**Path Integrals**

24 Quantum Mechanics

156 Advances in Quantum Mechanics

Tokyo, 2010.

quant-ph/061211.

Phys, 48 (1976), 119.

arXiv : 1108.2322 [math-ph].

I expect that the book will be translated into English.

semigroups of N–level systems, J. Math. Phys, 17 (1976), 821.

Mathematical Physics, World Scientific, Singapore, 1985.

This book is strongly recommended although it is thick.

I expect that this (crib) note will be published in some journal.

see http://www.hep.anl.gov/czachos/index.html.

This is a kind of dictionary of coherent states.

[11] H. S. Green : Matrix Mechanics, P. Noordhoff Ltd, Groningen, 1965. This is my favorite textbook of elementary Quantum Mechanics.

[8] K. Fujii and T. Suzuki : An Approximate Solution of the Jaynes–Cummings Model with Dissipation II : Another Approach, Int. J. Geom. Methods Mod. Phys, 9 (2012), 1250036,

[9] K. Fujii and et al ; Treasure Box of Mathematical Sciences (in Japanese), Yuseisha,

[10] V. Gorini, A. Kossakowski and E. C. G. Sudarshan ; Completely positive dynamical

[12] K. Hornberger : Introduction to Decoherence Theory, in "Theoretical Foundations of Quantum Information", Lecture Notes in Physics, 768 (2009), 221-276, Springer, Berlin,

[13] E. T. Jaynes and F. W. Cummings : Comparison of Quantum and Semiclassical Radiation

[14] J. R. Kauder and Bo-S. Skagerstam : Coherent States–Applications in Physics and

[15] G. Lindblad ; On the generator of quantum dynamical semigroups, Commun. Math.

[17] C. Zachos : Crib Notes on Campbell-Baker-Hausdorff expansions, unpublished, 1999,

Theories with Applications to the Beam Maser, Proc. IEEE, 51 (1963), 89.

[16] W. P. Schleich : Quantum Optics in Phase Space, WILEY–VCH, Berlin, 2001.

**Chapter 8**

**Provisional chapter**

**The Schwinger Action Principle and Its Applications to**

In physics it is generally of interest to understand the dynamics of a system. The way the dynamics is to be specified and studied invariably depends on the scale of the system, that is whether it is macroscopic or microscopic. The formal machinery with which the world is explained and understood depends at which of these two levels an experiment is conducted [1]. At the classical level, the dynamics of the system can be understood in terms of such

In fact, classical mechanics can be formulated in terms of a principle of stationary action to obtain the Euler-Lagrange equations. To carry this out, an action functional has to be defined.

> *L*(*q<sup>i</sup>* (*t*), *q*˙ *i*

The action depends on the Lagrangian, written *L* in (1). It is to be emphasized that the action is a functional, which can be thought of as a function defined on a space of functions. For any given trajectory or path in space, the action works out to be a number, so *S* maps paths

One way to obtain equations of motion is by means of Hamilton's principle. Hamilton's principle states that the actual motion of a particle with Lagrangian *L* is such that the action functional is stationary. This means the action functional achieves a minimum or maximum value. To apply and use this principle, Stationary action must result in the Euler-Lagrange equations of motion. Conversely, if the Euler-Lagrange equations are imposed, the action functional should be stationary. As is well known, the Euler-Lagrange equations provide a system of second-order differential equations for the path. This in turn leads to other

> ©2012 Bracken, 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. © 2013 Bracken; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Bracken; licensee InTech. This is a paper 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.

distribution, and reproduction in any medium, provided the original work is properly cited.

(*t*), *t*) *dt*. (1)

*<sup>S</sup>*[**q**(*t*)] = *<sup>t</sup>*<sup>2</sup>

*t*1

**The Schwinger Action Principle and Its Applications**

**Quantum Mechanics**

**to Quantum Mechanics**

http://dx.doi.org/10.5772/53472

It is written as *S* and given by

approaches to the same end.

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

things as trajectories in space or space-time.

Paul Bracken

**1. Introduction**

to real numbers.

Paul Bracken

**Provisional chapter**

### **The Schwinger Action Principle and Its Applications to Quantum Mechanics to Quantum Mechanics**

**The Schwinger Action Principle and Its Applications**

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

Paul Bracken

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53472

### **1. Introduction**

In physics it is generally of interest to understand the dynamics of a system. The way the dynamics is to be specified and studied invariably depends on the scale of the system, that is whether it is macroscopic or microscopic. The formal machinery with which the world is explained and understood depends at which of these two levels an experiment is conducted [1]. At the classical level, the dynamics of the system can be understood in terms of such things as trajectories in space or space-time.

