>M</sup> is a maximal extension of the intersection <sup>M</sup> ∩

be a system of Legendre submanifolds of *C* such that

When we take into account that the analogy between phases of <sup>Ψ</sup>(*x*) and <sup>Ψ</sup>(*p*) is seriously accepted, we see immediately the obvious quasiclassical relationship between Ψ(*x*) and

Let us also notice that all those concepts are invariant with respect to the special contact transformations in *C*. First of all, let us remind that *u* : *C* → *C* is a special contact transformation when it preserves <sup>Ω</sup>, *<sup>u</sup>*∗<sup>Ω</sup> = <sup>Ω</sup>. If *<sup>u</sup>* is such and *<sup>U</sup>* : H(*C*) → H(*C*) is

Now let us begin with the concept of the vertical distance, i.e., scalar product of Legendre submanifolds. Let us take a pair of Legendre manifolds <sup>M</sup>1,M<sup>2</sup> ∈ H(*C*) with the property that their Lagrange projections <sup>m</sup>1, <sup>m</sup><sup>2</sup> and also <sup>m</sup><sup>1</sup> ∩ <sup>m</sup><sup>2</sup> are connected and simply-connected. Then there exists exactly one number *t* ∈ **R** (or exp(*it*) ∈ SU(1)) of the property that (*gt*M1) ∩ <sup>M</sup><sup>2</sup> �= ∅. This number *<sup>t</sup>* or better its unitary exponent exp(*it*) is the scalar product of <sup>M</sup><sup>1</sup> and <sup>M</sup>2. To be more precise, the classical scalar product is exp(*it*), where *<sup>t</sup>* is its phase. More generally, we say that the vertical distance, or the Huygens scalar product [M1|M2] of the pair of Legendre submanifolds <sup>M</sup>1, <sup>M</sup><sup>2</sup> is a subset of **<sup>R</sup>** (or exponentially of SU(1)) such that if *<sup>t</sup>* ∈ [M1|M2], then <sup>M</sup><sup>2</sup> ∩ *gt* (M1) �= ∅. If [M1|M2] is empty, then we say that <sup>M</sup>1, <sup>M</sup><sup>2</sup> are orthogonal. Their Lagrange projections <sup>m</sup>1, <sup>m</sup><sup>2</sup> onto *<sup>P</sup>* <sup>=</sup> *<sup>T</sup>*∗*<sup>Q</sup>* are then disjoint. It is clear that any mapping *U* : H(*C*) → H(*C*) generated by a special contact transformation *u* : *C* → *C* is then "unitary" in the sense of the scalar product [·, ·], i.e., [*U*M1|*U*M2] = [M1|M2] for any

Let *<sup>M</sup>* <sup>⊂</sup> *<sup>P</sup>* <sup>=</sup> *<sup>T</sup>*∗*<sup>Q</sup>* be any co-isotropic submanifold and <sup>H</sup>*M*(*C*) ⊂ H(*C*) denote the set of Legendre submanifolds with Lagrange projections to *<sup>P</sup>* = *<sup>T</sup>*∗*<sup>Q</sup>* placed within *<sup>M</sup>*. Then the operations *EM* on Lagrangian submanifolds introduced above may be canonically lifted to the operations <sup>Π</sup>*<sup>M</sup>* acting on the horizontal lifts of <sup>m</sup> <sup>⊂</sup> *<sup>D</sup>*(*P*) = *<sup>D</sup>* (*T*∗*Q*). Namely, for any co-isotropic *<sup>M</sup>* there exists the canonical mapping <sup>Π</sup>*<sup>M</sup>* : H(*C*) → H*M*(*C*) with the property:

the natural projection induced by the fibre bundle projection *<sup>π</sup>* : *<sup>C</sup>* → *<sup>P</sup>*. In fact, <sup>Π</sup>*M*<sup>M</sup> is the

Ω-horizontal lift *K*Ω(*M*) = lift *K*(*M*). If m ∩ *M* = ∅ or if it is not regular, then it is assumed

Let us repeat again that Π*<sup>M</sup>* satisfy the properties (69)–(72) modified by the admitted empty-set values: <sup>Π</sup>*M*|H*M*(*C*) = IdH*<sup>M</sup>* (*C*), <sup>Π</sup>*<sup>M</sup>* ◦ <sup>Π</sup>*<sup>M</sup>* <sup>=</sup> <sup>Π</sup>*M*, <sup>Π</sup>*<sup>M</sup>* ◦ <sup>Π</sup>*<sup>N</sup>* <sup>=</sup> <sup>Π</sup>*<sup>N</sup>* ◦ <sup>Π</sup>*<sup>M</sup>* <sup>=</sup> <sup>Π</sup>*M*∩*<sup>N</sup>* if Cl *<sup>M</sup>* ∩ *<sup>N</sup>* = I. If <sup>Π</sup>*<sup>M</sup>* ◦ <sup>Π</sup>*<sup>N</sup>* = <sup>Π</sup>*<sup>N</sup>* ◦ <sup>Π</sup>*M*, then *<sup>M</sup>* ∩ *<sup>N</sup>*-compatible and the both sides equal <sup>Π</sup>*M*∩*N*. For any special contact transformation <sup>Π</sup>*f M* <sup>=</sup> *<sup>F</sup>* ◦ <sup>Π</sup>*<sup>M</sup>* ◦ *<sup>F</sup>*<sup>−</sup>1, where *<sup>F</sup>* is a

It is clear that every special contact transformation *u*, i.e., diffeomorphisms of *C* preserving Ω,

of *C* over *P*, such that the projections to *P*, m*<sup>q</sup>* form a polarization, i.e., a family of mutually

projects to *P* onto canonical transformation *u* preserving *γ*, *π* ◦ *u* = *u* ◦ *π*. Let

*<sup>π</sup>*−1(*M*) 

*<sup>π</sup>*−1(*M*)  *<sup>S</sup>*(*p*) + �*p*, *<sup>x</sup>*�

∩ <sup>M</sup>, where <sup>Π</sup> : H(*C*) → *<sup>D</sup>*(*P*) = *<sup>D</sup>* (*T*∗*Q*) is

*<sup>π</sup>*−1(*M*) 

. If m intersects *M* in a regular way,

*<sup>q</sup>*∈*<sup>Q</sup>* M*<sup>q</sup>* is an image of a cross-section

along the fibres of the

M*<sup>q</sup>* : *q* ∈ *Q*

. (90)

$$\text{L'}\mathfrak{M}\_{\mathfrak{q}} = E\_{\mathfrak{q}' \in \underline{Q}} \text{L'}\begin{pmatrix} q', q \end{pmatrix} \mathfrak{M}\_{\mathfrak{q}'} \qquad \text{L'}\begin{pmatrix} q', q \end{pmatrix} = \begin{bmatrix} \mathfrak{M}\_{\mathfrak{q}'} \vert \mathrm{L}\mathfrak{M}\_{\mathfrak{q}} \end{bmatrix} . \tag{91}$$

Then for any superposition-envelope <sup>M</sup> = *Eq*∈*<sup>Q</sup>* [*S*(*q*)] <sup>M</sup>*<sup>q</sup>* the following holds:

$$\mathsf{LIDR} = \mathsf{E}\_{q \in \mathsf{Q}} \left[ \mathsf{S}'(q) \right] \\ \mathfrak{M}\_{q} = \mathsf{E}\_{q \in \mathsf{Q}} \left[ \mathsf{S}(q) \right] \\ \mathsf{LIDR}\_{q} \tag{92}$$

where *<sup>S</sup>*′ (*q*) = Stat*q*′∈*<sup>Q</sup>* (*<sup>U</sup>* (*q*, *<sup>q</sup>*′ ) + *<sup>S</sup>* (*q*′ )). This is an obvious analogue and the phase classical limit of the linear rule for superposition of wave functions with the phase factors exp (*i*/¯*h*) *<sup>S</sup>* (*q*′ ). And *<sup>U</sup>* (*q*, *<sup>q</sup>*′ ) is just the generating function of the type *<sup>W</sup>*(*q*, *<sup>Q</sup>*) = *<sup>U</sup>* (*q*, *<sup>q</sup>*′ ). And a similar construction may be built for other types of generating functions.

Let us also mention that a similar "quasilinear" representation may be achieved for other operations on Legendre submanifolds, not necessarily ones induced by diffeomorphisms acting in *<sup>C</sup>*. For example, let us consider <sup>Π</sup>*M*, i.e., <sup>M</sup> = *Eq*∈*<sup>Q</sup>* [*S*(*q*)] <sup>M</sup>*q*, <sup>Π</sup>*M*<sup>M</sup> = *Eq*∈*<sup>Q</sup>* [*S*′ (*q*)] <sup>M</sup>*q*. Then we have *<sup>S</sup>*′ (*q*) = Stat*q*′∈*<sup>Q</sup>* (*<sup>S</sup>* (*q*′ ) <sup>+</sup> <sup>Π</sup>*<sup>M</sup>* (*q*′ , *<sup>q</sup>*)), where <sup>Π</sup>*<sup>M</sup>* : *<sup>Q</sup>* × *<sup>Q</sup>* → **<sup>R</sup>** is the Legendre propagator <sup>Π</sup>*<sup>M</sup>* (*q*′ , *q*) = <sup>M</sup>*q*′|Π*M*M*<sup>q</sup>* . This is again the envelope-like Huygens-quasilinear rule. Using the Stat-symbol one can also write a nice-looking analogue of the Feynman-Stückelberg "sum over paths" rule.

Incidentally, let us remind that by the *<sup>W</sup>*-type generating function *<sup>W</sup>* (*q*, *<sup>q</sup>*′ ) we mean such one that the corresponding canonical transformation (*q*, *<sup>p</sup>*) �→ (*q*′ , *p*′ ) is given by *pi* <sup>=</sup> *<sup>∂</sup><sup>W</sup>* (*q*, *<sup>q</sup>*′ ) /*∂q<sup>i</sup>* , *p*′ *<sup>i</sup>* <sup>=</sup> <sup>−</sup>*∂<sup>W</sup>* (*q*, *<sup>q</sup>*′ ) /*∂q*′*<sup>i</sup>* . Not every canonical transformation does possess such a function in the usual sense, but it may be replaced by a more general generating function. There is no place here to get deeper into details, cf. e.g. [11].

Let us also stress a few other facts connected with the notion of (generalized) envelope. Consider the compatible system of Hamilton-Jacobi equations: *Fa* (..., *<sup>x</sup>µ*,...;..., *<sup>∂</sup>S*/*∂xµ*,...) <sup>=</sup> 0. Any fibre of the cotangent bundle may be <sup>Λ</sup>*M*-projected onto *<sup>D</sup>*(*M*) — the set of Lagrange submanifolds of *<sup>M</sup>*, <sup>m</sup>*<sup>x</sup>* :<sup>=</sup> <sup>Λ</sup>*<sup>M</sup>* (*T*<sup>∗</sup> *<sup>x</sup> <sup>X</sup>*). Then, every (*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)-dimensional *<sup>π</sup>*−1(m*X*) (where *<sup>n</sup>* <sup>=</sup> dim *<sup>X</sup>*) is foliated by the family of Legendre lifts of <sup>m</sup>*X*. When *<sup>C</sup>* <sup>=</sup> *<sup>T</sup>*∗*<sup>X</sup>* <sup>×</sup> **<sup>R</sup>** or *<sup>C</sup>* <sup>=</sup> *<sup>T</sup>*∗*<sup>X</sup>* <sup>×</sup> <sup>U</sup>(1) those lifts are <sup>M</sup>(*x*,*c*) :<sup>=</sup> <sup>Π</sup>*<sup>M</sup>* (*T*<sup>∗</sup> *<sup>x</sup> <sup>X</sup>*, *<sup>c</sup>*). Then locally

$$\mathfrak{m}\_{\mathbf{x}} \cap T\_{\mathbf{y}}^{\*}\mathbf{X} = \left\{ d\sigma(\mathbf{x}, \cdot)\_{\mathbf{y}} \right\}, \quad \mathfrak{M}\_{\mathbf{x}} \cap \left( T\_{\mathbf{y}}^{\*}\mathbf{X} \times \mathbb{R} \right) = \left\{ d\sigma(\mathbf{x}, \cdot)\_{\mathbf{y}}, \sigma(\mathbf{x}, \mathbf{y}) \right\}. \tag{93}$$

For every pair of points *<sup>x</sup>*, *<sup>y</sup>* ∈ *<sup>X</sup>* we define the quantity *<sup>σ</sup>M*(*x*, *<sup>y</sup>*), namely

$$
\sigma\_M(\mathbf{x}, y) = \int\_{l(\mathbf{x}, y)} \omega = \int\_{\ell(\mathbf{x}, y)} p\_\mu d\mathbf{x}^\mu \,\mathrm{s}\tag{94}
$$

where ℓ(*x*, *<sup>y</sup>*) is any curve placed on a singular fibre containing *<sup>x</sup>*, *<sup>y</sup>* ∈ *<sup>X</sup>*. This *<sup>σ</sup><sup>M</sup>* is a fundamental solution, <sup>Π</sup>*M*M*<sup>S</sup>* <sup>=</sup> <sup>M</sup>*S*′ , <sup>Λ</sup>*M*m*<sup>S</sup>* <sup>=</sup> <sup>m</sup>*S*′ , where *<sup>S</sup>*′ (*x*) = Stat*<sup>y</sup>* (*S*(*y*) + *<sup>σ</sup>M*(*y*, *<sup>x</sup>*)). When Σ ⊂ *X* is a Cauchy surface for our Hamilton-Jacobi system, then *S*(*x*) =

Stat*q*∈<sup>Σ</sup> (*f*(*q*) + *<sup>σ</sup>M*(*q*, *<sup>x</sup>*)), where *<sup>f</sup>* : <sup>Σ</sup> → **<sup>R</sup>** are initial data. Therefore, the two-point characteristic function is a Hamilton-Jacobi propagator. The idempotence property of Λ*M*, <sup>Π</sup>*<sup>M</sup>* implies that *<sup>σ</sup>M*(*x*, *<sup>y</sup>*) = Stat*<sup>z</sup>* (*σM*(*x*, *<sup>z</sup>*) + *<sup>σ</sup>M*(*z*, *<sup>y</sup>*)). Let us quote an interesting example of the free material point in Galilean space-time. Then

$$\frac{1}{\hbar}\sigma\_M(\mathbf{x},y) = \frac{1}{\hbar}S(a,z;q,t) = \frac{m}{2\hbar(t-z)}g\_{ij}\left(q^i - a^i\right)\left(q^j - a^j\right). \tag{95}$$

When the Van Vleck determinant det *∂*2*S*/*∂q<sup>i</sup> ∂aj* is multiplied by some normalization constant, then the Van Vleck solution

$$\sqrt{\det\left[\frac{\partial^2 S}{\partial q^i \partial a^j}\right]} \exp\left(\frac{im}{2\hbar(t-z)} g\_{kl} \left(q^k - a^k\right) \left(q^l - a^l\right)\right) \tag{96}$$

becomes

$$\mathcal{K}\left(\overline{\xi},\tau\right) = \left(\frac{m}{2\pi i\hbar\tau}\right)^{n/2} \exp\left(\frac{im}{2\hbar\tau}\overline{\xi}^2\right) \tag{97}$$

where *τ* = *t* − *z*, *ξ<sup>k</sup>* = *q<sup>k</sup>* − *ak*, and *ξ* <sup>2</sup> <sup>=</sup> *gklξkξ<sup>l</sup>* . The normalization we have accepted is given by lim*τ*→<sup>0</sup> K *ξ*, *τ* = *δ ξ* , where K is the usual rigorous quantum propagator for the Schrödinger equations:

$$i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\Delta\Psi = -\frac{\hbar^2}{2m}g^{ij}\partial\_l\partial\_{\dot{l}}\Psi\_{\prime} \tag{98}$$

in spite of the fact that it was obtained in a purely classical way.

**Summary of Section 3:** In this section we have reminded some classical problems concerning classification of submanifolds in the classical phase space. Their classical interpretation in terms of symplectic and contact structures was discussed. Again it turns out that the classical limits are Huygens constructions based on the envelope concepts. The quantum and classical relationships between information and symmetry were discussed. This analysis shows again, without any use of the Weyl-Wigner-Moyal-Ville product that it is probability distributions concentrated on *n*-dimensional Lagrange manifolds that corresponds to the quantum pure states on the classical level. The homogeneous Van Vleck objects corresponding to the evolution problems were discussed. In particular, this may be used to the analysis of the Klein-Gordon equation. Discussed is the WKB-limit of certain quantum expressions like superpositions of wave functions and their scalar products. Again one obtains expressions based on the envelope concepts and the Huygens-like operations on the set of Lagrangian manifolds. Classical counterparts of the projection operators were found. The envelopes of diagrams of phases of wave functions are geometrically interpreted in terms of contact geometry. This is a geometric picture valid for all types of the eikonal equations. It enables one to interpret also the purely classical concepts like generating functions of canonical transformations in quantum-like form based on the envelopes of diagrams of phases. Roughly speaking, the envelope represents the phase of the classical superposition. It is shown that the quantum propagation for the free evolution Schrödinger equation may be smoothly guessed on the purely classical level, in terms of the Van Vleck determinant.

#### **4. Nonlinearity program in quantum mechanics**

24 Quantum Mechanics

becomes

given by lim*τ*→<sup>0</sup> K

Schrödinger equations:

Stat*q*∈<sup>Σ</sup> (*f*(*q*) + *<sup>σ</sup>M*(*q*, *<sup>x</sup>*)), where *<sup>f</sup>* : <sup>Σ</sup> → **<sup>R</sup>** are initial data. Therefore, the two-point characteristic function is a Hamilton-Jacobi propagator. The idempotence property of Λ*M*, <sup>Π</sup>*<sup>M</sup>* implies that *<sup>σ</sup>M*(*x*, *<sup>y</sup>*) = Stat*<sup>z</sup>* (*σM*(*x*, *<sup>z</sup>*) + *<sup>σ</sup>M*(*z*, *<sup>y</sup>*)). Let us quote an interesting example

2¯*h*(*t* − *z*)

*∂aj* 

> *gkl q<sup>k</sup>* − *a<sup>k</sup>*

*<sup>n</sup>*/2 exp

∆Ψ <sup>=</sup> <sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup>

**Summary of Section 3:** In this section we have reminded some classical problems concerning classification of submanifolds in the classical phase space. Their classical interpretation in terms of symplectic and contact structures was discussed. Again it turns out that the classical limits are Huygens constructions based on the envelope concepts. The quantum and classical relationships between information and symmetry were discussed. This analysis shows again, without any use of the Weyl-Wigner-Moyal-Ville product that it is probability distributions concentrated on *n*-dimensional Lagrange manifolds that corresponds to the quantum pure states on the classical level. The homogeneous Van Vleck objects corresponding to the evolution problems were discussed. In particular, this may be used to the analysis of the Klein-Gordon equation. Discussed is the WKB-limit of certain quantum expressions like superpositions of wave functions and their scalar products. Again one obtains expressions based on the envelope concepts and the Huygens-like operations on the set of Lagrangian manifolds. Classical counterparts of the projection operators were found. The envelopes of diagrams of phases of wave functions are geometrically interpreted in terms of contact geometry. This is a geometric picture valid for all types of the eikonal equations. It enables one to interpret also the purely classical concepts like generating functions of canonical transformations in quantum-like form based on the envelopes of diagrams of phases. Roughly speaking, the envelope represents the phase of the classical superposition. It is shown that the quantum propagation for the free evolution Schrödinger equation may be smoothly guessed on the purely classical level, in terms of the Van Vleck determinant.

<sup>2</sup> <sup>=</sup> *gklξkξ<sup>l</sup>*

*gij q<sup>i</sup>* − *a<sup>i</sup>*

 *q<sup>l</sup>* − *a<sup>l</sup>*

, where K is the usual rigorous quantum propagator for the

 *im* 2¯*h<sup>τ</sup> <sup>ξ</sup>* 2  *q<sup>j</sup>* − *a<sup>j</sup>* 

is multiplied by some normalization

. The normalization we have accepted is

<sup>2</sup>*<sup>m</sup> <sup>g</sup>ij∂i∂j*Ψ, (98)

, (97)

. (95)

(96)

*<sup>S</sup>*(*a*, *<sup>z</sup>*; *<sup>q</sup>*, *<sup>t</sup>*) = *<sup>m</sup>*

 *∂*2*S*/*∂q<sup>i</sup>*

 *im* 2¯*h*(*t* − *z*)

of the free material point in Galilean space-time. Then

*h*¯

 *∂*2*S ∂qi∂a<sup>j</sup>*

> K *ξ*, *τ* = *m* 2*πih*¯ *τ*

> > *ih*¯ *∂*Ψ *<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup> 2*m*

in spite of the fact that it was obtained in a purely classical way.

 exp

*<sup>σ</sup>M*(*x*, *<sup>y</sup>*) = <sup>1</sup>

1 *h*¯

When the Van Vleck determinant det

 det

where *τ* = *t* − *z*, *ξ<sup>k</sup>* = *q<sup>k</sup>* − *ak*, and *ξ*

*ξ*, *τ* = *δ ξ* 

constant, then the Van Vleck solution

Let us consider a finite-level quantum mechanical system. We try to interpret the Schrödinger equation as a usual self-adjoint equation of mathematical physics, just as if it was to be a classical one. If both the first- and second-order time derivatives of the state vector Ψ are to be admitted, the Lagrange function may be postulated as

$$\delta L(1,2) = \mathrm{i}a\Gamma\_{\overline{a}b} \left( \overline{\mathbf{Y}}^{\overline{a}} \dot{\mathbf{Y}}^{b} - \dot{\overline{\mathbf{Y}}}^{\overline{a}} \mathbf{Y}^{b} \right) + \beta \Gamma\_{\overline{a}b} \dot{\overline{\mathbf{Y}}}^{\overline{a}} \dot{\mathbf{Y}}^{b} - \gamma\_{\Gamma} H\_{\overline{a}b} \overline{\mathbf{Y}}^{\overline{a}} \mathbf{Y}^{b} \,\tag{99}$$

where *α*, *β*, *γ* are constants and Γ*ab* are components of the scalar product. <sup>Γ</sup>*Hab* are components of the Hamiltonian matrix in the covariant form,

$$
\Gamma\_{\mathbb{Z}} H\_{\overline{a}b} = \Gamma\_{\mathbb{Z}c} H^c{}\_{b\prime} \tag{100}
$$

whereas *H<sup>c</sup> b* are usual mixed tensor components. To be honest, in physics it is this mixed Hamilton operator that is considered as a primary quantity. From the Lagrangian point of view it is a twice covariant form that is primary. Because of this we decided to assume *H<sup>c</sup> b* as a primary quantity, but the Hermitian matrix *Hab* is assumed as the constitutive element in (99). When we are going to introduce a direct nonlinearity to the treatment, we introduce in addition a real-valued potential *V* Ψ, Ψ , e.g.,

$$V\left(\Psi,\overline{\Psi}\right) = f\left(\Gamma\_{\overline{a}b}\overline{\Psi}^{\overline{a}}\Psi^{b}\right),\tag{101}$$

where *f* is a real-valued function of the one real variable.

For the Lagrangian *L* = *L*(1, 2) − *V* we obtain the following "Schrödinger", or rather "Schrödinger-Klein-Gordon", equation:

$$2i\alpha \frac{d\Psi^a}{dt} - \beta \frac{d^2\Psi^a}{dt^2} = \gamma H^a{}\_b \Psi^b + f' \Psi^a. \tag{102}$$

The comparison with the usual Schrödinger equation tells us that *α* = *h*¯ /2, *γ* = 1. The energy function for *<sup>L</sup>* = *<sup>L</sup>*(1, 2) − *<sup>V</sup>* is given by E = *<sup>β</sup>*Γ*ab* ˙ Ψ*a* <sup>Ψ</sup>˙ *<sup>b</sup>* <sup>+</sup> *<sup>γ</sup>*Γ*Hab*Ψ*<sup>a</sup>* Ψ*<sup>b</sup>* + *V* Ψ, Ψ . Legendre transformation tells us that the corresponding Hamiltonian in the sense of analytical mechanics is:

$$\mathcal{H} = \frac{1}{\mathcal{\mathcal{B}}} \left[ \Gamma^{a\overline{b}} \pi\_a \overline{\pi}\_{\overline{b}} + \text{ia} \left( \pi\_a \mathbf{Y}^a - \overline{\pi}\_{\overline{a}} \overline{\mathbf{Y}}^{\overline{a}} \right) \right] + \left[ \frac{a^2}{\mathcal{\mathcal{B}}} \Gamma\_{\overline{a}b} + \gamma\_\Gamma H\_{\overline{a}b} \right] \overline{\mathbf{Y}}^{\overline{a}} \mathbf{Y}^b + V \left( \mathbf{Y}, \overline{\mathbf{Y}} \right), \tag{103}$$

where *<sup>π</sup>a*, *<sup>π</sup><sup>a</sup>* are canonical momenta conjugate to <sup>Ψ</sup>*a*, <sup>Ψ</sup>*<sup>a</sup>* . It is clear that E is always defined all over the state space. Unlike this, H is defined all over the phase space only when *β* �= 0. If *β* = 0, it is defined only on the constraints submanifold.

The possible nonlinearity of quantum mechanics is due to the term *V* Ψ, Ψ . As mentioned, this is a rather artificial, perturbative nonlinearity. It is definitely better to use the essential nonlinearity of the geometric, group-theoretic origin. To achieve this, one should follow the idea of transition from special to general relativity. The simplest way is to "de-absolutize" the scalar product. Namely, instead of being fixed once for all, the scalar product will be reinterpreted as a dynamical variable. It is to be self-interacting and free of any fixed absolute background. Therefore, in the finite-level theory, its dynamics will be GL(*n*, **C**)-invariant. The only natural Lagrangian will follow the structure of affinely-invariant kinetic energy of affinely-rigid body. So, for the metric Γ*ab* we postulate the following Lagrangian:

$$T = L[\Gamma] = \frac{A}{2} \Gamma^{b\overline{c}} \Gamma^{d\overline{a}} \dot{\Gamma}\_{\overline{a}b} \dot{\Gamma}\_{\overline{c}d} + \frac{B}{2} \Gamma^{b\overline{a}} \Gamma^{d\overline{c}} \dot{\Gamma}\_{\overline{a}b} \dot{\Gamma}\_{\overline{c}d}.\tag{104}$$

This is the only possibility which is not based on anything absolutely fixed. Obviously, the main term is the first one, controlled by *A*. The *B*-term is a correction, not very essential, but acceptable from the point of view of the assumed GL(*n*, **C**)-symmetry.

In the Lagrangian (100), (101) for the wave function the scalar product is also replaced by this new, dynamical version. Due to the resulting very essential nonlinearity following from (104) the quantum-classical gap in a sense becomes diffused. One can hope that in the resulting theory the decoherence phenomena may be explained. For example, if for simplicity we fix Ψ*<sup>a</sup>* as constant (non-excited), then we can show that differential equations for Γ obtained from (104) have the following solutions: Γ*rs* = *Grs* exp (*Et*) *z <sup>s</sup>* = exp (*Ft*)*<sup>r</sup> zGzs*, where *G* = Γ(0) is Hermitian and the forms *GErs* = *GrzE<sup>z</sup> <sup>s</sup>*, (*FG*)*rs* = *Fr zGzs* are also Hermitian. Let us observe that depending on the initial data *G*, *E*, *F* the scenarios of the evolution of *t* �→ Γ(*t*) may be quite different: oscillatory, exponentially increasing, exponentially decaying. This may suggest that in the rigorous total solutions for *t* �→ (Ψ(*t*), Γ(*t*)) also various phenomena may be predicted, e.g., oscillations, but perhaps also decoherence.

The maximal class of GL(*n*, **C**)-invariant Lagrangians *L*[Ψ, Γ] is relatively wide. It seems however that the simplest and at the same time most realistic subclass is given by the following expression:

$$L = ia\_1 \Gamma\_{\overline{\rm ll}b} \left( \overline{\mathbf{F}}^{\overline{\rm H}} \dot{\mathbf{Y}}^b - \dot{\overline{\mathbf{Y}}}^{\overline{\rm H}} \mathbf{Y}^b \right) + a\_2 \Gamma\_{\overline{\rm l}b} \dot{\overline{\mathbf{Y}}}^{\overline{\rm H}} \dot{\mathbf{Y}}^b + \left( a\_3 \Gamma\_{\overline{\rm l}b} + a\_4 H\_{\overline{\rm l}b} \right) \overline{\mathbf{Y}}^{\overline{\rm H}} \mathbf{Y}^b$$
 
$$+ \left. a\_5 \Gamma^{\overline{\rm d}\overline{\rm l}} \Gamma^{b\overline{\rm l}} \dot{\Gamma}\_{\overline{\rm d}b} \dot{\Gamma}\_{\overline{\rm d}d} + a\_6 \Gamma^{b\overline{\rm d}} \Gamma^{d\overline{\rm d}} \dot{\Gamma}\_{\overline{\rm d}b} \dot{\Gamma}\_{\overline{\rm d}d} - \mathcal{V} \left( \mathbf{Y}, \overline{\mathbf{Y}}; \Gamma \right) \,. \tag{105}$$

The quantities *α*1,..., *α*<sup>6</sup> are real constants. They control all the effects mentioned above. The separation of the *α*3- and *α*4-terms may look artificial. In fact, their superposition is as a matter of fact one term. Nevertheless, it seems reasonable to separate the true Hamiltonian effect from that following from the identity operator type. As mentioned formerly, to obtain the correct Schrödinger behaviour in the <sup>Ψ</sup>-sector we must put *<sup>α</sup>*<sup>1</sup> = *<sup>h</sup>*¯ /2, *<sup>α</sup>*<sup>4</sup> = −1. But of course if *<sup>α</sup>*<sup>2</sup> �= 0, then in addition to the Schrödinger behaviour we have also as usual in analytical mechanics, the acceleration term in Ψ. The scheme is a bit obscured because of our dealing with a finite-level system. The extension to the usual quantum mechanics, say in **R**3, is possible. Besides, let us remind our papers devoted to the study of the SU(2, 2)-gauge gravitation theory [12, 13]. There the problem of nonlinearity and the interplay between firstand second-order differential equations for the matter fields appear in a much more evident way.

We do not quote "Schrödinger equation" for the pairs (Ψ, Γ) ruled by Lagrangians (105). Their structure is very readable, nevertheless, their strong nonlinearity prevented us from finding their convincing full solutions, when the mutual interaction between Ψ- and Γ-degrees of freedom is taken into account.

**Summary of Section 4:** This section was one of the main parts of our study. We are aware that in spite of all similarities and analogies there is still some really quantum kernel of the theory which seems to be incompatible with any attempts of formulating the peaceful coexistence of the unitary "between measurement" and the "reduction-like" phenomena in quantum physics. As usual, the idea of nonlinearity in quantum physics turns out to be attractive. It seems to be the only way to coordinate the "between measurements" unitary evolution and the measurement reduction process. There were various more or less happy ways to introduce nonlinearity; some of them were rather artificial. Our idea resembles the transition from the special to general relativity. Namely, we give up the concept of scalar product fixed once for all and instead consider the scheme in which the wave function and scalar product are both dynamical objects in the mutual interaction. Lagrangian for the scalar product is geometric, invariant under the full linear group and so is the total Lagrangian for the system: wave function and the scalar product. The resulting scheme is nonlinear in an essential, non-perturbative way. There are some indications that the resulting nonlinear system of equations may describe both the "between measurements" evolution and the reduction of state process.

#### **5. Modifications of the WWMV approach**

26 Quantum Mechanics

The possible nonlinearity of quantum mechanics is due to the term *V*

*<sup>T</sup>* <sup>=</sup> *<sup>L</sup>*[Γ] = *<sup>A</sup>*

(104) have the following solutions: Γ*rs* = *Grs* exp (*Et*)

be predicted, e.g., oscillations, but perhaps also decoherence.

Ψ˙ *<sup>b</sup>* − ˙ Ψ*a* Ψ*b* 

*cd* <sup>+</sup> *<sup>α</sup>*6Γ*ba*Γ*dc*

Hermitian and the forms *GErs* = *GrzE<sup>z</sup>*

*<sup>L</sup>* = *<sup>i</sup>α*1Γ*ab*

<sup>+</sup> *<sup>α</sup>*5Γ*da*Γ*bc*

 Ψ*a*

> Γ˙ *ab*Γ˙

following expression:

affinely-rigid body. So, for the metric Γ*ab* we postulate the following Lagrangian:

Γ*da*Γ˙ *ab*Γ˙ *cd* + *B* <sup>2</sup> <sup>Γ</sup>*ba*Γ*dc* Γ˙ *ab*Γ˙

This is the only possibility which is not based on anything absolutely fixed. Obviously, the main term is the first one, controlled by *A*. The *B*-term is a correction, not very essential, but

In the Lagrangian (100), (101) for the wave function the scalar product is also replaced by this new, dynamical version. Due to the resulting very essential nonlinearity following from (104) the quantum-classical gap in a sense becomes diffused. One can hope that in the resulting theory the decoherence phenomena may be explained. For example, if for simplicity we fix Ψ*<sup>a</sup>* as constant (non-excited), then we can show that differential equations for Γ obtained from

*<sup>s</sup>*, (*FG*)*rs* = *Fr*

that depending on the initial data *G*, *E*, *F* the scenarios of the evolution of *t* �→ Γ(*t*) may be quite different: oscillatory, exponentially increasing, exponentially decaying. This may suggest that in the rigorous total solutions for *t* �→ (Ψ(*t*), Γ(*t*)) also various phenomena may

The maximal class of GL(*n*, **C**)-invariant Lagrangians *L*[Ψ, Γ] is relatively wide. It seems however that the simplest and at the same time most realistic subclass is given by the

> <sup>+</sup> *<sup>α</sup>*2Γ*ab* ˙ Ψ*a*

The quantities *α*1,..., *α*<sup>6</sup> are real constants. They control all the effects mentioned above. The separation of the *α*3- and *α*4-terms may look artificial. In fact, their superposition is as a matter of fact one term. Nevertheless, it seems reasonable to separate the true Hamiltonian effect from that following from the identity operator type. As mentioned formerly, to obtain the correct Schrödinger behaviour in the <sup>Ψ</sup>-sector we must put *<sup>α</sup>*<sup>1</sup> = *<sup>h</sup>*¯ /2, *<sup>α</sup>*<sup>4</sup> = −1. But of course if *<sup>α</sup>*<sup>2</sup> �= 0, then in addition to the Schrödinger behaviour we have also as usual in analytical mechanics, the acceleration term in Ψ. The scheme is a bit obscured because of our dealing with a finite-level system. The extension to the usual quantum mechanics, say in **R**3, is possible. Besides, let us remind our papers devoted to the study of the SU(2, 2)-gauge

*cd* − V

Γ˙ *ab*Γ˙ *z*

*<sup>s</sup>* = exp (*Ft*)*<sup>r</sup>*

<sup>Ψ</sup>˙ *<sup>b</sup>* <sup>+</sup> (*α*3Γ*ab* <sup>+</sup> *<sup>α</sup>*4*Hab*) <sup>Ψ</sup>*<sup>a</sup>*

Ψ, Ψ; Γ 

2 Γ*bc*

acceptable from the point of view of the assumed GL(*n*, **C**)-symmetry.

this is a rather artificial, perturbative nonlinearity. It is definitely better to use the essential nonlinearity of the geometric, group-theoretic origin. To achieve this, one should follow the idea of transition from special to general relativity. The simplest way is to "de-absolutize" the scalar product. Namely, instead of being fixed once for all, the scalar product will be reinterpreted as a dynamical variable. It is to be self-interacting and free of any fixed absolute background. Therefore, in the finite-level theory, its dynamics will be GL(*n*, **C**)-invariant. The only natural Lagrangian will follow the structure of affinely-invariant kinetic energy of

Ψ, Ψ 

. As mentioned,

*cd*. (104)

*zGzs*, where *G* = Γ(0) is

Ψ*b*

. (105)

*zGzs* are also Hermitian. Let us observe

We have seen that the mentioned approach was a very fruitful tool for studying the quasi-classical problems and the relationship between information and symmetry. Unfortunately, its literal version applies rigorously only to systems with affine geometry of the classical phase space. There are various ways to generalize those methods, usually based on group theory and deformation techniques. Some of those methods are applicable also to discrete structures.

Let *G* be a locally compact topological group. Its Haar measure element will be denoted by *dg*. To be honest, to avoid problems with the convergence of integrals, we may assume *G* to be compact. Let us introduce the following non-local product of functions on *G*:

$$A(A \perp B)(\mathbf{g}) = \int \mathcal{K}\left(\mathbf{g}; \mathbf{g}\_1, \mathbf{g}\_2\right) A(\mathbf{g}\_1) B(\mathbf{g}\_2) d\mathbf{g}\_1 d\mathbf{g}\_2. \tag{106}$$

To be honest, we think about the multiplication rule for functions not necessarily on *G* itself, but rather on its affine space, i.e., on the set on which *G* acts with trivial isotropy groups. Therefore, we assume the translational invariance, so that K (*g*; *<sup>g</sup>*1, *<sup>g</sup>*2) ≡ K *<sup>g</sup>*1*g*<sup>−</sup>1, *<sup>g</sup>*2*g*−<sup>1</sup> . The simultaneous assumption of associativity, *A* ⊥ (*B* ⊥ *C*)=(*A* ⊥ *B*) ⊥ *C* implies that K must satisfy the following functional equation:

$$
\int \mathcal{K}(\mathfrak{g}\_1, \mathfrak{g}) \mathcal{K}\left(\mathfrak{g}\_2 \mathfrak{g}^{-1}, \mathfrak{g}\_3 \mathfrak{g}^{-1}\right) d\mathfrak{g} = \int \mathcal{K}\left(\mathfrak{g}\_1 \mathfrak{g}^{-1}, \mathfrak{g}\_2 \mathfrak{g}^{-1}\right) \mathcal{K}(\mathfrak{g}, \mathfrak{g}\_3) d\mathfrak{g}.\tag{107}
$$

When *G* is a locally compact Abelian group, one can try to translate (107) into the language of Fourier transforms. This is suggested by the convolution-like structure of this condition. Let us remind that the dual group *<sup>G</sup>* is the multiplicative group of all continuous homomorphisms of *G* into **T** = SU(1) — the group of complex numbers of modulus 1. The Fourier transform of a complex function <sup>Ψ</sup> on *<sup>G</sup>* is the function <sup>Ψ</sup> on *<sup>G</sup>* given by:

$$
\hat{\Psi}(\chi) = \int \overline{\langle \chi | g \rangle} \Psi(g) dg,\tag{108}
$$

where �*χ*|*g*� denotes the value of *<sup>χ</sup>* <sup>∈</sup> *<sup>G</sup>* at *<sup>g</sup>* <sup>∈</sup> *<sup>G</sup>*. And conversely,

$$
\Psi(\mathbf{g}) = \int \langle \chi | \mathbf{g} \rangle \hat{\Psi}(\chi) d\chi. \tag{109}
$$

Performing the two-argument Fourier transformation on the equation (107) we obtain the following condition:

$$
\hat{\mathcal{K}}\left(\chi\_1,\chi\_2\chi\_3\right)\hat{\mathcal{K}}\left(\chi\_2,\chi\_3\right) = \hat{\mathcal{K}}\left(\chi\_1,\chi\_2\right)\hat{\mathcal{K}}\left(\chi\_1\chi\_2,\chi\_3\right).
\tag{110}
$$

This is an equation for the factor of ray representations. It is clear that the Fourier representation of (106) in the Abelian case is given by:

$$\left(\hat{A}\top\hat{B}\right)(\chi) = \int \hat{\mathcal{K}}\left(\chi\_1, \chi\_1^{-1}\chi\right) \hat{A}\left(\chi\_1\right) \hat{B}\left(\chi\_1^{-1}\chi\right) d\chi\_1. \tag{111}$$

This is the *<sup>K</sup>*-twisted convolution of functions. It becomes the usual convolution when *<sup>K</sup>* <sup>≡</sup> 1. In analogy to (111) one defines the twisted convolution of measures.

A similar operation, i.e., twisted convolution of functions, or more generally, one of measures, may be defined in any locally compact topological group,

$$A\left(A\top B\right)\left(\mathbf{g}\right) = \int \omega\left(h, h^{-1}\mathbf{g}\right) A\left(h\right) B\left(h^{-1}\mathbf{g}\right) dh,\tag{112}$$

again in the sense of Haar measure *dh*. This product is associative for any group *G*, not necessarily the Abelian one, if and only if the mentioned functional equation (110) holds, i.e., if *<sup>ω</sup>* behaves like the factor of the ray representation, *<sup>ω</sup>* (*g*1, *<sup>g</sup>*2) *<sup>ω</sup>* (*g*1*g*2, *<sup>g</sup>*3) = *<sup>ω</sup>* (*g*1, *<sup>g</sup>*2*g*3) *<sup>ω</sup>* (*g*2, *<sup>g</sup>*3). One can show that there is a relationship between the twisted convolutions of functions (or measures) over *G* and the usual ones in some *G*-extension of the circle group **T** = SU(1). The choice of the phase-space group *G* depends on the particular model.

Let us go back to the situation when *G* is the group which models the configuration space, not the phase space. And we assume *G* to be Abelian. The phase space G will be given by <sup>G</sup> <sup>=</sup> *<sup>G</sup>* <sup>×</sup> *<sup>G</sup>*, where the dual group *<sup>G</sup>* is to model the "space of momenta". In analogy to the natural symplectic two-form on the linear space *<sup>V</sup>* × *<sup>V</sup>*<sup>∗</sup> we introduce the following two-character on G, *ζ* : G×G → **C** (the two-character, because *ζ*(*ξ*, ·), *ζ*(·, *ξ*) are characters on G for any *ξ* ∈ G):

28 Quantum Mechanics

following condition:

particular model.

K(*g*1, *<sup>g</sup>*)K

*<sup>g</sup>*2*g*<sup>−</sup>1, *<sup>g</sup>*3*g*−<sup>1</sup>

 *dg* = K 

Fourier transform of a complex function <sup>Ψ</sup> on *<sup>G</sup>* is the function <sup>Ψ</sup> on *<sup>G</sup>* given by:

Performing the two-argument Fourier transformation on the equation (107) we obtain the

This is an equation for the factor of ray representations. It is clear that the Fourier

This is the *<sup>K</sup>*-twisted convolution of functions. It becomes the usual convolution when *<sup>K</sup>* <sup>≡</sup> 1.

A similar operation, i.e., twisted convolution of functions, or more generally, one of measures,

again in the sense of Haar measure *dh*. This product is associative for any group *G*, not necessarily the Abelian one, if and only if the mentioned functional equation (110) holds, i.e., if *<sup>ω</sup>* behaves like the factor of the ray representation, *<sup>ω</sup>* (*g*1, *<sup>g</sup>*2) *<sup>ω</sup>* (*g*1*g*2, *<sup>g</sup>*3) = *<sup>ω</sup>* (*g*1, *<sup>g</sup>*2*g*3) *<sup>ω</sup>* (*g*2, *<sup>g</sup>*3). One can show that there is a relationship between the twisted convolutions of functions (or measures) over *G* and the usual ones in some *G*-extension of the circle group **T** = SU(1). The choice of the phase-space group *G* depends on the

<sup>K</sup> (*χ*1, *<sup>χ</sup>*2*χ*3) <sup>K</sup> (*χ*2, *<sup>χ</sup>*3) <sup>=</sup> <sup>K</sup> (*χ*1, *<sup>χ</sup>*2) <sup>K</sup> (*χ*1*χ*2, *<sup>χ</sup>*3). (110)

*<sup>A</sup>* (*χ*1) *<sup>B</sup>*

 *χ*−<sup>1</sup> <sup>1</sup> *<sup>χ</sup>* 

<sup>Ψ</sup>(*χ*) =

Ψ(*g*) =

where �*χ*|*g*� denotes the value of *<sup>χ</sup>* <sup>∈</sup> *<sup>G</sup>* at *<sup>g</sup>* <sup>∈</sup> *<sup>G</sup>*. And conversely,

representation of (106) in the Abelian case is given by:

In analogy to (111) one defines the twisted convolution of measures.

 *ω <sup>h</sup>*, *<sup>h</sup>*−1*<sup>g</sup> A*(*h*)*B <sup>h</sup>*−1*<sup>g</sup>* 

may be defined in any locally compact topological group,

(*A*⊤*B*)(*g*) =

 *<sup>A</sup>*<sup>⊤</sup>*<sup>B</sup>* (*χ*) = K *<sup>χ</sup>*1, *<sup>χ</sup>*−<sup>1</sup> <sup>1</sup> *<sup>χ</sup>* 

When *G* is a locally compact Abelian group, one can try to translate (107) into the language of Fourier transforms. This is suggested by the convolution-like structure of this condition. Let us remind that the dual group *<sup>G</sup>* is the multiplicative group of all continuous homomorphisms of *G* into **T** = SU(1) — the group of complex numbers of modulus 1. The

*<sup>g</sup>*1*g*<sup>−</sup>1, *<sup>g</sup>*2*g*−<sup>1</sup>

�*χ*|*g*�Ψ(*g*)*dg*, (108)

�*χ*|*g*�<sup>Ψ</sup>(*χ*)*dχ*. (109)

K(*g*, *<sup>g</sup>*3)*dg*. (107)

*dχ*1. (111)

*dh*, (112)

$$
\zeta\left( (\mathbf{x}\_1, \pi\_1), (\mathbf{x}\_2, \pi\_2) \right) = \langle \pi\_1 | \mathbf{x}\_2 \rangle \overline{\langle \pi\_2 | \mathbf{x}\_1 \rangle} = \frac{\langle \pi\_1 | \mathbf{x}\_2 \rangle}{\langle \pi\_2 | \mathbf{x}\_1 \rangle}. \tag{113}
$$

It is non-singular in the sense that the mappings *ξ* �→ *ζ*(*ξ*, ·), *ξ* �→ *ζ*(·, *ξ*) are isomorphisms of G onto G. Wave functions in the position and momentum representations are defined as amplitudes on *<sup>G</sup>* and *<sup>G</sup>* respectively. The group actions of *<sup>G</sup>*, *<sup>G</sup>* on wave functions are given by the following unitary representations:

$$(\mathcal{U}(\mathfrak{x})\Psi)(y) = \Psi\left(\mathfrak{x}^{-1}y\right), \qquad (V(\pi)\Psi)(y) = \langle \pi|y\rangle\Psi(y). \tag{114}$$

The second operator is obviously equal to the argument translation when the momentum representation is used: (*V*(*π*)Ψ) <sup>∧</sup> (*λ*) = <sup>Ψ</sup> *π*−1*λ* . One can check easily that the following fundamental commutation relation is satisfied:

$$
\omega U(\mathbf{x}) V(\pi) U(\mathbf{x})^{-1} V(\pi)^{-1} = \overline{\langle \pi | \mathbf{x} \rangle} = \langle \pi | \mathbf{x} \rangle^{-1}.\tag{115}
$$

Following the ideas of the Weyl prescription we define the following unitary operators: *Wp*(*x*, *<sup>π</sup>*) = �*π*|*x*�*pU*(*x*)*V*(*π*) = �*π*|*x*�*p*−1*V*(*π*)*U*(*x*). If in *<sup>G</sup>* or *<sup>G</sup>* there exists a unique square root like in **R***n*, then we put *p* = 1/2 and then *W <sup>x</sup>*<sup>−</sup>1, *<sup>π</sup>*−1 = *<sup>W</sup>*(*x*, *<sup>π</sup>*)<sup>−</sup>1. But in general it does not exist and we retain *p* as a non-defined label. We take the linear closure:

$$\mathbf{A} = \int \widehat{A}(\mathbf{x}, \pi) \mathbf{W}\_p(\mathbf{x}, \pi) d\mathbf{x} d\pi,\tag{116}$$

*<sup>A</sup>* denoting the Fourier transform of *<sup>A</sup>*. The corresponding "multiplication" rule for the functions *A*, *B* is based on the kernel:

$$K\_p\left( (\mathbf{x}\_1, \pi\_1), (\mathbf{x}\_2, \pi\_2) \right) = \int \langle \pi\_1 | \tilde{\xi} \rangle \langle \eta | \mathbf{x}\_1 \rangle \langle \pi\_2 | \tilde{\zeta} \rangle \langle \theta | \mathbf{x}\_2 \rangle \langle \eta | \tilde{\zeta} \rangle^{1-p} \langle \theta | \tilde{\xi} \rangle^{-p} d\tilde{\xi} d\eta d\tilde{\zeta} d\theta. \tag{117}$$

One can ask about the analogue of the "continuous canonical basis" of the usual *<sup>H</sup>*+-algebra over **R**2*n*:

$$\rho\_{\overline{\eta}\_1, \overline{\eta}\_2}(\overline{\eta}, \overline{p}) = \delta\left(\overline{\eta} - \frac{1}{2} \left(\overline{\eta}\_1 + \overline{\eta}\_2\right)\right) \exp\left(\frac{i}{\hbar}\overline{p} \cdot \left(\overline{\eta}\_2 - \overline{\eta}\_1\right)\right),\tag{118}$$

$$\rho\_{\overline{p}\_1, \overline{p}\_2}(\overline{q}, \overline{p}) = \delta\left(\overline{p} - \frac{1}{2}\left(\overline{p}\_1 + \overline{p}\_2\right)\right) \exp\left(\frac{i}{\hbar}\left(\overline{p}\_1 - \overline{p}\_2\right) \cdot \overline{q}\right). \tag{119}$$

Those bases satisfied:

$$q^l \* \rho\_{\overline{\eta}\_1, \overline{\eta}\_2} = q\_1^{\iota} \rho\_{\overline{\eta}\_1, \overline{\eta}\_2 \prime} \qquad \rho\_{\overline{\eta}\_1, \overline{\eta}\_2} \* q^l = q\_2^{\iota} \rho\_{\overline{\eta}\_1, \overline{\eta}\_2 \prime} \tag{120}$$

$$
\rho\_1 \* \rho \mathbf{p}\_1 \mathbf{p}\_2 = p\_{1i} \rho \mathbf{p}\_1 \mathbf{p}\_2 \prime \qquad \rho \mathbf{p}\_1 \mathbf{p}\_2 \* \mathbf{p}\_i = p\_{2i} \rho \mathbf{p}\_1 \mathbf{p}\_2 \tag{121}
$$

It turns out, however, that when there is no square-rooting in *<sup>G</sup>*, *<sup>G</sup>*, there are some problems. Namely, in **R**2*<sup>n</sup>* we could use both *ζ* and *ζ*<sup>2</sup> as kernels. But in a general Abelian group it is essential that we use *ζ*, not *ζ*<sup>2</sup> as a kernel. The analogue of (120), (121) reads: *A* ∗ *ρx*,*<sup>y</sup>* = *<sup>A</sup>*(*x*)*ρx*,*y*, *<sup>ρ</sup>x*,*<sup>y</sup>* ∗ *<sup>A</sup>* = *<sup>ρ</sup>x*,*yA*(*y*). We obtain *<sup>ρ</sup>x*1,*x*<sup>2</sup> (*x*, *<sup>π</sup>*) = *<sup>δ</sup> <sup>x</sup>*1*x*2*x*−<sup>1</sup> �*π*|*x*1*x*−1�. If *<sup>x</sup>*1*x*<sup>2</sup> fails to be a square, then *gx*<sup>1</sup> *<sup>x</sup>*<sup>2</sup> = 0. It is not yet clear for us if our procedure was improper or if we deal with the real superselection rule.

Let us observe that the group commutator does not feel the choice of *p*:

$$\mathcal{W}\_{\mathcal{P}}(\mathbf{x}\_{1},\boldsymbol{\pi}\_{1})\mathcal{W}\_{\mathcal{P}}(\mathbf{x}\_{2},\boldsymbol{\pi}\_{2})\mathcal{W}\_{\mathcal{P}}(\mathbf{x}\_{1},\boldsymbol{\pi}\_{1})^{-1}\mathcal{W}\_{\mathcal{P}}(\mathbf{x}\_{2},\boldsymbol{\pi}\_{2})^{-1} = \mathsf{f}\left((\mathbf{x}\_{1},\boldsymbol{\pi}\_{1}),(\mathbf{x}\_{2},\boldsymbol{\pi}\_{2})\right)\mathrm{Id}.\tag{122}$$

If *G* = **Z***<sup>n</sup>* or **T***<sup>n</sup>* = (SU(1)) *<sup>n</sup>*, then the mentioned problem with *ζ* may be connected with what in solid state physics is known as so-called Umklapp-Prozessen.

Let us observe also that in a sense one can use the following kernel of the non-local product of functions over the discrete group **<sup>Z</sup>**2*n*: *<sup>K</sup>*(*n*, *<sup>m</sup>*) = exp *iBabnamb* , where [*Bab*] is the real skew-symmetric matrix. Nevertheless, the resulting product will have then some strange features.

Let us finish with some remarks concerning the asymptotics of "large quantum numbers" in the quasi-classical limit transition. For simplicity we consider only the wave functions of the planar rotators, Ψ*n*(*ϕ*) = exp(*inϕ*), *n* ∈ **Z**, where *ϕ* is the angular variable. Ψ*<sup>n</sup>* is proportional to the eigenfunction of the angular momentum with the eigenvalue *nh*¯,

$$\frac{\hbar}{i}\frac{\partial}{\partial\rho}\Psi\_n = \hbar n \Psi\_n. \tag{123}$$

$$\text{Clearly}\tag{12}$$

$$\Psi\_{\hbar} \simeq \exp\left(\frac{i}{\hbar}(n\hbar)\varrho\right);\tag{124}$$

it is just *ln* = *nh*¯ that is interpreted as the physical value of the angular momentum. But the analysis ¯*h* → 0 does not work directly. We must take superpositions of quickly-oscillating eigenequations,

$$\Psi(\boldsymbol{\uprho}) = \sum\_{\boldsymbol{\uprho}} c\_{\boldsymbol{\uprho}} \exp(in\boldsymbol{\uprho}),\tag{125}$$

where the sequence **Z** ∋ *n* �→ *cn* ∈ **C** is concentrated in a range

$$
\hbar \mathfrak{n}\_0 - \Delta \mathfrak{n} \ll \mathfrak{n} \ll \mathfrak{n}\_0 + \Delta \mathfrak{n}.\tag{126}
$$

It is assumed here that *<sup>n</sup>*<sup>0</sup> ≫ <sup>∆</sup>*<sup>n</sup>* ≫ 1 and the sequence is assumed to be slowly-varying in the range (126), so that

$$\frac{|c\_{n+1} - c\_{\hbar}|}{|c\_{\hbar}|} \ll 1.\tag{127}$$

It follows from the Fourier theory that approximately (125) may be replaced by:

$$\Psi(\varphi) = \int \mathfrak{c}(k) \exp(ik\varphi) dk,\tag{128}$$

where at the discrete values of *k* = *n c*(*k*) equals *cn* and changes slowly, e.g., linearly between them. Then Ψ(*ϕ*) is well concentrated and one can consider it as a quickly-vanishing at infinity function on **R**. And then one substitutes *k* = *p*/¯*h* and further on the previous asymptotics ¯*h* → 0 may be used. The same when there are more degrees of freedom. Let us remind that it was just this limit transition we have used in the theory of angular momentum [14–16]. By the way, the conditions (126), (127) enabled one to remove artificial picks of the basic wave functions at *k* = 2*π* in SU(2). Namely, the subsequent picks have opposite signs and mutually cancel when (126), (127) are satisfied.

**Summary of Section 5:** We have mentioned here about some generalizations of the Weyl-Wigner-Moyal-Ville procedure. They are based on some group-theoretic models and may be perhaps helpful in the formally "classical", although in fact quantum, approach to dynamics.

#### **6. Conclusions**

30 Quantum Mechanics

Those bases satisfied:

*<sup>q</sup><sup>i</sup>* <sup>∗</sup> *<sup>ρ</sup>q*1,*q*<sup>2</sup> <sup>=</sup> *<sup>q</sup>*<sup>1</sup>

*<sup>A</sup>*(*x*)*ρx*,*y*, *<sup>ρ</sup>x*,*<sup>y</sup>* ∗ *<sup>A</sup>* = *<sup>ρ</sup>x*,*yA*(*y*). We obtain *<sup>ρ</sup>x*1,*x*<sup>2</sup> (*x*, *<sup>π</sup>*) = *<sup>δ</sup>*

of functions over the discrete group **Z**2*n*: *K*(*n*, *m*) = exp

Clearly <sup>Ψ</sup>*<sup>n</sup>* <sup>≃</sup> exp

deal with the real superselection rule.

If *G* = **Z***<sup>n</sup>* or **T***<sup>n</sup>* = (SU(1))

features.

eigenequations,

*pi* <sup>∗</sup> *<sup>ρ</sup>p*1,*p*<sup>2</sup> <sup>=</sup> *<sup>p</sup>*1*iρp*1,*p*<sup>2</sup>

Let us observe that the group commutator does not feel the choice of *p*:

what in solid state physics is known as so-called Umklapp-Prozessen.

*i <sup>ρ</sup>q*1,*q*<sup>2</sup> , *<sup>ρ</sup>q*1,*q*<sup>2</sup> <sup>∗</sup> *<sup>q</sup><sup>i</sup>* <sup>=</sup> *<sup>q</sup>*<sup>2</sup>

It turns out, however, that when there is no square-rooting in *<sup>G</sup>*, *<sup>G</sup>*, there are some problems. Namely, in **R**2*<sup>n</sup>* we could use both *ζ* and *ζ*<sup>2</sup> as kernels. But in a general Abelian group it is essential that we use *ζ*, not *ζ*<sup>2</sup> as a kernel. The analogue of (120), (121) reads: *A* ∗ *ρx*,*<sup>y</sup>* =

be a square, then *gx*<sup>1</sup> *<sup>x</sup>*<sup>2</sup> = 0. It is not yet clear for us if our procedure was improper or if we

Let us observe also that in a sense one can use the following kernel of the non-local product

skew-symmetric matrix. Nevertheless, the resulting product will have then some strange

Let us finish with some remarks concerning the asymptotics of "large quantum numbers" in the quasi-classical limit transition. For simplicity we consider only the wave functions of the planar rotators, Ψ*n*(*ϕ*) = exp(*inϕ*), *n* ∈ **Z**, where *ϕ* is the angular variable. Ψ*<sup>n</sup>* is

> *i h*¯ (*nh*¯)*ϕ*

it is just *ln* = *nh*¯ that is interpreted as the physical value of the angular momentum. But the analysis ¯*h* → 0 does not work directly. We must take superpositions of quickly-oscillating

*n*

proportional to the eigenfunction of the angular momentum with the eigenvalue *nh*¯,

*h*¯ *i ∂*

Ψ(*ϕ*) = ∑

where the sequence **Z** ∋ *n* �→ *cn* ∈ **C** is concentrated in a range

*Wp*(*x*1, *<sup>π</sup>*1)*Wp*(*x*2, *<sup>π</sup>*2)*Wp*(*x*1, *<sup>π</sup>*1)−1*Wp*(*x*2, *<sup>π</sup>*2)−<sup>1</sup> <sup>=</sup> *<sup>ζ</sup>* ((*x*1, *<sup>π</sup>*1),(*x*2, *<sup>π</sup>*2))Id. (122)

, *<sup>ρ</sup>p*1,*p*<sup>2</sup> <sup>∗</sup> *pi* <sup>=</sup> *<sup>p</sup>*2*iρp*1,*p*<sup>2</sup>

*i <sup>ρ</sup>q*1,*q*<sup>2</sup>

*<sup>x</sup>*1*x*2*x*−<sup>1</sup>

*<sup>n</sup>*, then the mentioned problem with *ζ* may be connected with

*iBabnamb*

*∂ϕ* <sup>Ψ</sup>*<sup>n</sup>* <sup>=</sup> *hn*¯ <sup>Ψ</sup>*n*. (123)

*cn* exp(*inϕ*), (125)

*<sup>n</sup>*<sup>0</sup> − <sup>∆</sup>*<sup>n</sup>* ≪ *<sup>n</sup>* ≪ *<sup>n</sup>*<sup>0</sup> + <sup>∆</sup>*n*. (126)

; (124)

, (120)

. (121)

�*π*|*x*1*x*−1�. If *<sup>x</sup>*1*x*<sup>2</sup> fails to

, where [*Bab*] is the real

We have discussed certain problems concerning the relationship between classical and quantum theories. Analyzed are both differences and formal similarities between them. What concerns similarities, we show that in contrast to some popular views, it is not points in the classical phase space but rather *n*-dimensional Lagrangian submanifolds in the phase space that corresponds to the quantum pure states. More precisely, the classical "pure state" is a probability distribution on the Lagrange manifold, or rather on its horizontal Legendre lift to the contact space. Here *n* is the number of degrees of freedom and the contact space is, roughly speaking, the Cartesian product of the phase space by **R** or U(1) with geometry given by *pidq<sup>i</sup>* − *dz*. It was shown that superpositions, scalar products, etc. are defined in the set of Legendre manifolds and have some formal properties of the corresponding quantum concepts. They are based on the Huygens notion of envelope of the wave fronts. This was shown both directly on the basis of limit transition in the Weyl-Wigner-Moyal-Ville formalism and on the basis of general symplectic language. Nevertheless, it is clear that quantum mechanics with its reduction and decoherence problems is something completely different than the classical theory. We try to show that unlike this view, there is a nonlinear modification of quantum theory which perhaps would be free of the mentioned paradoxes. It is based on the classical language of variational principles and on the concept of dynamical scalar product. The system consisting of wave function and scalar product satisfies an essentially nonlinear, non-perturbative dynamical equation. Its characteristic nonlinearity seems to be able to describe analytically the decoherence process. Finally, we review some generalization of the Weyl-Wigner-Moyal-Ville formalism and discuss the quasi-classical limit in terms of "large quantum numbers".

The general conclusion/hypothesis is that perhaps there is no such a gap between classics and quanta as one commonly believes.

#### **Acknowledgements**

This paper partially contains results obtained within the framework of the research project N N501 049 540 financed from the Scientific Research Support Fund in the years 2011-2014. The authors are greatly indebted to the Polish Ministry of Science and Higher Education for this financial support.

#### **Author details**

J. J. Sławianowski<sup>∗</sup> and V. Kovalchuk<sup>∗</sup>

Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland

<sup>∗</sup>Address all correspondence to: jslawian@ippt.pan.pl, vkoval@ippt.pan.pl

#### **References**


[9] Moyal JE. Stochastic Processes and Statistical Physics. Journal of the Royal Society B 1949;11 150–210.

32 Quantum Mechanics

in terms of "large quantum numbers".

and quanta as one commonly believes.

J. J. Sławianowski<sup>∗</sup> and V. Kovalchuk<sup>∗</sup>

Publishing Company, Inc.; 1978.

Ordnung. Leipzig: B.G. Teubner; 1956.

1999;40 49–63; quant-ph:9709036.

Society 1949;45 99–124.

Kong-Barcelona-Budapest: Springer-Verlag; 1989.

Benjamin-Cummings Publishing Company, Inc.; 1963.

**Acknowledgements**

this financial support.

**Author details**

Poland

**References**

generalization of the Weyl-Wigner-Moyal-Ville formalism and discuss the quasi-classical limit

The general conclusion/hypothesis is that perhaps there is no such a gap between classics

This paper partially contains results obtained within the framework of the research project N N501 049 540 financed from the Scientific Research Support Fund in the years 2011-2014. The authors are greatly indebted to the Polish Ministry of Science and Higher Education for

Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw,

[1] Abraham R., Marsden JE. Foundations of Mechanics (second edition). London-Amsterdam-Don Mills-Ontario-Sydney-Tokyo: The Benjamin-Cummings

[2] Arnold VI. Mathematical Methods of Classical Mechanics (second edition). Graduate Texts in Mathematics 60, New York-Berlin-Heidelberg-London-Paris-Tokyo-Hong

[3] Caratheodory C. Variationsrechnung und Partielle Differential-Gleidungen Erster

[4] Doebner HD., Goldin GA. Introducing Nonlinear Gauge Transformations in a Family

[5] Doebner HD., Goldin GA., Nattermann P. Gauge Transformations in Quantum Mechanics and the Unification of Nonlinear Schrödinger Equations. J. Math. Phys.

[7] Mackey GW. The Mathematical Foundations of Quantum Mechanics. New York: The

[8] Moyal JE. Quantum Mechanics as a Statistical Theory. Proc. Cambridge Philosophical

[6] Landau LD., Lifshitz EM. Quantum Mechanics. London: Pergamon Press; 1958.

of Nonlinear Schrödinger Equations. Phys. Rew. A 1996;54 3764–3771.

<sup>∗</sup>Address all correspondence to: jslawian@ippt.pan.pl, vkoval@ippt.pan.pl


## **Photons and Signals in the Age of Information**

Cynthia Kolb Whitney

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59067

#### **1. Introduction**

The history of Physics contains some inexplicable mysteries, and one of them is this: back in the early 20th century, Einstein was working on the idea of the 'Photon' and on the idea of the 'Signal' at essentially the same time [1,2], but he did not relate them to each other. They seem to have arisen totally separate in his mind, and they led to totally separate subsequent developments. The Photon played a central role in the development of Quantum Mechanics (QM), and the Signal played the central role in the development of Special Relativity Theory (SRT).

QM and SRT are now the two great pillars of early 20th century Physics, but they seem to be in conflict over the issue of communication. In QM, Schrödinger's Equation is basically the Fourier transform (just a restatement in terms of different variables) of a statement from Classical Mechanics: Total energy = kinetic energy + potential energy. This statement has no signal propagation speed involved in it. So QM appears to allow instantaneous communication over arbitrary distances. But in SRT, Einstein's Second Postulate limits *all* communication to light speed, *c*.

Since QM and SRT conflict so dramatically on the issue of communication, at least one of them must, in some sense, be wrong. So we are left unsure about what to believe, or take as a foundation for future development.

Can we find out anything decisive from experiments or observations?

The foundation for QM is usually called the Quantum Hypothesis, and not the Quantum Postulate, because the granularity aspect of QM is experimentally testable in various ways. So far, the granularity aspect of QM never seems to fail. This fact stands in favor of QM. But many people still do not believe in the instantaneous communication aspect of QM that is manifest in Schrödinger's Equation.

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The foundation for SRT can be called the Light-Speed Postulate, but not the Light-Speed Hypothesis, because light speed really is *not* experimentally testable, since a test would involve at least two different spacetime points, and the correlation of data from two different spacetime points would involve the Light-Speed Postulate itself. Despite numerous claims to the contrary, SRT has *not* been tested in a way that actually could have falsified it, and only very indirect testing appears even feasible for SRT.

Einstein's General Relativity Theory (GRT) flows from SRT, and GRT is testable, at least observationally, although not experimentally. But the new hypotheses that it offers for observational test are few in number, and great in technical difficulty. So even indirect testing of SRT through GRT does not look very promising.

An altogether different approach therefore seems needed: instead of demanding experiments or observations, we should be reviewing the founding ideas themselves, in light of new insights gathered in the intervening century. Evidently, at least one of the founding ideas, and possibly both of them, need to be updated, or else retired and replaced. This paper aims to identify possible update(s)/replacement(s) that may help.

Some of the needed insights come from engineering practice, rather than from theoretical physics. In the mid 20th century there was a flowering of Information Theory (IT), first in connection with wartime code breaking and code making, and then in connection with the post-war communication industry. All of that development led in turn to our modern computation industry, and our current 'Age of Information'.

IT uses the concept of Entropy, taken from classical Thermodynamics, and with the application of a minus sign, provides a quantitative mathematical measure for Information. This measure can be used in support of all sorts of engineering concept analyses and design decisions, *etc*. A convenient reference about the IT concepts and their general applications is Leon Brillouin's wonderful little book *Science and Information Theory* [3]. Flores Gallegos [4] discusses some particular applications in QM.

Viewed from our vantage point here in the early 21st century, IT actually provides a clear disqualifier for Einstein's Second Postulate. The problem is this: the Second Postulate is based on the behavior of a classical infinite plane wave, and an infinite plane wave cannot convey any information whatsoever!

The reason for this perhaps startling assertion is that an infinite plane wave is to electromag‐ netic communication what a steady hum is to auditory communication: background at best. There is no music in the monotonous hum, and there is no message in the infinite plane wave.

Information requires structure: amplitude modulation, or frequency modulation, or on-off switching. An infinite plane wave does not have any such structure. Because of this deficit, we certainly need a new Signal model as the foundation for an updated SRT.

Possibly we also need a better Photon model at the foundation of QM. We do not actually have a detailed and universally accepted model for photons in QM. Attention has focused more on material systems, which are said to change state, and in so doing, emit or absorb photons that carry packets of energy and angular momentum. As for electric and magnetic fields, what we have is the notion from Quantum Electrodynamics of a 'virtual photon'. This terminology reveals the desirability of having some unified, photon-like, approach for both Coulomb-Ampère and radiation fields, but it does not actually provide details.

The foundation for SRT can be called the Light-Speed Postulate, but not the Light-Speed Hypothesis, because light speed really is *not* experimentally testable, since a test would involve at least two different spacetime points, and the correlation of data from two different spacetime points would involve the Light-Speed Postulate itself. Despite numerous claims to the contrary, SRT has *not* been tested in a way that actually could have falsified it, and only very

Einstein's General Relativity Theory (GRT) flows from SRT, and GRT is testable, at least observationally, although not experimentally. But the new hypotheses that it offers for observational test are few in number, and great in technical difficulty. So even indirect testing

An altogether different approach therefore seems needed: instead of demanding experiments or observations, we should be reviewing the founding ideas themselves, in light of new insights gathered in the intervening century. Evidently, at least one of the founding ideas, and possibly both of them, need to be updated, or else retired and replaced. This paper aims to identify

Some of the needed insights come from engineering practice, rather than from theoretical physics. In the mid 20th century there was a flowering of Information Theory (IT), first in connection with wartime code breaking and code making, and then in connection with the post-war communication industry. All of that development led in turn to our modern

IT uses the concept of Entropy, taken from classical Thermodynamics, and with the application of a minus sign, provides a quantitative mathematical measure for Information. This measure can be used in support of all sorts of engineering concept analyses and design decisions, *etc*. A convenient reference about the IT concepts and their general applications is Leon Brillouin's wonderful little book *Science and Information Theory* [3]. Flores Gallegos [4] discusses some

Viewed from our vantage point here in the early 21st century, IT actually provides a clear disqualifier for Einstein's Second Postulate. The problem is this: the Second Postulate is based on the behavior of a classical infinite plane wave, and an infinite plane wave cannot convey

The reason for this perhaps startling assertion is that an infinite plane wave is to electromag‐ netic communication what a steady hum is to auditory communication: background at best. There is no music in the monotonous hum, and there is no message in the infinite plane wave. Information requires structure: amplitude modulation, or frequency modulation, or on-off switching. An infinite plane wave does not have any such structure. Because of this deficit, we

Possibly we also need a better Photon model at the foundation of QM. We do not actually have a detailed and universally accepted model for photons in QM. Attention has focused more on material systems, which are said to change state, and in so doing, emit or absorb photons that carry packets of energy and angular momentum. As for electric and magnetic fields, what we

certainly need a new Signal model as the foundation for an updated SRT.

indirect testing appears even feasible for SRT.

38 Selected Topics in Applications of Quantum Mechanics

of SRT through GRT does not look very promising.

possible update(s)/replacement(s) that may help.

particular applications in QM.

any information whatsoever!

computation industry, and our current 'Age of Information'.

Without detail to argue against, we are apparently free to develop a Photon model *de novo*, in a way that serves not only as the Photon model for QM, but also provides the more realistic Signal model needed for an updated SRT.

The new Photon/Signal model need *not* involve yet another new Postulate. Remember what Euclid taught the world through his Geometry: use no more Postulates than absolutely necessary. The reason is that unnecessary Postulates can conflict with other Postulates already in place, and so lead to Paradoxes.

In SRT, we do indeed have many Paradoxes, involving rods, clocks, trains, lightening strikes, snakes, barns, twins, and so on, and on. A prime suspect for their root cause is the unnecessary Second Postulate. In QM, Schrödinger's Equation seemingly came full-formed from heaven, and to that extent was also a Postulate, and indeed one that conflicted with Einstein's Second Postulate. The mysterious feel of quantum duality, for example, may suggest a possible QM Paradox yet to be fully articulated. If so, the new Photon/Signal model can offer a candidate approach to solve the problem.

The Photon/Signal model is just very familiar, old-fashioned mathematics: **1)** partial differen‐ tial equations (Maxwell's first order coupled field equations), **2)** their family of solutions (starting with Gaussian pulses, and generating by differentiations the higher-and-higher order Hermite polynomials multiplying the Gaussian pulses), and **3)** boundary conditions (no backflow of energy behind the source, no overflow of energy beyond the receiver). This formulation is enough to determine the particular solution that fits any particular problem.

One general rule in Science is this: when a venerable and well-tested mathematical approach exists, try it **first**, before abandoning it in favor of a new approach. The irony is that Einstein chose not to apply usual the mathematical approach, and instead to introduce his additional Postulate. But he did it so long ago that, despite the long list of Paradoxes generated by SRT, his Second Postulate has itself become 'venerable', and therefore nearly impossible to unseat.

But with SRT updated with the new Photon/Signal model, based entirely on the old-fashioned mathematical approach, QM no longer needs to conflict with SRT. Atoms can have solution states in which energy loss by radiation is countered by an energy gain mechanism newly identified with the updated SRT. It is no longer necessary to postulate the Schrödinger equation just to prevent atomic death by loss of orbit energy to far-field radiation.

With n new starting point for QM, there can be a new development for QM. Some type of Quantum Gravity has long been sought, but with GRT being founded in SRT, with *c*-speedonly communication, and with QM being founded in Classical Physics, with instantaneous communication, that goal has been hard to reach. But the combination of an updated SRT and traditional Statistical Mechanics (SM) offers a way forward.

The new protocol for treating gravity is this: **1)** First notice that gravitational attraction formally resembles a statistical residue from magnetic interactions between elements of charge-neutral matter that are carrying electrical currents, 'current elements' for short. Current elements were well described by Ampère before Maxwell ever came along. **2)** Then apply SM to pairs of current elements.

QM also connects to modern technology problems. Chemistry is wonderfully rich application area for QM. But current-day Quantum Chemistry (QC) is not very easy to use, mainly because of heavy computation loads associated with numerical integrations. The new photon/signal model leads to a different approach for QC, one that is algebraic, rather than integral, in character. This paper includes some recent results from the application of Algebraic Chemistry (AC) to research on the chemistry of water; namely, the form of water known as 'EZ water', because it excludes positive ions.

Finally, QM may connect to Elementary Particle Physics in a manner yet to be fully developed. Just as the myriad compounds in Chemistry arises from not-very-many chemical elements, some significant part of the myriad of currently understood 'elementary' particles may arise from just two of them: the electron and the positron.

The suggestion for this idea lies in Chemistry's Periodic Table (PT). The way that the electron spin states fill up with increasing nuclear charge suggests the existence of not only electron pairs that have opposing spins, but also electron rings that have aligned spins. Electron rings involve up to three, five, even seven, electrons. Basically, electrons within atoms form, not only opposite-spin couples, but also same-spin teams. Since electron rings apparently do occur in atoms, they may well also occur separated from atomic nuclei, and therefore looking like exotic elementary particles. And the same may be said of positrons. Thus we have a rich array of possibilities yet to explore.

#### **2. The photon / signal model**

The first part of the Photon/Signal model consists of the governing partial differential equa‐ tions. These are Maxwell's four first-order coupled field equations. Jackson [5] gives Maxwell's equations in modern notation and Gaussian units as:

$$
\nabla \cdot \mathbf{B} = 0,\\
\nabla \cdot \mathbf{D} = 4\pi \rho,\\
\nabla \times \mathbf{E} + \frac{1}{c} \hat{c} \mathbf{B} / \hat{c}t = 0,\\
\nabla \times \mathbf{H} - \frac{1}{c} \hat{c} \mathbf{D} / \hat{c}t = \frac{4\pi}{c} \mathbf{J}.\tag{1}
$$

Here **B** is magnetic field and **E** is electric field. The constant 1 / *c* = *ε*0*μ*0, where *ε*0 is electric permittivity and *μ*0 is magnetic permeability. In free space, **D**=*ε*0**E**, **H**=**B**/*μ*0, and charge density *ρ* and current density **J** are zero. Free space will be the case of interest henceforth in this paper.

The second part of the Photon/Signal model is the family of suitable finite-energy solutions. This is developed as follows. Let the word 'pulse' be the short description for a field profile that is rounded on top and sloping down on the sides, and fading gradually to zero; for example, a Gaussian function.

The mechanism for the waveform development is that the spatial derivatives applied in Maxwell's equations change the original Gaussian pulse into successively longer wavelets consisting of successively higher order Hermite polynomials multiplying the original Gaus‐ sian. As a result, the energy in the wavelet gets more and more spread out along the propa‐ gation path.

Observe that waveform development is inexorable, just like ever-growing entropy in Ther‐ modynamics. This is, I believe, where Entropy really enters into Physics. That is, Maxwell's first order coupled field equations are what give to Physics its obvious Arrow of Time. In citing electromagnetism as the cause for irreversibility, this idea follows Bentwich [6]. (It does *not*, however, give any hope for *reversing* anything.) *4* Observe that waveform development is inexorable, just like ever-growing entropy in Thermodynamics. This is, I believe, where Entropy really enters into Physics. That is, Maxwell's first order cou-

Let the spatial variable argument for the pulse be *x*. Let the direction of the initial pulse be *y*. Figure 1 illustrates this Gaussian pulse, along with a snapshot showing how it evolves over one complete cycle through Maxwell's first-order coupled field equations. Series 1 is the input pulse, and Series 2 is the waveform developed from it. A more complicated Figure was given in Whitney [7]. This simplified Figure 1 focuses on just the one issue: waveform development. The wavelet is shown bold because it has not been given sufficient attention before. pled field equations are what give to Physics its obvious Arrow of Time. In citing electromagnetism as the cause for irreversibility, this idea follows Bentwich [6]. (It does *not*, however, give any hope for *reversing* anything.) Let the spatial variable argument for the pulse be *x* . Let the direction of the initial pulse be *y* . Figure 1 illustrates this Gaussian pulse, along with a snapshot showing how it evolves over one complete cycle through Maxwell's first-order coupled field equations. Series 1 is the input pulse, and Series 2 is the waveform developed from it. A more complicated Figure was given in Whitney [7].

This simplified Figure 1 focuses on just the one issue: waveform development. The wavelet is shown

Now, to solve the stated propagation problem, one can pose a primary pair of pulses in **E** and **B** to **Figure 1.** Illustration of waveform development.

say more.

bold because it has not been given sufficient attention before.

matter that are carrying electrical currents, 'current elements' for short. Current elements were well described by Ampère before Maxwell ever came along. **2)** Then apply SM to pairs of

QM also connects to modern technology problems. Chemistry is wonderfully rich application area for QM. But current-day Quantum Chemistry (QC) is not very easy to use, mainly because of heavy computation loads associated with numerical integrations. The new photon/signal model leads to a different approach for QC, one that is algebraic, rather than integral, in character. This paper includes some recent results from the application of Algebraic Chemistry (AC) to research on the chemistry of water; namely, the form of water known as 'EZ water',

Finally, QM may connect to Elementary Particle Physics in a manner yet to be fully developed. Just as the myriad compounds in Chemistry arises from not-very-many chemical elements, some significant part of the myriad of currently understood 'elementary' particles may arise

The suggestion for this idea lies in Chemistry's Periodic Table (PT). The way that the electron spin states fill up with increasing nuclear charge suggests the existence of not only electron pairs that have opposing spins, but also electron rings that have aligned spins. Electron rings involve up to three, five, even seven, electrons. Basically, electrons within atoms form, not only opposite-spin couples, but also same-spin teams. Since electron rings apparently do occur in atoms, they may well also occur separated from atomic nuclei, and therefore looking like exotic elementary particles. And the same may be said of positrons. Thus we have a rich array of

The first part of the Photon/Signal model consists of the governing partial differential equa‐ tions. These are Maxwell's four first-order coupled field equations. Jackson [5] gives Maxwell's

1 14 0, 4 , / 0, *t t* / .

Here **B** is magnetic field and **E** is electric field. The constant 1 / *c* = *ε*0*μ*0, where *ε*0 is electric permittivity and *μ*0 is magnetic permeability. In free space, **D**=*ε*0**E**, **H**=**B**/*μ*0, and charge density *ρ* and current density **J** are zero. Free space will be the case of interest henceforth in this paper.

The second part of the Photon/Signal model is the family of suitable finite-energy solutions. This is developed as follows. Let the word 'pulse' be the short description for a field profile that is rounded on top and sloping down on the sides, and fading gradually to zero; for

**B D EB HD J**

Ñ = Ñ = Ñ´ + ¶ ¶ = Ñ´ - ¶ ¶ = g g

*c cc*

(1)

p

current elements.

because it excludes positive ions.

40 Selected Topics in Applications of Quantum Mechanics

possibilities yet to explore.