In fact, classical mechanics can be formulated in terms of a principle of stationary action to obtain the Euler-Lagrange equations. To carry this out, an action functional has to be defined. It is written as *S* and given by

$$S[\mathbf{q}(t)] = \int\_{t\_1}^{t\_2} L(q^i(t), \dot{q}^i(t), t) \, dt. \tag{1}$$

The action depends on the Lagrangian, written *L* in (1). It is to be emphasized that the action is a functional, which can be thought of as a function defined on a space of functions. For any given trajectory or path in space, the action works out to be a number, so *S* maps paths to real numbers.

One way to obtain equations of motion is by means of Hamilton's principle. Hamilton's principle states that the actual motion of a particle with Lagrangian *L* is such that the action functional is stationary. This means the action functional achieves a minimum or maximum value. To apply and use this principle, Stationary action must result in the Euler-Lagrange equations of motion. Conversely, if the Euler-Lagrange equations are imposed, the action functional should be stationary. As is well known, the Euler-Lagrange equations provide a system of second-order differential equations for the path. This in turn leads to other approaches to the same end.

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. © 2013 Bracken; licensee InTech. This is an open access article 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. © 2013 Bracken; licensee InTech. This is a paper 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.

©2012 Bracken, licensee InTech. This is an open access chapter distributed under the terms of the Creative

As an illustration, the momentum canonically conjugate to the coordinate *q<sup>i</sup>* is defined by

$$p\_i = \frac{\partial L}{\partial \dot{q}^i}.\tag{2}$$

with any vector space *<sup>V</sup>* is the dual space *<sup>V</sup>*<sup>∗</sup> whose elements are referred to as bras in Dirac's terminology. A basis for *<sup>V</sup>*<sup>∗</sup> is denoted by {�*a*|} and is dual to {|*a*�}. The quantities satisfy

<sup>|</sup>*a*′′� <sup>=</sup> *<sup>δ</sup>*(*a*′

continuous. The choice of a complete set of mutually compatible observables is not unique. Suppose {|*b*�} also provides a basis for *V* relevant to another set of mutually compatible observables *<sup>B</sup>*1, *<sup>B</sup>*2, ··· . Since {|*a*�} and {|*b*�} are both bases for *<sup>V</sup>*, this means that one set of

and the �*a*|*b*� coefficients are some set of complex numbers, so that �*b*|*a*�<sup>∗</sup> = �*a*|*b*�. If there is a third basis for *V* provided by {|*c*�}, then these complex numbers are related by means of

*b*

Schwinger's action principle is based on the types of transformation properties of the

Suppose the transformation function is subjected to, as Schwinger asserted, any conceivable infinitesimal variation. Then, by performing an arbitrary variation of (7), it follows that

Now a new operator can be defined which evaluates this actual variation when it is placed between the relevant state vectors. Define this operator to be *δWab*, so that it has the following

*h*¯

Including the factor of ¯*h* gives the operator units of action. Using (10) in (8) produces,

*<sup>δ</sup>*�*a*|*b*� <sup>=</sup> *<sup>i</sup>*

<sup>|</sup>*b*� <sup>=</sup> <sup>∑</sup>*<sup>a</sup>*

�*a*|*c*� = ∑

transformation functions which can be constructed from this basis set [4].

*b*

*<sup>δ</sup>*�*a*|*c*� <sup>=</sup> ∑

, *<sup>a</sup>*′′) is the Kronecker delta if *<sup>a</sup>*′ is a discrete set, and the Dirac delta if it is

, *<sup>a</sup>*′′), (5)

http://dx.doi.org/10.5772/53472

161

The Schwinger Action Principle and Its Applications to Quantum Mechanics


�*a*|*b*��*b*|*c*�. (7)

[(*δ*�*a*|*b*�)�*b*|*c*� + �*a*|*b*�(*δ*�*b*|*c*�)], (8)

*<sup>δ</sup>*�*a*|*b*� <sup>=</sup> *<sup>δ</sup>*�*b*|*a*�∗. (9)

[�*a*|*δWab*|*b*��*b*|*c*� + �*a*|*b*��*b*|*δWbc*|*c*�] = �*a*|*δWab* + *δWbc*|*c*�, (11)

*δWac* = *δWab* + *δWbc*. (12)

�*a*|*δWab*|*b*�. (10)