**2. The photon / signal model**

example, a Gaussian function.

from just two of them: the electron and the positron.

equations in modern notation and Gaussian units as:

pr

guarantee travel, and, for more realism, add a second such pair, offset a quarter cycle in time and perpendicular in space, to model circular polarization, like a real physical photon exhibits. The third part of the Photon/Signal model is the pair of propagation boundary conditions: no backflow of energy behind the source, and no overflow of energy beyond the receiver. To guarantee these boundary conditions, one can demand **E 0** at these boundaries. Now, to solve the stated propagation problem, one can pose a primary pair of pulses in **E** and **B** to guarantee travel, and, for more realism, add a second such pair, offset a quarter cycle in time and perpendicular in space, to model circular polarization, like a real physical photon exhibits.

**Figure 1.** Illustration of waveform development.

One can fulfill the required zero **E** fields by matching the signal leaving the source toward the receiver with **1)** another fictitious signal going from the source in the opposite direction, so as to make the **E** field zero at the source, and **2)** another fictitious signal approaching the receiver from the opposite direction, so as to make the **E** field zero at the receiver. One can even continue this The third part of the Photon/Signal model is the pair of propagation boundary conditions: no backflow of energy behind the source, and no overflow of energy beyond the receiver. To guarantee these boundary conditions, one can demand **E**=0 at these boundaries.

boundary-fixing process, to clean up each tiny new departure from zero **E** at a boundary that each additional fictitious signal creates at the end of the path opposite to the end that it is meant to correct. Carried to an infinite sum of corrections upon corrections, this is certainly a very complicated picture about electromagnetic fields. How then can we extract from it some simple statement about propa-One can fulfill the required zero **E** fields by matching the signal leaving the source toward the receiver with **1)** another fictitious signal going from the source in the opposite direction, so as to make the **E** field zero at the source, and **2)** another fictitious signal approaching the receiver

gation speed? We have all been trained to just say *c* . "But relative to what?" we might ask. If we knew the source and receiver were stationary relative to each other, we could be sure that *c c*relativeÊtoÊreceiver , consistent with Einstein's Second Postulate. Otherwise, we would have to

If we would focus attention to moments when the bulk of the energy is very near the receiver, we could be fairly confident that *c c*relativeÊtoÊreceiver , again consistent with Einstein's Second Postulate. But if we would focus attention to moments when the bulk of the energy is still very near the source, moments when the receiver is nothing more than a distant phantom, we would be hard pressed to argue against the proposition that *c c*relativeÊtoÊsource . (This would be consistent with from the opposite direction, so as to make the **E** field zero at the receiver. One can even continue this boundary-fixing process, to clean up each tiny new departure from zero **E** at a boundary that each additional fictitious signal creates at the end of the path opposite to the end that it is meant to correct.

Carried to an infinite sum of corrections upon corrections, this is certainly a very complicated picture about electromagnetic fields. How then can we extract from it some simple statement about propagation speed? We have all been trained to just say *c*. "But relative to what?" we might ask. If we knew the source and receiver were stationary relative to each other, we could be sure that *c* =*c*relative to receiver, consistent with Einstein's Second Postulate. Otherwise, we would have to say more.

If we would focus attention to moments when the bulk of the energy is very near the receiver, we could be fairly confident that *c* = *c*relative to receiver, again consistent with Einstein's Second Postulate. But if we would focus attention to moments when the bulk of the energy is still very near the source, moments when the receiver is nothing more than a distant phantom, we would be hard pressed to argue against the proposition that *c* = *c*relative to source. (This would be consis‐ tent with the 1908 Ritz Proposal [8], which was much investigated in the early to mid 20th century as a candidate alternative to Einstein's Second Postulate, but was ultimately rejected.)

If we would need to characterize an entire propagation scenario, we would have to go even further, and consider *all* moments along the way. We would have neither the Einstein Second Postulate, nor the Ritz Proposal, but rather something else more complicated. It certainly must conform to Einstein's Second Postulate late in the scenario, when the bulk of the energy is near the receiver. But early in the scenario, when the bulk of the energy is still near the source, it must conform to Ritz's Proposal. And in between, it must represent in some mathematically appropriate way an idea not previously considered: a transition from one reference for *c* to the other.

Here is one way to formulate the problem. Let variable *x* represent distance along the propa‐ gation path, and let variable *t* represent time into the propagation process. At any point *x*, *t* there are fields **E** (*x*, *t*) and **B** (*x*, *t*) with magnitudes *E*(*x*, *t*) and *B*(*x*, *t*), and from them a local energy density:

$$S(\mathbf{x},t) = \left[E^2(\mathbf{x},t) + B^2(\mathbf{x},t)\right] \Big/ \text{2}.\tag{2}$$

One way to characterize the propagation is with path integrals. [9-13] Consider the following ratio of two path integrals:

$$r(t) = \int\_{\text{path}} \mathbf{x} \mathbf{S}(\mathbf{x}, t) d\mathbf{x} \Big/ \int\_{\text{path}} \mathbf{S}(\mathbf{x}, t) d\mathbf{x} \,. \tag{3}$$

This ratio provides a function that begins at the source, and ends at the receiver, and at the temporal midpoint of the scenario, gives equal weight to both the source and the receiver. This behavior captures the proposed transition of the reference to which the light speed *c* is assumed relative.

#### **3. Using the changing reference for** *c* **changes the results**

The concept of the changing reference for *c* to be relative to appears relevant for an important scenario that was considered even before Einstein arrived on the scene. The scenario involves the potentials and fields created by rapidly moving sources, and it was addressed starting in the late nineteenth century and very early twentieth centuries. Researchers then made the same Assumption that Einstein later made his Second Postulate in founding SRT, but they were not attentive enough to see that there indeed *was* an Assumption, and to call it out as his Second Postulate. So Einstein is to be commended for calling attention to this Assumption.

The sources generally cited for this early (1898 to 1901) problem are A. Liénard [14] and E. Wiechert [15]. Although they worked at about the same time, they worked separately. They got the same results, as did all contemporary and subsequent investigators, because all persons working from then up until now have used the same input Assumption; namely, that the speed of light is always *c* with respect to the receiver of the light.

The Liénard- Wiechert results are given in [5], and in every other standard EM book. They are displayed in [7], and that short passage is quoted again here, for review and subsequent further discussion:

"The standard scalar and vector potentials are:

from the opposite direction, so as to make the **E** field zero at the receiver. One can even continue this boundary-fixing process, to clean up each tiny new departure from zero **E** at a boundary that each additional fictitious signal creates at the end of the path opposite to the end that it is

Carried to an infinite sum of corrections upon corrections, this is certainly a very complicated picture about electromagnetic fields. How then can we extract from it some simple statement about propagation speed? We have all been trained to just say *c*. "But relative to what?" we might ask. If we knew the source and receiver were stationary relative to each other, we could be sure that *c* =*c*relative to receiver, consistent with Einstein's Second Postulate. Otherwise, we

If we would focus attention to moments when the bulk of the energy is very near the receiver, we could be fairly confident that *c* = *c*relative to receiver, again consistent with Einstein's Second Postulate. But if we would focus attention to moments when the bulk of the energy is still very near the source, moments when the receiver is nothing more than a distant phantom, we would be hard pressed to argue against the proposition that *c* = *c*relative to source. (This would be consis‐ tent with the 1908 Ritz Proposal [8], which was much investigated in the early to mid 20th century as a candidate alternative to Einstein's Second Postulate, but was ultimately rejected.) If we would need to characterize an entire propagation scenario, we would have to go even further, and consider *all* moments along the way. We would have neither the Einstein Second Postulate, nor the Ritz Proposal, but rather something else more complicated. It certainly must conform to Einstein's Second Postulate late in the scenario, when the bulk of the energy is near the receiver. But early in the scenario, when the bulk of the energy is still near the source, it must conform to Ritz's Proposal. And in between, it must represent in some mathematically appropriate way an idea not previously considered: a transition from one reference for *c* to the

Here is one way to formulate the problem. Let variable *x* represent distance along the propa‐ gation path, and let variable *t* represent time into the propagation process. At any point *x*, *t* there are fields **E** (*x*, *t*) and **B** (*x*, *t*) with magnitudes *E*(*x*, *t*) and *B*(*x*, *t*), and from them a local

One way to characterize the propagation is with path integrals. [9-13] Consider the following

This ratio provides a function that begins at the source, and ends at the receiver, and at the temporal midpoint of the scenario, gives equal weight to both the source and the receiver. This behavior captures the proposed transition of the reference to which the light speed *c* is assumed

ë û (2)

*r t xS x t dx S x t dx* <sup>=</sup> ò ò (3)

2 2 *Sxt E xt B xt* ( , ) ( , ) ( , ) 2. = + é ù

() ( ,) ( ,) . path path

meant to correct.

42 Selected Topics in Applications of Quantum Mechanics

would have to say more.

other.

energy density:

relative.

ratio of two path integrals:

$$\Phi(\mathbf{r},t) = e\left[\mathbf{1} / \kappa \mathbf{R}\right]\_{\text{retarded}} \text{ and } \mathbf{A}(\mathbf{r},t) = e\left[\mathbf{B} / \kappa \mathbf{R}\right]\_{\text{retarded}}\tag{4}$$

"where *κ* =1− **n⋅β**, **β** is source velocity normalized by light speed *c*, and **n**=**R**/*R* (a unit vector), and **R**=**r**source (*t-R/c*)-**r**receiver (*t*) (an implicit definition for the terminology 'retarded').

"The LW fields obtained from those potentials are then:

$$\begin{aligned} \mathbf{E}(\mathbf{r},t) &= e \left\{ \frac{1}{\kappa^3 R^2} (\mathbf{n} - \mathbf{\not\!\mathbf{B}}) (1 - \beta^2) + \frac{1}{c\kappa^3 R} \mathbf{n} \times \left[ (\mathbf{n} - \mathbf{\not\!\mathbf{B}}) \times (d\mathbf{\not\!\mathbf{B}} / \, dt) \right] \right\}\_{\text{retarded}}, \\ \mathbf{B}(\mathbf{r},t) &= \mathbf{n}\_{\text{retarded}} \times \mathbf{E}(\mathbf{r},t). \end{aligned} \tag{5}$$

"The 1 / *R* fields are radiation fields, and they make a Poynting vector (energy flow per unit area per unit time) that lies along **n**retarded:

$$\begin{split} \mathbf{P} &= \frac{c}{4\pi} \mathbf{E}\_{\text{radiative}} \times \mathbf{B}\_{\text{radiative}}\\ &= \frac{c}{4\pi} \mathbf{E}\_{\text{radiative}} \times \left(\mathbf{n}\_{\text{retarded}} \times \mathbf{E}\_{\text{radiative}}\right) = \frac{c}{4\pi} \left(E\_{\text{radiative}}\right)^2 \mathbf{n}\_{\text{retarded}} \quad . \end{split} \tag{6}$$

"The 1 / *R* <sup>2</sup> fields are Coulomb-Ampère fields, and the Coulomb field

$$\mathbf{E}(\mathbf{r},t) = e \left\langle (\mathbf{n} - \mathbf{\beta})(1 - \beta^2) \Big/ \kappa^3 R^2 \right\rangle\_{\text{retarded}}.\tag{7}$$

"does *not* lie along **n**retarded as one might initially expect; instead, it lies along (**n**-**β**)retarded. Assume that **β** does not change much over the total field propagation time, in which case (**n**-**β**)retarded is virtually indistinguishable from **n**present."

Thus the Coulomb attraction/repulsion and the radiation Poynting vector have distinctly different directions. This result does *not* look right physically. It looks as though advance information is being provided on one, but not the other, of two information channels.

Now consider the same problem using the new and more nuanced definition for the light speed reference. Observe that a line that connects the source position at the temporal midpoint of the scenario to the receiver position at the temporal midpoint of the scenario defines both the distance and the direction that the energy must travel in order to achieve the proposed transmission of the signal from the source to the receiver. This specification implies that we need potentials and fields to be, not *retarded*, but **half***-retarded*. Now the potentials become:

$$\Phi(\mathbf{r},t) = e\left[\mathbf{l} / \kappa \mathbf{R}\right]\_{\text{half \text{\textquotedblleft}t\text{-treated}}} \text{ and } \mathbf{A}(\mathbf{r},t) = e\left[\mathbf{B} / \kappa \mathbf{R}\right]\_{\text{half \textquotedblright}t\text{-treated}}.\tag{8}$$

The fields become:

$$\mathbf{E}(\mathbf{r},t) = e \left\{ \frac{(\mathbf{n}-\mathbf{\dot{\beta}})(1-\beta^2)}{\kappa^3 R^2} + \frac{\mathbf{n}}{c\kappa^3 R} \times \left[ (\mathbf{n}-\mathbf{\dot{\beta}}) \times d\mathbf{\dot{\beta}} \wedge dt \right] \right\}\_{\text{half- retarded}} \quad \text{and} \ \mathbf{B}(\mathbf{r},t) = \mathbf{n}\_{\text{half- retarded}} \times \mathbf{E}(\mathbf{r},t). \tag{9}$$

The Poynting vector becomes:

$$\begin{split} \mathbf{P} &= \frac{c}{4\pi} \mathbf{E}\_{\text{radiative}} \times \mathbf{B}\_{\text{radiative}}\\ &= \frac{c}{4\pi} \mathbf{E}\_{\text{radiative}} \times \left( \mathbf{n}\_{\text{half- retarded}} \times \mathbf{E}\_{\text{radiative}} \right) = \frac{c}{4\pi} (E\_{\text{radiative}})^2 \mathbf{n}\_{\text{half- retarded}}. \end{split} \tag{10}$$

The direction of the Coulomb field becomes:

$$(\mathbf{n} - \mathbf{j})\_{\text{half } - \text{rotarded}} \approx (\mathbf{n}\_{\text{present}})\_{\text{half } - \text{rotarded}} \triangleq \mathbf{n}\_{\text{half } - \text{rotated}}.\tag{11}$$

This direction is the same as the direction of the radiation Poynting vector. That is, the Coulomb field and the Poynting vector are now reconciled to the *same* direction, instead of conflicting with each other.

### **4. The corrected force direction means the hydrogen atom can survive classically**

{ } 2 32 ( , ) ( )(1 ) . retarded **Er n** *t e* = -- <sup>b</sup> b k

virtually indistinguishable from **n**present."

44 Selected Topics in Applications of Quantum Mechanics

The fields become:

with each other.

"does *not* lie along **n**retarded as one might initially expect; instead, it lies along (**n**-**β**)retarded. Assume that **β** does not change much over the total field propagation time, in which case (**n**-**β**)retarded is

Thus the Coulomb attraction/repulsion and the radiation Poynting vector have distinctly different directions. This result does *not* look right physically. It looks as though advance

Now consider the same problem using the new and more nuanced definition for the light speed reference. Observe that a line that connects the source position at the temporal midpoint of the scenario to the receiver position at the temporal midpoint of the scenario defines both the distance and the direction that the energy must travel in order to achieve the proposed transmission of the signal from the source to the receiver. This specification implies that we need potentials and fields to be, not *retarded*, but **half***-retarded*. Now the potentials become:

information is being provided on one, but not the other, of two information channels.

[ ] [ ] ( , ) 1/ (,) / . retarded retarded *te R* and *te R* - - F =

( )(1 ) (,) ()/ (,) ( , ). retarded

ì ü - - <sup>=</sup> í ý + ´ -´ = ´

*t e d dt* and *t t R cR* -

( ) <sup>2</sup>

radiative retarded radiative radiative retarded

This direction is the same as the direction of the radiation Poynting vector. That is, the Coulomb field and the Poynting vector are now reconciled to the *same* direction, instead of conflicting


 p**half half**

() ( ) . **nn n** - » b **half** - -- retarded present retarded **half** @ **half** retarded (11)

() . 4 4

**En E n**

retarded


 **half half n n E r <sup>n</sup> Br n Er** <sup>b</sup> <sup>b</sup> <sup>b</sup> (9)

 k= **half half r A r** b (8)

k

3 2 3

b

k

The Poynting vector becomes:

4

p

*c*

p

= ´

The direction of the Coulomb field becomes:

**PE B**

[ ] <sup>2</sup>

*c c*

= ´ ´=

î þ

 k

radiative radiative

*R* (7)

(10)

Many authors have expressed the opinion that really explaining the Hydrogen atom requires some presently-unknown short-range repulsive force between the electron and the proton; see, for example, Lokajicek, *et al* [16]. But given the results just presented, no mysterious new repulsive force is needed. With the direction of the Coulomb field being **nhalf**-retarded, there is a tiny tangential component of Coulomb force aligned with the orbit velocity. So there is a torque on the atom, and the torque pumps energy into the atom, and that process can work to balance the energy loss due to radiation.

That is to say: having a more nearly correct model for potentials and fields created by rapidly moving charges makes it possible to *explain* the immortality of the Hydrogen atom without first *postulating* the immortality of the Hydrogen atom; *i.e*., postulating Schrödinger's equation.

That is to say: we need not postulate Schrödinger's equation; w can instead just carry out the old-fashioned math.

In my 2012 and 2013 Intech papers, I listed just the pertinent results. Here is more detail and derivation:

Let the masses of the electron and the proton be *m*e and *m*p. Note that *m*e< <*m*p, but *m*p is *not* infinite.

Let the orbit radii of the electron and the proton be *r*e and *r*p. Note that *r*<sup>p</sup> < <*r*e, but *r*p is *not* zero.

Let the charges on the electron and the proton be −*e* and +*e*.

The magnitude of the nominally attractive force within the atom is *<sup>F</sup>* <sup>=</sup>*<sup>e</sup>* <sup>2</sup> / (*r*<sup>e</sup> <sup>+</sup> *<sup>r</sup>*p)2.

Let the orbit frequency be *Ω*. The orbit speed of the electron is *v*e=*r*e*Ω* and that of the proton is *v*<sup>p</sup> =*r*p*Ω*.

The magnitude of the tiny tangential force on the electron is *F*e= *F v*<sup>p</sup> / 2*c* = *F r*p*Ω* / 2*c*.

The magnitude of the tiny tangential force on the proton is *F*<sup>p</sup> = *Fv*<sup>e</sup> / 2*c* = *F r*e*Ω* / 2*c*.

The magnitudes of the torques on the electron and on the proton are *T*e=*r*e*F*<sup>e</sup> and *T*<sup>p</sup> =*r*p*F*p, both equal to *F r*e*r*p*Ω* / 2*c*.

The total torque on the electron-proton system is *T*total=*T*<sup>e</sup> <sup>+</sup> *<sup>T</sup>*<sup>p</sup> =2*<sup>T</sup>* <sup>=</sup> *<sup>F</sup> <sup>r</sup>*e*r*p*<sup>Ω</sup>* <sup>2</sup> / *<sup>c</sup>*.

The torque power delivered to the system is *P*torque=*T<sup>Ω</sup>* <sup>=</sup> *<sup>F</sup> rerp<sup>Ω</sup>* <sup>2</sup> / *<sup>c</sup>*.

The squared orbit frequency is determined from either *<sup>F</sup>* <sup>=</sup>*m*e*r*e*<sup>Ω</sup>* <sup>2</sup> or *<sup>F</sup>* <sup>=</sup>*m*p*r*p*<sup>Ω</sup>* <sup>2</sup> .

The more convenient of the two options is *<sup>Ω</sup>* <sup>2</sup> <sup>=</sup> *<sup>F</sup>* / *<sup>m</sup>*e*r*e. With that expression, the approxima‐ tion *r*e≈*r*<sup>e</sup> + *r*p yields:

$$P\_{\text{torque}} = F r\_{\text{e}} r\_{\text{p}} \Omega^2 \int \mathbf{c} = F^2 r\_{\text{p}} \Big/ m\_{\text{e}} \mathbf{c} = \left( e^4 r\_{\text{p}} / m\_{\text{e}} \mathbf{c} \right) \Big/ \left( r\_{\text{e}} + r\_{\text{p}} \right)^4 \approx \left( e^4 \Big/ m\_{\text{p}} \mathbf{c} \right) \Big/ \left( r\_{\text{e}} + r\_{\text{p}} \right)^3. \tag{12}$$

This is a reasonably simple expression. But more important than its simplicity is its very *existence*. The existence of any such expression means that there exists an energy *gain* mecha‐ nism to balance against the known energy loss mechanism, *i.e*. radiation. This situation provides a chance for balance, allowing the Hydrogen atom to avoid death by energy loss to radiation.

The *P*torque is actually quite large, and so it changes the whole emphasis of worry concerning the Hydrogen atom. The question becomes, not why does the Hydrogen atom not radiate and collapse to death, but rather why does the Hydrogen atom not torque itself up and ex‐ pand way beyond its known size?

The fact is: there exists much, much more radiation than was previously worried about, and it is enough to produce the proper balance between radiation and torque.

In [17], I said that the extra radiation arises from finite signal propagation speed, which results in circular motion of the center of mass of the Hydrogen atom, which in turn produces Thomas rotation, and thereby scales up by a factor of 2 the overall rotation rate generating the radiation, which increases the radiation power by a factor of 2<sup>4</sup> .

But what should one say about that center-of-mass circular motion? In Newtonian physics, where signal propagation speed is infinite, there is no such thing. In Maxwell physics, the emphasis is on fields, and the responses of individual charges, but not as much on the responses of whole charge systems, such as atoms. So the issue doesn't come up there. In Einstein's relativity physics, the emphasis is often on the observers of events more than on the events themselves. System center-of-mass circulation seems not to come up, although Thomas rotation does.

In [17], I noted that Thomas rotation is generally believed to be a result of the properties of Lorentz transformations, and hence of SRT. That is the belief because one can think of Lorentz transformations, not only in the usual, passive sense, as conversion from an observer in one inertial coordinate frame to another observer in another inertial coordinate frame, but also in the active sense, as the application of a 'boost' in velocity, the result of an acceleration, the result of a physical force. A series of non-co-linear boosts does indeed produce Thomas rotation.

But in [17] I also remarked that Thomas rotation does arise, not just from Lorentz transforma‐ tions, but also from Galilean Transformations. That fact can be demonstrated in detail as follows:

For simplicity, let all motion be in the *x*, *y* plane. The scenario begins at time coordinate *ct*<sup>0</sup> with one of the particles, say the electron, at rest at spatial coordinates *x*0, *y*0. Let an attraction from another particle act in the *x* direction. Let an increment of velocity Δ**V***<sup>x</sup>* =*ΔV* be imposed, and let an increment of time *Δt* elapse. The coordinates of the electron then become:

$$\mathbf{x}\_{1} = \mathbf{c}(t\_{0} + \Delta t), \mathbf{x}\_{1} = \mathbf{x}\_{0} + \Delta V \Delta t \text{ and } \mathbf{y}\_{1} = \mathbf{y}\_{0}. \tag{13}$$

Now let an attraction from another particle act in the *y* direction. Let an increment of velocity Δ**V***<sup>y</sup>* =*ΔV* be imposed, and let another increment of time *Δt* elapse. The coordinates of the electron then become:

( ) ( ) ( ) ( ) 22 4 4 4 <sup>3</sup> . *P Fr r c F r m c e r m c r r e m c r r* torque e p = W= = + » + pe pe ep p ep (12)

This is a reasonably simple expression. But more important than its simplicity is its very *existence*. The existence of any such expression means that there exists an energy *gain* mecha‐ nism to balance against the known energy loss mechanism, *i.e*. radiation. This situation provides a chance for balance, allowing the Hydrogen atom to avoid death by energy loss to

The *P*torque is actually quite large, and so it changes the whole emphasis of worry concerning the Hydrogen atom. The question becomes, not why does the Hydrogen atom not radiate and collapse to death, but rather why does the Hydrogen atom not torque itself up and ex‐

The fact is: there exists much, much more radiation than was previously worried about, and

In [17], I said that the extra radiation arises from finite signal propagation speed, which results in circular motion of the center of mass of the Hydrogen atom, which in turn produces Thomas rotation, and thereby scales up by a factor of 2 the overall rotation rate generating the radiation,

But what should one say about that center-of-mass circular motion? In Newtonian physics, where signal propagation speed is infinite, there is no such thing. In Maxwell physics, the emphasis is on fields, and the responses of individual charges, but not as much on the responses of whole charge systems, such as atoms. So the issue doesn't come up there. In Einstein's relativity physics, the emphasis is often on the observers of events more than on the events themselves. System center-of-mass circulation seems not to come up, although Thomas

In [17], I noted that Thomas rotation is generally believed to be a result of the properties of Lorentz transformations, and hence of SRT. That is the belief because one can think of Lorentz transformations, not only in the usual, passive sense, as conversion from an observer in one inertial coordinate frame to another observer in another inertial coordinate frame, but also in the active sense, as the application of a 'boost' in velocity, the result of an acceleration, the result of a physical force. A series of non-co-linear boosts does indeed produce Thomas

But in [17] I also remarked that Thomas rotation does arise, not just from Lorentz transforma‐ tions, but also from Galilean Transformations. That fact can be demonstrated in detail as

For simplicity, let all motion be in the *x*, *y* plane. The scenario begins at time coordinate *ct*<sup>0</sup> with one of the particles, say the electron, at rest at spatial coordinates *x*0, *y*0. Let an attraction from another particle act in the *x* direction. Let an increment of velocity Δ**V***<sup>x</sup>* =*ΔV* be imposed,

and let an increment of time *Δt* elapse. The coordinates of the electron then become:

.

it is enough to produce the proper balance between radiation and torque.

which increases the radiation power by a factor of 2<sup>4</sup>

radiation.

rotation does.

rotation.

follows:

pand way beyond its known size?

46 Selected Topics in Applications of Quantum Mechanics

$$\begin{aligned} t\_2 &= c(t\_0 + \Delta t + \Delta t), \\ \mathbf{x}\_2 &= \mathbf{x}\_0 + \Delta V(\Delta t + \Delta t) = \mathbf{x}\_0 + 2\Delta V \Delta t, \\ \text{and } y\_2 &= y\_0 + \Delta V \Delta t. \end{aligned} \tag{14}$$

Observe that, if the Galilean velocity boosts had been applied in the opposite order, then the ending coordinates of the electron would have been:

$$\begin{aligned} t\_2 &= c(t\_0 + \Delta t + \Delta t), \\ \mathbf{x}\_2 &= \mathbf{x}\_0 + \Delta V \Delta t, \\ \text{and } y\_2 &= y\_0 + 2\Delta V \Delta t. \end{aligned} \tag{15}$$

Observe that: the squared incremental length changes have the same magnitude either way: (2*ΔVΔt*) <sup>2</sup> + (*ΔVΔt*) <sup>2</sup> ≡ (*ΔVΔt*) <sup>2</sup> + (2*ΔVΔt*) <sup>2</sup> =5(*ΔVΔt*) 2 . That fact means the two possible sequences of Galilean boost applications differ only by a rotation. That means each one individually contains a rotation equal to half that total angle difference. This is the Thomas rotation.

Let me now go further, and assert that Thomas rotation will arise from *any* kind of velocity transformation – Lorentz, or Galilean, or any other new kind that may not have a name yet. Thomas rotation is a property of actual reality, not of any particular mathematical model for reality.

With the Thomas rotation included, the total radiation from the atomic system is:

$$P\_{\text{total radiated}} = \underline{\mathfrak{L}}^4 \frac{\mathfrak{L}e^2}{3c^3} a\_\text{e}^2 = \left(\underline{\mathfrak{L}}^5 \underline{e}^6 \;/\, m\_\text{e}^2\right) \Big/ \mathfrak{G} c^3 \left(r\_\text{e} + r\_\text{p}\right)^4. \tag{16}$$

The value of the separation *r*<sup>e</sup> + *r*p for which *P*total radiated =*P*torque is:

$$r\_{\rm e} + r\_{\rm p} = 32 m\_{\rm p} e^2 \int 3 m\_{\rm e}^2 c^2 = 5.5 \times 10^{-9} \,\text{cm}.\tag{17}$$

In the traditional approach to QM, *r*<sup>e</sup> <sup>+</sup> *<sup>r</sup>*<sup>p</sup> <sup>=</sup>*<sup>h</sup>* <sup>2</sup> / <sup>4</sup>*<sup>π</sup>* <sup>2</sup> *μe* <sup>2</sup> , where *μ* is the reduced mass, defined by *<sup>μ</sup>* <sup>−</sup><sup>1</sup> <sup>=</sup>*m*<sup>e</sup> <sup>−</sup><sup>1</sup> + *m*<sup>p</sup> <sup>−</sup>1, and very nearly equal to *m*e, and *h* is Planck's constant, 6.626176×10<sup>−</sup><sup>34</sup> Joulesec, a fundamental constant given by Nature.

The present analysis does not require Planck's constant as an input. Instead, it provides an estimate of Planck's constant as an *output*:

$$\begin{split} \hbar \hbar &= \sqrt{4\pi^2 \mu e^2 (r\_e + r\_p)} \approx \sqrt{4\pi^2 m\_e e^2 \Im 2m\_p e^2 \int 3m\_e^2 c^2} \\ &\approx \frac{\pi e^2}{c} \sqrt{128 m\_p / \Im m\_e} \approx 6.77 \times 10^{-34} \text{ Joule-sec}} \end{split} \tag{18}$$

and, for convenience, also an estimate of the often-seen reduced Planck's constant:

$$h = h / 2\,\pi \approx (e^2 / c) \sqrt{32m\_p / 3m\_o} \approx 1.08 \times 10^{34} \,\text{Joule-sec.} \tag{19}$$

These estimates can be improved by taking due account of the fact that the sines of small angles are not exactly equal to the angles themselves, and the cosines of small angles are not exactly equal to unity. The sine corrections are third order in angle, while the cosine corrections are second order in angle, which is more significant. Including those corrections reduces the radiation power slightly, and so reduces the solution *r*<sup>e</sup> + *r*p slightly, and so reduces the estimates of *h* and ℏ slightly – a step in the right direction.

Observe that, in making *h* an output *from*, rather than an input *to*, a theory, the present work follows both Enders [18] and Ralston [19].

#### **5. EM interactions between neutral atoms: A candidate model for gravity?**

Our current best understanding of gravity comes from GRT, which is founded in, and developed from, SRT. The parameter *c* from SRT appears in GRT, and reveals the lineage. Like electromagnetic signals, gravitational signals must have the finite propagation speed *c*. So might gravitational signals actually be electromagnetic signals? If we choose to update SRT with a more realistic signal model in place of the number *c*, does that require a similar update for GRT? This Section explores such questions.

First let it be noted that the Einstein gravitational field equations, like Maxwell's coupled field equations, are a description at the microscopic scale, but the observable phenomena exist at an extremely macroscopic scale.

Figure 2 illustrates a fairly typical barred spiral galaxy. What this image seems to suggest is: there are mass concentrations at the two armpits of this galaxy. Maybe they are mega stars, or black holes. Maybe they orbit another mass concentration at the center of the galaxy, or maybe they just orbit each other. In any event, the two mass concentrations together create a rather structured field of gravitational potential, into which millions, or billions, of smaller stars are entrained, or temporarily detained, as they orbit the galaxy.

*11*

Figure 3 shows the skeleton of a potential field created by two super-massive bodies orbiting at half the signal propagation speed. The lines mark minima in gravitational potential as a function of angle around the galaxy. Observe that this skeleton approximately matches this galaxy image.

**Figure 2.** A typical barred spiral galaxy: "A Barred Spiral Galaxy ngc 1300 hubble photo".

The present analysis does not require Planck's constant as an input. Instead, it provides an

2 2 2 2 2 22

These estimates can be improved by taking due account of the fact that the sines of small angles are not exactly equal to the angles themselves, and the cosines of small angles are not exactly equal to unity. The sine corrections are third order in angle, while the cosine corrections are second order in angle, which is more significant. Including those corrections reduces the radiation power slightly, and so reduces the solution *r*<sup>e</sup> + *r*p slightly, and so reduces the

Observe that, in making *h* an output *from*, rather than an input *to*, a theory, the present work

**5. EM interactions between neutral atoms: A candidate model for gravity?**

Our current best understanding of gravity comes from GRT, which is founded in, and developed from, SRT. The parameter *c* from SRT appears in GRT, and reveals the lineage. Like electromagnetic signals, gravitational signals must have the finite propagation speed *c*. So might gravitational signals actually be electromagnetic signals? If we choose to update SRT with a more realistic signal model in place of the number *c*, does that require a similar update

First let it be noted that the Einstein gravitational field equations, like Maxwell's coupled field equations, are a description at the microscopic scale, but the observable phenomena exist at

Figure 2 illustrates a fairly typical barred spiral galaxy. What this image seems to suggest is: there are mass concentrations at the two armpits of this galaxy. Maybe they are mega stars, or black holes. Maybe they orbit another mass concentration at the center of the galaxy, or maybe they just orbit each other. In any event, the two mass concentrations together create a rather structured field of gravitational potential, into which millions, or billions, of smaller stars are

p

e p epe

p e Joule-sec .;

4 ( ) 4 32 3

*h e r r me me mc*

and, for convenience, also an estimate of the often-seen reduced Planck's constant:

<sup>2</sup> <sup>34</sup> / 2 ( / ) 32 / 3 1.08 10 p e h = » *h ec m m*

128 / 3 6.77 10

*m m*

» » ´

= +»

34

(18)

» ´ Joule-sec. (19)


estimate of Planck's constant as an *output*:

48 Selected Topics in Applications of Quantum Mechanics

2

p m

*e*

p

*c*

p

estimates of *h* and ℏ slightly – a step in the right direction.

follows both Enders [18] and Ralston [19].

for GRT? This Section explores such questions.

entrained, or temporarily detained, as they orbit the galaxy.

an extremely macroscopic scale.

Note 1: To persist over time, the orbit speed of the driving two-body system in Figs. 2 and 3 must be such that the rates of energy loss by gravitational radiation and energy gain by torquing balance each **Figure 3.** The skeleton for a barred spiral disc galaxy. Originally computed for [20].

detail is suggested in Fig. 2.

does.

locity **v** :

Note 2: Since most of the captive stars in Fig 2 orbit at lesser speed than the driving two bodies, and their potential pattern, with its skeleton, Fig. 3, the individual stars in the outer reaches do not keep up with the rotating spiral potential pattern. An individual star sees a recurring 'density wave' of neighbor stars first approaching, then receding. Such density waves have long been known, but not well explained. Note 3: Fig. 3 is constructed using brute-force calculation of potential for lots and lots of location points, and then using numerical search over angles for local minimum values, and then fitting a function to the minima. Note the slightly sinuous bar between the two driving bodies. Even this Note 1: To persist over time, the orbit speed of the driving two-body system in Figs. 2 and 3 must be such that the rates of energy loss by gravitational radiation and energy gain by torquing balance each other. Like the Hydrogen atom, the galactic-size two-body problem can be solved this way.

**Figure 3.** The skeleton for a barred spiral disc galaxy. Originally computed for [20].

other. Like the Hydrogen atom, the galactic-size two-body problem can be solved this way.

Now, in order to model gravity in terms of electromagnetic interactions, we need an expression of electromagnetic interaction that is appropriate for neutral atoms. The best expression appears to be one that was known even *before* Maxwell. André Marie Ampère already had a well-developed theory about forces between what he called 'current elements'. This term referred to charge-neutral material increments in electrical circuits. In modern times, P. Graneau wrote extensively about Ampère's theory and experiments; see for example Graneau [21] Ampère's theory works perfectly well for ordinary closed circuits, as well as for incomplete broken circuits, such as may exist momentarily in transient situations, like explosive rupture of circuits. Ampère's theory ought not be forgotten solely on the basis that more modern theory also works per-Note 2: Since most of the captive stars in Fig 2 orbit at lesser speed than the driving two bodies, and their potential pattern, with its skeleton, Fig. 3, the individual stars in the outer reaches do not keep up with the rotating spiral potential pattern. An individual star sees a recurring 'density wave' of neighbor stars first approaching, then receding. Such density waves have long been known, but not well explained.

> fectly well for closed and stable electrical circuits. Indeed, in some technological applications involving transient situations like ruptures, Ampère's theory explains more than the modern theory

> One reason why Ampère's formulation can sometimes be more powerful than a formulation based on Maxwell's equations is that Ampère's formulation can describe a multi-participant scenario straight away, whereas Maxwell's equations require iteration through successive steps involving both Maxwell's equations, and the Lorentz force law for the force **F** acting on a charge *q* moving with ve

Note 3: Fig. 3 is constructed using brute-force calculation of potential for lots and lots of location points, and then using numerical search over angles for local minimum values, and then fitting a function to the minima. Note the slightly sinuous bar between the two driving bodies. Even this detail is suggested in Fig. 2.

Now, in order to model gravity in terms of electromagnetic interactions, we need an expres‐ sion of electromagnetic interaction that is appropriate for neutral atoms. The best expression appears to be one that was known even *before* Maxwell. André Marie Ampère already had a well-developed theory about forces between what he called 'current elements'. This term referredtocharge-neutralmaterialincrements inelectrical circuits.Inmoderntimes,P.Graneau wrote extensively about Ampère's theory and experiments; see for example Graneau [21]

Ampère's theory works perfectly well for ordinary closed circuits, as well as for incomplete broken circuits, such as may exist momentarily in transient situations, like explosive rupture of circuits. Ampère's theory ought not be forgotten solely on the basis that more modern theory also works perfectly well for closed and stable electrical circuits. Indeed, in some technological applications involving transient situations like ruptures, Ampère's theory explains more than the modern theory does.

One reason why Ampère's formulation can sometimes be more powerful than a formulation based on Maxwell's equations is that Ampère's formulation can describe a multi-participant scenario straight away, whereas Maxwell's equations require iteration through successive steps involving both Maxwell's equations, and the Lorentz force law for the force **F** acting on a charge *q* moving with velocity **v**:

$$\mathbf{F} = q \left[ \mathbf{E} + \mathbf{v} \times \mathbf{B} \right]. \tag{20}$$

The iteration goes as follows: first, the input charge positions and motions generate **E** and **B** fields; second, the Lorentz force law tells how each charge *q* responds to the fields from the other charges; another step through Maxwell's equations tells how all the fields change, and so on.

One particular scenario illustrates the difference between the approaches especially well. Consider a current-carrying wire. It is tedious to use the Maxwell-Lorentz-Maxwell-Lorentz iterative approach to arrive at the understanding that the moving electrons favor the surface of the wire, or even the exterior neighborhood near the wire, leaving the interior of the wire depleted of electrons, and therefore in a state of internal repulsion between the remaining positive nuclei. The Ampère's formulation skips over all this detail, and describes the resulting consequence: the wire experiences internal longitudinal force, and in fact might even rupture. If it does rupture, one can easily tell that the event was *not* due to ordinary resistive heating and melting, since the fragments are found to be neither hot to touch nor melted in appearance.

The Ampère approach looks promising for gravity problems because *any* gravity problem is definitely a multi-participant scenario. And for the same reason, the following analysis also invokes ideas from modern Statistical Mechanics.

Ampère's force formula can be written:

Note 3: Fig. 3 is constructed using brute-force calculation of potential for lots and lots of location points, and then using numerical search over angles for local minimum values, and then fitting a function to the minima. Note the slightly sinuous bar between the two driving bodies. Even

Now, in order to model gravity in terms of electromagnetic interactions, we need an expres‐ sion of electromagnetic interaction that is appropriate for neutral atoms. The best expression appears to be one that was known even *before* Maxwell. André Marie Ampère already had a well-developed theory about forces between what he called 'current elements'. This term referredtocharge-neutralmaterialincrements inelectrical circuits.Inmoderntimes,P.Graneau wrote extensively about Ampère's theory and experiments; see for example Graneau [21]

Ampère's theory works perfectly well for ordinary closed circuits, as well as for incomplete broken circuits, such as may exist momentarily in transient situations, like explosive rupture of circuits. Ampère's theory ought not be forgotten solely on the basis that more modern theory also works perfectly well for closed and stable electrical circuits. Indeed, in some technological applications involving transient situations like ruptures, Ampère's theory explains more than

One reason why Ampère's formulation can sometimes be more powerful than a formulation based on Maxwell's equations is that Ampère's formulation can describe a multi-participant scenario straight away, whereas Maxwell's equations require iteration through successive steps involving both Maxwell's equations, and the Lorentz force law for the force **F** acting on

The iteration goes as follows: first, the input charge positions and motions generate **E** and **B** fields; second, the Lorentz force law tells how each charge *q* responds to the fields from the other charges; another step through Maxwell's equations tells how all the fields change, and

One particular scenario illustrates the difference between the approaches especially well. Consider a current-carrying wire. It is tedious to use the Maxwell-Lorentz-Maxwell-Lorentz iterative approach to arrive at the understanding that the moving electrons favor the surface of the wire, or even the exterior neighborhood near the wire, leaving the interior of the wire depleted of electrons, and therefore in a state of internal repulsion between the remaining positive nuclei. The Ampère's formulation skips over all this detail, and describes the resulting consequence: the wire experiences internal longitudinal force, and in fact might even rupture. If it does rupture, one can easily tell that the event was *not* due to ordinary resistive heating and melting, since the fragments are found to be neither hot to touch nor melted in appearance.

The Ampère approach looks promising for gravity problems because *any* gravity problem is definitely a multi-participant scenario. And for the same reason, the following analysis also

**F E vB** = +´ *q*[ ]. (20)

this detail is suggested in Fig. 2.

50 Selected Topics in Applications of Quantum Mechanics

the modern theory does.

so on.

a charge *q* moving with velocity **v**:

invokes ideas from modern Statistical Mechanics.

$$
\Delta F\_{m,u} = +i\_m i\_u \left[ \Delta m \, \Delta n \left( \left( r\_{m,u} \right)^2 \right] \left( 3 \cos \alpha \cos \beta - 2 \cos \gamma \right) . \tag{21}
$$

The indices *m* and *n* identify two interacting currents. The *i <sup>m</sup>* and *i <sup>n</sup>* are current magnitudes. The *Δm* and *Δn* are magnitudes of tiny directed length increments Δ**m** and Δ**n** through which the currents flow. The products of currents and directed length increments, *i <sup>m</sup>* Δ**m** and *i <sup>n</sup>* Δ**m**, are the current elements. The *rm*,*<sup>n</sup>* is the length of the vector separation **r***m,n* between the current elements. The *α*, *β*, and *γ* are angles with respect to the connecting line between the two current elements, and with respect to each other. Current element *i <sup>m</sup>* Δ**m** is at angle *α* from the connecting line, and current element *i <sup>n</sup>* Δ**n** is at angle *β* from the connecting line. The *γ* is the angle between the two planes defined by the connecting line and each of the two current elements, as if the distance *rm*,*<sup>n</sup>* did not separate them. The value ranges are all full circle: 0<*α* <2*π*,

One can get a feel for the general behavior of Ampère's force formula by considering the angle factor 3cos*α*cos*β* −2cos*γ* for a few special cases:


The 1 / (*rm*,*n*)2 aspect of the Ampère force law is just like Newton's law for gravity. Ampère designed his law that way, because, in his time, the greatest prior achievement in Science was Newton's conquest of gravity. Now we wish to return the favor, and exploit the Ampère Force Law to understand something novel about gravity.

What makes the Ampère current element so potentially appropriate for application to gravity? First of all, it is charge-neutral, like the masses in a gravity scenario. Secondly, its electrons are moving, and although its nucleus is moving too, that motion is not anywhere near as fast. So at all times and all places where matter exists, at the microscopic level a net electron current flows.

The concept that current elements generate forces that can attract or repel each other suggest that pairs of current elements – or pairs of atoms – can be regarded as a system that can have positive or negative total energy. The kinetic part of the energy may be disregarded, since the current elements may be essentially static, but the potential part of the energy is worth paying attention to.

The main novel feature that gravity presents is that we usually have, not two current elements, but huge numbers of atoms, and each atom must have some relationship with *all* other atoms. The complexity of the situation naturally conjures up ideas from Statistical Mechanics. Here, ideas from Statistical Mechanics are applied to gravity described in terms of Ampère forces between atoms that are viewed as current elements. Some atom-to-atom relationships are momentarily attractive, and some relationships are momentarily repulsive, and all relation‐ ships must vary over time. We can look at the population of atom pairs as a whole, and think of it as a statistical ensemble, in which every condition of attraction/repulsion is represented somewhere.

Every area of physics that has statistical ensembles has Gaussian probability functions. In Classical Thermodynamics, a Gaussian probability density function for a random variable, such as a component of a particle momentum vector, implies maximum entropy, subject to a prescribed value for the standard deviation of that random variable. In Quantum Mechanics, a Gaussian probability density function (the squared wave function amplitude) is associated with minimum uncertainty, meaning minimum product of standard deviations in Fourier conjugate variables, like position and momentum.

Sometimes it is not immediately obvious that a problem involves the equivalent of a Gaussian function, because the Gaussian itself involves a squared variable, such as *x* <sup>2</sup> or *p* <sup>2</sup> . The squared variable is proportional to some energy *E*. So one sees probability density functions expressed in the form exp(− *E* / < *E* >) / <*E* > where <*E* > is the average value of energy *E*, usually some‐ thing like *kT* , where *k* is Boltzmann's consistent and *T* is absolute temperature.

In the case of gravity, that energy *E* of interest is gravitational potential energy. It can have both positive *and* negative values. The central concept in Statistical Mechanics is that lowerenergy states are populated more richly than higher-energy states are. This concept means that any two atoms, viewed as current elements, will be with respect to each other in a state of negative potential energy more often than in a state of positive potential energy. So they will, on average, attract each other more than repel each other. Therefore, <*E* > is negative. Since temperature cannot be negative, this is something novel.

To deal with negative <*E* > we really need a probability density function with a Gaussian factor of the form exp − *E* <sup>2</sup> / 2< *E* <sup>2</sup> > , where <*E* <sup>2</sup> > is another parameter, positive, but also not related to temperature.

A few examples can illustrate how to find the parameters.

Let the two energies be *E*max and *E*min = − | *E*max |. With the attractive–force, negative-energy state dominating the scenario, <*E* > is negative, <*E* > = − | < *E* > |. The two Boltzmann factors are:

$$\text{Exp}(+E\_{\text{max}} \mid \mid ) \\ \text{for the state with negative energy } E\_{\text{min}} = -E\_{\text{max}} \tag{22}$$

and

The concept that current elements generate forces that can attract or repel each other suggest that pairs of current elements – or pairs of atoms – can be regarded as a system that can have positive or negative total energy. The kinetic part of the energy may be disregarded, since the current elements may be essentially static, but the potential part of the energy is worth paying

The main novel feature that gravity presents is that we usually have, not two current elements, but huge numbers of atoms, and each atom must have some relationship with *all* other atoms. The complexity of the situation naturally conjures up ideas from Statistical Mechanics. Here, ideas from Statistical Mechanics are applied to gravity described in terms of Ampère forces between atoms that are viewed as current elements. Some atom-to-atom relationships are momentarily attractive, and some relationships are momentarily repulsive, and all relation‐ ships must vary over time. We can look at the population of atom pairs as a whole, and think of it as a statistical ensemble, in which every condition of attraction/repulsion is represented

Every area of physics that has statistical ensembles has Gaussian probability functions. In Classical Thermodynamics, a Gaussian probability density function for a random variable, such as a component of a particle momentum vector, implies maximum entropy, subject to a prescribed value for the standard deviation of that random variable. In Quantum Mechanics, a Gaussian probability density function (the squared wave function amplitude) is associated with minimum uncertainty, meaning minimum product of standard deviations in Fourier

Sometimes it is not immediately obvious that a problem involves the equivalent of a Gaussian

variable is proportional to some energy *E*. So one sees probability density functions expressed in the form exp(− *E* / < *E* >) / <*E* > where <*E* > is the average value of energy *E*, usually some‐

In the case of gravity, that energy *E* of interest is gravitational potential energy. It can have both positive *and* negative values. The central concept in Statistical Mechanics is that lowerenergy states are populated more richly than higher-energy states are. This concept means that any two atoms, viewed as current elements, will be with respect to each other in a state of negative potential energy more often than in a state of positive potential energy. So they will, on average, attract each other more than repel each other. Therefore, <*E* > is negative. Since

To deal with negative <*E* > we really need a probability density function with a Gaussian factor of the form exp − *E* <sup>2</sup> / 2< *E* <sup>2</sup> > , where <*E* <sup>2</sup> > is another parameter, positive, but also not related

Let the two energies be *E*max and *E*min = − | *E*max |. With the attractive–force, negative-energy state dominating the scenario, <*E* > is negative, <*E* > = − | < *E* > |. The two Boltzmann factors

or *p* <sup>2</sup>

. The squared

function, because the Gaussian itself involves a squared variable, such as *x* <sup>2</sup>

thing like *kT* , where *k* is Boltzmann's consistent and *T* is absolute temperature.

attention to.

52 Selected Topics in Applications of Quantum Mechanics

somewhere.

to temperature.

are:

conjugate variables, like position and momentum.

temperature cannot be negative, this is something novel.

A few examples can illustrate how to find the parameters.

$$\exp(-E\_{\text{max}} \mid ) \text{for the state with positive energy, } E\_{\text{max}}.\tag{23}$$

The average energy *E*avg must satisfy the definition:

$$E\_{\text{avg}} = \frac{-E\_{\text{max}} \exp(+E\_{\text{max}} \mid ) + E\_{\text{max}} \exp(-E\_{\text{max}} \mid )}{\exp(+E\_{\text{max}} \mid ) + \exp(-E\_{\text{max}} \mid )}. \tag{24}$$

Simple trial calculations and numerical search of the results works well enough to solve the problem at hand. The solution is approximately:

$$E\_{\text{avg}} \approx -E\_{\text{max}} / 1.2 = -0.8333 E\_{\text{max}} = -\frac{s}{6} E\_{\text{max}}.\tag{25}$$

(There is also, of course, a positive, and presently irrelevant, solution of the same magnitude.) For this case we have we have line integration in place of point evaluation. *E*avg becomes:

$$E\_{\rm avg} = -\int\_{-E\_{\rm max}}^{0} E \sinh(E/|\lhd E>) dE \Big/ \int\_{-E\_{\rm max}}^{0} \cosh(E/|\lhd E>) dE. \tag{26}$$

Because the denominator is simpler, begin with that. It is:

$$\begin{split} \left| \left| E\_{\text{avg}} \mid \sinh(E \mid E\_{\text{avg}} \mid) \right| \right|\_{-E\_{\text{max}}}^{0} &= - \left[ \left| \left| E\_{\text{avg}} \mid \sinh(-E\_{\text{max}} \mid \mid E\_{\text{avg}} \mid) \right| \right] \\ &= + \left| \left| E\_{\text{avg}} \mid \sinh(E\_{\text{max}} \mid \mid E\_{\text{avg}} \mid) \right| \right. \end{split} \tag{27}$$

This is a positive number.

The numerator is more complicated, but it can be evaluated using integration by parts:

$$\int\_{-E\_{\text{max}}}^{0} \mathbf{L}d\mathbf{V} = \mathbf{L}\mathbf{V} \Big|\_{-E\_{\text{max}}}^{0} - \int\_{-E\_{\text{max}}}^{0} \mathbf{V}d\mathbf{U} \quad . \tag{28}$$

where *U* =*E* and *dV* =sinh(*E* / | *E*avg |)*dE*, so that *V* = | *E*avg |cosh(*E* / | *E*avg |). The first term in the numerator evaluation is:

$$\left| LVV \right|\_{-E\_{\text{max}}}^0 = -LI(-E\_{\text{max}})V(-E\_{\text{max}}) = -\left[ -E\_{\text{max}} \mid  \left| \cosh(-E\_{\text{max}} \,/|\, \epsilon \to \!> \!> \!> \!> \!
$$= E\_{\text{max}} \left|  \left| \cosh(E\_{\text{max}} \,/|\, \epsilon \to \!> \!> \!> \!$$
$$

The second term in the numerator evaluation is:

.

$$\begin{split} -\int\_{-E\_{\text{max}}}^{0} V dL &= -\int\_{-E\_{\text{max}}}^{0} | |\cosh(E/)| dE = -|\,E\_{\text{avg}}|^2 \sinh(E/)\Big|\_{-E\_{\text{max}}}^{0} \\ &= ||^2 \sinh(-E\_{\text{max}}/|) = -||^2 \sinh(E\_{\text{max}}/|) \quad. \end{split} \tag{30}$$

The sought numerator divided by denominator for <*E* > is then:

$$\begin{split} \epsilon < E > & \frac{E\_{\text{max}} \left|  \right| \cosh(E\_{\text{max}} \left|  \right) - \left|  \right|)}{\left|  \right|)} + \left|  \right| \ . \end{split} \tag{31}$$

Since the sought <*E* > is negative, equal to − | <*E* > |, we have:

$$|\mathcal{Q} < E> = -\mathcal{Q}| < E> = \frac{-E\_{\text{max}}}{\tanh(E\_{\text{max}} / |)}. \tag{32}$$

Again, numerical search is a practical approach for finding a solution. We find:

$$
\omega < E > \approx -0.48 SE\_{\text{max}}.\tag{33}
$$

Observe that, as should be expected, this solution is significantly smaller in magnitude than was the solution with only two energy values, ±*E*max, to balance between, which came in at −0.833*E*max.

The continuous Gaussian profile,

$$\left| \exp(-E^2/2 < E^2 >) \right| \sqrt{2\pi} < E^2 >,\tag{34}$$

extends to infinity in both directions. This attribute is inappropriate for the problem at hand, which definitely possesses limits ±*E*max beyond which the modeling problem does not extend. Therefore, let us turn to discrete approximations for a Gaussian. These are based on the binomial expansion for an arbitrary (*a* + *b*) *<sup>n</sup>*. The binomial coefficients are familiar to many people from Pascal's famous triangle:

[ ] max

max max

2 2

The sought numerator divided by denominator for <*E* > is then:

max max

*E E*


= +< > < >

Since the sought <*E* > is negative, equal to − | <*E* > |, we have:

*<sup>E</sup> <sup>E</sup>*

 | | cosh( / | |) . *<sup>E</sup> UV U E V E E E E E EE E E* - =- - - =- - < > - < > = <> <>

max max max 0 0 <sup>0</sup> <sup>2</sup>

 | | sinh( / | |) | | sinh( / | |) . *E E* avg *<sup>E</sup> VdU E E E dE E E E E EE E EE*

max max max

2 2| | . tanh( / | |) *<sup>E</sup> E E*

Observe that, as should be expected, this solution is significantly smaller in magnitude than was the solution with only two energy values, ±*E*max, to balance between, which came in at

22 2 exp( / 2 ) 2 , - <> <> *EE E*

p

extends to infinity in both directions. This attribute is inappropriate for the problem at hand, which definitely possesses limits ±*E*max beyond which the modeling problem does not extend.


Again, numerical search is a practical approach for finding a solution. We find:

*EE E E E E E*

*E EE*

max

avg avg


< > < >- < >


max max


ò ò (30)

2

avg

max max

< > (32)

(34)

max < >» - *E E* 0.485 . (33)

*E E*

max max max max

( ) ( ) | | cosh( / | |)

(29)

(31)

0

54 Selected Topics in Applications of Quantum Mechanics

*E*

< >=

.

−0.833*E*max.

The continuous Gaussian profile,

The second term in the numerator evaluation is:

$$
\begin{array}{cccccc}
1\\ & 1 & 1\\ 1 & 2 & 1\\ 1 & 3 & 3 & 1\\ & & & & & & \end{array}
\tag{35}
$$

The numbers in Pascal's triangle are constructed with addition of neighboring numbers above. This is easy for small *n*, but small *n* means *crude*, and we need *refined*. So we want *large n*. So we need a formula involving multiplication instead of addition. That would be:

$$n!, n, \frac{n(n-1)}{1\times 2}, \frac{n(n-1)(n-2)}{3!}, \dots, \frac{(n)!}{(n')!(n-n')!}, \dots, \frac{n(n-1)(n-2)}{3!}, \frac{n(n-1)}{1\times 2}, n, 1. \tag{36}$$

Observe that the binomial coefficients are symmetric around the middle of the list, like a Gaussian function is symmetric around zero argument. If *n* is an even number, the number of binomial coefficients is 2*n* −1, odd, and the middle one, the maximum one, is *n* ! / (*n* / 2)! <sup>2</sup> . If *n* is an odd number, the number of binomial coefficients is 2*n*, an even number, and the middle two numbers, the maximum two, are both *n* ! / { (*n* + 1) / 2 ! (*n* −1) / 2 !}.

For our modeling problem, let *n* be an odd number. Let the binomial coefficients be represented as *Bb* with *b* = −*n* to *b* =1. Let them be associated with equal energy increments *ΔE* =*E*max / *n* starting from −*E*max and covering the range to zero. Associate the minimum binomial coeffi‐ cient *B*−*<sup>n</sup>* with the increment starting with −*E*max, and the maximal binomial coefficient with the increment ending with *E* =0, and associate the other coefficients with the increments between those limits. The problem to solve is:

$$ \approx \sum\_{b=-n}^{1} B\_b E\_b \sinh(E\_b/) \Big/ \sum\_{b=-n}^{1} B\_b \cosh(E\_b/). \tag{37}$$

Numerical investigations done to date suggest that the solution comes at approximately <*E* > = − *E*max / 2 *n*. Here the *n* for this discrete model is analogous to the standard deviation *σ* for the corresponding continuous Gaussian.

The problem of modeling gravity therefore reduces to the problem of determining what value of *n* should be used. Here is the most pertinent fact: compared to anything electromagnetic, gravity is extremely weak. Consider two Hydrogen atoms at a given separation distance. Let us compare the gravitation force with the maximum Ampère force between them.

The gravitational attraction is proportional to *G*(*m*p)2, where *G* is the universal gravitation constant, about 6.6×10<sup>−</sup><sup>11</sup> Newton× meter<sup>2</sup> per kilogram<sup>2</sup> , and *m*p is the mass of the proton, about 1.66×10<sup>−</sup>27kg, so (*m*p)2 is about 2.76×10<sup>−</sup><sup>54</sup> kg<sup>2</sup> . Overall,

$$\frac{1}{4\pi\varepsilon\_0}e^2(v/c)^2 \approx \frac{2.56 \times 10^{-38} \times 0.45 \times 10^{-4}}{1.113 \times 10^{-10}} \approx 1.035 \times 10^{-32} \,\text{Newton} \times \text{meter}^2.\tag{38}$$

Clearly, the maximum Ampère force between atoms viewed as current elements is generously larger than the gravitational force between atoms viewed as charge-neutral masses – by about 32 orders of magnitude!

But the typical Ampère force is nowhere near as big as the maximum Ampère force, due to the fact that it depends on three angles, any one of which can spoil it. Occurrence of the maximum Ampère force is very rare indeed. And occurrence of the *minimum* (negative) Ampère force is equally rare. Only the piddling near-zero Ampère forces are common, and even then, each tiny attractive force is mostly cancelled with the tiny repulsive force of equal magnitude but less frequent occurrence. All we have is a tiny residue of attractive force, due to the nature of Boltzmann factors and Statistical Mechanics.

Of course, being tiny does not mean being insignificant. Like the tiny residue that is microwave background radiation, the tiny residue that is gravity is a possible key to understanding something about the Universe in which we live. Here is an example problem: at present, we know the actual particle radius of the electron is something extremely tiny, but we do *not* know what its numerical value is. There exist a number of length-dimensioned quantities associated with the electron, all called 'radius', but distinguished by specific names and numerical values. MacGregor [22] lists seven of them. Most are on the order of 10<sup>−</sup><sup>13</sup> cm, although one is much smaller, and is presently only upper-bounded at <10<sup>−</sup>16 cm.

The radius attributed to the electron can have a role in the gravity problem. The ratio of an atomic radius to the electron radius can imply a candidate level of discretization for the binomial approximation to the Gaussian factor involved in the gravity problem. For an atomic radius, let us consider the first orbit radius of Hydrogen, *r*H1 =0.529×10−<sup>8</sup> cm. For the electron radius, let us consider two of the possibilities from MacGregor [22].

One of the electron radii is called *classical*. This one captures the Coulomb energy equivalence

$$\frac{1}{4\pi\varepsilon\_0}e^2 \mid r\_{\text{o-classical}} = m\_\text{e}c^2,\tag{39}$$

which implies

$$r\_{\text{e clascalal}} = \frac{1}{4\pi\varepsilon\_0} \left(e^2 \Big/ m\_\circ c^2\right) = 2.82 \times 10^{-13} \text{ cm}.\tag{40}$$

The ratio *ρ* =*r*H1 /*r*e classical is then:

$$
\ln n = 0.529 \times 10^{-8} \Big/ 2.82 \times 10^{-13} = 5.29 \times 10^{-9} \Big/ 2.82 \times 10^{-13} \approx 1.86 \times 10^{4},\tag{41}
$$

which implies

The gravitational attraction is proportional to *G*(*m*p)2, where *G* is the universal gravitation

2 2 32 2

´ ´´ » »´ ´

Clearly, the maximum Ampère force between atoms viewed as current elements is generously larger than the gravitational force between atoms viewed as charge-neutral masses – by about

But the typical Ampère force is nowhere near as big as the maximum Ampère force, due to the fact that it depends on three angles, any one of which can spoil it. Occurrence of the maximum Ampère force is very rare indeed. And occurrence of the *minimum* (negative) Ampère force is equally rare. Only the piddling near-zero Ampère forces are common, and even then, each tiny attractive force is mostly cancelled with the tiny repulsive force of equal magnitude but less frequent occurrence. All we have is a tiny residue of attractive force, due

Of course, being tiny does not mean being insignificant. Like the tiny residue that is microwave background radiation, the tiny residue that is gravity is a possible key to understanding something about the Universe in which we live. Here is an example problem: at present, we know the actual particle radius of the electron is something extremely tiny, but we do *not* know what its numerical value is. There exist a number of length-dimensioned quantities associated with the electron, all called 'radius', but distinguished by specific names and numerical values. MacGregor [22] lists seven of them. Most are on the order of 10<sup>−</sup><sup>13</sup> cm, although one is much

The radius attributed to the electron can have a role in the gravity problem. The ratio of an atomic radius to the electron radius can imply a candidate level of discretization for the binomial approximation to the Gaussian factor involved in the gravity problem. For an atomic

One of the electron radii is called *classical*. This one captures the Coulomb energy equivalence

2 2

( ) 2 2 <sup>13</sup>

<sup>1</sup> 2.82 10 <sup>4</sup> e classical <sup>e</sup> *<sup>r</sup> e mc* cm. - = =´

(39)

(40)

<sup>1</sup> / , <sup>4</sup> e classical e *e r mc* <sup>=</sup>

<sup>1</sup> 2.56 10 0.45 10 (/) 1.035 10 . <sup>4</sup> 1.113 10 *evc* Newton meter

38 4



to the nature of Boltzmann factors and Statistical Mechanics.

smaller, and is presently only upper-bounded at <10<sup>−</sup>16 cm.

radius, let us consider the first orbit radius of Hydrogen, *r*H1 =0.529×10−<sup>8</sup>

0

0

pe

pe

radius, let us consider two of the possibilities from MacGregor [22].

10

per kilogram<sup>2</sup>

. Overall,


´ (38)

, and *m*p is the mass of the proton, about

cm. For the electron

constant, about 6.6×10<sup>−</sup><sup>11</sup> Newton× meter<sup>2</sup>

56 Selected Topics in Applications of Quantum Mechanics

0

32 orders of magnitude!

which implies

pe

1.66×10<sup>−</sup>27kg, so (*m*p)2 is about 2.76×10<sup>−</sup><sup>54</sup> kg<sup>2</sup>

$$m = 0.529 \times 10^{-8} \not{2.82} \times 10^{-13} = 5.29 \times 10^{-9} \not{2.82} \times 10^{-13} \approx 1.86 \times 10^{4}.\tag{42}$$

The square root of this number would then be the dimensionless *n* for the discretization:

$$
\sqrt{n} = \sqrt{1.86 \times 10^4} \approx 1.37 \times 10^2 = 137.\tag{43}
$$

This is a number already famous in Physics, but in a context other than gravity. It is the inverse of the so-called 'fine structure constant' *α*, defined as:

$$
\alpha = 2\pi e^2 / \text{ch}.\tag{44}
$$

This number plays a role in spectroscopy, where spectral lines occur in families, closely spaced but clearly distinguishable. There, the explanation comes from QM; clearly, another manifes‐ tation of natural discretization.

But if the classical radius of the electron were used in the gravity problem, the ratio of the average Ampère force magnitude to the maximum Ampère force magnitude would be approximately

$$
\sqrt{n}/2n = 1/2\sqrt{n} \approx 3.65 \times 10^{-3}.\tag{45}
$$

This ratio is *not* appropriately small, so this is *not* the right discretization level for the gravity problem.

Another one of the electron radii given by MacGregor [22] is called *actual*. It characterizes results of scattering experiments, and is the one presently only upper-bounded, at *r*e actual <10−<sup>16</sup> cm. No one knows how much smaller it could eventually turn out to be. So how much smaller would it *have* to be, in order to account for the extreme weakness of gravity? Gravity requires 2 / *n* ≈10<sup>−</sup>32, or.

$$
\sqrt{\mathfrak{n}} \approx 2 \times 10^{\ast 32}, \text{ or } \mathfrak{n} \approx 4 \times 10^{64}. \tag{46}
$$

That in turn requires:

$$r\_{\text{e actual}} = r\_{\text{Hil}} / \rho = 0.529 \times 10^{-8} \,\text{cm} / 4 \times 10^{64} \approx 10^{-73} \,\text{cm}.\tag{47}$$

At present, such a value for *r*e actual certainly looks impossible to test with any kind of meas‐ urement. It is smaller than anything we yet know about any elementary particle. But that circumstance may be a good thing, because an extremely small electron makes it easier to understand what data from Chemistry reflects, discussed next. And an extremely small electron, along with a correspondingly small positron, helps explain aspects Elementary Particle Physics, discussed after that.

#### **6. Algebraic chemistry and EZ water**

Prof. Gerald Pollack of U. Washington wrote the authoritative book [23] about the physical phenomenon called 'EZ water'. The EZ is short for Exclusion Zone, with the word 'exclusion' referring to a surface phenomenon that expels positive hydronium ions.

Prof. Pollack gave a talk about EZ Water at the 2013 meeting of the Natural Philosophy Alliance at College Park, MD, USA. All the phenomena he described were surprising; some were truly puzzling. EZ water apparently makes extended orderly arrays of hexagonal units. How can *that* behavior comport with our understanding that Nature maximizes entropy? Explanations then available were not at all quantitative. That fact suggested a real need for a more quanti‐ tative approach.

I had recently written my book about Algebraic Chemistry (AC). [24] The name reflects the fact that the technique has no integrals or other complicated math operations that would demand capabilities beyond those of a hand calculator. The worst operation is square root. So the AC approach looked promising for quick application to EZ water.

The fundamental idea behind AC is that all atoms share some similarities with Hydrogen atoms: **1)** They have a nucleus that is similar to a proton, but scaled up to nuclear charge *Z* and nuclear mass *M* ; **2)** They have a population of electrons that is not entirely unlike a single electron; *i.e.,* an interacting community that is somewhat coherent, and somewhat like one *big* electron orbiting the nucleus; **3)** It is possible for the electron count to be different from the nuclear charge. This last possibility is what characterizes ions, and thereby creates all of Chemistry.

We begin with a clue: Eq. (17) indicates that the radius of the Hydrogen atom scales with the mass of the proton. This fact suggests that the base orbit energy of the Hydrogen atom scales with the inverse of proton mass. It further suggests that for element with nuclear charge *Z* and mass *M* , the base orbit energy may scale with *Z* / *M* . If so, then when first-order ionization potentials for all elements are scaled by the inverse factor, *M* /*Z*, then the scaled first-order ionization potentials (called *I P*1,*<sup>Z</sup>* ) might fall into some pattern.

We proceed with an observation: A pattern indeed emerges: the rise on *every* period in the Periodic Table is *exactly the same factor*, 7 / 2.

That in turn requires:

tative approach.

Chemistry.

Particle Physics, discussed after that.

58 Selected Topics in Applications of Quantum Mechanics

**6. Algebraic chemistry and EZ water**

8 64 73

(47)

<sup>1</sup> 0.529 10 4 10 10 e actual H *r r* cm cm. - - = = ´ ´»

At present, such a value for *r*e actual certainly looks impossible to test with any kind of meas‐ urement. It is smaller than anything we yet know about any elementary particle. But that circumstance may be a good thing, because an extremely small electron makes it easier to understand what data from Chemistry reflects, discussed next. And an extremely small electron, along with a correspondingly small positron, helps explain aspects Elementary

Prof. Gerald Pollack of U. Washington wrote the authoritative book [23] about the physical phenomenon called 'EZ water'. The EZ is short for Exclusion Zone, with the word 'exclusion'

Prof. Pollack gave a talk about EZ Water at the 2013 meeting of the Natural Philosophy Alliance at College Park, MD, USA. All the phenomena he described were surprising; some were truly puzzling. EZ water apparently makes extended orderly arrays of hexagonal units. How can *that* behavior comport with our understanding that Nature maximizes entropy? Explanations then available were not at all quantitative. That fact suggested a real need for a more quanti‐

I had recently written my book about Algebraic Chemistry (AC). [24] The name reflects the fact that the technique has no integrals or other complicated math operations that would demand capabilities beyond those of a hand calculator. The worst operation is square root. So

The fundamental idea behind AC is that all atoms share some similarities with Hydrogen atoms: **1)** They have a nucleus that is similar to a proton, but scaled up to nuclear charge *Z* and nuclear mass *M* ; **2)** They have a population of electrons that is not entirely unlike a single electron; *i.e.,* an interacting community that is somewhat coherent, and somewhat like one *big* electron orbiting the nucleus; **3)** It is possible for the electron count to be different from the nuclear charge. This last possibility is what characterizes ions, and thereby creates all of

We begin with a clue: Eq. (17) indicates that the radius of the Hydrogen atom scales with the mass of the proton. This fact suggests that the base orbit energy of the Hydrogen atom scales with the inverse of proton mass. It further suggests that for element with nuclear charge *Z* and mass *M* , the base orbit energy may scale with *Z* / *M* . If so, then when first-order ionization potentials for all elements are scaled by the inverse factor, *M* /*Z*, then the scaled first-order

referring to a surface phenomenon that expels positive hydronium ions.

the AC approach looked promising for quick application to EZ water.

ionization potentials (called *I P*1,*<sup>Z</sup>* ) might fall into some pattern.

r

We make a Hypothesis: All *I P*1,*<sup>Z</sup>* contain information valuable for all other elements: *population generic information*. Each *I P*1,*<sup>Z</sup>* contains a universal baseline contribution *I P*1,1 about interaction between the nucleus and the population of electrons as a whole. For all elements beyond Hydrogen, there is also a contribution *ΔI P*1,*<sup>Z</sup>* about interactions among the electrons.

The *ΔI P*1,2 can be very significant. For Helium, *ΔI P*1,2 is *huge*, meaning that two electrons bond together very *strongly*. And for Lithium, *ΔI P*1,3 is *negative*, meaning that two electrons actively work together to try to exclude a third electron.

Over the periods, there is obvious detail about the electron-electron interactions. Within each period, there are obvious sub-periods keyed to the nominal angular-momentum quantum number that is being filled. Plotted on a log scale, all sub-period rises are straight lines. The slopes all appear to be rational fractions. We can display these rational fractions in a Table, as was done in [24] and [25]:


A non-traditional parameter *N* is included in the display because, for *l* >0, it is possible to write a simple formula for the fraction:

$$\text{fraction} = \left[ (2l+1) / \text{N}^2 \right] \left[ (N-l) / l \right]. \tag{48}$$

Also, all periods in the Periodic Table have length 2*N* <sup>2</sup> .

All this numerical regularity suggests that there really is a reliable pattern here, and we can reasonably seek to exploit it. Here is the first exploitation that suggests itself: Given first-order ionization potentials of many elements, we can estimate the additional energy required to remove a second electron from each, and then a third, and so on. This was first done in [24]. Formulae were given for each *individual* electron removal or addition, and evaluated for a large number of elements.

One point that Ref. [24] emphasized was that the energy to remove a second electron, or a third, and so on, is *not* the same thing as the so- called 'second-order ionization potential', 'third-order ionization potential', and so on. Those energies are very *large*, which implies that those events are very *violent*: ripping two, or three, or more, electrons off an atom *all at once*. Those energies do exhibit a lot of numerical regularity, but that isn't important for under‐ standing typical lab-bench chemistry, which is all about *gentle* events that occur *one-at-a-time*.

Ref [25] presented the equivalent summed formulae for removal of, or addition of, one, two, and three electrons, removed or added *one-at-a-time*. Basically, use of these formulae save the user some repetitive arithmetic that would be incurred using the formulae from Ref. [24].

#### **Development of More Formulae:**

For the present paper, the formulae from [25] are extended from the illustrative cases of removing, or adding, one, two, and three electrons, to the general case of removing, or adding, *N* electrons.

Ref. [25] used symbols *W* and *H* to distinguish between energy increments associated with electron-nucleus interaction, and energy increments associated with electron-electron inter‐ actions. That distinction is analogous to the distinction between work and heat in thermody‐ namics: the work part is something a human can control, and the heat part is something that Nature simply does, regardless of what the human does.

We had:

$$\mathcal{W}\_{\text{removing }\mathbf{e}\_1 \text{ from the neutral atom}} = \text{IP}\_{1,1}(\mathbf{Z}/\mathbf{M}\_{\mathbf{Z}}).\tag{49}$$

The ion produced has a little less attraction between the nucleus and the now reduced electron cloud. So removing another electron should take a little less work:

$$\mathcal{W}\_{\text{removing }o\_1 \text{ é } o\_2} = \text{IP}\_{1,1} \left[ Z + \sqrt{Z(Z-l)} \right] \Big/ \mathcal{M}\_Z \,. \tag{50}$$

And then:

$$\mathcal{W}\_{\text{memory}\,\text{e}\_1,\text{e}\_2,\text{f}\,\text{e}\_3} = \text{IP}\_{1,1} \Big[ Z + \sqrt{Z(Z-1)} + \sqrt{Z(Z-2)} \Big] \Big/ M\_Z. \tag{51}$$

This pattern generalizes to:

$$\mathcal{W}\_{\text{memory\\_e\\_through\\_e}} = I P\_{1,l} \left[ \sum\_{l=1}^{N} \sqrt{Z(Z+l-i)} \right] \Big| \mathcal{M}\_{\text{Z}}.\tag{52}$$

We also had:

$$H\_{\text{removing }e\_1 \text{ from the natural atom}} = (\Delta IP\_{1,Z} - \Delta IP\_{1,Z-1})(Z / M\_Z). \tag{53}$$

We inferred in [25] that:

Photons and Signals in the Age of Information http://dx.doi.org/10.5772/59067 61

$$\mathbf{H}\_{\text{nonoving }a\_1, \text{Re}\_2} = \left[ \Delta \mathbf{I} \mathbf{P}\_{1, \mathbf{Z}} \mathbf{Z} - \Delta \mathbf{I} \mathbf{P}\_{1, \mathbf{Z} - 2} (\mathbf{Z} - \mathbf{2}) \right] \mathbf{M}\_{\mathbf{Z}}.\tag{54}$$

and

Those energies do exhibit a lot of numerical regularity, but that isn't important for under‐ standing typical lab-bench chemistry, which is all about *gentle* events that occur *one-at-a-time*. Ref [25] presented the equivalent summed formulae for removal of, or addition of, one, two, and three electrons, removed or added *one-at-a-time*. Basically, use of these formulae save the user some repetitive arithmetic that would be incurred using the formulae from Ref. [24].

For the present paper, the formulae from [25] are extended from the illustrative cases of removing, or adding, one, two, and three electrons, to the general case of removing, or adding,

Ref. [25] used symbols *W* and *H* to distinguish between energy increments associated with electron-nucleus interaction, and energy increments associated with electron-electron inter‐ actions. That distinction is analogous to the distinction between work and heat in thermody‐ namics: the work part is something a human can control, and the heat part is something that

The ion produced has a little less attraction between the nucleus and the now reduced electron

1,1 ( 1) . 1 2 *<sup>W</sup>*removing e & e *<sup>Z</sup>* = +- *IP Z Z Z M* é ù

1,1 ( 1) ( 2) . 12 3 *<sup>W</sup>*removing e , e , & e *<sup>Z</sup>* = + -+ - *IP Z Z Z Z Z M* é ù

1,1 <sup>1</sup> ( 1) . removing e through e 1 N *N*

*<sup>W</sup> <sup>i</sup> <sup>Z</sup> IP Z Z i M* <sup>=</sup> <sup>=</sup> é ù + - ë û å (52)

<sup>1</sup> 1, 1, 1 ( )( / ). *H*removing e from the neutral atom *ZZ Z* = D -D *IP IP Z M* - (53)

<sup>1</sup> 1,1( / ). *W*removing e from the neutral atom *<sup>Z</sup>* = *IP Z M* (49)

ë û (50)

ë û (51)

**Development of More Formulae:**

60 Selected Topics in Applications of Quantum Mechanics

Nature simply does, regardless of what the human does.

cloud. So removing another electron should take a little less work:

*N* electrons.

We had:

And then:

We also had:

This pattern generalizes to:

We inferred in [25] that:

$$\mathbf{H}\_{\text{nonoving }a\_1, a\_2, \dots, a\_3} = \left[ \Delta \mathbf{I} \mathbf{P}\_{1, \mathbb{Z}} \mathbf{Z} - \Delta \mathbf{I} \mathbf{P}\_{1, \mathbb{Z}-3} (\mathbf{Z} - \mathbf{3}) \right] \big/ \mathbf{M}\_{\mathbb{Z}}.\tag{55}$$

This pattern generalizes to:

$$\mathbf{H}\_{\text{memory}\,\text{e}\_1\,\text{through }\mathbf{e}\_N} = \left[\Delta\text{IP}\_{1,Z}\mathbf{Z} - \Delta\text{IP}\_{1,Z-N}(\mathbf{Z}-\mathbf{N})\right]\Big\|\mathbf{M}\_Z.\tag{56}$$

Finally, in [25] we had:

$$\begin{aligned} \left(\mathcal{W} + H\right)\_{\text{removeding }a\_1, \text{a}, a\_2} &= \\ \left[\Lambda \text{IP}\_{1,1}\right] \left[Z + \sqrt{Z(Z-1)}\right] \left[\mathcal{M}\_Z + \left[\Lambda \text{IP}\_{1,Z} Z - \Lambda \text{IP}\_{1,Z-2}(Z-2)\right] \right] \mathcal{M}\_Z \end{aligned} \tag{57}$$

and

$$\begin{aligned} \left(\mathcal{W} + \mathcal{H}\right)\_{\text{remaining }o\_1, e\_2, \text{As}\_1} &= \\ \left[\mathcal{HP}\_{1,1}\right] \left[Z + \sqrt{Z(Z-1)} + \sqrt{Z(Z-2)}\right] \left\{\mathcal{M}\_Z + \left[\Delta \text{IP}\_{1,2} Z - \Delta \text{IP}\_{1,2-3} (Z-3)\right] \right\} \mathcal{M}\_Z & , \end{aligned} \tag{58}$$

This pattern generalizes to:

$$\begin{aligned} \left(\mathcal{W} + \mathcal{H}\right)\_{\text{removing }\mathbf{e}\_1 \text{ through } \mathbf{e}\_N} &= \\ \left[\mathcal{H}\_{1,1}\right] \sum\_{l=1}^N \sqrt{\mathcal{Z}(\mathcal{Z}+1-l)} \left[\mathcal{M}\_{\mathcal{Z}} + \left[\Delta \text{IP}\_{1,\mathcal{Z}} \mathcal{Z} - \Delta \text{IP}\_{1,\mathcal{Z}-\mathcal{N}}(\mathcal{Z}-\mathbf{N})\right] \right] \mathcal{M}\_{\mathcal{Z}} \end{aligned} \tag{59}$$

Now let us turn to *adding* electrons. First, use the formula for the energy for removing an electron from a neutral atom of element *Z* to describe instead *removing* an electron from the *singly charged negative ion* of element *Z*, which has *Z* + 1 electrons to start with:

$$\left(\left(\mathcal{W} + H\right)\_{\text{nonoving }o\_1 \text{ from negative ion}} = \left[\text{IP}\_{1,1}\sqrt{\mathcal{Z}(\mathcal{Z}+\mathbf{l})} + \Delta\text{IP}\_{1,\mathcal{Z}\approx 1}(\mathcal{Z}+\mathbf{l}) - \Delta\text{IP}\_{1,\mathcal{Z}}\overline{\mathcal{Z}}\right] \not\!\!/ \mathbf{M}\_{\mathcal{Z}}.\tag{60}$$

Reversing the direction of the operation:

$$(\mathcal{W} + H)\_{\text{adding } \mathbf{e}\_l \text{ to neutral atom}} = -\left[IP\_{1,1}\sqrt{\mathcal{Z}(\mathcal{Z}+\mathbf{l})} + \Lambda \text{IP}\_{1,\mathcal{Z}\text{ to } \mathbf{l}}(\mathcal{Z}+\mathbf{l}) - \Lambda \text{IP}\_{1,\mathcal{Z}}\mathcal{Z}\right] \Big/ \mathcal{M}\_{\mathcal{Z}}.\tag{61}$$

This means the work for adding one electron into the nuclear field is:

$$\mathcal{W}\_{\text{adding }\mathbf{e}\_{\text{l to neutral atom}}} = -\mathrm{IP}\_{\mathbb{L},\mathbf{l}} \sqrt{\mathcal{Z}(\mathbb{Z}+\mathbf{l})} \,\mathrm{M}\_{\mathbb{Z}}.\tag{62}$$

And the heat for re-adjusting the electron population is:

$$\delta H\_{\text{adding }\mathbf{e}\_1 \text{ to neutral atom}} = -\left[\Delta I \mathbf{P}\_{1,Z+1}(Z+\mathbf{l}) + \Delta I \mathbf{P}\_{1,Z}Z\right] \Big/ \mathbf{M}\_Z. \tag{63}$$

Now let us add a second electron. This will require additional work:

$$\mathcal{W}\_{\text{adding }\mathbf{e}\_2\text{ after }\mathbf{e}\_1} = -\mathrm{IP}\_{1,1}\sqrt{\mathbf{Z}(\mathbf{Z}+\mathbf{2})} \Big/ \mathcal{M}\_{\mathbf{Z}}.\tag{64}$$

And it will cause another heat adjustment:

$$H\_{\text{adding }\mathbf{e}\_2\text{ after }\mathbf{e}\_l} = -\Delta \text{IP}\_{\mathbf{1},\mathbf{Z}+\mathbf{2}}(\mathbf{Z}+\mathbf{2})/M\_Z + \Delta \text{IP}\_{\mathbf{1},\mathbf{Z}+\mathbf{1}}(\mathbf{Z}+\mathbf{1})/M\_Z. \tag{65}$$

This means total energy involved in adding two electrons is:

$$\begin{aligned} &(\mathcal{W}+\mathcal{H})\_{\text{adding }\mathbf{e}\_1 \triangleq \mathbf{e}\_2} = \\ &-\mathrm{IP}\_{1,1}\Big[\sqrt{\mathcal{Z}(\mathcal{Z}+\mathbf{l})}+\sqrt{\mathcal{Z}(\mathcal{Z}+\mathbf{2})}\Big]\Big|\mathcal{M}\_{\text{Z}}+\Big[\Delta\mathrm{IP}\_{1,\mathcal{Z}}\mathcal{Z}-\Delta\mathrm{IP}\_{1,\mathcal{Z}\ast\mathbf{2}}(\mathcal{Z}+\mathbf{2})\Big]\Big|\mathcal{M}\_{\text{Z}}\Big] \end{aligned} \tag{66}$$

Likewise, the total energy involved in adding three electrons is:

$$\begin{split} & (W+H)\_{\text{adding }o\_1, o\_2 \text{ for } o\_1} \\ & - \mathrm{IP}\_{1,1} \Big[ \sqrt{Z(Z+1)} + \sqrt{Z(Z+2)} + \sqrt{Z(Z+3)} \Big] \Big| M\_Z + \Big[ \Delta \mathrm{IP}\_{1,2} Z - \Delta \mathrm{IP}\_{1,2 \times 3} (Z+3) \Big] \Big| M\_Z \Big] \end{split} \tag{67}$$

This patten generalizes to:

$$\begin{aligned} \left(\mathcal{W} + H\right)\_{\text{adding } \mathbf{e}\_i \text{ through } \mathbf{e}\_N} &= \\ \left[\mathcal{W}\_{1,1}\right] \sum\_{l=1}^N \sqrt{\mathcal{Z}(\mathbf{Z} + \bar{\imath})} \left[\mathcal{M}\_{\mathbf{Z}} + \left[\Delta \text{IP}\_{1,\mathbf{Z}} \mathbf{Z} - \Delta \text{IP}\_{1,\mathbf{Z} \star N}(\mathbf{Z} + \mathbf{N})\right] \right] \mathcal{M}\_{\mathbf{Z}} \end{aligned} \tag{68}$$

#### **Numerical Data to Insert in Formulae:**

<sup>1</sup> 1,1 1, 1 1, ( ) ( 1) ( 1) . *W H* adding e to neutral atom *<sup>Z</sup> Z Z IP Z Z IP Z IP Z M* <sup>+</sup> <sup>+</sup> =- + +D + -D é ù

<sup>1</sup> 1, 1 1, ( 1) . *H*adding e to neutral atom *<sup>Z</sup> Z Z IP Z IP Z M* <sup>+</sup> =- D + +D é ù

This means the work for adding one electron into the nuclear field is:

Now let us add a second electron. This will require additional work:

This means total energy involved in adding two electrons is:

Likewise, the total energy involved in adding three electrons is:

1,1 1, 1, 2

1,1 1, 1, 3

1,1 1, 1, 1



*IP Z Z Z Z M IP Z IP Z M* <sup>+</sup>

*IP Z Z Z Z Z Z M IP Z IP Z M* <sup>+</sup>

( 1) ( 2) ( 2) .