�*a*′

basis vectors can be expressed in terms of the other set

where *<sup>δ</sup>*(*a*′

and moreover,

action between states

�*a*|*δWac*|*c*� <sup>=</sup> ∑

*b*

using the completeness relation (7). Now it follows from (11) that

The dependence on the velocity components *q*˙ *<sup>i</sup>* can be eliminated in favor of the canonical momentum. This means that (2) must be solved for the *q*˙ *<sup>i</sup>* in terms of the *q<sup>i</sup>* and *pi*, and the inverse function theorem states this is possible if and only if (*∂pi*/*∂q*˙ *j* ) � 0. Given a non-singular system all dependence on *q*˙ *<sup>i</sup>* can be eliminated by means of a Legendre transformation

$$H(\mathbf{q}, \mathbf{p}\_\prime t) = p\_i \dot{q}^i - L(\mathbf{q}\_\prime \dot{\mathbf{q}}\_\prime t). \tag{3}$$

The Hamiltonian equations are obtained by considering the derivatives of the Hamiltonian with respect to *q<sup>i</sup>* and *pi*. The action in terms of *H* is written

$$S[\mathbf{q}\_\prime \mathbf{p}] = \int\_{t\_1}^{t\_2} dt \, [p\_i \dot{q}^i - H(\mathbf{q}\_\prime \mathbf{p}\_\prime t)]\_\prime \tag{4}$$

such that the action now depends on both **q** and **p**. Using this the principle of stationary action can then be modified so that Hamilton's equations result.

In passing to the quantum domain, the concept of path or trajectory is of less importance, largely because it has no meaning. In quantum physics it can no longer be assumed that the interaction between system and measuring device can be made arbitrarily small and that there are no restrictions on what measurements can be made on the system either in terms of type or in accuracy. Both these assumptions tend to break down at the scales of interest here, and one is much more interested in states and observables, which replace the classical idea of a trajectory with well defined properties [2,3].

Now let us follow Dirac and consider possible measurements on a system as observables. Suppose *Ai* denotes any observable with *ai* as a possible outcome of any measurement of this observable. As much information as possible can be extracted with regard to a quantum mechanical system by measuring some set of observables {*Ai*}*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> without restriction. Thus, the observables should be mutually compatible because the measurement of any observable in the set does not affect the measurement of any of the other observables. The most information that can be assembled about a system is the collection of numbers {*ai*}*<sup>n</sup> <sup>i</sup>*=1, which are possible values for the set of mutually compatible observables, and this set specifies the state of the system.

#### **2. Schwinger's action principle**

In Dirac's view of quantum mechanics, the state of a system is associated with a vector in a complex vector space *V*. The knowledge of the values for a complete set of mutually compatible observables gives the most information about a state. It can then be assumed that {|*a*�}, where |*a*� = |*a*1, ··· , *an*�, the set of all possible states, forms a basis for *<sup>V</sup>*. Associated with any vector space *<sup>V</sup>* is the dual space *<sup>V</sup>*<sup>∗</sup> whose elements are referred to as bras in Dirac's terminology. A basis for *<sup>V</sup>*<sup>∗</sup> is denoted by {�*a*|} and is dual to {|*a*�}. The quantities satisfy

$$
\langle a'|a''\rangle = \delta(a',a''),
\tag{5}
$$

where *<sup>δ</sup>*(*a*′ , *<sup>a</sup>*′′) is the Kronecker delta if *<sup>a</sup>*′ is a discrete set, and the Dirac delta if it is continuous. The choice of a complete set of mutually compatible observables is not unique. Suppose {|*b*�} also provides a basis for *V* relevant to another set of mutually compatible observables *<sup>B</sup>*1, *<sup>B</sup>*2, ··· . Since {|*a*�} and {|*b*�} are both bases for *<sup>V</sup>*, this means that one set of basis vectors can be expressed in terms of the other set

$$|b\rangle = \sum\_{a} |a\rangle \langle a|b\rangle\_{\prime} \tag{6}$$

and the �*a*|*b*� coefficients are some set of complex numbers, so that �*b*|*a*�<sup>∗</sup> = �*a*|*b*�. If there is a third basis for *V* provided by {|*c*�}, then these complex numbers are related by means of

$$
\langle \langle a|c \rangle = \sum\_{b} \langle a|b \rangle \langle b|c \rangle. \tag{7}
$$

Schwinger's action principle is based on the types of transformation properties of the transformation functions which can be constructed from this basis set [4].

Suppose the transformation function is subjected to, as Schwinger asserted, any conceivable infinitesimal variation. Then, by performing an arbitrary variation of (7), it follows that

$$
\delta \langle a|c \rangle = \sum\_{b} \left[ \left( \delta \langle a|b \rangle \right) \langle b|c \rangle + \langle a|b \rangle \left( \delta \langle b|c \rangle \right) \right] \tag{8}
$$

and moreover,

2 Quantum Mechanics

transformation

state of the system.