( 1) ( 2) ( 3) ( 3) .

( ) ( ) ,

ë û ë û <sup>å</sup> (68)

*i Z Z Z N Z*

*IP Z Z i M IP Z IP Z N M* <sup>+</sup> <sup>=</sup>

é ù + + D -D + é ù

1 2 &

1

+ =

adding e through e*<sup>N</sup>*

adding e e

+ =

12 3 ,

adding e e & e

( )

*W H*

*N*

+ =

This patten generalizes to:

( )

*W H*

( )

*W H*

And the heat for re-adjusting the electron population is:

And it will cause another heat adjustment:

62 Selected Topics in Applications of Quantum Mechanics

ë û (61)

ë û (63)

<sup>1</sup> 1,1 ( 1) . *W*adding e to neutral atom *<sup>Z</sup>* =- + *IP Z Z M* (62)

2 1 1,1 ( 2) . *W*adding e after e *<sup>Z</sup>* =- + *IP Z Z M* (64)

2 1 1, 2 1, 1 ( 2) / ( 1) / . *H*adding e after e *Z ZZ Z* = -D + + D + *IP Z M IP Z M* + + (65)

*ZZ Z Z*

*ZZ Z Z*

(66)

(67)

Numerical data for elements up to number 118 are given in [24]. The numerical analysis of EZ water requires at most the data for the first ten elements. Expressed in electron volts, eV, these numerical data are:


#### **Ordinary Water:**

Here are some example calculations concerning possible ionic configurations of ordinary, normal water.

Most people would guess that water is 2 H<sup>+</sup> + O2<sup>−</sup> . But let us evaluate that ionic configuration. The transition H→ H<sup>+</sup> takes:

$$\text{IP}\_{1,1} \mid \text{M}\_1 = 14.250 \, / 1.008 = 14.1369 \, \text{eV} \, \text{.} \tag{69}$$

So 2H<sup>+</sup> takes 2×14.1369=28.2738 eV.

The transition O→O2<sup>−</sup> takes:

$$\begin{aligned} &-\text{IP}\_{1,\text{[}}\left[\sqrt{\text{Z}(\text{Z}+1)}+\sqrt{\text{Z}(\text{Z}+2)}\right]\Big|\,\text{M}\_{\text{Z}}+\left[\sqrt{\text{A}\text{IP}\_{1,\text{Z}}\text{Z}-\text{A}\text{IP}\_{1,\text{Z}+1}(\text{Z}+2)}\right]\Big|\,\text{M}\_{\text{Z}} \\ &=-14.250\Big[\sqrt{8\times9}+\sqrt{8\times10}\Big]\Big|\,15.999+\Big[\Delta\text{IP}\_{1,\text{8}}\times8-\Delta\text{IP}\_{1,\text{10}}\times10\Big]\Big|\,15.999 \\ &=-14.250\Big[\sqrt{72}+\sqrt{80}\Big]\Big|\,15.999+\Big[13.031\times8-29.391\times10\Big]\Big|\,15.999 \\ &=-14.250\Big[8.4853+8.9443\Big]\Big|\,15.999+\Big[104.248-293.910\Big]\Big|\,15.999 \\ &=-14.250\Big[\Big{]}\Big{1}.4296\Big[\Big{]}\Big{1}.5999+\Big[-189.662\Big]\Big{1}5.999 \\ &=-15.5242-11.8546=-27.3788\,\text{eV} \end{aligned} \tag{70}$$

So the ionic configuration 2 H<sup>+</sup> + O2<sup>−</sup> requires 28.2738−27.3788=0.8950 eV. This is a positive energy requirement, which implies that some external assistance is needed to create this ionic configuration. So normal water may *not* be 2 H<sup>+</sup> + O2<sup>−</sup> after all!

Another possibility is readily at hand though. The ionic configuration for normal water could be 2 H<sup>−</sup> + O2+ . The transition H→ H<sup>−</sup> takes:

$$\begin{aligned} &-IP\_{1,1}\sqrt{\mathcal{Z}(\mathbb{Z}+\mathbb{I})}/M\_{\mathbb{Z}}-\Delta IP\_{1,\mathbb{Z}\times\mathbb{I}}(\mathbb{Z}+\mathbb{I})/M\_{\mathbb{Z}}+\Delta IP\_{1,\mathbb{Z}}Z/M\_{\mathbb{Z}} \\ &=-IP\_{1,1}\sqrt{\mathbb{I}\times\mathbb{Z}}/M\_{\mathbb{I}}-\Delta IP\_{1,\mathbb{Z}}\times 2/M\_{\mathbb{I}}+\Delta IP\_{1,\mathbb{I}}\times 1/M\_{\mathbb{I}} \\ &=-14.250\times 1.4142/1.008-35.625\times 2/1.008+0\times 1/1.008 \\ &=-19.9924-70.6845+0=-90.6769\,\text{eV} \ . \end{aligned} \tag{71}$$

So 2 H<sup>−</sup> takes 2×(−90.6769)= −181.3538 eV. Notice that this is a *huge* negative energy. It reflects the fact that electrons *really* like to make pairs. Indeed, their propensity to do so motivated the invention of the so-called 'spin' quantum number. Without spin, many electrons in atoms would be violating the 'Pauli Exclusion Principle', which says that only *one* electron can be in any particular quantum state. Electron pairs are famous in Condensed Matter Physics too, under the name 'Cooper pairs'.

Proceeding now to the transition O→O2+ , that takes:

$$\begin{aligned} \text{IP}\_{1,1} \left[ \text{Z} + \sqrt{\text{Z}(\text{Z}-1)} \right] \text{M}\_{\text{Z}} + \left[ \Delta \text{IP}\_{1,\text{Z}} \text{Z} - \Delta \text{IP}\_{1,\text{Z}-2} (\text{Z}-2) \right] \text{M}\_{\text{Z}} \\ = \text{IP}\_{1,1} \left[ \text{8} + \sqrt{\text{8} \times 7} \right] \left[ \text{M}\_{\text{8}} + \left[ \Delta \text{IP}\_{1,\text{8}} \times \text{8} - \Delta \text{IP}\_{1,\text{8}} \times \text{6} \right] \right] \text{M}\_{\text{8}} \\ = \text{14.250} \left[ \text{8} + \sqrt{\text{56}} \right] \left[ \text{15.999} + \left[ \text{13.031} \times \text{8} - \text{7.320} \times \text{6} \right] \right] \text{15.999} \\ = \text{13.790} \text{7} + \text{3.770} \text{7} = \text{17.5614} \text{eV} \end{aligned} \tag{72}$$

Thus the creation of the water molecule in the ionic configuration 2 H<sup>−</sup> + O2+ demands alto‐ gether −181.3538 + 17.5614= −163.7924 eV. This energy is solidly negative, which means that ordinary water is overwhelmingly in this ionic configuration, 2 H<sup>−</sup> + O2+ . This *is* normal water.

However, in situations where more than one version of anything can exist, both generally *do* exist, in proportions determined by their so-called Boltzmann factors, exp(− *E* / *kT* ). Here *E* is energy, *k* is Boltzmann's constant, and *T* is absolute temperature. Boltzmann factors are the result of entropy maximization at work. Because of non-zero Boltzmann factors, there will exist a tiny, tiny fraction of the first ionic configuration, 2 H<sup>+</sup> + O2<sup>−</sup> .

This analysis of normal water shows how quantitative approaches can sometimes unseat longstanding, but never-justified, assumptions in Chemistry.

#### **About EZ Water:**

The following transition is generally thought to represent the creation of EZ water:


So the ionic configuration 2 H<sup>+</sup> + O2<sup>−</sup> requires 28.2738−27.3788=0.8950 eV. This is a positive energy requirement, which implies that some external assistance is needed to create this ionic

Another possibility is readily at hand though. The ionic configuration for normal water could

1,1 1 1,2 1 1,1 1

 takes 2×(−90.6769)= −181.3538 eV. Notice that this is a *huge* negative energy. It reflects the fact that electrons *really* like to make pairs. Indeed, their propensity to do so motivated the invention of the so-called 'spin' quantum number. Without spin, many electrons in atoms would be violating the 'Pauli Exclusion Principle', which says that only *one* electron can be in any particular quantum state. Electron pairs are famous in Condensed Matter Physics too,

, that takes:

1,1 8 1,8 1,6 8

8 87 8 6

*IP M IP IP M* - é ù + - + D -D - é ù ë û ë û

= + ´ + D ´ -D ´ é ù é ù ë û ë û

*ZZ Z Z IP Z Z Z M IP Z IP Z M*

14.250 8 56 15.999 13.031 8 7.320 6 15.999 13.7907 3.7707 17.5614eV .

( 1) ( 2)

gether −181.3538 + 17.5614= −163.7924 eV. This energy is solidly negative, which means that

However, in situations where more than one version of anything can exist, both generally *do* exist, in proportions determined by their so-called Boltzmann factors, exp(− *E* / *kT* ). Here *E* is energy, *k* is Boltzmann's constant, and *T* is absolute temperature. Boltzmann factors are the result of entropy maximization at work. Because of non-zero Boltzmann factors, there will exist

This analysis of normal water shows how quantitative approaches can sometimes unseat long-

The following transition is generally thought to represent the creation of EZ water:

[ ]

.

( 1) ( 1) / / 1 2 2 / 1 / 14.250 1.4142 1.008 35.625 2 / 1.008 0 1 / 1.008 19.9924 70.6845 0 90.6769eV .

*Z Z Z ZZ IP Z Z M IP Z M IP Z M IP M IP M IP M*

takes:


1,1 1, 1, 2

= + é ù + ´- ´ ë û = +=

Thus the creation of the water molecule in the ionic configuration 2 H<sup>−</sup> + O2+

ordinary water is overwhelmingly in this ionic configuration, 2 H<sup>−</sup> + O2+

a tiny, tiny fraction of the first ionic configuration, 2 H<sup>+</sup> + O2<sup>−</sup>

standing, but never-justified, assumptions in Chemistry.

**About EZ Water:**

1,1 1, 1 1,

after all!

(71)

(72)

demands alto‐

. This *is* normal water.

configuration. So normal water may *not* be 2 H<sup>+</sup> + O2<sup>−</sup>

. The transition H→ H<sup>−</sup>

64 Selected Topics in Applications of Quantum Mechanics

be 2 H<sup>−</sup> + O2+

So 2 H<sup>−</sup>

under the name 'Cooper pairs'.

Proceeding now to the transition O→O2+

Here parentheses are used to avoid implying anything about what charge the individual atoms within any ion or radical may carry. A full numerical analysis should consider all possible, or at least all plausible, ionic configurations of every molecule or radical involved.

One possible ionic configuration for the EZ water ion (H3O2)<sup>−</sup> is 3 <sup>H</sup><sup>+</sup> <sup>+</sup> <sup>2</sup> <sup>O</sup>2<sup>−</sup> . The 3 H<sup>+</sup> takes total energy 3 <sup>×</sup> *<sup>I</sup> <sup>P</sup>*1,1 / *Mz* <sup>=</sup> 3×14.250 / 1.008=42.4107 eV. The 2 <sup>O</sup>2<sup>−</sup> takes total energy 2×(−27.3788)= −54.7576 eV. So the ionic configuration 3 H<sup>+</sup> + 2 O2<sup>−</sup> altogether takes 42.4107−54.7576= −12.3469 eV. This is a negative energy, so this ionic configuration certainly can occur.

But there is also another possibility for the EZ water ion (H3O2)<sup>−</sup>. It could have the ionic configuration 3 H<sup>−</sup> + 2 O<sup>+</sup> . An H<sup>−</sup> takes −90.6769 eV, so 3 H<sup>−</sup> takes 3×(−90.6769)= −272.0307 eV. An O<sup>+</sup> takes energy:

$$(IP\_{1,l} \times 8 + \Delta IP\_{1,8} \times 8 - \Delta IP\_{1,7} \times 7) / M\_8$$

$$= (14.250 \times 8 + 13.031 \times 8 - 13.031 \times 7) / 15.999$$

$$= (114.000 + 104.248 - 91.217) / 15.999 = 7.9399 \text{eV } \,\text{ .}$$

So 2 O<sup>+</sup> takes 2×7.9399=15.8798 eV. Then the ionic configuration 3 H<sup>−</sup> + 2 O<sup>+</sup> takes −272.0307 + 15.8798= −256.1509 eV. This energy is much more negative than that of the first candidate ionic configuration for EZ water, 3 H<sup>+</sup> + 2 O2<sup>−</sup> . This fact means EZ water is nearly always in this second candidate ionic configuration, 3 H<sup>−</sup> + 2 O<sup>+</sup> .

One possible ionic configuration for the hydronium ion (H3O)+ is 3 <sup>H</sup><sup>+</sup> <sup>+</sup> <sup>O</sup>2<sup>−</sup> . This, I believe, is what most people would guess. But from the study of regular water, we know the candidate 3 H<sup>+</sup> would take 3×14.250=42.4107 eV, and that the candidate O<sup>2</sup><sup>−</sup> would take −27.3788 eV, so the candidate ionic configuration 3 H<sup>+</sup> + O2<sup>−</sup> for hydronium would take 42.4107−27.3788=15.0319 eV. This energy is positive, *so this ionic configuration for the hydronium ion is not promising*.

However, as was the case with the EZ water ion, there is another possibility for the hydronium ion. It (H3O)+ could have the ionic configuration 3 <sup>H</sup><sup>−</sup> <sup>+</sup> <sup>O</sup>4+ . The 3 H<sup>−</sup> would take 3×(−90.6769)= −272.0307 eV, and the O4+ would take:

$$\begin{aligned} \text{IP}\_{1,1} \left[ Z + \sqrt{Z(Z-1)} + \sqrt{Z(Z-2)} + \sqrt{Z(Z-3)} \right] \Big| M\_{\mathbb{Z}} + \left[ \Delta \text{IP}\_{1,\mathbb{Z}} Z - \Delta \text{IP}\_{1,\mathbb{Z}-4} (Z-4) \right] \Big| M\_{\mathbb{Z}} \\ = 14.250 \times \left[ 8 + \sqrt{56} + \sqrt{48} + \sqrt{40} \right] \Big| 15.999 + \left[ 13.031 \times 8 - 9.077 \times 4 \right] \Big| 15.999 \\ = 14.250 \times 28.7361 / 15.999 + 67.940 / 15.999 \\ = 25.5947 + 4.2465 = 29.8411 \text{eV} \end{aligned} \tag{74}$$

So for the hydronium ion (H3O)+, the second candidate ionic configuration 3 <sup>H</sup><sup>−</sup> <sup>+</sup> <sup>O</sup>4+ would take −272.0307 + 29.8411= −242.1895 eV. This very negative energy explains why the reaction product that accompanies EZ water *is* a hydronium ion, rather than a naked proton plus a normal water molecule, which would take 14.1369 −163.7925= −149.6556eV, which is not *as* negative.

The EZ water ion and the hydronium ion together take −268.6204−242.1895= −510.8099 eV. Compare this energy to the energy taken by three normal water molecules: 3×(−163.7925)= −491.3775 eV. The EZ water ion with the hydronium ion has lower energy than the three normal water molecules. That means that Nature will take any opportunity to make EZ water ions and hydronium ions.

It appears that what creates the opportunity is a surface, plus a little energy to separate the ions. Any material body provides some gravity to create a surface, and if there is also some small energy source, such as sunlight, to help separate ions, and if there is also some normal water, the situation will automatically create EZ water too. Even an icy comet might be able to create some EZ water.

#### **7. Microphysics**

Just as the myriad compounds in Chemistry arise from not-very-many chemical elements, some significant part of the m *26*

called Series 2, is the energy gain rate due to torquing, here bolder because it was not previously

Figure 4. Log energy loss and gain rates *vs*. log system radius. **Figure 4.** Log energy loss and gain rates *vs*. log system radius.

will *not* find many isolated Hydrogen atoms.

states'. Mills discusses these in [27].

that, then there is no particle speed limit.

Observe that radiation dominates for radii below the crossing point, whereas torquing dominates for radii above the crossing point. This means the balance between the two effects is *unstable*: a small excursion from balance in either direction causes more excursion in the same direction. This is interesting. It means that Hydrogen does not like to exist as an isolated atom. It wants to engage in chemical reactions. In the Universe at large, you will find Hydrogen in H2 molecules, or other mol-Observe that radiation dominates for radii below the crossing point, whereas torquing dominates for radii above the crossing point. This means the balance between the two effects is *unstable*: a small excursion from balance in either direction causes more excursion in the same

always non-negative, so the torquing curve is non-negative too at the balance points.

as crossings because of the finite resolution of the plot, but they are certainly present.

**Objects cannot be created from editing field codes.**

ecules, or you will find Hydrogen plasma, consisting of naked protons and free electrons, but you

**2)** We can discover even more about the Hydrogen atom if we include the appropriate angle sine and cosine factors in the calculations. These factors are oscillatory. So negative numbers sometimes occur with the torquing curve, and they cannot be plotted on a logarithmic scale. However, we are mainly interested in the points of balance between torquing and radiation, and the radiation curve is

Figure 5 shows that we have not just one balance point, but *many* balance points. The balance points occur in close pairs, one stable and one unstable. The two pairs furthest left on the plot do not show

More solution pairs are to be found, off the Figure to the left. Indeed, the solution pairs continue *indefinitely*, into smaller and smaller system radii. So Hydrogen has an infinite family of 'sub-

The smaller and smaller radii of the balance points in Fig. 5 correspond to higher and higher orbit speeds. This idea conflicts with a prohibition imposed by SRT: no physical particle possessing mass is allowed to move at a speed matching or exceeding light speed *c* . What does this conflict mean? I believe it means the prohibition should be understood more precisely to say: no physical particle possessing mass *can be perceived* to move at a speed matching or exceeding light speed *c* , *if we agree to process all received data in accord with Einstein's Second Postulate*. If we do not agree to

Series1 Series2

direction. This is interesting. It means that Hydrogen does not like to exist as an isolated atom. It wants to engage in chemical reactions. In the Universe at large, you will find Hydrogen in H2 molecules, or other molecules, or you will find Hydrogen plasma, consisting of naked protons and free electrons, but you will *not* find many isolated Hydrogen atoms.

So for the hydronium ion (H3O)+, the second candidate ionic configuration 3 <sup>H</sup><sup>−</sup> <sup>+</sup> <sup>O</sup>4+ would take −272.0307 + 29.8411= −242.1895 eV. This very negative energy explains why the reaction product that accompanies EZ water *is* a hydronium ion, rather than a naked proton plus a normal water molecule, which would take 14.1369 −163.7925= −149.6556eV, which is not *as*

The EZ water ion and the hydronium ion together take −268.6204−242.1895= −510.8099 eV. Compare this energy to the energy taken by three normal water molecules: 3×(−163.7925)= −491.3775 eV. The EZ water ion with the hydronium ion has lower energy than the three normal water molecules. That means that Nature will take any opportunity to make

It appears that what creates the opportunity is a surface, plus a little energy to separate the ions. Any material body provides some gravity to create a surface, and if there is also some small energy source, such as sunlight, to help separate ions, and if there is also some normal water, the situation will automatically create EZ water too. Even an icy comet might be able

Just as the myriad compounds in Chemistry arise from not-very-many chemical elements,

1 20 39 58 77 96 115134153172191210229248267286

**Objects cannot be created from editing field codes.** Figure 4. Log energy loss and gain rates *vs*. log system radius. Observe that radiation dominates for radii below the crossing point, whereas torquing dominates for radii above the crossing point. This means the balance between the two effects is *unstable*: a small excursion from balance in either direction causes more excursion in the same direction. This is interesting. It means that Hydrogen does not like to exist as an isolated atom. It wants to engage in chemical reactions. In the Universe at large, you will find Hydrogen in H2 molecules, or other molecules, or you will find Hydrogen plasma, consisting of naked protons and free electrons, but you

Observe that radiation dominates for radii below the crossing point, whereas torquing dominates for radii above the crossing point. This means the balance between the two effects is *unstable*: a small excursion from balance in either direction causes more excursion in the same

> **2)** We can discover even more about the Hydrogen atom if we include the appropriate angle sine and cosine factors in the calculations. These factors are oscillatory. So negative numbers sometimes occur with the torquing curve, and they cannot be plotted on a logarithmic scale. However, we are mainly interested in the points of balance between torquing and radiation, and the radiation curve is

> Figure 5 shows that we have not just one balance point, but *many* balance points. The balance points occur in close pairs, one stable and one unstable. The two pairs furthest left on the plot do not show

> More solution pairs are to be found, off the Figure to the left. Indeed, the solution pairs continue *indefinitely*, into smaller and smaller system radii. So Hydrogen has an infinite family of 'sub-

> The smaller and smaller radii of the balance points in Fig. 5 correspond to higher and higher orbit speeds. This idea conflicts with a prohibition imposed by SRT: no physical particle possessing mass is allowed to move at a speed matching or exceeding light speed *c* . What does this conflict mean? I believe it means the prohibition should be understood more precisely to say: no physical particle possessing mass *can be perceived* to move at a speed matching or exceeding light speed *c* , *if we agree to process all received data in accord with Einstein's Second Postulate*. If we do not agree to

always non-negative, so the torquing curve is non-negative too at the balance points.

as crossings because of the finite resolution of the plot, but they are certainly present.

called Series 2, is the energy gain rate due to torquing, here bolder because it was not previously

Series1 Series2

**Error!**

negative.

EZ water ions and hydronium ions.

66 Selected Topics in Applications of Quantum Mechanics

to create some EZ water.

some significant part of the m

recognized.


will *not* find many isolated Hydrogen atoms.

**Figure 4.** Log energy loss and gain rates *vs*. log system radius.

states'. Mills discusses these in [27].

that, then there is no particle speed limit.


0

5

10

15

**7. Microphysics**

*26*

**1.** We can discover even more about the Hydrogen atom if we include the appropriate angle sine and cosine factors in the calculations. These factors are oscillatory. So negative numbers sometimes occur with the torquing curve, and they cannot be plotted on a logarithmic scale. However, we are mainly interested in the points of balance between torquing and radiation, and the radiation curve is always non-negative, so the torquing curve is non-negative too at the balance points.

Figure 5 shows that we have not just one balance point, but *many* balance points. The balance points occur in close pairs, one stable and one unstable. The two pairs furthest left on the plot do not show as crossings because of the finite resolution of the plot, but they are certainly present.

More solution pairs are to be found, off the Figure to the left. Indeed, the solution pairs continue *indefinitely*, into smaller and smaller system radii. So Hydrogen has an infinite family of 'substates'. Mills discusses these in [27].

The smaller and smaller radii of the balance points in Fig. 5 correspond to higher and higher orbit speeds. This idea conflicts with a prohibition imposed by SRT: no physical particle possessing mass is allowed to move at a speed matching or exceeding light speed *c*. What does this conflict mean? I believe it means the prohibition should be understood more precisely to say: no physical particle possessing mass *can be* **perceived** to move at a speed matching or exceeding light speed *c*, **if** *we agree to process all received data in accord with Einstein's Second Postulate*. If we do not agree to that, then there is no particle speed limit. *27*

Figure 5. More and more balance points below the Hydrogen ground state.

As with Hydrogen, the oscillatory angle factors create a family of solutions for this system too. Half of them are stable, and half are unstable, and uncountably many of them occur at small radii and high speeds well in excess of *c* . Due to the finite resolution of the plot, only one pair of solutions is clearly visible in Fig. 6. But two more, at smaller radius and higher speed, are also certainly present. We can characterize these, and all high-speed solutions, without even knowing exactly what the radiation curve is like - its exact amplitude, or its *r* 4 dependence. The one low-speed solution is just *v*<sup>0</sup> 0.02 *c* . The many high-speed solutions have to occur in pairs just above and below

Figure 6. Positronium solutions.

orbit speeds of the form *vn v*<sup>0</sup> *n* 2*c* , where *n* is an arbitrary positive integer.

**3)** We can also study positronium (a system consisting of one electron and one positron). See Fig. 6. **Figure 5.** More and more balance points below the Hydrogen ground state.

0

0

5

10

15

20

25

30

35

10

20

25

30

35

**2.** We can also study positronium (a system consisting of one electron and one positron). See Fig. 6. As with Hydrogen, the oscillatory angle factors create a family of solutions for this system too. Half of them are stable, and half are unstable, and uncountably many of them occur at small radii and high speeds well in excess of *c*. Due to the finite resolution of the plot, only one pair of solutions is clearly visible in Fig. 6. But two more, at smaller radius and higher speed, are also certainly present. We can characterize these, and all high-speed solutions, without even knowing exactly what the radiation curve is like - its exact amplitude, or its *r* <sup>−</sup><sup>4</sup> dependence. The one low-speed solution is just *v*<sup>0</sup> ≈0.02×*c*. The many high-speed solutions have to occur in pairs just above and below orbit speeds of the form *vn* =*v*<sup>0</sup> + *n* ×2*πc*, where *n* is an arbitrary positive integer. 0 5 Figure 5. More and more balance points below the Hydrogen ground state. **3)** We can also study positronium (a system consisting of one electron and one positron). See Fig. 6. As with Hydrogen, the oscillatory angle factors create a family of solutions for this system too. Half of them are stable, and half are unstable, and uncountably many of them occur at small radii and high speeds well in excess of *c* . Due to the finite resolution of the plot, only one pair of solutions is clearly visible in Fig. 6. But two more, at smaller radius and higher speed, are also certainly present. We can characterize these, and all high-speed solutions, without even knowing exactly what the radiation curve is like - its exact amplitude, or its *r* 4 dependence. The one low-speed solution is just *v*<sup>0</sup> 0.02 *c* . The many high-speed solutions have to occur in pairs just above and below

orbit speeds of the form *vn v*<sup>0</sup> *n* 2*c* , where *n* is an arbitrary positive integer.

*27*

Series1 Series2

**Figure 6.** Positronium solutions.

0

A parenthetical note applies for Fig. 6: for same-mass systems, the amplitude of the radiation curve should be less by a factor of 4 because there is no center-of-mass motion. This sort of numerical detail does not significantly affect where the solutions fall. That is determined almost entirely by the cosine factors that produce the deep dips that intersect the peaks in the curve for rate of energy gain due torquing.

Figure 6. Positronium solutions.

**3.** The oscillatory nature of the angle factors can turn a situation of seeming repulsion into a situation of actual attraction. This phenomenon of sign reversal due to signal delay is well known to engineers, who often deal with oscillating signals in feedback control systems For the present application, consider two electrons in a circular orbit, and suppose they move at speed *πc*. One electron launches its signal radially outward. By the time this electron has executed half an orbit, this signal has expanded a distance equal to the orbit diameter. By then, the two electrons have exchanged places. So the expanding signal first contacts the second electron at exactly the signal launch point. Then the two electrons complete their orbit. At the end, the second electron finally understands its signal: it is to move radially. But by now, the two electrons have changed places again, and for the second electron, the direction commanded is *inward*. That situation is equivalent to *attraction*.

Given this mechanism for attraction, we can also study homogeneous systems: two electrons, or two positrons, for example. Again, there exist both stable and unstable solutions, and there are infinitely many of each, corresponding to orbit speeds of the form *vn* =*πc* + *n* ×2*πc* for arbitrary positive integer *n*. See Fig. 7. places. So the expanding signal first contacts the second electron at exactly the signal launch point. Then the two electrons complete their orbit. At the end, the second electron finally understands its signal: it is to move radially. But by now, the two electrons have changed places again, and for the second electron, the direction commanded is *inward*. That situation is equivalent to *attraction*. Given this mechanism for attraction, we can also study homogeneous systems: two electrons, or two positrons, for example. Again, there exist both stable and unstable solutions, and there are infinitely

many of each, corresponding to orbit speeds of the form *vn c n* 2*c* for arbitrary positive

has expanded a distance equal to the orbit diameter. By then, the two electrons have exchanged

A parenthetical note applies for Fig. 6: for same-mass systems, the amplitude of the radiation curve should be less by a factor of 4 because there is no center-of-mass motion. This sort of numerical detail does not significantly affect where the solutions fall. That is determined almost entirely by the cosine factors that produce the deep dips that intersect the peaks in the curve for rate of energy gain

**4)** The oscillatory nature of the angle factors can turn a situation of seeming repulsion into a situation of actual attraction. This phenomenon of sign reversal due to signal delay is well known to engi-

An additional parenthetical note applies for Fig. 7: for same-charge systems, the angular pattern of **Figure 7.** Same-charge solutions.

0

*28*

due torquing.

integer *n* . See Fig. 7.

**2.** We can also study positronium (a system consisting of one electron and one positron). See Fig. 6. As with Hydrogen, the oscillatory angle factors create a family of solutions for this system too. Half of them are stable, and half are unstable, and uncountably many of them occur at small radii and high speeds well in excess of *c*. Due to the finite resolution of the plot, only one pair of solutions is clearly visible in Fig. 6. But two more, at smaller radius and higher speed, are also certainly present. We can characterize these, and all high-speed solutions, without even knowing exactly what the radiation curve is like - its exact

Figure 5. More and more balance points below the Hydrogen ground state. **3)** We can also study positronium (a system consisting of one electron and one positron). See Fig. 6. As with Hydrogen, the oscillatory angle factors create a family of solutions for this system too. Half of them are stable, and half are unstable, and uncountably many of them occur at small radii and high speeds well in excess of *c* . Due to the finite resolution of the plot, only one pair of solutions is clearly visible in Fig. 6. But two more, at smaller radius and higher speed, are also certainly present. We can characterize these, and all high-speed solutions, without even knowing exactly what the radiation curve is like - its exact amplitude, or its *r* 4 dependence. The one low-speed solution is just *v*<sup>0</sup> 0.02 *c* . The many high-speed solutions have to occur in pairs just above and below

high-speed solutions have to occur in pairs just above and below orbit speeds of the form

Figure 6. Positronium solutions.

A parenthetical note applies for Fig. 6: for same-mass systems, the amplitude of the radiation curve should be less by a factor of 4 because there is no center-of-mass motion. This sort of numerical detail does not significantly affect where the solutions fall. That is determined almost entirely by the cosine factors that produce the deep dips that intersect the peaks in the

**3.** The oscillatory nature of the angle factors can turn a situation of seeming repulsion into a situation of actual attraction. This phenomenon of sign reversal due to signal delay is well known to engineers, who often deal with oscillating signals in feedback control systems For the present application, consider two electrons in a circular orbit, and suppose they move at speed *πc*. One electron launches its signal radially outward. By the time this electron has executed half an orbit, this signal has expanded a distance equal to the orbit diameter. By then, the two electrons have exchanged places. So the expanding signal first contacts the second electron at exactly the signal launch point. Then the two electrons complete their orbit. At the end, the second electron finally understands its signal: it is to move radially. But by now, the two electrons have changed places again, and for the second electron, the direction commanded is *inward*. That situation is equivalent to

orbit speeds of the form *vn v*<sup>0</sup> *n* 2*c* , where *n* is an arbitrary positive integer.

*vn* =*v*<sup>0</sup> + *n* ×2*πc*, where *n* is an arbitrary positive integer.

dependence. The one low-speed solution is just *v*<sup>0</sup> ≈0.02×*c*. The many

*27*

Series1 Series2

Series1 Series2

amplitude, or its *r* <sup>−</sup><sup>4</sup>

0

curve for rate of energy gain due torquing.

**Figure 6.** Positronium solutions.

*attraction*.

5

10

15

20

25

30

35

0

5

10

15

68 Selected Topics in Applications of Quantum Mechanics

20

25

30

35

the radiation is quadrupole, rather than dipole, so the amplitude of the radiation energy loss curve should decline as *r* 6 rather than *r* <sup>4</sup> . Again, this sort of numerical detail does not significantly affect where the solutions fall, which is determined almost entirely by the cosine factors that produce the deep dips that intersect the peaks in the curve for rate of energy gain due torquing. The stable solutions for two electrons bring to mind the situation that is so well known in Chemistry: electron pairs. They are everywhere in Chemistry. The most famous case occurs for Helium. Helium is a noble gas, and it reacts with other elements only under extreme duress. Helium has two electrons, and pulling one electron away is very costly: Helium has the highest ionization potential of any element. The message is: two electrons definitely do form a stable subsystem within an atom. An additional parenthetical note applies for Fig. 7: for same-charge systems, the angular pattern of the radiation is quadrupole, rather than dipole, so the amplitude of the radiation energy loss curve should decline as *r* <sup>−</sup><sup>6</sup> rather than *r* <sup>−</sup><sup>4</sup> . Again, this sort of numerical detail does not significantly affect where the solutions fall, which is determined almost entirely by the cosine factors that produce the deep dips that intersect the peaks in the curve for rate of energy gain due torquing.

Figure 7. Same-charge solutions.

The standard QM explanation for this invokes the concept of electron spin, with two possible values, The stable solutions for two electrons bring to mind the situation that is so well known in Chemistry: electron pairs. They are everywhere in Chemistry. The most famous case occurs for Helium. Helium is a noble gas, and it reacts with other elements only under extreme duress. Helium has two electrons, and pulling one electron away is very costly: Helium has the highest ionization potential of any element. The message is: two electrons definitely do form a stable subsystem within an atom. The standard QM explanation for this invokes the concept of electron spin, with two possible values, ±ℏ / 2, allowing two electrons in the same overall energy state. Electron pairing also occurs famously in Solid State Physics, under the name of Cooper Pairs.

#### **8. Conclusion**

In the present Chapter, the new concept applied is the more realistic signal model for use in an improved version of SRT. The realistic signal model is based on Information Theory,

The new concept is implemented with very standard mathematics: differential equations, their family of solutions, and the particular problem boundary conditions. These mathematical ingredients for a proper signal model were all available in 1905, but they were not used in SRT. Why? I believe the fundamental reason is the history: Information Theory was not yet available, so no researcher at that time would have been likely to detect the inadequacy of the infinite plane wave as a signal model.

The present paper has shown that there are rewards for instead using the realistic photon/ signal model. They include more insight into Quantum Mechanics, and into Gravity Theory, and potentially into Elementary Particle Physics. These are all subjects to be studied much more fully in the future.

Many textbook treatments of SRT devote a lot of space to Lorentz Transformations (LT's). The present work has not mentioned LT's at all. To this author, LT's just seem to describe the wrongly informed opinions of different observers. So I don't really want to focus on LT's. But I have to mention them, because repairing SRT to take proper account of the concepts of IT casts doubt on Einstein's SRT, and hence on LT's. Therefore, I hereby relegate the unavoidable discussion of coordinate transformations, LT's and others, to the following Appendix.

#### **9. Appendix**

The situation in the late nineteenth century included the following fact: Maxwell's first order coupled field equations appeared *not* to be invariant under Galilean transformation of coordinates (GT's). Phipps [28] has written extensively about this apparent conflict between Maxwell's Electromagnetic Theory and Newton's Mechanics. In the early twentieth century, SRT brought in LT's, and the conflict seemed to be resolved: Maxwell's electromagnetic theory was clearly invariant under LT's. This fact was taken as evidence in favor of Einstein's SRT over Newton's Mechanics.

But there is a puzzle left to resolve: Maxwell's first order coupled field equations appear to qualify as *tensor* equations. Mathematicians had developed tensors in the first place to enable the articulation of mathematical statements that would be *coordinate-free*. So tensor equations are *by definition* invariant to *all* invertible coordinate transformations.

So what had happened here? I believe two circumstances had collided to create a very bad situation. One circumstance was that Mathematics had such a long history of developing *eternal* truths: the focus had been on arithmetic, geometry, and trigonometry – all of them eternal in character. Even archeo-astronomy was largely about the eternal *repetition* of events, and *not* about temporal *evolution* of events. Eternal truths really need not have a time dimension. They can, however, have as many spatial dimensions as may be desired, and that became the focus for much of tensor analysis. The other circumstance was that time became a really significant variable with the advent of modern Physics: Kepler, Galileo, Newton, and Maxwell. And time is a really different kind of variable than space is. Maxwell was very well aware of the difference, as he developed his electromagnetic theory in terms of Hamilton's quaternions. The modern equivalent of the quaternion tool is the set of four 2×2 complex Pauli spin matrices:

$$
\sigma\_{ct} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \sigma\_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \sigma\_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, \sigma\_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}. \tag{75}
$$

The first one, the time-like one, is the identity matrix. The other three, the space-like ones, produce the identity matrix when squared. When two of them are cross-multiplied, they generate a factor of *i* = −1 times the third one, corresponding to the vector cross product in three-dimensional space.

The new concept is implemented with very standard mathematics: differential equations, their family of solutions, and the particular problem boundary conditions. These mathematical ingredients for a proper signal model were all available in 1905, but they were not used in SRT. Why? I believe the fundamental reason is the history: Information Theory was not yet available, so no researcher at that time would have been likely to detect the inadequacy of the infinite

The present paper has shown that there are rewards for instead using the realistic photon/ signal model. They include more insight into Quantum Mechanics, and into Gravity Theory, and potentially into Elementary Particle Physics. These are all subjects to be studied much

Many textbook treatments of SRT devote a lot of space to Lorentz Transformations (LT's). The present work has not mentioned LT's at all. To this author, LT's just seem to describe the wrongly informed opinions of different observers. So I don't really want to focus on LT's. But I have to mention them, because repairing SRT to take proper account of the concepts of IT casts doubt on Einstein's SRT, and hence on LT's. Therefore, I hereby relegate the unavoidable discussion of coordinate transformations, LT's and others, to the following Appendix.

The situation in the late nineteenth century included the following fact: Maxwell's first order coupled field equations appeared *not* to be invariant under Galilean transformation of coordinates (GT's). Phipps [28] has written extensively about this apparent conflict between Maxwell's Electromagnetic Theory and Newton's Mechanics. In the early twentieth century, SRT brought in LT's, and the conflict seemed to be resolved: Maxwell's electromagnetic theory was clearly invariant under LT's. This fact was taken as evidence in favor of Einstein's SRT

But there is a puzzle left to resolve: Maxwell's first order coupled field equations appear to qualify as *tensor* equations. Mathematicians had developed tensors in the first place to enable the articulation of mathematical statements that would be *coordinate-free*. So tensor equations

So what had happened here? I believe two circumstances had collided to create a very bad situation. One circumstance was that Mathematics had such a long history of developing *eternal* truths: the focus had been on arithmetic, geometry, and trigonometry – all of them eternal in character. Even archeo-astronomy was largely about the eternal *repetition* of events, and *not* about temporal *evolution* of events. Eternal truths really need not have a time dimension. They can, however, have as many spatial dimensions as may be desired, and that became the focus for much of tensor analysis. The other circumstance was that time became a really significant variable with the advent of modern Physics: Kepler, Galileo, Newton, and Maxwell. And time is a really different kind of variable than space is. Maxwell was very well aware of the difference, as he developed his electromagnetic theory in terms of Hamilton's quaternions. The modern equivalent of the quaternion tool is the set of four 2×2 complex Pauli spin matrices:

are *by definition* invariant to *all* invertible coordinate transformations.

plane wave as a signal model.

70 Selected Topics in Applications of Quantum Mechanics

more fully in the future.

**9. Appendix**

over Newton's Mechanics.

The collision between circumstances came in the formulation of differential operators. People were familiar with the scalar chain rule,*d/dt= ∂ / ∂<sup>t</sup> + v∂ / ∂<sup>x</sup> ,* and did not realize that more information was needed in the context of vector and tensor applications with time as well as space dimensions.

The 2×2 Pauli matrices are easy to appreciate visually, so I will also discuss transformations of coordinates in terms of 2×2 matrices also - but only real ones, not complex ones. Let *s* stand for any spatial coordinate. A general coordinate transformation involving *ct* and *s* has the form:

$$
\begin{bmatrix} ct^\prime \\ s^\prime \end{bmatrix} = \frac{1}{\sqrt{1 - AB}} \begin{bmatrix} 1 & B \\ A & 1 \end{bmatrix} \begin{bmatrix} ct \\ s \end{bmatrix} . \tag{76}
$$

For the familiar Lorentz transformation, *A*= *B* = −*v* / *c*, where *v* is the speed of the new coordi‐ nate frame relative to the old one. The letter *v* is lower case to remind us that *v* <*c* ; *i.e*. *v* / *c* <1. We have:

$$
\begin{bmatrix} ct \\ s \end{bmatrix} = \frac{1}{\sqrt{1 - v^2 / c^2}} \begin{bmatrix} 1 & -v/c \\ -v/c & 1 \end{bmatrix} \begin{bmatrix} ct \\ s \end{bmatrix}. \tag{77}
$$

For the long discarded Galilean transformation, *A*= −*V* / *c* and *B* =0. The letter *V* is upper case to remind us that *V* is *not* limited, and might exceed *c*. So we have:

$$
\begin{bmatrix} ct^\prime \\ s \end{bmatrix} = \frac{1}{1} \begin{bmatrix} 1 & 0 \\ -V/c & 1 \end{bmatrix} \begin{bmatrix} ct \\ s \end{bmatrix} \tag{78}
$$

For all such general coordinate transformations, there also exists a complement transforma‐ tion:

$$
\begin{bmatrix} ct \\ -s \end{bmatrix} = \frac{1}{\sqrt{1 - AB}} \begin{bmatrix} 1 & -A \\ -B & 1 \end{bmatrix} \begin{bmatrix} ct \\ -s \end{bmatrix}. \tag{79}
$$

Its purpose is to preserve inner products; for example:

$$
\begin{bmatrix} ct & s \\ \end{bmatrix} \begin{bmatrix} ct \\ -s \\ \end{bmatrix} = \begin{pmatrix} ct \\ \end{pmatrix}^2 - s^2 \tag{80}
$$

Observe that:

$$
\begin{bmatrix} ct & s \\ \end{bmatrix} \begin{bmatrix} ct \\ -s \\ \end{bmatrix} = \begin{bmatrix} ct & s \\ \end{bmatrix} \frac{1}{\sqrt{1 - AB}} \begin{bmatrix} 1 & A \\ B & 1 \end{bmatrix} \frac{1}{\sqrt{1 - AB}} \begin{bmatrix} 1 & -A \\ -B & 1 \end{bmatrix} \begin{bmatrix} ct \\ -s \\ \end{bmatrix} \tag{81}
$$
 
$$
= \begin{bmatrix} ct & s \\ \end{bmatrix} \frac{1}{1 - AB} \begin{bmatrix} 1 - AB & 0 \\ 0 & -BA + 1 \end{bmatrix} \begin{bmatrix} ct \\ -s \\ \end{bmatrix} \equiv \begin{pmatrix} ct \end{pmatrix}^2 - s^2 \quad . \tag{81}
$$

For Lorentz transformation, the complement transformation is the inverse, or equivalently, the *re*verse transformation:

$$
\begin{bmatrix} ct'' \\ -s'' \end{bmatrix} = \frac{1}{\sqrt{1 - v^2 / c^2}} \begin{bmatrix} 1 & +v/c \\ +v/c & 1 \end{bmatrix} \begin{bmatrix} ct \\ -s \end{bmatrix} . \tag{82}
$$

But for Galilean transformation, the complement transformation is:

$$
\begin{bmatrix} ct \ \text{''} \\ -s \ \text{''} \end{bmatrix} = \frac{1}{1} \begin{bmatrix} 1 & +V/c \\ 0 & 1 \end{bmatrix} \begin{bmatrix} ct \\ -s \end{bmatrix}. \tag{83}
$$

This *is* the *in*verse transformation, but *not* the *re*verse transformation. It is rather the *trans‐ pose* of the reverse transformation. It looks so very strange because, for more than a century now, *only* Lorentz transformations of velocity have been used in mainstream theoretical Physics, and transposition does not change them.

The vital role for this strange new thing lies with the differential operators. The story is much like it was for the coordinates: there are *two* complementing transformations, and they involve, not only inversion/reversal, but also transposition. Let ∂*ct* represent differentiation with respect to the time-like coordinate, and ∂*s* represent differentiation with respect to the spatial variable. Let us demand invariance of inner products involving differential operators; for example, like:

$$
\begin{bmatrix} \boldsymbol{\partial}\_{ct} & \boldsymbol{\partial}\_{s} \end{bmatrix} \begin{bmatrix} ct \\ -s \end{bmatrix} = \mathbf{1} - \mathbf{1} = \mathbf{0} \text{ and } \begin{bmatrix} \boldsymbol{\partial}\_{ct} & -\boldsymbol{\partial}\_{s} \end{bmatrix} \begin{bmatrix} ct \\ s \end{bmatrix} = \mathbf{1} - \mathbf{1} = \mathbf{0}
$$
 
$$
\text{or } \begin{bmatrix} \boldsymbol{\partial}\_{ct} & \boldsymbol{\partial}\_{s} \end{bmatrix} \begin{bmatrix} \boldsymbol{\partial}\_{ct} \\ -\boldsymbol{\partial}\_{s} \end{bmatrix} = \boldsymbol{\partial}\_{ct}^{2} - \boldsymbol{\partial}\_{s}^{2} \text{ and } \mathbf{d} = \begin{bmatrix} \boldsymbol{\partial}\_{ct} & -\boldsymbol{\partial}\_{s} \end{bmatrix} \begin{bmatrix} \boldsymbol{\partial}\_{ct} \\ \boldsymbol{\partial}\_{s} \end{bmatrix} = \boldsymbol{\partial}\_{ct}^{2} - \boldsymbol{\partial}\_{s}^{2}. \tag{84}
$$

*i.e*., always *two* statements – not just *one* statement. This level of detail was missing from the *scalar* chain rule, and that omission caused people to believe that Maxwell's equations could not be shown to be invariant under GT. And so they welcomed LT instead. This is not to say we should now revert to using GT again. Indeed, because of the half-retardation issue discussed in Sect. 3, the best transformation to use may involve, not *V* / *c*, but rather *V* /2*c*. This question needs detailed future study.

The use of 2×2 matrices can make the detail needed in such future study very clear. However, many mathematicians tend to prefer tensor notation. But current-day tensor notation uses only two index positions, both on the right: down called 'covariant', up, called 'contravariant'. To represent the transformations needed for Physics, it would be helpful, and maybe necessary, to add two more index locations, up and down on the *left*, to acknowledge transposition, and using words like 'trans-covariant' and 'trans-contravariant' to emphasize what putting indices in those positions means.

#### **Acknowledgements**

Its purpose is to preserve inner products; for example:

72 Selected Topics in Applications of Quantum Mechanics

*ct s ct s*

Physics, and transposition does not change them.

Observe that:

*re*verse transformation:

2 2 () .

é ù é ùê ú = - ë û -ë û (80)

(81)

(84)

2 2

é ù é ùé ù <sup>+</sup> ê ú <sup>=</sup> ê úê ú - +- ë û - ë ûë û (82)

é ù é ùé ù <sup>+</sup> ê ú ê úê ú <sup>=</sup> - - ë û ë ûë û (83)

*ct ct s ct s s*

" 11 1 1 ' '

1 1 0

<sup>é</sup> - ùé ù <sup>=</sup> é ù <sup>ê</sup> úê ú º - ë û - -+- <sup>ë</sup> ûë û

But for Galilean transformation, the complement transformation is:

" 11 1 1

For Lorentz transformation, the complement transformation is the inverse, or equivalently, the

*ct A A ct*

*s AB B Bs AB*

() . <sup>1</sup> 0 1

é ù é ù é ùé ù - é ù éù ë û ëû ê ú <sup>=</sup> ê ú ê úê ú - - - - - ë û ë û ë ûë û

*AB ct ct s ct s AB BA s*

> 2 2 " 1 <sup>1</sup> . " 1 1 / *ct v c ct s vc s v c*

> > " 1 <sup>1</sup> . " 01 <sup>1</sup> *ct V c ct s s*

This *is* the *in*verse transformation, but *not* the *re*verse transformation. It is rather the *trans‐ pose* of the reverse transformation. It looks so very strange because, for more than a century now, *only* Lorentz transformations of velocity have been used in mainstream theoretical

The vital role for this strange new thing lies with the differential operators. The story is much like it was for the coordinates: there are *two* complementing transformations, and they involve, not only inversion/reversal, but also transposition. Let ∂*ct* represent differentiation with respect to the time-like coordinate, and ∂*s* represent differentiation with respect to the spatial variable. Let us demand invariance of inner products involving differential operators; for example, like:

1 1 0 and 110

or and .

é ù ¶ ¶é ù é ù ¶ ¶ = ¶ - ¶ = ¶ -¶ = ¶ - ¶ é ù ë ûê ú ë ûê ú -¶ ¶ ë û ë û

*ct ct s s* é ù é ù é ù ¶ ¶ = - = ¶ -¶ = - = é ù ë ûê ú ë ûê ú -ë û ë û

> *ct ct ct s ct s ct s ct s s s*

*ct s ct s*

2 2 2 2

This Chapter is dedicated to the memory of a most courageous researcher in theoretical and applied electrodynamics: Dr. Peter Graneau, 1921-2014. He encouraged me, and many other researchers, to give serious attention to the History behind Physics.

### **Author details**

Cynthia Kolb Whitney\*

Address all correspondence to: galilean\_electrodynamics@comcast.net

Galilean Electrodynamics, USA

#### **References**


[18] Enders, P., Quantization as Selection Rather than Eigenvalue Problem, Chapt 23, pp. 543-564, *Advances in Quantum Mechanics*, Intech (2013).

[4] Flores-Gallegos, N., Shannon Informational Entropies and Chemical Reactivity,

[5] Jackson, J.D., *Classical Electrodynmics*, Second Edition, John Wiley & Sons, New York,

[6] Bentwich, J., The Theoretical Ramifications of the Computational Unified Field Theo‐

[7] Whitney, C.K., Better Unification for Physics in General Through Quantum Mechan‐ ics in Particular, Chapter 7, pp. 127-160 in *Theoretical Concepts of Quantum Mechanics*,

[8] Ritz, W., Researches critiques sur l'electrodynamique generale, Ann. Chim. et Phys.

[9] Putz, M.V., Path Integrals for Electronic Densities, Reactivity Indices, and Localiza‐ tion Functions in Quantum Systems, *International Journal of Molecular Sciences* 10, pp.

[10] Bracken, P., Quantum Mechanics Entropy and a Quantum Version of the H-Theo‐ rem, Chapt. 21, pp. 469-488 in *Theoretical Concepts of Quamtum* Mechanics 469-488

[11] Bracken, P., The Schwinger Action Principle and its Applications to Quantum Me‐ chanics, Chapt. 8, pp 159-182 in *Advances in Quantum Mechanics*, Intech (2013).

[12] Spitnev, V., Generalized Path Intergral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference, Chapt 9., pp. 183-212 in *Advances in Quantum Me‐*

[13] Bulnes, F., Correction, Restoration and Re-Composition if Quantum Mechanical Fields of Particles by Path Integrals and Their Applications, 489-514 in *Advances in*

[14] Liénard, A., Champ Electrique et Magnétique produit par une Charge Electrique Concentrée en un Point et Animée d'un Movement Quelconque, *L'Eclairage Electri‐*

[15] Wiechert, E., Elektrodynamische Elementargesetze, *Archives Néerlandesises des Scien‐*

[16] Lokajicek, M., V. Kundrát, and J. Procházka, Schrödinger Equation and (Future) Quantum Physics, Chapter 6 in Advances in Quantum Mechanics, Ed. Paul Bracken,

[17] Whitney, C.K., On the Dual Concepts of 'Quantum State' and 'Quantum Process', Chapter 17 in *Advances in Quantum Mechanics*, Ed. Paul Bracken, InTech. (2013).

Chapt. 29, pp. 683-722 in *Advances in Quantum Mechanics,* Intech (2013).

ry, Chapt. 28, pp. 671-681, *Advances in Quantum Mechanics*, Intech (2013).

NY. (1975).

Ed. M.R. Pahlavani, InTech. (2012).

13, pp. 145-275, (1908).

74 Selected Topics in Applications of Quantum Mechanics

4816-4940 (2009).

*chanics*, Intech (2013).

InTech. (2013).

*Quantum Mechanics*, Intech (2013).

*que*, *XVI*, pp. 5-14, 53-59, and 106-112 (1898).

*ces Exactes et Naturelles*, *série II, Tome IV*, pp. 549 – 573 (1901).

(2012).


#### **Path Integral Methods in Generalized Uncertainty Principle Path Integral Methods in Generalized Uncertainty Principle**

Hadjira Benzair, Mahmoud Merad and Taher Boudjedaa Hadjira Benzair1, Mahmoud Merad2 and Taher Boudjedaa3

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

#### **1. Introduction**

10.5772/59106

As we known the modern physics is based on the two fundamental pillars of physics. The first is the general relativity theory, discovered by Albert Einstein, which gave us a detailed explanation of the macro-dimension world; for example, planets, stars, galaxies and clusters of galaxies and even the extra-universe, that explains the force exerted by the gravitational field of a massive object on any body within the vicinity of its surface. It mainly uses the Riemannian geometry as a mathematical formalism. The second perspective is the quantum mechanics, that describes the micro-dimensions, such as; molecules, atoms and even the smallest components of the latter, like the electrons and quarks, which explains the three principal forces in the micro-world, (like the weak force, electromagnetic force and strong force). It uses the operator theory acting on a Hilbert space algebra (von Neumann algebras). After the mid-twentieth century a new theory in physics has been emerged called the Non-commutative (NC) geometry. It came to unify the four fundamental forces, And its roots go back to the inability of classical physics to explain certain macroscopic phenomena. Mathematically described by a Poisson manifold *M*, and denoted by *F*(*M*) algebra (commutative) regular functions on *M*, called observable. In this case, it is important to quantify these Poisson varieties (quantum mechanics) in order to obtain results more "precise" than classical mechanics. Many studies have focused on the possibility of quantification of such varieties and the idea of using the theory of algebraic deformations, called "deformation of quantization" is due to (Bayen et al, 1978). And as has been creativity in this mathematical aspect, through a group of researchers ( for example, (Bordemann et al., (2005); Makhlouf, 2007)). The motivations to the occurrence of this deformation theory are multiple, in a string theory, (see for example (Veneziano, (1986); Amati et al, (1987); Konishi et al, (1990); Kato, (1990) and Guida et al, (1991) also Gross et al, 1988) in a quantum gravity, (Garay, 1995) in a non-commutative geometry, (Capozziello et al, 2000) and in a black hole physics (Scardigli, (1999); Scardigli & Casadio, (2003)).

©2012 prezimena autora, kod vise prvi et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

In the same context, the innovations physicists were prominent by inserting this study algebra on several applications in physics. The first of these applications is the papers of (Kempf et al, 1995), which is based on introducing a parameter of deformation *β* in the Heisenberg incertitude principle, given by:

$$
\Delta x \Delta p \geqslant \frac{\hbar}{2} \left[ 1 + \beta \left( \Delta p \right)^2 + \beta \left< p \right>^2 \right], \tag{1}
$$

Where the commutation between the position and the momentum operators in one dimension, can be written as:

$$\left[\widehat{\mathbf{x}}, \widehat{p}\right] = i\hbar \left(1 + \beta p^2\right),\tag{2}$$

This example, we have found several applications in the non-relativistic quantum mechanics; such as, the harmonic oscillator of arbitrary dimensions (cf., e.g., Refs (Chang et al, (2002); Hinrichsen & Kempf, (1996); Kempf, (1994), Kempf et al, (1995) and Kempf, (1997)), the problem of the cosmological constant has been studied (Pet & Polchinski, (1999)), the effect of the minimal length on the 3-D Coulomb potential has also been studied in (Brau, (1999) and Akhoury, (2003)), the one-dimensional box (Nozari & Azizi, (2006))*,* the study of the dynamics of a non-relativistic particle with mass variable *m*(*t*) (cf., e.g., Ref Merad & Falek, (2009)), and also in the relativistic extension of this problem has some limited attempts, among them we mention: the Dirac equation in the presence of a minimum length in (Nouicer, (2006)), where the Dirac oscillator in one dimension has been solved exactly, the generalized Dirac equation was recently studied by Nozari (Nozari & Karami, (2005)), the bosonic oscillator DKP (spin 0 and 1)-dimensional and three-dimensional that were treated respectively in (Falek & Merad, (2009); (2010)).

On the other hand, the path integral is an alternative technical of Heisenberg and Schrödinger methods. This approach is based on the Lagrangian form, which offers an alternative view of quantum mechanics, that has quickly established itself in theoretical physics, with its extension on quantum field theory and gauge theories. The extension of this technique within the framework of deformed algebra was applied to the relativistic and non-relativistic quantum mechanics. For example, the harmonic oscillator in one dimension (Nouicer, 2006) and in D dimensions (Chergui et al, 2010) and the (1+1)-dimensional Klein–Gordon equation with mixed vector-scalar linear potentials (Merad et al, 2010) but recently it is shown that the problem concerning the choice of point discretization in the path integral is not yet resolved and this arbitrariness is fixed by comparing the discrete action in its infinitesimal form with the corresponding wave equation by which judicious choice of the discretization parameter is indicated by the order of operators ( cf., e.g., Refs Benzair et al, (2012); (2014)). This resembles the case of curved spaces in which the mid-point (i.e. *x*¯ = *<sup>x</sup>* + *<sup>x</sup>*−<sup>1</sup> /2 ) was privileged to have correct quantum correction due to the curvature. Similar arguments in the case of space-time transformations (cf., e.g., Ref (Khandekar et al, 1993)) are presented. In ( Ref (Kleinert, 1990)), an outcome considers all points of the interval in an equivalent manner but unfortunately with minimal length deformation. The problem is raised and we will say that it is more like that of the quantization with constraints (see, e.g., Ref (Lecheheb et al, 2007)).

In this chapter we propose to construct the path integral formalism in the momentum space representation to adapt this type of deformation, defined in Eqs. (1) and (2). Then, we

describe in detail the method of calculating the quantum corrections according to Feynman approach (cf., e.g., Ref (Khandekar et al, 1993)). As it is shown in (Benzair et al, (2012); (2014)), different methods gave different results, where the quantum correction *CT* depends on the *α*-point discretization interval and there are specific options for the choice of the discretization *α*−parameter, which coincides with the equation method, and this leads to the vanishing of the term *CT* and this corresponds to *<sup>α</sup>* = 0 and *<sup>α</sup>* = 1/2 within the method of (Kleinert, 1990) and to *α* = <sup>1</sup> 2 <sup>1</sup> <sup>±</sup> 1/√<sup>2</sup> within the standard method of (Khandekar et al, 1993).

#### **2. Brief review of a minimal length relation**

2

In the same context, the innovations physicists were prominent by inserting this study algebra on several applications in physics. The first of these applications is the papers of (Kempf et al, 1995), which is based on introducing a parameter of deformation *β* in the

1 + *β* (∆*p*)

Where the commutation between the position and the momentum operators in one

 1 + *βp*<sup>2</sup> 

This example, we have found several applications in the non-relativistic quantum mechanics; such as, the harmonic oscillator of arbitrary dimensions (cf., e.g., Refs (Chang et al, (2002); Hinrichsen & Kempf, (1996); Kempf, (1994), Kempf et al, (1995) and Kempf, (1997)), the problem of the cosmological constant has been studied (Pet & Polchinski, (1999)), the effect of the minimal length on the 3-D Coulomb potential has also been studied in (Brau, (1999) and Akhoury, (2003)), the one-dimensional box (Nozari & Azizi, (2006))*,* the study of the dynamics of a non-relativistic particle with mass variable *m*(*t*) (cf., e.g., Ref Merad & Falek, (2009)), and also in the relativistic extension of this problem has some limited attempts, among them we mention: the Dirac equation in the presence of a minimum length in (Nouicer, (2006)), where the Dirac oscillator in one dimension has been solved exactly, the generalized Dirac equation was recently studied by Nozari (Nozari & Karami, (2005)), the bosonic oscillator DKP (spin 0 and 1)-dimensional and three-dimensional that were treated

On the other hand, the path integral is an alternative technical of Heisenberg and Schrödinger methods. This approach is based on the Lagrangian form, which offers an alternative view of quantum mechanics, that has quickly established itself in theoretical physics, with its extension on quantum field theory and gauge theories. The extension of this technique within the framework of deformed algebra was applied to the relativistic and non-relativistic quantum mechanics. For example, the harmonic oscillator in one dimension (Nouicer, 2006) and in D dimensions (Chergui et al, 2010) and the (1+1)-dimensional Klein–Gordon equation with mixed vector-scalar linear potentials (Merad et al, 2010) but recently it is shown that the problem concerning the choice of point discretization in the path integral is not yet resolved and this arbitrariness is fixed by comparing the discrete action in its infinitesimal form with the corresponding wave equation by which judicious choice of the discretization parameter is indicated by the order of operators ( cf., e.g., Refs Benzair et al, (2012); (2014)). This resembles

to have correct quantum correction due to the curvature. Similar arguments in the case of space-time transformations (cf., e.g., Ref (Khandekar et al, 1993)) are presented. In ( Ref (Kleinert, 1990)), an outcome considers all points of the interval in an equivalent manner but unfortunately with minimal length deformation. The problem is raised and we will say that it is more like that of the quantization with constraints (see, e.g., Ref (Lecheheb et al, 2007)). In this chapter we propose to construct the path integral formalism in the momentum space representation to adapt this type of deformation, defined in Eqs. (1) and (2). Then, we

[*<sup>x</sup>*, *<sup>p</sup>*] <sup>=</sup> *ih*¯

<sup>2</sup> + *β* �*p*�

2 

, (1)

, (2)

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

/2 ) was privileged

Heisenberg incertitude principle, given by:

respectively in (Falek & Merad, (2009); (2010)).

the case of curved spaces in which the mid-point (i.e. *x*¯ =

dimension, can be written as:

∆*x*∆*p <sup>h</sup>*¯ 2 

> As it seems that in the Kempf's work (cf., e.g., Ref (Kempf et al, 1995)), there is a minimal value of (∆*x*)min different zero which is given by:

$$(\Delta x)\_{\min} \left( \langle p \rangle \right) = \hbar \sqrt{\beta} \sqrt{1 + \beta \left\langle p \right\rangle^2}. \tag{3}$$

= *h*¯ *β* corresponds to �*p*� = 0. (4)

The operators (*x*, *<sup>p</sup>*) that verifies the commutation relation amended (2) may be considered as the functions of *q* and *p* operators, satisfying the relationship of canonical commutation: [*q*ˆ, *p*ˆ] = *ih*¯, as follow

$$
\hat{\mathfrak{X}} = i\hbar \left( 1 + \beta p^2 \right) \mathfrak{f}\_{\prime} \ \mathfrak{f} = p. \tag{5}
$$

In the momentum space representation, we define the expressions of *x*ˆ and *p*ˆ act on the functions Ψ (*p*) defined by:

$$
\hat{p}.\Psi\left(p\right) = p\Psi\left(p\right), \ \hat{\mathbf{x}}\Psi\left(p\right) = i\hbar \left(1 + \beta p^2\right) \frac{\partial}{\partial p}\Psi\left(p\right). \tag{6}
$$

The most important condition to be satisfied by the representation (2), is the preservation of the operators symmetry *x* and *p*, where their values are real. Despite the fact that *p* is not modified, then its symmetry is obvious; it is not the case for the *x* operator. Indeed, the symmetry condition is written

$$\left( \left( \Psi \right| \widehat{\mathfrak{x}} \right) \left| \Phi \right\rangle = \left\langle \Psi \right| \left( \widehat{\mathfrak{x}} \left| \Phi \right\rangle \right). \tag{7}$$

The scalar product should be defined as

$$
\langle \Psi \mid \Phi \rangle = \int\_{-\infty}^{+\infty} \frac{dp}{1 + \beta p^2} \Psi^\* \left( p \right) \Phi \left( p \right) \,. \tag{8}
$$

The modification of this product implies a new closure relation, which is written as

$$\int\_{-\infty}^{+\infty} \frac{dp}{1 + \pounds p^2} \, |p\rangle \, \langle p| = 1. \tag{9}$$

Inserting the latter relation in the scalar product of two momentum eigenvectors operator, we get:

$$
\langle \begin{pmatrix} p \ \vert \ p' \end{pmatrix} \rangle = \left( 1 + \beta p^2 \right) \delta \left( p - p' \right) \,, \tag{10}
$$

also is given by

$$
\langle p|\left|p'\right> = \delta\left(\frac{1}{\sqrt{\beta}}\arctan\sqrt{\beta}p - \frac{1}{\sqrt{\beta}}\arctan\sqrt{\beta}p'\right).\tag{11}
$$

In this case the Schrödinger equation for the particle in the harmonic oscillator of momentum space representation in one dimension, can be written as

$$\hat{H} = \left(\frac{\hat{p}^2}{2m} + \frac{m\omega^2}{2}\hat{\mathbf{x}}^2\right) = \left[\frac{p^2}{2m} + \frac{m\omega^2}{2}\left(i\hbar\left(1 + \beta p^2\right)\frac{\partial}{\partial p}\right)^2\right].\tag{12}$$

Exact solutions of spectrum energy and the normalized eigenfunctions of the bound states are defined in (Chang et al, (2002):

$$E\_{\rm nl} = \hbar\omega \left[ \left( n + \frac{1}{2} \right) \sqrt{1 + \left( \frac{\beta\hbar m\omega}{2} \right)^2} + \left( n^2 + n + 1 \right) \left( \frac{\beta\hbar m\omega}{2} \right) \right]. \tag{13}$$

and

$$\Psi\_n(p) = \sqrt{\frac{2^{2\lambda - 1}(\lambda + n)n!\sqrt{\beta}}{\pi \Gamma(2\lambda + n)[\Gamma(\lambda)]^{-2}}} \left[\frac{1}{\sqrt{1 + \beta p^2}}\right]^\lambda \mathcal{C}\_n^\lambda \left(\frac{\sqrt{\beta}p}{\sqrt{1 + \beta p^2}}\right),\tag{14}$$

with *C<sup>λ</sup> <sup>n</sup>* are Gegenbauer polynomials.

#### **3. Construction propagators with generalized Heisenberg principle**

The purpose of this section is to discuss the propagators and the quantum corrections via the standard Feynman approach, for a non-relativistic quantum mechanics in the context of the deformed and non-deformed space at *α*-point discretization. Then we will circulate this study on the relativistic problems through the two applications chosen below.

#### **3.1. Ordinary quantum mechanics case**

In this subsection, we will illustrate the spatio-temporal technique and the method of calculating the quantum corrections according to the standard Feynman approach in the

ordinary quantum mechanics. So, in one dimension, we consider the propagator expression of ordinary non-relativistic quantum mechanics to the discontinuous form path integral

$$K\_N = A\_N \int \exp\left\{ \frac{i}{\hbar} \sum\_{n=1}^{N+1} S\_{\hbar} \right\} \prod\_{n=1}^N dz\_{\hbar \nu} \tag{15}$$

with *AN* = *m* 2*πih*¯*ε N*+<sup>1</sup> and *Sn* is the discrete action into intervals [*n* − 1, *n*] , takes the form follows:

$$S\_{\mathfrak{N}} = \frac{\mathfrak{m}}{2\varepsilon} \left( z\_{\mathfrak{n}} - z\_{\mathfrak{n}-1} \right)^2 - \varepsilon V\left( \mathfrak{x}\_{\mathfrak{n}}\right). \tag{16}$$

According to the standard method of Feynman (cf., e.g., Ref (Khandekar et al, 1993)), we will apply the spatio-temporal method of processing *z*¯ (*α*) *<sup>n</sup>* = *<sup>f</sup>*(*q*¯ (*α*) *<sup>j</sup>* ) to *<sup>α</sup>*-point discretization defined as:

$$
\bar{z}\_n^{(\mathfrak{a})} = \mathfrak{a}z\_n + \left(1 - \mathfrak{a}\right)z\_{n-1}.\tag{17}
$$

Two terms of quantum corrections have appeared (*Cact*, *Cmes*). Let's start by calculating the correction to the *Cact* action. Developing ∆*zn* to *α*-point discretization, we have:

$$
\Delta z\_{\rm nl} = \Delta q\_{\rm nl} \overline{f}\_{\rm{n}}^{(a)} \left( 1 + \frac{(1 - 2a)}{2!} \frac{f\_{\rm{n}}^{(a)} \overline{f}\_{\rm{n}}^{(a)}}{\overline{f}\_{\rm{n}}^{(a)}} \Delta q\_{\rm{n}} + \frac{(1 - a)^3 + a^3}{3!} \frac{f\_{\rm{n}}^{(a)} \overline{f}\_{\rm{n}}^{(a)}}{\overline{f}\_{\rm{n}}^{(a)}} \Delta q\_{\rm{n}}^2 \right), \tag{18}
$$

where the prime on the function ¯ *f* (*α*) *<sup>j</sup>* indicates the derivative ¯ *f* (*α*) *<sup>j</sup>* over *q*¯ (*α*) *<sup>j</sup>* . So, the kinetic energy term in the action is:

$$\frac{\Lambda z\_n^2}{4\varepsilon} = $$

$$\frac{\Lambda q\_n^2}{4\varepsilon} \left( \tilde{f}\_n^{(a)} \right)^2 \left( 1 + (1 - 2a) \frac{\tilde{f}\_n^{(a)} \prime}{\tilde{f}\_n^{(a)} \prime} \Delta q\_n + \left[ \frac{(1 - 2a)^2}{4} \left( \frac{\tilde{f}\_n^{(a)} \prime}{\tilde{f}\_n^{(a)} \prime} \right)^2 + \frac{\left\{1 - a^3\right\} + a^3}{3} \frac{\tilde{f}\_n^{(a)} \prime}{\tilde{f}\_n^{(a)} \prime} \right] \Delta q\_n^2 \right). \tag{19}$$

The potential energy takes a simple form:

4

we get:

and

with *C<sup>λ</sup>*

also is given by

The modification of this product implies a new closure relation, which is written as

*dp*

Inserting the latter relation in the scalar product of two momentum eigenvectors operator,

arctan *β<sup>p</sup>* − √

In this case the Schrödinger equation for the particle in the harmonic oscillator of momentum

Exact solutions of spectrum energy and the normalized eigenfunctions of the bound states

 *<sup>β</sup>hm*¯ *<sup>ω</sup>* 2 2 + 

√*β*

**3. Construction propagators with generalized Heisenberg principle**

 √ 1 1+*βp*<sup>2</sup>

The purpose of this section is to discuss the propagators and the quantum corrections via the standard Feynman approach, for a non-relativistic quantum mechanics in the context of the deformed and non-deformed space at *α*-point discretization. Then we will circulate this

In this subsection, we will illustrate the spatio-temporal technique and the method of calculating the quantum corrections according to the standard Feynman approach in the

1 *β* arctan *βp*′

*n*<sup>2</sup> + *n* + 1

*λ Cλ n*

 *∂ ∂p* 2 

 *<sup>β</sup>hm*¯ *<sup>ω</sup>* 2

 <sup>√</sup>*β<sup>p</sup>* √1+*βp*<sup>2</sup> <sup>1</sup>+*βp*<sup>2</sup> <sup>|</sup>*p*� �*p*<sup>|</sup> <sup>=</sup> 1. (9)

, (10)

. (11)

. (12)

. (13)

, (14)

 <sup>+</sup><sup>∞</sup> −∞

 *<sup>p</sup>* <sup>|</sup> *<sup>p</sup>*′ = 1 + *βp*<sup>2</sup> *δ <sup>p</sup>* <sup>−</sup> *<sup>p</sup>*′ 

� = *δ* √ 1 *β*

space representation in one dimension, can be written as

<sup>2</sup> *<sup>x</sup>*<sup>2</sup> = *p*2 <sup>2</sup>*<sup>m</sup>* <sup>+</sup> *<sup>m</sup>ω*<sup>2</sup> 2 *ih*¯ 1 + *βp*<sup>2</sup>

<sup>2</sup>2*λ*−1(*λ*+*n*)*n*!

*<sup>π</sup>*Γ(2*λ*+*n*)[Γ(*λ*)]−<sup>2</sup>

study on the relativistic problems through the two applications chosen below.

�*p*<sup>|</sup> *<sup>p</sup>*′

*H*ˆ = *<sup>p</sup>*<sup>2</sup> <sup>2</sup>*<sup>m</sup>* <sup>+</sup> *<sup>m</sup>ω*<sup>2</sup>

are defined in (Chang et al, (2002):

*En* = *h*¯ *ω*

 *n* + <sup>1</sup> 2 1 +

Ψ*<sup>n</sup>* (*p*) =

*<sup>n</sup>* are Gegenbauer polynomials.

**3.1. Ordinary quantum mechanics case**

$$
\varepsilon V(z\_n) = \varepsilon V(\bar{q}\_n^{(s)}) + \mathcal{O}(\varepsilon^2) = \varepsilon V(\bar{q}\_n^{(s)}).\tag{20}
$$

In addition, we note that the transformation *z* = *f*(*q*) made the path integral rather complicated, where the mass parameter is transformed into *m* ¯ *f* (*α*)′ *<sup>n</sup>* . At this point, we would apply the transformation over time parameter in order to overcome this difficulty:

$$
\varepsilon = \sigma\_n f'\left(q\_n\right) f'\left(q\_{n-1}\right), \text{ where } \sigma\_n = s\_n - s\_{n-1}.\tag{21}
$$

Development *<sup>f</sup>* ′ (*qn*) and *<sup>f</sup>* ′ (*qn*−1) at *<sup>α</sup>*-point discretization in order two of <sup>∆</sup>*qn*, is written as:

$$\varepsilon = \sigma\_n \left( \bar{f}\_n^{(a)} \right)^2 \left( 1 + (1 - 2\mathfrak{a}) \frac{\bar{f}\_n^{(a)} \, \_n}{\bar{f}\_n^{(a)} \,} \Delta q\_n + \left( \frac{(1 - \mathfrak{a})^2 + \mathfrak{a}^2}{2} \frac{\bar{f}\_n^{(a)} \, \_n}{\bar{f}\_n^{(a)} \,} - \mathfrak{a} \left( 1 - \mathfrak{a} \right) \left( \frac{\bar{f}\_n^{(a)} \, \_n}{\bar{f}\_n^{(a)} \,} \right)^2 \right) \Delta q\_n^2 \right). \tag{22}$$

From expressions (19) and (22), we can deduce the quantum correction from the action:

$$\exp\left[\frac{i}{\hbar}\frac{m}{2\varepsilon}\left(\Delta z\_n\right)^2\right] = \exp\left[\frac{i}{\hbar}\frac{m}{2\sigma\_n}\left(\Delta q\_n\right)^2\right] \left(1 + \mathbb{C}\_{act}\right) \tag{23}$$

with

$$\mathcal{C}\_{\text{act}} = \frac{i}{\hbar} \frac{m}{8\sigma\_{\text{u}}} \Delta q\_{\text{u}}^{4} \left[ \left( -16a^{2} + 16a - 3 \right) \left( \frac{\bar{f}\_{\text{u}}^{(a)} \prime}{\bar{f}\_{\text{u}}^{(a)} \prime} \right)^{2} - \frac{2}{3} \frac{\bar{f}\_{\text{u}}^{(a)} \prime \prime}{\bar{f}\_{\text{u}}^{(a)} \prime} \right]. \tag{24}$$

Turning now to calculate the second correction *Cmes*, we have:

$$A\_N \prod\_{n=1}^N dz\_n = \left(\sqrt{\frac{m}{2\pi i \hbar \varepsilon}}\right)^{N+1} \prod\_{n=1}^N dq\_n f'\left(q\_n\right). \tag{25}$$

This can be achieved by rewriting:

$$A\_N \prod\_{n=1}^N dz\_n = \left[ f'(q\_b) \, f'(q\_d) \right]^{-1/2} \prod\_{n=1}^{N+1} \left( \sqrt{\frac{m f'(q\_n) f'(q\_{n-1})}{2 \pi i \hbar \varepsilon}} \right) \prod\_{n=1}^N dq\_{n.} \tag{26}$$

Then, we develop *<sup>f</sup>* ′ (*qn*) and *<sup>f</sup>* ′ (*qn*−1) to the second order of <sup>∆</sup>*qn* as follows:

$$\left[f'\left(q\_n\right)f'\left(q\_{n-1}\right)\right]^{1/2} = \tilde{f}\_n^{(a)} \left(1 + \frac{(1-2a)}{2} \frac{\tilde{f}\_n^{(a)} \, ''}{f\_n^{(a)} \, '} \Delta q\_n + \left(\frac{(1-a)^2 + a^2}{4} \frac{\tilde{f}\_n^{(a)} \, '''}{f\_n^{(a)} \, '} - \frac{a(1-a)}{2} \left(\frac{\tilde{f}\_n^{(a)} \, '}{f\_n^{(a)} \, '}\right)^2\right) \Delta q\_n^2\right) \tag{27}$$

From the expressions (26), (27) and (22) we can deduce *Cmes*,

$$A\_N \prod\_{n=1}^N dz\_n = \prod\_{n=1}^{N+1} \left( \sqrt{\frac{m}{2\pi i \hbar \sigma\_n}} \right) \left( 1 + \mathbb{C}\_{m \text{cs}} \right) \prod\_{n=1}^N dq\_{n\text{.}} \tag{28}$$

to the form

$$\mathbb{C}\_{\text{mes}} = \frac{(1 - 2\alpha)^2}{4} \left( \frac{f\_{\boldsymbol{l}}^{(a)} \boldsymbol{\prime}}{f\_{\boldsymbol{l}}^{(a)} \boldsymbol{\prime}} \right)^2 \Delta q\_n^2. \tag{29}$$

.

To calculate the total quantum corrections *CT*, we use the following expression

$$
\left\langle \left( \Delta q \right)^{2n} \right\rangle = \left( \frac{i\hbar \sigma}{m} \right)^n \left( 2n - 1 \right)!! \,, \tag{30}
$$

The result is

6

*ε* = *σ<sup>n</sup>*

with

*<sup>f</sup>* ′ (*qn*) *<sup>f</sup>* ′ (*qn*−1)

to the form

 ¯ *f* (*α*)′ *n* 2 

<sup>1</sup> <sup>+</sup> (<sup>1</sup> <sup>−</sup> <sup>2</sup>*α*) ¯

exp *i h*¯ *m* <sup>2</sup>*<sup>ε</sup>* (∆*zn*) 2 = exp *i h*¯ *m* <sup>2</sup>*σ<sup>n</sup>* (∆*qn*)

*Cact* = *<sup>i</sup>*

This can be achieved by rewriting:

*AN N* ∏*n*=1 *h*¯ *m* <sup>8</sup>*σ<sup>n</sup>* <sup>∆</sup>*q*<sup>4</sup> *n* 

*AN N* ∏*n*=1

*dzn* =

1/2 = ¯ *f* (*α*)′ *n* 

> *AN N* ∏*n*=1

Turning now to calculate the second correction *Cmes*, we have:

*dzn* =

*<sup>f</sup>* ′ (*qb*) *<sup>f</sup>* ′ (*qa*)

<sup>1</sup> <sup>+</sup> (1−2*α*) 2

From the expressions (26), (27) and (22) we can deduce *Cmes*,

*dzn* =

*N*+1 ∏*n*=1

*Cmes* <sup>=</sup> (1−2*α*)

Then, we develop *<sup>f</sup>* ′ (*qn*) and *<sup>f</sup>* ′ (*qn*−1) to the second order of <sup>∆</sup>*qn* as follows:

¯ *f* (*α*)′′ *n* ¯ *f* (*α*)′ *n*

*f* (*α*)′′ *n* ¯ *f* (*α*)′ *n*

∆*qn* +

 (1−*α*) 2 +*α*<sup>2</sup> 2

From expressions (19) and (22), we can deduce the quantum correction from the action:

−16*α*<sup>2</sup> + 16*α* − 3

 *m* 2*πih*¯*ε*

> −1/2 *<sup>N</sup>*+<sup>1</sup> ∏*n*=1

> > ∆*qn* +

*<sup>m</sup>* <sup>2</sup>*πih*¯ *<sup>σ</sup><sup>n</sup>*

2 4

 ¯ *f* (*α*)′′ *j* ¯ *f* (*α*)′ *j*

 (1−*α*) 2 +*α*<sup>2</sup> 4

(1 + *Cmes*)

2 ∆*q*<sup>2</sup>

Development *<sup>f</sup>* ′ (*qn*) and *<sup>f</sup>* ′ (*qn*−1) at *<sup>α</sup>*-point discretization in order two of <sup>∆</sup>*qn*, is written as:

¯ *f* (*α*)′′′ *n* ¯ *f* (*α*)′ *n*

> 2

> > 2 − 2 3 ¯ *f* (*α*)′′′ *n* ¯ *f* (*α*)′ *n*

 ¯ *f* (*α*)′′ *n* ¯ *f* (*α*)′ *n*

*m f* ′

(*qn*)*<sup>f</sup>* ′

(*qn*<sup>−</sup>1) 2*πih*¯*ε*

> ¯ *f* (*α*)′′′ *n* ¯ *f* (*α*)′ *n*

*N* ∏*n*=1  *<sup>N</sup>* ∏*n*=1

− *<sup>α</sup>*(1−*α*) 2

 ¯ *f* (*α*)′′ *n* ¯ *f* (*α*)′ *n*

*dqn*., (28)

*<sup>n</sup>*. (29)

*<sup>N</sup>*+<sup>1</sup> *<sup>N</sup>* ∏*n*=1 − *α* (1 − *α*)

 ¯ *f* (*α*)′′ *n* ¯ *f* (*α*)′ *n*

2 ∆*q*<sup>2</sup> *n* 

(1 + *Cact*), (23)

. (24)

*dqn*.. (26)

2 ∆*q*<sup>2</sup> *n* .

(27)

*dqn <sup>f</sup>* ′ (*qn*). (25)

. (22)

$$
\left\langle \left( \Delta q \right)^{2} \right\rangle = \left( \frac{i\hbar \sigma}{m} \right) \Big/ \left\langle \left( \Delta q \right)^{4} \right\rangle = \left( \frac{i\hbar \sigma}{m} \right)^{2} \left( \mathfrak{Z} \right)!! = -\mathfrak{Z} \left( \frac{\hbar \sigma}{m} \right)^{2} . \tag{31}
$$

By combining all these results, we arrive at:

$$\mathcal{C}\_{T} = V\_{eff} = -\sigma\_{n}\frac{i\hbar^{2}}{4m} \left[ \left( \frac{11}{2} - 28a^{2} + 28a \right) \left( \frac{f\_{\text{u}}^{(a)} \prime}{\bar{f}\_{\text{u}}^{(a)} \prime} \right)^{2} - \frac{f\_{\text{u}}^{(a)} \prime}{\bar{f}\_{\text{u}}^{(a)} \prime} \right]. \tag{32}$$

When we used the standard formalism of Feynman (cf., e.g., Ref (Khandekar et al, 1993)) to *α*-point discretization, a single case of *α*−point gave the same result of the method equation, this value is *α* = 1/2., where the effective potential is

$$V\_{eff} = \sigma\_n \frac{i\hbar^2}{4m} \left[ \frac{3}{2} \left( \frac{\vec{f}\_n^{(a)\_{\prime\prime}}}{\vec{f}\_n^{(a)\_{\prime\prime}}} \right)^2 - \frac{\vec{f}\_n^{(a)\_{\prime\prime}}}{\vec{f}\_n^{(a)\_{\prime\prime}}} \right]. \tag{33}$$

But in the presence *β* parameter deformation, the value of *α*-point discretization is different according to what has been explained by the Feynman approach.

#### **3.2. The non-relativistic QM with minimal length case**

We illustrate the use of the path integral formalism of the transition amplitude in the momentum space representation for a quantum time-independent quadratic systems with the presence of nonzero minimum position uncertainty. We start with the propagator expressed as:

$$K^{(\beta)}\left(p\_b, t\_b, p\_a, t\_a\right) = \lim\_{N \to \infty} \left\langle p\_b \left| \prod\_{l=1}^N \exp(\frac{i\varepsilon}{\hbar} \hat{H}) \right| p\_a \right\rangle. \tag{34}$$

Inserting the closure relation for the momentum states (9) between each pair of infinitesimal evolution operators (*U*(*j*, *<sup>j</sup>* − <sup>1</sup>) = exp(*i<sup>ε</sup> <sup>h</sup>*¯ *<sup>H</sup>*<sup>ˆ</sup> )), we obtain

$$K^{(\beta)}\left(p\_b, t\_b, p\_a, t\_a\right) = \lim\_{N \to \infty} \prod\_{l=1}^{N} \int \frac{dp\_l}{\left(1 + \beta p\_l^2\right)} \prod\_{l=1}^{N+1} \left[1 + \frac{i\varepsilon}{\hbar} \hat{H}\right] \left< p\_l \mid p\_{l-1} \right>\,\tag{35}$$

where the projection relation ( *<sup>p</sup>* | *<sup>p</sup>*−<sup>1</sup> ) is defined in eq.(10). Then we inject the Hamiltonian operator of a particle with nonzero minimum position uncertainty on the projection relation for any systems studied. It is clear that there are only few cases where it is exactly solvable; namely, the case of a linear potential (*V*(*x*) = *gx*) and the case of a harmonic potential � *<sup>V</sup>*(*x*) = *<sup>m</sup>ω*<sup>2</sup> <sup>2</sup> *<sup>x</sup>*<sup>2</sup> � . In this chapter, we will study only the form quadratic, for example, the harmonic oscillator potential in one dimension and the spinorial relativistic particle.

The construction of momentum space path integral representation of the transition amplitude for a particle moving in the potential of the harmonic oscillator in one dimension with nonzero minimum position uncertainty. Following the well-known steps to construct a quantum propagator *<sup>K</sup>*(*β*), we write:

$$K^{(\delta)}\left(p\_{b\cdot}t\_{b\cdot}p\_{a'}t\_{a}\right) = \lim\_{N\to\infty} \prod\_{j=1}^{N} \int \frac{dp\_{j}}{\left(1+\beta p\_{j}^{2}\right)} \prod\_{j=1}^{N+1} \int \frac{dq\_{j}}{2\pi\hbar} \left(1+\beta p\_{j}^{2}\right) \exp\left\{\frac{i}{\hbar} \sum\_{j=1}^{N+1} \left[q\_{j}\Delta p\_{j} - \varepsilon \frac{p\_{j}^{2}}{2m}\right] \right\}.\tag{36}$$

$$-\varepsilon \frac{m\omega^{2}}{2} \left(\left(1+\beta p\_{j}^{2}\right)^{2} q\_{j}^{2} - 6i\hbar\beta p\_{j} q\_{j} \left(1+\beta p\_{j}^{2}\right) - 2\hbar^{2}\beta \left(1+3\beta p\_{j}^{2}\right)\right)\right\}.\tag{37}$$

The form of expression (36) shows that the path integral over the variables *qj* is Gaussian, so the result is simply written as:

$$K^{(\mathcal{S})}\left(p\_b, t\_b, p\_a, t\_a\right) = \lim\_{N \to \infty} \prod\_{j=1}^{N} \int \frac{dp\_j}{\left(1 + \beta p\_j^2\right)} \prod\_{j=1}^{N+1} \frac{1}{\sqrt{2\pi i \hbar m \omega^2 \varepsilon}}$$

$$\exp\left\{\frac{i}{\hbar} \sum\_{j=1}^{N+1} \left[\frac{\left(\Delta p\_j\right)^2}{2m\omega^2 \varepsilon \left(1 + \beta p\_j^2\right)} + \frac{3i\hbar \beta p\_j \Delta p\_j}{\left(1 + \beta p\_j^2\right)} - \varepsilon \frac{p\_j^2}{2m} - \varepsilon \frac{m\omega^2 \hbar^2 \beta}{2} \left(-2 + 3\beta p\_j^2\right)\right]\right\}.\tag{37}$$

This latter expression shows that the kinetic term is similar a system of space dependent mass and can be removed by an *α*−point coordinate transformation method (see, Ref (Khandekar et al, 1993)), where the *α*-point discretization interval defined by

$$
\bar{p}\_{\!\!\!\!\!/}^{(a)} = \mathfrak{a}\!\!\!p\_{\!\!\!/} + (1 - \mathfrak{a})\,\,\bar{p}\_{\!\!\!/} \,\_{\!\!\!/} . \tag{38}
$$

So, we will introduce the function *f*(*p*), where the first derivative of *f*(*p*¯ (*α*) ) is equal to 1/(1 + *βp*¯ (*α*)2 ). Thus, there are three quantum corrections obtained in expression (37).


Expanding the exponential ( *i h*¯ (∆*p*) 2 <sup>2</sup>*mω*2*ε*(1+*βp*<sup>2</sup> ) <sup>2</sup> ) of the *α*−point discretization interval, we find

$$\exp\left[\frac{i}{\hbar}\sum\_{l=1}^{N+1}\left(\frac{\left(\Delta p\_{l}\right)^{2}}{2m\omega^{2}\epsilon\left(1+\beta p\_{l}^{2}\right)^{2}}\right)\right] = \exp\left[\frac{i}{\hbar}\sum\_{j=1}^{N+1}\left(\frac{\left(\vec{f}^{(\nu)}\right)^{2}\left(\Delta p\_{l}\right)^{2}}{2m\omega^{2}\epsilon}\right)\right]\left(1+\mathsf{C}\_{\mathrm{act}}^{(1)}\right),\tag{39}$$

where

8

harmonic potential

particle.

�

quantum propagator *<sup>K</sup>*(*β*), we write:

*<sup>K</sup>*(*β*) (*pb*, *tb*; *pa*, *ta*) <sup>=</sup> lim

−*ε <sup>m</sup>ω*<sup>2</sup> 2 ��

the result is simply written as:

*N*+1 ∑ =1

exp � *i h*¯

(*α*)2

Expanding the exponential (



> *i h*¯

1/(1 + *βp*¯

*<sup>V</sup>*(*x*) = *<sup>m</sup>ω*<sup>2</sup>

*N*→∞

1 + *βp*<sup>2</sup> *j* �2 *q*2

*N* ∏ *j*=1

*<sup>K</sup>*(*β*) (*pb*, *tb*, *pa*, *ta*) <sup>=</sup> lim

2

et al, 1993)), where the *α*-point discretization interval defined by

*p*¯ (*α*)

*act*

(∆*p*) 2

<sup>2</sup>*mω*2*ε*(1+*βp*<sup>2</sup>

)

So, we will introduce the function *f*(*p*), where the first derivative of *f*(*p*¯

)

� (∆*p*)

<sup>2</sup>*mω*2*ε*(1+*βp*<sup>2</sup>

� *dpj* � 1+*βp*<sup>2</sup> *j* �

*<sup>j</sup>* <sup>−</sup> <sup>6</sup>*ih*¯ *<sup>β</sup>pj*

<sup>2</sup> *<sup>x</sup>*<sup>2</sup> �

it is exactly solvable; namely, the case of a linear potential (*V*(*x*) = *gx*) and the case of a

for example, the harmonic oscillator potential in one dimension and the spinorial relativistic

The construction of momentum space path integral representation of the transition amplitude for a particle moving in the potential of the harmonic oscillator in one dimension with nonzero minimum position uncertainty. Following the well-known steps to construct a

> *N*+1 ∏ *j*=1

> > *qj* � 1 + *βp*<sup>2</sup> *j* � − 2¯*h*2*β* �

The form of expression (36) shows that the path integral over the variables *qj* is Gaussian, so

*N* ∏ =1

This latter expression shows that the kinetic term is similar a system of space dependent mass and can be removed by an *α*−point coordinate transformation method (see, Ref (Khandekar

). Thus, there are three quantum corrections obtained in expression (37).

*f*

� *dp* (1+*βp*<sup>2</sup> )

) <sup>−</sup> *<sup>ε</sup> <sup>p</sup>*<sup>2</sup> 

*N*+1 ∏ =1 √ 1 2*πihm*¯ *ω*2*ε*

�

= *<sup>α</sup>p* + (<sup>1</sup> − *<sup>α</sup>*) *<sup>p</sup>*−1. (38)

<sup>2</sup> ) of the *α*−point discretization interval, we find

−2 + 3*βp*<sup>2</sup> ���

(*α*)

) is equal to

<sup>2</sup>*<sup>m</sup>* <sup>−</sup> *<sup>ε</sup> <sup>m</sup>ω*2*h*¯ <sup>2</sup>*<sup>β</sup>* 2

*N*→∞

<sup>2</sup> <sup>+</sup> <sup>3</sup>*ih*¯ *<sup>β</sup>p*∆*p* (1+*βp*<sup>2</sup>

� *dqj* 2*πh*¯ � 1 + *βp*<sup>2</sup> *j* � exp *i h*¯

. In this chapter, we will study only the form quadratic,

*N*+1 ∑ *j*=1

1 + 3*βp*<sup>2</sup> *j* ����

�

*qj*∆*pj* <sup>−</sup> *<sup>ε</sup>*

*p*2 *j* 2*m*

. (36)

. (37)

$$\begin{split} \mathcal{L}\_{\text{act}}^{(1)} &= \frac{i}{2\hbar m \omega^{2} \varepsilon} \Bigg[ \frac{2(1-a)f\_{l}^{(a)\prime} \left(f\_{l}^{(a)\prime}\right)^{2}}{f\_{l}^{(a)\prime}} \left(\Delta p\_{l}\right)^{3} + (1-a)^{2} \left[ \left(\frac{f\_{l}^{(a)\prime}}{f\_{l}^{(a)\prime}}\right)^{2} + \frac{f\_{l}^{(a)\prime\prime}}{f\_{l}^{(a)\prime}} \right] (\vec{f}\_{l}^{(a)\prime})^{2} \left(\Delta p\_{l}\right)^{4} \Bigg] \\ & \quad - \frac{2(1-a)^{2}}{\left(2\hbar m \omega^{2} \varepsilon\right)^{2}} \Bigg( \frac{f\_{l}^{(a)\prime}}{f\_{l}^{(a)\prime}} \Bigg)^{2} \left(\vec{f}\_{l}^{(a)\prime}\right)^{4} \left(\Delta p\_{l}\right)^{6}, \end{split} \tag{40} \end{split}$$

and the measure term will be developed as.

$$\prod\_{j=1}^{N} \int \frac{dp\_j}{1 + \beta p\_j^2} = \sqrt{\left(1 + \beta p\_b^2\right)\left(1 + \beta p\_d^2\right)} \prod\_{j=1}^{N} \int dp\_n \prod\_{j=1}^{N+1} \frac{1}{\sqrt{\left(1 + \beta p\_j^2\right)\left(1 + \beta p\_{j-1}^2\right)}}$$

$$= \left[\frac{1}{f\_s^2 f\_s^2}\right] \prod\_{n=1}^{N} \int dp\_j \prod\_{n=1}^{N+1} f\_j^{(a) \prime} \left(1 + C\_m^{(1)}\right),\tag{41}$$

where

$$\mathbb{C}\_{m}^{(1)} = \frac{(1 - 2a)}{2} \frac{\bar{f}\_{n}^{(a)} \, ^{\prime} \!\!\!u}{\bar{f}\_{n}^{(a)} \, ^{\prime} \!\!\!u} \Delta p\_{n} + \left( \frac{(1 - a)^{2} + a^{2}}{4} \frac{\bar{f}\_{n}^{(a)} \, ^{\prime} \!\!u}{\bar{f}\_{n}^{(a)} \, ^{\prime} \!\!u} - \frac{a(1 - a)}{2} \left( \frac{\bar{f}\_{n}^{(a)} \, ^{\prime} \!\!u}{\bar{f}\_{n}^{(a)} \, ^{\prime} \!\!u} \right)^{2} \right) \Delta p\_{n}^{2} . \tag{42}$$

and the *f*-factor term will be developed as

$$\exp\left(-\frac{3\beta p\_{\rangle}\Delta p\_{\rangle}}{1+\beta p\_{\rangle}^2}\right) = 1 + \mathbb{C}\_{f'}^T \tag{43}$$

where

$$\mathbf{C}\_{f}^{T} = \frac{3}{2} \left( \frac{\vec{f}\_{l}^{(a)\prime}}{\vec{f}\_{l}^{(a\prime)}} \right) \Delta p\_{l} + \frac{9}{8} \left( \frac{\vec{f}\_{l}^{(a)\prime}}{\vec{f}\_{l}^{(a\prime)}} \right)^{2} \left( \Delta p\_{l} \right)^{2} + \frac{3}{2} \left( 1 - a \right) \left( \frac{\vec{f}\_{l}^{(a)\prime}}{\vec{f}\_{l}^{(a\prime)}} - \left( \frac{\vec{f}\_{l}^{(a)\prime}}{\vec{f}\_{l}^{(a\prime)}} \right)^{2} \right) \left( \Delta p\_{l} \right)^{2},\tag{44}$$

Now, in order to convert this expression to the usual form of Feynman path integral, let us bring the kinetic term to the conventional one, with a constant mass term by using the following coordinate transformation *p* = *g*(*k*), this transformation generates two corrections:



The *α*-point expansion of ∆*p* reads at each ()

$$
\Delta p\_{\rangle} = g(k\_{\rangle}) - g(k\_{\rangle - 1}) = \Delta k\_{\rangle} \overline{\mathbf{g}}\_{\rangle}^{(a)} \left( 1 + \frac{(1 - 2a)}{2!} \frac{\mathbf{g}\_{\rangle}^{(a)} \mathbf{g}\_{\rangle}}{\overline{\mathbf{g}}\_{\rangle}^{(a)}} \Delta k\_{\rangle} + \frac{(1 - a)^3 + a^3}{3!} \frac{\mathbf{g}\_{\rangle}^{(a)} \mathbf{g}\_{\rangle}}{\overline{\mathbf{g}}\_{\rangle}^{(a)}} \Delta k\_{\rangle}^2 \right). \tag{45}
$$

The choice of *g* is fixed by the following condition ((*∂g*/*∂k*) = (*∂ f* /*∂p*) −1 ), that makes the transformation *<sup>g</sup>*(*k*) = tan√*β<sup>k</sup>* <sup>√</sup>*<sup>β</sup>* where the region *<sup>p</sup>* <sup>∈</sup> [−∞, <sup>+</sup>∞] is mapped to *<sup>k</sup>* <sup>∈</sup> � −*π*/2�*β*, *π*/2�*β* � . Thereafter, we develop the exponential of the kinetic term as

$$\exp\left[\frac{i}{\hbar}\sum\_{l=1}^{N+1}\left(\frac{\left(\Lambda p\_l\right)^2}{2m\omega^2\varepsilon\left(1+\beta p\_l^2\right)}\right)\right] = \exp\left\{\frac{i}{\hbar}\sum\_{l=1}^{N+1}\left[\frac{\Lambda k\_l^2}{2m\omega^2\varepsilon}\right]\right\}\left[1+\mathcal{C}\_{act}^{(1)}\right]\left[1+\mathcal{C}\_{act}^{(2)}\right],\tag{46}$$

where *<sup>C</sup>*(2) *act* is given by

$$\begin{split} \mathbf{C}\_{\text{act}}^{(2)} &= \left\{ \frac{i}{2\hbar m\omega^{2}\varepsilon} \left[ (1-2a) \frac{\hat{\mathbf{g}}\_{l}^{(a)} \prime \prime}{\hat{\mathbf{g}}\_{l}^{(a) \prime} \prime} \left( \hat{\mathbf{g}}\_{l}^{(a) \prime} \right)^{2} \left( \hat{f}\_{l}^{(a) \prime} \right)^{2} \Delta k\_{l}^{3} \right. \\ &+ \left[ \frac{(1-2a)^{2}}{4} \left( \frac{\hat{\mathbf{g}}\_{l}^{(a) \prime} \prime}{\hat{\mathbf{g}}\_{l}^{(a) \prime} \prime} \right)^{2} + \frac{(1-a)^{3} + a^{3}}{3} \frac{\hat{\mathbf{g}}\_{l}^{(a) \prime \prime} \prime}{\hat{\mathbf{g}}\_{l}^{(a) \prime} \prime} \right] \left( \hat{f}\_{l}^{(a) \prime} \right)^{2} \left( \hat{\mathbf{g}}\_{l}^{(a) \prime} \right)^{2} \Delta k\_{l}^{4} \right] \\ &- \frac{(1-2a)^{2}}{2(2\hbar m\omega^{2}\varepsilon)^{2}} \left( \frac{\hat{\mathbf{g}}\_{l}^{(a) \prime \prime}}{\hat{\mathbf{g}}\_{l}^{(a) \prime} \prime} \right)^{2} \left( \hat{\mathbf{g}}\_{l}^{(a) \prime} \right)^{4} \left( \hat{f}\_{l}^{(a) \prime \prime} \right)^{4} \Delta k\_{l}^{6} + ... \right\}. \tag{47} \end{split} \tag{48}$$

The measure induce also a correction

$$\prod\_{j=1}^{N} \int \frac{dp\_j}{1 + \beta p\_j^2} = \sqrt{\frac{1}{f\_i' f\_i' g\_i \xi\_i'}} \prod\_{n=1}^{N} \int dk\_j \prod\_{j=1}^{N+1} f\_j^{(\*)} \, ^t \hat{g}\_j^{(t)} \left(1 + \mathbb{C}\_m^{(2)}\right) \left(1 + \mathbb{C}\_m^{(1)}\right)$$

$$= \prod\_{j=1}^{N} \int dk\_j \prod\_{j=1}^{N+1} \left(1 + \mathbb{C}\_m^{(1)}\right) \left(1 + \mathbb{C}\_m^{(2)}\right),\tag{48}$$

where *<sup>C</sup>*(2) *<sup>m</sup>* is given by

$$\mathbb{C}\_{m}^{(2)} = \frac{(1-2a)}{2} \frac{\mathbb{g}\_{/}^{(a)} \iota}{\mathfrak{g}\_{/}^{(a)} \iota} \Delta k\_{\circ} + \left[ -a \frac{(1-a)}{2} \left( \frac{\mathfrak{g}\_{/}^{(a)} \iota}{\mathfrak{g}\_{/}^{(a)} \iota} \right)^{2} + \frac{(1-a)^{2} + a^{2}}{4} \frac{\mathfrak{g}\_{/}^{(a)} \iota}{\mathfrak{g}\_{/}^{(a)} \iota} \right] \Delta k\_{\circ}^{2} . \tag{49}$$

,

By combining all these corrections as follows:

10

�


(*α*)′ �

The choice of *g* is fixed by the following condition ((*∂g*/*∂k*) = (*∂ f* /*∂p*)

= exp

(<sup>1</sup> <sup>−</sup> <sup>2</sup>*α*) *<sup>g</sup>*¯

+ (1−*α*) 3 +*α*<sup>3</sup> 3

*N* ∏*n*=1 � *dkj*

*N*+1 ∏ *j*=1 � <sup>1</sup> <sup>+</sup> *<sup>C</sup>*(1) *<sup>m</sup>*

−*<sup>α</sup>* (1−*α*) 2

<sup>∆</sup>*kj* +

�

� *i h*¯

<sup>1</sup> <sup>+</sup> (1−2*α*) 2! *g*¯ (*α*)′′ *g*¯ (*α*)′

. Thereafter, we develop the exponential of the kinetic term as

� ∆*k*<sup>2</sup> 2*mω*2*ε*

� � ¯ *f* (*α*)′ �<sup>2</sup> � *g*¯ (*α*)′ �2 ∆*k*<sup>4</sup> �

*N*+1 ∑ =1

(*α*)′′ *g*¯ (*α*)′ � *g*¯ (*α*)′ �<sup>2</sup> � ¯ *f* (*α*)′ �2 ∆*k*<sup>3</sup> 

*g*¯ (*α*)′′′ *g*¯ (*α*)′

*N*+1 ∏ *j*=1 ¯ *f* (*α*)′ *<sup>j</sup> g*¯ (*α*)′ *j* � <sup>1</sup> <sup>+</sup> *<sup>C</sup>*(2) *<sup>m</sup>*

� *g*¯ (*α*)′′ *j g*¯ (*α*)′ *j*

� �

<sup>1</sup> <sup>+</sup> *<sup>C</sup>*(2) *<sup>m</sup>*

�2

�

+ (1−*α*) 2 +*α*<sup>2</sup> 4

<sup>∆</sup>*k* <sup>+</sup> (1−*α*)

<sup>√</sup>*<sup>β</sup>* where the region *<sup>p</sup>* <sup>∈</sup> [−∞, <sup>+</sup>∞] is mapped to *<sup>k</sup>* <sup>∈</sup>

�� �

3 +*α*<sup>3</sup> 3!

<sup>1</sup> <sup>+</sup> *<sup>C</sup>*(1) *act* � �

� �

*g*¯ (*α*)′′′ *j g*¯ (*α*)′ *j*

 <sup>∆</sup>*k*<sup>2</sup>

<sup>1</sup> <sup>+</sup> *<sup>C</sup>*(1) *<sup>m</sup>*

�

*<sup>j</sup>* . (49)

, (48)

*g*¯ (*α*)′′′ *g*¯ (*α*)′ ∆*k*<sup>2</sup> �

−1

<sup>1</sup> <sup>+</sup> *<sup>C</sup>*(2) *act* �

. (47)

. (45)

, (46)

), that makes

The *α*-point expansion of ∆*p* reads at each ()

<sup>∆</sup>*p* = *<sup>g</sup>*(*k*) − *<sup>g</sup>*(*k*−1) = <sup>∆</sup>*kg*¯

� (∆*p*)

*<sup>C</sup>*(2) *act* <sup>=</sup> � *i* 2¯*hmω*2*ε*

<sup>2</sup>*mω*2*ε*(1+*βp*<sup>2</sup>

2

� *g*¯ (*α*)′′ *g*¯ (*α*)′ �2

2 2(2¯*hmω*2*ε*) 2 � *g*¯ (*α*)′′ *g*¯ (*α*)′ �2 � *g*¯ (*α*)′ �<sup>4</sup> � ¯ *f* (*α*)′ �4 ∆*k*<sup>6</sup> + ... �

− (1−2*α*)

= *N* ∏ *j*=1 � *dkj*

) 2 ��

the transformation *<sup>g</sup>*(*k*) = tan√*β<sup>k</sup>*

*N*+1 ∑ =1

*act* is given by

+ � (1−2*α*) 2 4

The measure induce also a correction

� *dpj* 1+*βp*<sup>2</sup> *j* = � 1 *f* ′ *b f* ′ *ag*′ *bg*′ *a*

*<sup>C</sup>*(2) *<sup>m</sup>* <sup>=</sup> (1−2*α*) 2 *g*¯ (*α*)′′ *j g*¯ (*α*)′ *j*

*N* ∏ *j*=1

where *<sup>C</sup>*(2) *<sup>m</sup>* is given by

�

−*π*/2�*β*, *π*/2�*β*

exp � *i h*¯

where *<sup>C</sup>*(2)

$$\mathbf{1} + \mathbf{C}\_{T} = \left(\mathbf{1} + \mathbf{C}\_{\rm act}^{(1)}\right) \left(\mathbf{1} + \mathbf{C}\_{\rm act}^{(2)}\right) \left(\mathbf{1} + \mathbf{C}\_{\rm m}^{(1)}\right) \left(\mathbf{1} + \mathbf{C}\_{\rm m}^{(2)}\right) \left(\mathbf{1} + \mathbf{C}\_{f}^{T}\right). \tag{50}$$

where the corrections terms are evaluated perturbatively, using the expectation values

$$
\left\langle \left( \Delta k \right)^{2\ell} \right\rangle = \left( i\hbar m\omega^2 \varepsilon \right)^{\ell} \left( 2\ell - 1 \right)!! . \tag{51}
$$

So through all of these corrections, one can conclude the correction total *CT* depending on the *α*-point discretization can be obtained as

$$\mathcal{C}\_{T} = \frac{i}{\hbar} \hbar^2 m \omega^2 \varepsilon \beta \left[ -1 + \left( 8a^2 - 8a + \frac{5}{2} \right) \tan^2 \sqrt{\beta} k\_{\parallel} \right] \tag{52}$$

Furthermore, a judicious choice of the discretization parameter is indicated by the order of operators present in the wave equation method. The result coincides with the path integral approach when *CT* equals

$$\mathbf{C}\_{T} = i\hbar\varepsilon m\omega^{2}\mathcal{J}\left[-1 + \frac{3}{2}\tan^{2}\sqrt{\beta}k\right],\tag{53}$$

We note that the different *α*-point discretization value (i.e. *α* = <sup>1</sup> 2 <sup>1</sup> <sup>±</sup> 1/√<sup>2</sup> ) are obtained compared with the ordinary quantum mechanics case (*α* = 1/2). So the propagator (37) becomes,

$$K^{(\beta)}\left(p\_{\nu},t\_{b};p\_{a^{\prime}},t\_{a}\right) = \lim\_{N \to \infty} \prod\_{n=1}^{N} \int dk\_{n} \prod\_{n=1}^{N+1} \frac{1}{\sqrt{2\pi i \hbar m \omega^{2} \varepsilon}} \exp\left\{ \frac{i}{\hbar} \sum\_{n=1}^{N+1} \left[\frac{(\Delta k\_{a})^{2}}{2m\omega^{2}\varepsilon} - \varepsilon \frac{\tan^{2}\left(\sqrt{\beta}k\_{a}\right)}{2m\beta}\right] \right\},\tag{54}$$

This expression is exactly the path integral representation of the transition amplitude of a particle, moving in the symmetric Pöschl-Teller potential (cf., e.g., Ref (Grouche & Steiner, 1998)):

$$K^{(\mathfrak{F})} = \lim\_{N \to \infty} \prod\_{n=1}^{N} \int dk\_n \prod\_{n=1}^{N+1} \frac{1}{\sqrt{2\pi i \hbar m \omega^2 \varepsilon}} \exp\left\{ \frac{i}{\hbar} \sum\_{n=1}^{N+1} \left[ \frac{(\Delta k\_n)^2}{2m\omega^2 \varepsilon} - \varepsilon \frac{\hbar \hbar^2 m \omega^2}{2} \lambda \left(\lambda - 1\right) \tan^2 \left(\sqrt{\beta} k\_n\right) \right] \right\} \tag{55}$$

with (*λ* = <sup>1</sup> + (<sup>1</sup> + (2/*βhm*¯ *<sup>ω</sup>*)2)1/2 /2). The solution of this path integral is given by:

$$K^{(\delta)}\left(p\_b, t\_b; p\_{a^t} t\_a\right) = \sum\_{n=0}^{\infty} \frac{2^{2\lambda - 1} (\lambda + n) n! \sqrt{\beta}}{\pi \Gamma(2\lambda + n) \Gamma(\lambda)^{1-2}} \exp\left[-\frac{i}{\hbar} \frac{\beta \hbar^2 m \omega^2 (t\_b - t\_a)}{2} \left(n^2 + 2\left(n + 1\right)\lambda\right)\right]$$

$$\cos^{\lambda}\left(\sqrt{\beta} k\_b\right) \cos^{\lambda}\left(\sqrt{\beta} k\_a\right) \mathcal{C}\_n^{\lambda}\left(\sin\left(\sqrt{\beta} k\_b\right)\right) \mathcal{C}\_n^{\lambda}\left(\sin\left(\sqrt{\beta} k\_b\right)\right). \tag{56}$$

We finally obtain the spectral decomposition of the transition amplitude for the one-dimensional harmonic oscillator with nonzero minimum position uncertainty

$$K^{\left(\beta\right)}\left(p\_{\flat},t\_{\flat};p\_{a},t\_{a}\right) = \sum\_{n=0}^{\infty} \Psi\_{n}\left(p\_{\flat}\right) \Psi\_{n}^{\*}\left(p\_{a}\right) e^{-\frac{i}{\hbar}E\_{n}\left(t\_{\flat}-t\_{a}\right)}.\tag{57}$$

The energy spectrum is obtained from the poles of the Green function (57):

$$E\_n = \frac{\beta \hbar^2 m \omega^2}{2} \left( n^2 + 2 \left( n + 1 \right) \lambda \right). \tag{58}$$

Using the expression of *λ*, one finds:

$$E\_n = \hbar\omega \left[ \left( n + \frac{1}{2} \right) \sqrt{1 + \left( \frac{\hbar \hbar m \omega}{2} \right)^2} + \left( n^2 + n + 1 \right) \left( \frac{\hbar \hbar m \omega}{2} \right) \right]. \tag{59}$$

Also, the normalized eigenfunctions of the bound states can be easily deduced

$$\Psi\_n(p) = \sqrt{\frac{2^{2\lambda - 1}(\lambda + n)n!\sqrt{\beta}}{\pi \Gamma(2\lambda + n) \left[\Gamma(\lambda)\right]^{-2}}} \left[\frac{1}{\sqrt{1 + \beta p^2}}\right]^\lambda \mathcal{C}\_n^\lambda \left(\frac{\sqrt{\beta}p}{\sqrt{1 + \beta p^2}}\right). \tag{60}$$

We note that equations (59) and (60) coincide exactly with those obtained in (Chang et al, 2002). Also we can verify these results when *β* → 0, which transform to these results:

$$E\_{\boldsymbol{\pi}} = \underset{\boldsymbol{\theta} \to 0}{\hbar \omega} \left( \boldsymbol{n} + \frac{1}{2} \right), \quad \boldsymbol{\Psi}\_{\boldsymbol{n}} \left( \boldsymbol{p} \right)\_{\boldsymbol{\theta} \to 0} = \left[ \frac{1}{2^n n! \sqrt{\pi}} \right]^{1/2} \left( \frac{1}{m \hbar \omega} \right)^{1/4} e^{-\frac{\boldsymbol{p}^2}{2m \hbar \omega}} H\_{\boldsymbol{n}} \left( \sqrt{\frac{1}{m \hbar \omega}} \boldsymbol{p} \right). \tag{61}$$

For the one dimensional harmonic oscillator in the framework non-commutative geometry represented by Eqs. (1) and (2), the quantum corrections from the viewpoint of Feynman (cf., e.g., Ref (Khandekar et al, 1993)) at the *α*-point discretization interval, we found only two points discretization (*α* = <sup>1</sup> 2 <sup>1</sup> <sup>±</sup> 1/√<sup>2</sup> ) consistent with differential equation (see, Ref (Chang, et al, 2002)) which gives different value of ordinary quantum mechanics.

So, in the following subsection we aim to expand this type deformation for relativistic systems.

#### **3.3. The relativistic QM with minimal length**

12

*<sup>K</sup>*(*β*) (*pb*, *tb*; *pa*, *ta*) <sup>=</sup>

cos*<sup>λ</sup> βkb*

Using the expression of *λ*, one finds:

*En* = *h*¯ *ω*

 *n* + <sup>1</sup> 2 1 +

Ψ*<sup>n</sup>* (*p*) =

*En* = *β*→0 *h*¯ *ω n* + <sup>1</sup> 2 

systems.

two points discretization (*α* = <sup>1</sup>

∞ ∑ *n*=0

22*λ*−1(*λ*+*n*)*n*!

cos*<sup>λ</sup> βka*

*<sup>K</sup>*(*β*) (*pb*, *tb*; *pa*, *ta*) <sup>=</sup>

√*β <sup>π</sup>*Γ(2*λ*+*n*)[Γ(*λ*)]−<sup>2</sup> exp

We finally obtain the spectral decomposition of the transition amplitude for the

 *Cλ n* sin

one-dimensional harmonic oscillator with nonzero minimum position uncertainty

The energy spectrum is obtained from the poles of the Green function (57):

*mω*<sup>2</sup> 2

Also, the normalized eigenfunctions of the bound states can be easily deduced

*<sup>π</sup>*Γ(2*λ*+*n*)[Γ(*λ*)]−<sup>2</sup>

<sup>2</sup>2*λ*−1(*λ*+*n*)*n*!

*β*→0

 1 2*nn*! √*π*

<sup>1</sup> <sup>±</sup> 1/√<sup>2</sup>

(Chang, et al, 2002)) which gives different value of ordinary quantum mechanics.

, Ψ*<sup>n</sup>* (*p*) =

2   *<sup>β</sup>hm*¯ *<sup>ω</sup>* 2 2 + 

√*β*

 √ 1 1+*βp*<sup>2</sup>

We note that equations (59) and (60) coincide exactly with those obtained in (Chang et al, 2002). Also we can verify these results when *β* → 0, which transform to these results:

For the one dimensional harmonic oscillator in the framework non-commutative geometry represented by Eqs. (1) and (2), the quantum corrections from the viewpoint of Feynman (cf., e.g., Ref (Khandekar et al, 1993)) at the *α*-point discretization interval, we found only

So, in the following subsection we aim to expand this type deformation for relativistic

1/2 <sup>1</sup> *mh*¯ *ω*

*En* <sup>=</sup> *<sup>β</sup>h*¯ <sup>2</sup>

∞ ∑ *n*=0  − *i h*¯ *βh*¯ <sup>2</sup>

<sup>Ψ</sup>*<sup>n</sup>* (*pb*) <sup>Ψ</sup><sup>∗</sup>

*n*<sup>2</sup> + 2 (*n* + 1) *λ*

*βkb*

 *Cλ n* sin

*<sup>n</sup>* (*pa*)*<sup>e</sup>*

− *<sup>i</sup>*

*n*<sup>2</sup> + *n* + 1

*λ Cλ n*

> 1/4 *e* − *<sup>p</sup>*<sup>2</sup> <sup>2</sup>*mh*¯ *<sup>ω</sup> Hn*

*<sup>h</sup>*¯ *En*(*tb*−*ta* )

 *<sup>β</sup>hm*¯ *<sup>ω</sup>* 2

 <sup>√</sup>*β<sup>p</sup>* √1+*βp*<sup>2</sup>

*mω*2(*tb*−*ta* ) 2

*βkb*

*n*<sup>2</sup> + 2 (*n* + 1) *λ*

. (56)

. (57)

. (59)

. (60)

. (61)

. (58)

 <sup>1</sup> *mh*¯ *<sup>ω</sup> p* 

) consistent with differential equation (see, Ref

Our interest in the following section is to construct the propagator for two applications relativistic quantum mechanics in the presence of a minimal length, the first is (1+1)-dimensional Dirac oscillator, where the momentum component is shifting *p* by *<sup>p</sup>* − *imωγ*0*<sup>x</sup>* (cf., e.g., Ref (Szmytkowski & Gruchowski, 2001), and the second is a spinorial relativistic particle under the action of a Lorentz potential *V*(*x*), → **A** = 0 plus a scalar potential *S*(*x*), described by the (1+1)-dimensional Dirac equation:

$$\left(\gamma^{\mu}\text{\text{\textquotedblleft}\Gamma}\_{\mu} - m + \imath\varepsilon\right)\hat{S}^{(\beta)} = I,\tag{62}$$

where *µ* = 0, 1, *γµ* are the Dirac matrices in the 2-dimentional Minkowski space. So, via the same procedure in the our previous work (Benzair et al, 2012 and 2014), we can obtain the standard propagator result for both systems, where there are two types of propagation, one with positive energy (+*E*(*β*) *<sup>n</sup>* ) propagation and the other with negative energy (−*E*(*β*) *<sup>n</sup>* ) propagation

$$S^{(\beta)}\left(p\_b, p\_a, t\_b - t\_a\right) = -\sum\_{n=0}^{\infty} \begin{bmatrix} \Theta\left(t\_b - t\_a\right) \Psi\_n^{(\beta)+}\left(p\_b\right) \Psi\_n^{(\beta)+}\left(p\_a\right) e^{-iE\_n^{(\beta)}(t\_b - t\_a)} + \\ \Theta\left(-\left(t\_b - t\_a\right)\right) \Psi\_n^{(\beta)-}\left(p\_b\right) \Psi\_n^{(\beta)-}\left(p\_a\right) e^{iE\_n^{(\beta)}(t\_b - t\_a)} \end{bmatrix},\tag{63}$$

For one-dimensional Dirac oscillator in the momentum space representation with the presence of minimal length uncertainty, can be expressed the energy spectrum as follows

$$E\_{n,\pm}^{(\beta)} = \pm \sqrt{m^2 + \beta \left(m\omega\right)^2 n^2 + 2n\left(m\omega\right)}.\tag{64}$$

and the corresponding wave functions

$$\Psi\_n^{(\beta)+}\left(p\right) = \begin{pmatrix} f\_n^{(\beta)+}\left(p\right) \\ g\_n^{(\beta)+}\left(p\right) \end{pmatrix}, \text{and } \Psi\_n^{(\beta)-}\left(p\right) = \begin{pmatrix} f\_n^{(\beta)-}\left(p\right) \\ g\_n^{(\beta)-}\left(p\right) \end{pmatrix} \tag{65}$$

where the components of the wave functions *f* (*β*)± *<sup>n</sup>* (*p*) and *<sup>g</sup>* (*β*)± *<sup>n</sup>* (*p*) are respectively

$$f\_n^{(\beta)+}\left(p\right) = \sqrt{\Gamma\left(\eta\right)^2 \frac{2^{2\eta-1}(n+1)!(n+\eta)\sqrt{\beta}\left(\mathcal{E}\_n^{(\beta)} + m\right)}{\pi\Gamma(n+2\eta)2\mathcal{E}\_n^{(\beta)}}} \left(\frac{1}{1+\beta p^2}\right)^{\eta} \mathcal{E}\_n^{\eta}\left(\frac{\sqrt{\beta}p}{1+\beta p^2}\right). \tag{66}$$

$$\mathcal{G}\_{n}^{(\notin)+}\left(p\right) = \frac{1}{\sqrt{\beta}}\sqrt{\Gamma\left(\eta\right)^{2}\frac{2^{2q-1}n!\left(n+\eta\right)\sqrt{\beta}}{\pi\Gamma\left(n+2\eta\right)2E\_{n}^{(\notin)}\left(E\_{n}^{(\notin)}+m\right)}}\left(\frac{1}{1+\beta p^{2}}\right)^{\eta+1}\mathcal{C}\_{n-1}^{\eta+1}\left(\frac{\sqrt{\beta}p}{1+\beta p^{2}}\right).\tag{67}$$

and

$$f\_{n}^{(\beta)-}\left(p\right) = \sqrt{\Gamma\left(\eta\right)^{2} \frac{2^{2\eta-1}n!(n+\eta)\sqrt{\beta}\left(E\_{n}^{(\beta)}-m\right)}{\pi\Gamma(n+2\eta)2E\_{n}^{(\beta)}}} \left(\frac{1}{1+\beta p^{2}}\right)^{\eta} \mathcal{C}\_{n}^{\eta}\left(\frac{\sqrt{\beta}p}{1+\beta p^{2}}\right). \tag{68}$$

$$\mathcal{G}\_{n}^{(\notin)-}\left(p\right) = \frac{2i}{\sqrt{\beta}}\sqrt{\Gamma\left(\eta\right)^{2}\frac{2^{2q-1}n!\left(n+\eta\right)\sqrt{\beta}}{\pi\Gamma\left(n+2\eta\right)2E\_{n}^{(\notin)}\left(E\_{n}^{(\notin)}-m\right)}}\left(\frac{1}{1+\beta p^{2}}\right)^{\eta+1}\mathcal{C}\_{n-1}^{\eta+1}\left(\frac{\sqrt{\beta}p}{1+\beta p^{2}}\right). \tag{69}$$

and in the second application we can express these results; the energy spectrum are

$$E\_n^{(\beta)\pm} = -\frac{m\_0 V\_0}{S\_0} \pm \omega\_n^{(\beta)}; T = t\_b - t\_a. \tag{70}$$

with

$$
\omega\_n^{(\beta)} = \frac{\left(S\_0^2 - V\_0^2\right)}{S\_0} \sqrt{\beta \left(n^2 + \frac{2n}{\beta \sqrt{S\_0^2 - V\_0^2}}\right)}.\tag{71}
$$

and the wave functions appropriate to the energy spectrum *<sup>E</sup>*(*β*)<sup>±</sup> *<sup>n</sup>* :

$$\begin{split} \mathbf{Y}^{(\beta)\pm}(k) &= \exp\left(i\frac{\left(\mathbf{E}\_{a}^{(\beta)\pm}\mathbf{V}\_{0} + m\_{0}\mathbf{S}\_{0}\right)}{\left(\sqrt{\frac{\mathbf{N}\_{a}\left(\mathbf{S}\_{0}^{(\beta)} - \mathbf{V}\_{0}^{2}\right)}{\left(\mathbf{S}\_{0} - \mathbf{V}\_{0}^{2}\right)}}\right)k\right) \\ \times \left(\sqrt{\frac{\mathbf{N}\_{a}\left(\mathbf{S}\_{0}^{(\beta)} - \mathbf{V}\_{0}^{2}\right)}{4\mathbf{S}\_{0}\left(\mathbf{E}\_{a}^{(\beta)\pm}\mathbf{S}\_{0} + m\_{0}\mathbf{V}\_{0}\right)}}\right)\left[\sqrt{\frac{\left(\frac{\mathbf{E}\_{a}^{(\beta)} + \mathbf{S}\_{0} + m\_{0}\mathbf{V}\_{0}\right)}{\left(\mathbf{S}\_{0} + \mathbf{V}\_{0}\right)}}}{\left(\sqrt{\frac{\mathbf{N}\_{a}\left(\mathbf{S}\_{0}^{(\beta)} - \mathbf{V}\_{0}^{2}\right)}{\left(\mathbf{S}\_{0} - \mathbf{V}\_{0}\right)}}}\,\upsilon^{\eta}\mathbf{C}\_{\alpha}^{\eta}\left(u\right) + \frac{2i}{\sqrt{\beta}}\sqrt{\frac{\left(\mathbf{S}\_{0} - \mathbf{V}\_{0}\right)}{\left(\mathbf{E}\_{a}^{(\beta)\pm}\mathbf{S}\_{0} + m\_{0}\mathbf{V}\_{0}\right)}}\,\upsilon^{\eta+1}\mathbf{C}\_{\alpha-1}^{\eta+1}\left(u\right)\right]\right)\right]. \end{split} \tag{7.2}$$

To use the old variables, we need the following relations

$$
\mu = \frac{p\sqrt{\beta}}{\sqrt{1+\beta p^2}}, \upsilon = \frac{1}{\sqrt{1+\beta p^2}} \text{ and } k = \frac{\arctan}{\sqrt{\beta}} \left(\sqrt{\beta}p\right). \tag{73}
$$

In the end, in order to separate the *β* dependent contribution, let us consider a very small *β*. The form of (71) can easily expand to first-order in *β*, be written as

$$
\omega\_{\rm nl}^{\ll \mathcal{B}} = \sqrt{2n} \frac{\left(S\_0^2 - V\_0^2\right)^{3/4}}{S\_0} + \beta \frac{\left(S\_0^2 - V\_0^2\right)^{5/4} \left(n^2\right)}{2S\_0 \sqrt{2n}} + O\left(\beta^2\right). \tag{74}
$$

Then we get

$$E\_{\rm II}^{(\beta)} = -\frac{m\_0 V\_0}{S\_0} \pm \sqrt{2n} \frac{\left(S\_0^2 - V\_0^2\right)^{3/4}}{S\_0} \pm \beta \frac{\left(S\_0^2 - V\_0^2\right)^{5/4} \left(n^2\right)}{2S\_0 \sqrt{2n}} + O\left(\beta^2\right),\tag{75}$$

The first term in (75) is the energy spectrum of the ordinary Dirac equation in the presence of electromagnetic field and the second term represents the correction due to the presence of the minimal length.

#### **3.4. Resolution of (1+1)-dimensional Dirac equation in position space representation**

In this subsection, we'll examine the same above second system in the position space representation, and using the properties of the Hermite polynomial. We can calculate the corrections in the values of spectrum energy and this will be seen in this regard. This system is described by the (1+1)-dimensional Dirac equation

$$\left\{\sigma\_2\mathfrak{H} + \sigma\_3\left(m\_0 + \mathcal{S}\left(\mathfrak{X}\right)\right) - \left(i\partial\_l - V\left(\mathfrak{X}\right)\right)\right\}\Psi(\mathfrak{x}, t) = 0,\tag{76}$$

where *σ*<sup>2</sup> and *σ*<sup>3</sup> are the standard Pauli matrices

14

*f* (*β*)− *<sup>n</sup>* (*p*) =

*<sup>n</sup>* (*p*) <sup>=</sup> <sup>√</sup>

*g* (*β*)−

with

×

� *Nn*(*S*<sup>2</sup>

4*S*<sup>0</sup> �

� *Nn*(*S*<sup>2</sup>

4*S*<sup>0</sup> �

Then we get

0−*V*<sup>2</sup> 0 )

*<sup>E</sup>*(*β*)<sup>±</sup> *<sup>n</sup> <sup>S</sup>*0+*m*0*V*<sup>0</sup>

0−*V*<sup>2</sup> 0 )

*<sup>E</sup>*(*β*)<sup>±</sup> *<sup>n</sup> <sup>S</sup>*0+*m*0*V*<sup>0</sup>

� 

� −

��

��

To use the old variables, we need the following relations

√*β* √

<sup>1</sup>+*βp*<sup>2</sup> , *<sup>υ</sup>* <sup>=</sup> <sup>√</sup>

The form of (71) can easily expand to first-order in *β*, be written as

<sup>2</sup>*<sup>n</sup>* (*S*<sup>2</sup> 0−*V*<sup>2</sup> 0 ) 3/4 *<sup>S</sup>*<sup>0</sup> <sup>+</sup> *<sup>β</sup>* (*S*<sup>2</sup>

*<sup>S</sup>*<sup>0</sup> <sup>±</sup> <sup>√</sup>

<sup>2</sup>*<sup>n</sup>* (*S*<sup>2</sup> 0−*V*<sup>2</sup> 0 ) 3/4 *<sup>S</sup>*<sup>0</sup> <sup>±</sup> *<sup>β</sup>* (*S*<sup>2</sup>

*<sup>u</sup>* = *<sup>p</sup>*

*ω* ≪*β <sup>n</sup>* <sup>=</sup> <sup>√</sup>

*<sup>E</sup>*(*β*) *<sup>n</sup>* <sup>=</sup> <sup>−</sup> *<sup>m</sup>*0*V*<sup>0</sup>

� Γ (*η*)

2*ı β* � Γ (*η*)

<sup>2</sup> <sup>2</sup>2*η*−1*n*!(*n*+*η*)

<sup>2</sup> 22*<sup>η</sup>*−1*n*!(*n*+*η*)

*<sup>π</sup>*Γ(*n*+2*η*)2*E*(*β*) *<sup>n</sup>*

*<sup>E</sup>*(*β*)<sup>±</sup> *<sup>n</sup>* <sup>=</sup> <sup>−</sup> *<sup>m</sup>*0*V*<sup>0</sup>

*<sup>ω</sup>*(*β*) *<sup>n</sup>* <sup>=</sup> (*S*<sup>2</sup>

and the wave functions appropriate to the energy spectrum *<sup>E</sup>*(*β*)<sup>±</sup> *<sup>n</sup>* :

<sup>Ψ</sup>(*β*)<sup>±</sup> (*k*) <sup>=</sup> exp

*<sup>E</sup>*(*β*)<sup>±</sup> *<sup>n</sup> <sup>S</sup>*0+*m*0*V*<sup>0</sup>

*<sup>E</sup>*(*β*)<sup>±</sup> *<sup>n</sup> <sup>S</sup>*0+*m*0*V*<sup>0</sup>

and in the second application we can express these results; the energy spectrum are

0−*V*<sup>2</sup> 0 ) *S*0

�

� *i* �

� (*S*0+*V*0) *<sup>υ</sup>ηC<sup>η</sup>*

� (*S*0−*V*0) *<sup>υ</sup>ηC<sup>η</sup>*

1

In the end, in order to separate the *β* dependent contribution, let us consider a very small *β*.

*β*(*n*<sup>2</sup> + <sup>2</sup>*<sup>n</sup> β* √*S*2 0−*V*<sup>2</sup> 0

*<sup>E</sup>*(*β*)<sup>±</sup> *<sup>n</sup> <sup>V</sup>*0+*m*0*S*<sup>0</sup>

(*S*2 0−*V*<sup>2</sup> <sup>0</sup> ) *<sup>k</sup>*

*<sup>n</sup>* (*u*) <sup>+</sup> <sup>√</sup> 2*i β*

*<sup>n</sup>* (*u*) <sup>+</sup> <sup>√</sup> 2*i β*

<sup>1</sup>+*βp*<sup>2</sup> and *<sup>k</sup>* <sup>=</sup> arctan

0−*V*<sup>2</sup> 0 ) 5/4 (*n*2 )

2*S*<sup>0</sup> √

> 0−*V*<sup>2</sup> 0 ) 5/4 (*n*2 )

2*S*<sup>0</sup> √

√*β*

<sup>2</sup>*<sup>n</sup>* <sup>+</sup> *<sup>O</sup>*

�

�

� (*S*0−*V*0) �

� (*S*0+*V*0) �

> ��*βp* �

> > � *β*2 �

<sup>2</sup>*<sup>n</sup>* <sup>+</sup> *<sup>O</sup>*

� *β*2 �

*<sup>E</sup>*(*β*)<sup>±</sup> *<sup>n</sup> <sup>S</sup>*0+*m*0*V*<sup>0</sup>

*<sup>E</sup>*(*β*)<sup>±</sup> *<sup>n</sup> <sup>S</sup>*0+*m*0*V*<sup>0</sup>

√*β* � *<sup>E</sup>*(*β*) *<sup>n</sup>* <sup>−</sup>*<sup>m</sup>* �

> � *<sup>E</sup>*(*β*) *<sup>n</sup>* <sup>−</sup>*<sup>m</sup>* � � 1 1+*βp*<sup>2</sup>

√*β*

� 1 1+*βp*<sup>2</sup> �*η Cη n*

�*η*+<sup>1</sup>

*<sup>S</sup>*<sup>0</sup> <sup>±</sup> *<sup>ω</sup>*(*β*) *<sup>n</sup>* ; *<sup>T</sup>* <sup>=</sup> *tb* <sup>−</sup> *ta*. (70)

� <sup>√</sup>*β<sup>p</sup>* 1+*βp*<sup>2</sup> �

> � <sup>√</sup>*β<sup>p</sup>* 1+*βp*<sup>2</sup> �

). (71)

� *<sup>υ</sup>η*+1*Cη*+<sup>1</sup>

� *<sup>υ</sup>η*+1*Cη*+<sup>1</sup>

. (73)

. (74)

, (75)

*<sup>n</sup>*−<sup>1</sup> (*u*)

*<sup>n</sup>*−<sup>1</sup> (*u*)

   .

(72)

 

*<sup>C</sup>η*+<sup>1</sup> *n*−1 . (68)

. (69)

*<sup>π</sup>*Γ(*n*+2*η*)2*E*(*β*) *<sup>n</sup>*

$$
\sigma\_2 = \begin{pmatrix} 0 \ -i \\ i \ 0 \end{pmatrix}, \quad \sigma\_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. \tag{77}
$$

We note that in (1+1) dimensions, the Dirac algebra is represented by the Pauli matrices. These reflect the invariant character of the parity of the Dirac equation. In fact in this dimension, there are no spin properties. This looks meaningless. However, in second quantization, we are obliged to use anticommutation relations to take into account the statistics of particles and to have a stable theory. At this level and even in (1+1) dimensions, the spin is an intrinsic characteristic of the particles in connection with the Wigner representation of relativistic particles.

As the potentials are time-independent, we have then to find the stationary states of this equation. Accordingly, let us choose for Ψ(*x*, *t*) the form exp (−*iEt*) Φ(*x*); we then get the following eigenvalue equation

$$\left\{\sigma\_2\mathfrak{H} + \sigma\_3\left(m\_0 + \mathcal{S}\left(\mathfrak{X}\right)\right) - \left(E - V\left(\mathfrak{X}\right)\right)\right\}\Phi(\mathfrak{x}) = 0. \tag{78}$$

In the position space acts as

$$
\hat{\mathbf{x}} = \mathbf{x}, \qquad \hat{\mathbf{p}} = -i\partial\_{\mathbf{x}} \left( 1 - \frac{\mathcal{G}}{3} \partial\_{\mathbf{x}}^2 \right). \tag{79}
$$

Then, a modified Dirac equation can be written as,

$$
\left[-i\sigma\_2 \partial\_\mathbf{x} \left(1 - \frac{\beta}{3} \partial\_\mathbf{x}^2\right) + \sigma\_3 \left(m\_0 + \mathcal{S}\left(\mathbf{x}\right)\right) - \left(E - V\left(\mathbf{x}\right)\right)\right] \Phi(\mathbf{x}) = 0. \tag{80}
$$

By using the following ansatz

$$\Phi = \left\{-i\sigma\_2\partial\_\mathbf{x}\left(1-\frac{\beta}{3}\partial\_\mathbf{x}^2\right) + \sigma\_3\left(m\_0 + \mathcal{S}\left(\mathbf{x}\right)\right) + \left(E - V\left(\mathbf{x}\right)\right)\right\}\chi.\tag{81}$$

where *χ* = ( *χ*1 *χ*2 ) is a two-component function spinor, Eq.(80) becomes a differential equation of fourth order whose solution is very complicated in the presence of potentials.

Now, by suggesting that the system is subjected to the action of linear vector plus scalar potentials,

$$V(\mathbf{x}) = V\_0 \mathbf{x}, \quad \mathbf{S}(\mathbf{x}) = \mathbf{S}\_0 \mathbf{x} \tag{82}$$

with *S*<sup>0</sup> and *V*<sup>0</sup> being arbitrary constants, we find that the Dirac spinor satisfies:

$$\left\{-\frac{2}{5}\beta\hat{\boldsymbol{\sigma}}\_{\text{x}}^{4} + \hat{\boldsymbol{\sigma}}\_{\text{x}}^{2} + (\boldsymbol{E} - \boldsymbol{V}\_{0}\mathbf{x})^{2} - (\boldsymbol{m}\_{0} + \mathbf{S}\_{0}\mathbf{x})^{2} - \left(1 - \beta\hat{\boldsymbol{\sigma}}\_{\text{x}}^{2}\right)(\mathbf{S}\_{0}\boldsymbol{\sigma}\_{1} + \mathbf{i}\boldsymbol{V}\_{0}\mathbf{\sigma}\_{2})\right\}\boldsymbol{\chi} = \mathbf{0},\tag{83}$$

We note that the linear potential, such a uniform external electromagnetic field plays a significant role in various domains of physics. For example, in particle physics, it can be regarded as a model to describe quark confinement (cf., e.g., Ref (Ferreira et al, 1971)). Further, the linear potential well has potential applications in electronics (in semiconductor devices), where the electrons are confined in almost linear quantum wells (cf., e.g., Ref (Singh, 1997)). Quantum Mechanics-Fundamentals and Applications to Technology).

Now in order to decouple the system (83), we introduce the following canonical transformation

$$\chi\left(\mathbf{x}\right) = \mathsf{U}\mathfrak{f}\left(\mathbf{x}\right) \,. \tag{84}$$

where *U* is given by

$$
\Delta U = \left[ \frac{(V\_0 + S\_0)}{\sqrt{(S\_0^2 - V\_0^2)} - \sqrt{(S\_0^2 - V\_0^2)}} \right]. \tag{85}
$$

Then the function *ξ* (*x*) satisfies the following equation

$$\begin{cases} -\frac{2\mathcal{E}}{3a}\partial\_x^4 + \frac{(1+\varepsilon\beta a)}{a}\partial\_x^2 - a\left[\mathbf{x}^2 + 2\frac{(m\_0\mathbf{S}\_0 + EV\_0)}{a^2}\mathbf{x}\right] + \frac{(E^2 - m\_0^2)}{a} - \varepsilon\right]\tilde{\xi}\_\varepsilon(\mathbf{x}) = 0,\qquad(86)$$
  $\text{with } a = \sqrt{(S\_0^2 - V\_0^2)} \text{ and } \varepsilon = \pm 1.$ 

As it has been mentioned previously that the solution is complicated, we try to find via the usual perturbation method of quantum mechanics the first energy correction at order 1 in *β* and point out how the introduction of the modified Heisenberg algebra affects the physical results. To do this, let us first suppose in this case that *α*<sup>2</sup> > 0 so as to avoid

complex eigenvalues and arrange equation (86) as a sum of two terms, one of which being the perturbative term, as follows,

$$\left[H^{0}\left(z,\partial\_{z}\right) + H^{pert}\left(\partial\_{z}\right)\right]\xi\_{\varepsilon}\left(z\right) = 0,\tag{87}$$

by setting

$$z = \left(S\_0^2 - V\_0^2\right)^{1/4} \left(\mathbf{x} + \frac{\left(m\_0 S\_0 + EV\_0\right)}{\left(S\_0^2 - V\_0^2\right)}\right). \tag{88}$$

and

16

By using the following ansatz

*χ*1 *χ*2

where *χ* = (

potentials,

 −2 <sup>3</sup> *β∂*<sup>4</sup> *<sup>x</sup>* + *∂*<sup>2</sup>

transformation

where *U* is given by

 −2*<sup>β</sup>* <sup>3</sup>*<sup>α</sup> <sup>∂</sup>*<sup>4</sup>

 (*S*2 <sup>0</sup> <sup>−</sup> *<sup>V</sup>*<sup>2</sup>

with *α* =

Φ = −*iσ*2*∂<sup>x</sup>* <sup>1</sup> <sup>−</sup> *<sup>β</sup>* 3 *∂*2 *x* 

*<sup>x</sup>* + (*<sup>E</sup>* − *<sup>V</sup>*0*x*)

*U* = 

Then the function *ξ* (*x*) satisfies the following equation

<sup>0</sup> ) and *<sup>ε</sup>* <sup>=</sup> <sup>±</sup>1.

*<sup>x</sup>* <sup>+</sup> (1+*εβα*) *<sup>α</sup> ∂*<sup>2</sup> *<sup>x</sup>* − *α*  (*S*2 <sup>0</sup> <sup>−</sup> *<sup>V</sup>*<sup>2</sup> <sup>0</sup> ) <sup>−</sup> (*S*2 <sup>0</sup> <sup>−</sup> *<sup>V</sup>*<sup>2</sup> 0 ) 

+ *<sup>σ</sup>*<sup>3</sup> (*m*<sup>0</sup> + *<sup>S</sup>* (*x*)) + (*<sup>E</sup>* − *<sup>V</sup>* (*x*))

*<sup>V</sup>*(*x*) = *<sup>V</sup>*0*x*, *<sup>S</sup>*(*x*) = *<sup>S</sup>*0*x*, (82)

(*S*0*σ*<sup>1</sup> + *iV*0*σ*2)

*χ* (*x*) = *Uξ* (*x*), (84)

. (85)

*ξε* (*x*) = 0, (86)

) is a two-component function spinor, Eq.(80) becomes a differential equation

of fourth order whose solution is very complicated in the presence of potentials.

with *S*<sup>0</sup> and *V*<sup>0</sup> being arbitrary constants, we find that the Dirac spinor satisfies:

<sup>2</sup> − (*m*<sup>0</sup> + *<sup>S</sup>*0*x*)

Now, by suggesting that the system is subjected to the action of linear vector plus scalar

2 − 1 − *β∂*<sup>2</sup> *x* 

We note that the linear potential, such a uniform external electromagnetic field plays a significant role in various domains of physics. For example, in particle physics, it can be regarded as a model to describe quark confinement (cf., e.g., Ref (Ferreira et al, 1971)). Further, the linear potential well has potential applications in electronics (in semiconductor devices), where the electrons are confined in almost linear quantum wells (cf., e.g., Ref (Singh, 1997)). Quantum Mechanics-Fundamentals and Applications to Technology).

Now in order to decouple the system (83), we introduce the following canonical

(*V*<sup>0</sup> + *<sup>S</sup>*0) (*V*<sup>0</sup> + *<sup>S</sup>*0)

*<sup>x</sup>*<sup>2</sup> <sup>+</sup> <sup>2</sup> (*m*0*S*0+*EV*0)

As it has been mentioned previously that the solution is complicated, we try to find via the usual perturbation method of quantum mechanics the first energy correction at order 1 in *β* and point out how the introduction of the modified Heisenberg algebra affects the physical results. To do this, let us first suppose in this case that *α*<sup>2</sup> > 0 so as to avoid

*<sup>α</sup>*<sup>2</sup> *x*  <sup>+</sup> (*E*2−*m*<sup>2</sup>

<sup>0</sup>) *<sup>α</sup>* <sup>−</sup> *<sup>ε</sup>*

*χ*, (81)

*χ* = 0, (83)

$$\begin{aligned} H^0 &= \partial\_z^2 - z^2 + z\_{1\prime} \\ H^{\text{pert}} &= -\frac{2}{3} \beta a \partial\_z^4 + \varepsilon \beta a \partial\_z^2 \end{aligned} \tag{89}$$

where

$$z\_1 = \frac{\left(m\_0 S\_0 + E V\_0\right)^2}{\left(S\_0^2 - V\_0^2\right)^{\frac{3}{2}}} + \frac{\left(E^2 - m\_0^2\right)}{\sqrt{\left(S\_0^2 - V\_0^2\right)}} - \varepsilon. \tag{90}$$

Now, in case where *Hpert* (*∂z*) vanishes, (i.e. when *β* → 0), equation (87) becomes that of the harmonic oscillator whose solution is known'

$$\xi\_{\varepsilon}^{\beta=0} \left( z \right) = \mathbb{C}\_{n'} \exp \left( -\frac{1}{2} z^2 \right) H\_{n'} \left( z \right) \; , \; n' = n + \frac{1}{2} + \frac{\varepsilon}{2} \; , \tag{91}$$

with *z*<sup>1</sup> verifying

$$z\_1 = 2\mathfrak{n} + 1. \quad \mathfrak{n} = 0, 1, 2, \dots \,, \tag{92}$$

where *<sup>n</sup>*′ = (*<sup>n</sup>* + 1, *<sup>n</sup>*). Hence from (90) and (92), we obtain the following energy levels for our Dirac equation:

$$E\_{n,\pm}^{\mathcal{B}=0} = -\frac{m\_0 V\_0}{S\_0} \pm \sqrt{2\pi} \frac{\left(S\_0^2 - V\_0^2\right)^{3/4}}{S\_0}.\tag{93}$$

We note the existence of the two signs in (93) which is a characteristic property of energies in relativistic quantum mechanics. Now, to find the first correction in the energy levels, we take the expectation value of the perturbation operator by using eigenfunctions (91)

$$
\Delta z\_{\eta1} = \frac{\left< \mathfrak{F}^{\mathfrak{F}^{\mathfrak{g}=0}}(z) |H^{\mu \nu t}| \mathfrak{F}^{\mathfrak{g}=0}(z) \right>}{\langle \mathfrak{F}^{\mathfrak{f}^{\mathfrak{g}=0}}(z) | \mathfrak{F}^{\mathfrak{g}=0}(z) \rangle}. \tag{94}
$$

With the help the properties of Hermite polynomial (Gradshteyn & Ryzhik, 2000), we obtain this result:

$$\Delta z\_{n1} = \frac{\beta \mathfrak{a} \int \exp\left(-\frac{1}{2}z^2\right) H\_n(z) \left[-\frac{2}{3}\partial\_z^4 + \varepsilon \partial\_z^2\right] \exp\left(-\frac{1}{2}z^2\right) H\_n(z) dz}{\int \exp\left(-\frac{1}{2}z^2\right) H\_n(z) \exp\left(-\frac{1}{2}z^2\right) H\_n(z) dz} = -\frac{\beta \mathfrak{a}}{2} \left(n^2\right),\tag{95}$$

From the relation (90), we derive the expression of ∆*En*<sup>1</sup> as a function of ∆*zn*1, and we write'

$$
\Delta E\_n^1 = \frac{\left(S\_0^2 - V\_0^2\right)^{3/2} \Delta z\_{n1}}{2S\_0 \left(E\_{n,\pm}^{\theta - 0} S\_0 + m\_0 V\_0\right)}.\tag{96}
$$

Then, by substituting (95) and (93) in (96), we find'

$$
\Delta E\_n^1 = \pm \beta \frac{\left(S\_0^2 - V\_0^2\right)^2 \left(n^2\right)}{2S\_0 \sqrt{\left(2n\right)} \left(S\_0^2 - V\_0^2\right)^{3/4}}.\tag{97}
$$

The energy spectrum of this study at order 1 in *β* can be rewritten as

$$E\_n\left(\boldsymbol{\beta}\right) = E\_{n,\pm}^{\beta=0} + \Delta E\_n^1 + \mathcal{O}\left(\boldsymbol{\beta}^2\right),\tag{98}$$

which is equal to

$$E\_{\rm n} \left( \beta \right) = -\frac{m\_0 V\_0}{S\_0} \pm \sqrt{\left( 2n \right)} \frac{\left( \mathbf{S}\_0^2 - V\_0^2 \right)^{3/4}}{S\_0} \pm \beta \frac{\left( \mathbf{S}\_0^2 - V\_0^2 \right)^{5/4} \left( n^2 \right)}{2S\_0 \sqrt{\left( 2n \right)}} + O \left( \beta^2 \right). \tag{99}$$

We note that the same correction spectrum energy obtained where using the momentum space representation defined in eq. (75).

#### **4. Conclusion**

We have discussed in this chapter the path integral formalism in the case of the appearance of the parameter of deformation *β* in the generalized Heisenberg principle (1), where we calculated the quantum corrections according to the Feynman approach (cf., e.g., Ref (Khandekar et al, 1993)) for Harmonic oscillator particle in one dimension. And we have shown that the different *α* values obtained for the ordinary quantum mechanics, that make us wonder about these results. In addition, we have generalized the study of relativistic particles which have one half (1/2) spin for example Dirac oscillator and relativistic spinning particle subjected to the action of combined vector and scalar linear potentials, with a deformed commutation relation for the Heisenberg principle. We have obtained the same *α* values for Harmonic oscillator. This has been explained in our previous works (Benzair et al, 2012; 2014). Using the residue theorem, the energy spectrum and corresponding eigenfunctions expressed in terms of Gegenbauer polynomials are then deduced as a function of the deformation parameter *β*. It has been noted above the energy spectrum of the relativistic

spinorial particle is dependent on term quadratic in *n* that is similar to the energy levels of a particle confined in a potential well. In addition, we studied in the second relativistic application, the energy spectrum of the Dirac equation for a spin 1/2 subjected to the action of combined vector and scalar linear potentials, with a deformed commutation relation for the Heisenberg principle, where we have used the old variable *p* in the position space representation, we have obtained a differential equation of fourth order whose analytic solution is complicated. We have calculated the energy correction in first order for *β* by using a usual approximation technique of quantum mechanics. We note that the two methods gave the same results for the first energy correction at order 1 in *β*. In this study, we did not take the case *S*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>V</sup>*<sup>2</sup> <sup>0</sup> <sup>&</sup>lt; 0, so as to avoid the complex eigenvalues. But when *<sup>S</sup>*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>V</sup>*<sup>2</sup> <sup>0</sup> <sup>=</sup> 0, the calculation is very simple and we can obtain physical results.

Finally, let us signal that the problems of choosing *α*−point discretization in the case of deformed space are under consideration.

#### **Author details**

18

<sup>∆</sup>*zn*<sup>1</sup> <sup>=</sup> *βα* exp(<sup>−</sup> <sup>1</sup>

Then, by substituting (95) and (93) in (96), we find'

*En* (*β*) = − *<sup>m</sup>*0*V*<sup>0</sup>

space representation defined in eq. (75).

*<sup>S</sup>*<sup>0</sup> <sup>±</sup> (2*n*) (*S*<sup>2</sup> 0−*V*<sup>2</sup> 0 ) 3/4 *<sup>S</sup>*<sup>0</sup> <sup>±</sup> *<sup>β</sup>* (*S*<sup>2</sup>

which is equal to

**4. Conclusion**

2 *z*2

exp(<sup>−</sup> <sup>1</sup>

)*Hn*(*z*)[− <sup>2</sup>

∆*E*1

∆*E*1

The energy spectrum of this study at order 1 in *β* can be rewritten as

*En* (*β*) <sup>=</sup> *<sup>E</sup>β*=<sup>0</sup>

3 *∂*4 *<sup>z</sup>*+*ε∂*<sup>2</sup>

From the relation (90), we derive the expression of ∆*En*<sup>1</sup> as a function of ∆*zn*1, and we write'

0−*V*<sup>2</sup> 0 ) 3/2∆*zn*<sup>1</sup>

> 0−*V*<sup>2</sup> 0 ) 2 (*n*<sup>2</sup> )

<sup>√</sup>(2*n*)(*S*<sup>2</sup>

0−*V*<sup>2</sup> 0 )

*<sup>n</sup>* + *O β*2 

> 0−*V*<sup>2</sup> 0 ) 5/4 (*n*<sup>2</sup> )

<sup>√</sup>(2*n*) <sup>+</sup> *<sup>O</sup>*

2*S*<sup>0</sup>

<sup>2</sup> *<sup>z</sup>*2)*Hn*(*z*) exp(<sup>−</sup> <sup>1</sup>

*<sup>n</sup>* <sup>=</sup> (*S*<sup>2</sup>

*<sup>n</sup>* <sup>=</sup> <sup>±</sup>*<sup>β</sup>* (*S*<sup>2</sup>

2*S*<sup>0</sup>

*<sup>n</sup>*,<sup>±</sup> <sup>+</sup> <sup>∆</sup>*E*<sup>1</sup>

We note that the same correction spectrum energy obtained where using the momentum

We have discussed in this chapter the path integral formalism in the case of the appearance of the parameter of deformation *β* in the generalized Heisenberg principle (1), where we calculated the quantum corrections according to the Feynman approach (cf., e.g., Ref (Khandekar et al, 1993)) for Harmonic oscillator particle in one dimension. And we have shown that the different *α* values obtained for the ordinary quantum mechanics, that make us wonder about these results. In addition, we have generalized the study of relativistic particles which have one half (1/2) spin for example Dirac oscillator and relativistic spinning particle subjected to the action of combined vector and scalar linear potentials, with a deformed commutation relation for the Heisenberg principle. We have obtained the same *α* values for Harmonic oscillator. This has been explained in our previous works (Benzair et al, 2012; 2014). Using the residue theorem, the energy spectrum and corresponding eigenfunctions expressed in terms of Gegenbauer polynomials are then deduced as a function of the deformation parameter *β*. It has been noted above the energy spectrum of the relativistic

2*S*<sup>0</sup> *Eβ*=<sup>0</sup> *<sup>n</sup>*,<sup>±</sup> *<sup>S</sup>*0+*m*0*V*<sup>0</sup>

*<sup>z</sup>* ] exp(<sup>−</sup> <sup>1</sup> 2 *z*2

)*Hn*(*z*)*dz*

2 *n*2 

. (96)

3/4 . (97)

, (98)

. (99)

 *β*2  , (95)

<sup>2</sup> *<sup>z</sup>*2)*Hn*(*z*)*dz* <sup>=</sup> <sup>−</sup> *βα*

Hadjira Benzair1, Mahmoud Merad2 and Taher Boudjedaa<sup>3</sup>

1 Laboratoire LRPPS, Faculté des Sciences et de la Technologie et des Sciences de la Matière, Université Kasdi Merbah Ouargla, Ouargla, Algeria

2 Laboratoire SDC, Département des sciences de la matière, Université de Oum-El-Bouaghi, Algeria

3 Laboratoire de Physique Théorique, Département de Physique, Université de Jijel, BP 98, Ouled Aissa, Jijel, Algeria

#### **References**


[23] Kempf, A. (1997). Non-pointlike particles in harmonic oscillators, J. Phys. Vol. A 30: 2093-2102.

20

A 32: 7691-7696

[6] Bordemann, M.,.Makhlouf, A. & Petit, T. (2005). Déformation par quantification et

[7] Brau, F. (1999) Minimal length uncertainty relation and the hydrogen atom, J. Phys. Vol

[8] Capozziello, S., Lambiase, G. & Scarpetta, G. (2000). Generalized uncertainty principle

[9] Chang, L. N., Minic, D., Okamura, N. & Takeuchi, T. (2002). Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations,

[10] Chargui, Y., Chetouani, L. & Trabelsi, A. (2010). Path integral approach to the D-dimensional harmonic oscillator with minimal length, Phys. Scr. Vol. 81: 015005.

[11] Falek, M. & Merad, M. (2009). Bosonic oscillator in the presence of minimal lengths, J.

[12] Falek, M. & Merad, M. (2010). Generalization of Bosonic oscillator in the presence of

[14] Garay, L. J. (1995). Quantum gravity and minimum length, Int. J. Mod. Phys. Vol. A 10:

[15] Gross, D. J. & Mende, P. F. (1988). String theory beyond the Planck scale, Nucl. Phys.

[16] Guida, R., Konishi, K. & Provero, P. (1991) On the short distance behavior of string

[17] Gradshteyn, I. S. & Ryzhik, I. M. (2000). Table of Integrals Series and Products

[18] Grosche, C. & Steiner, F. (1998). Handbook of Feynman Path Integrals (Springer, Berlin).

[19] Hinrichsen, H. & Kempf, A. (1996). Maximal localization in the presence of minimal uncertainties in positions and in momenta, J. Math. Phys. Vol. 37: 2121-2137.

[20] Kato, M. (1990). Particle theories with minimum observable length and open string

[21] Kempf, A. (1994). Uncertainty relation in quantum mechanics with quantum group symmetry, Journal of Mathematical Physics, J. Math. Phys. Vol. 35: 4483-4495.

[22] Kempf, A., Mangano, G. & . Mann, R. B. (1995). Hilbert space representation of the

minimal length uncertainty relation, Phys. Rev. Vol. D 52: 1108-1118.

[13] Ferreira, P. L., Helayel, J. A. & Zagury, N. (1971). Il Nuovo Cimento Vol: A 2 215.

rigidité des algèbres enveloppantes, Journal of Algebra, Vol 285: 623-648.

from quantum geometry, Int. J. Theor. Phys. Vol. 39: 15-22.

Phys. Rev. Vol. D 65: 125027-125035.

minimal lengths, J. Math. Phys. Vol 51: 033516.

theories,. Mod. Phys. Lett. Vol. A 6: 1487-1504.

theory, Physics Letters B, vol. 245, no. 1, pp. 43–47, 1990.

Math. Phys. Vol 50: 023508.

145-165.

Vol. B 303: 407–454.

(Academic Press, New York).


[23] Kempf, A. (1997). Non-pointlike particles in harmonic oscillators, J. Phys. Vol. A 30: 2093-2102.

Path Integral Methods in Generalized Uncertainty Principle 21

10.5772/59106


**Chapter 4**