**2. Schwinger's action principle**

The dependence on the velocity components *q*˙

a non-singular system all dependence on *q*˙

momentum. This means that (2) must be solved for the *q*˙

with respect to *q<sup>i</sup>* and *pi*. The action in terms of *H* is written

*S*[**q**, **p**] =

action can then be modified so that Hamilton's equations result.

mechanical system by measuring some set of observables {*Ai*}*<sup>n</sup>*

idea of a trajectory with well defined properties [2,3].

the inverse function theorem states this is possible if and only if (*∂pi*/*∂q*˙

*H*(**q**, **p**, *t*) = *piq*˙

 *<sup>t</sup>*<sup>2</sup> *t*1

As an illustration, the momentum canonically conjugate to the coordinate *q<sup>i</sup>* is defined by

*pi* <sup>=</sup> *<sup>∂</sup><sup>L</sup> ∂q*˙*<sup>i</sup>*

The Hamiltonian equations are obtained by considering the derivatives of the Hamiltonian

*dt* [*piq*˙

such that the action now depends on both **q** and **p**. Using this the principle of stationary

In passing to the quantum domain, the concept of path or trajectory is of less importance, largely because it has no meaning. In quantum physics it can no longer be assumed that the interaction between system and measuring device can be made arbitrarily small and that there are no restrictions on what measurements can be made on the system either in terms of type or in accuracy. Both these assumptions tend to break down at the scales of interest here, and one is much more interested in states and observables, which replace the classical

Now let us follow Dirac and consider possible measurements on a system as observables. Suppose *Ai* denotes any observable with *ai* as a possible outcome of any measurement of this observable. As much information as possible can be extracted with regard to a quantum

the observables should be mutually compatible because the measurement of any observable in the set does not affect the measurement of any of the other observables. The most

are possible values for the set of mutually compatible observables, and this set specifies the

In Dirac's view of quantum mechanics, the state of a system is associated with a vector in a complex vector space *V*. The knowledge of the values for a complete set of mutually compatible observables gives the most information about a state. It can then be assumed that {|*a*�}, where |*a*� = |*a*1, ··· , *an*�, the set of all possible states, forms a basis for *<sup>V</sup>*. Associated

information that can be assembled about a system is the collection of numbers {*ai*}*<sup>n</sup>*

. (2)

*<sup>i</sup>* can be eliminated in favor of the canonical

*<sup>i</sup>* can be eliminated by means of a Legendre

*<sup>i</sup>* − *L*(**q**, **˙q**, *t*). (3)

*<sup>i</sup>* − *H*(**q**, **p**, *t*)], (4)

*<sup>i</sup>* in terms of the *q<sup>i</sup>* and *pi*, and

*j*

*<sup>i</sup>*=<sup>1</sup> without restriction. Thus,

*<sup>i</sup>*=1, which

) � 0. Given

$$
\delta \langle a|b \rangle = \delta \langle b|a \rangle^\*. \tag{9}
$$

Now a new operator can be defined which evaluates this actual variation when it is placed between the relevant state vectors. Define this operator to be *δWab*, so that it has the following action between states

$$
\delta \langle a|b \rangle = \frac{\dot{l}}{\hbar} \langle a|\delta \mathcal{W}\_{ab}|b \rangle. \tag{10}
$$

Including the factor of ¯*h* gives the operator units of action. Using (10) in (8) produces,

$$
\langle a|\delta \mathcal{W}\_{\rm ac}|c\rangle = \sum\_{b} \left[ \langle a|\delta \mathcal{W}\_{\rm ab}|b\rangle \langle b|c\rangle + \langle a|b\rangle \langle b|\delta \mathcal{W}\_{\rm bc}|c\rangle \right] = \langle a|\delta \mathcal{W}\_{\rm ab} + \delta \mathcal{W}\_{\rm bc}|c\rangle,\tag{11}
$$

using the completeness relation (7). Now it follows from (11) that

$$
\delta \mathcal{W}\_{\rm{ac}} = \delta \mathcal{W}\_{\rm{ab}} + \delta \mathcal{W}\_{\rm{bc}}.\tag{12}
$$

In the case in which the *<sup>a</sup>* and *<sup>b</sup>* descriptions are identified then using *<sup>δ</sup>*�*a*|*a*′ � = 0, there results

$$
\delta \mathcal{W}\_{aa} = 0.\tag{13}
$$

eigenstates of the transformed operator, it must be that states transform as |*a*� → *U*†|*a*�. Thinking of the transformation as being infinitesimal in nature, the operator *U* can be written

*h*¯

Here operator *G* is a Hermitian operator and depends only on the observables *A*<sup>1</sup> at the time

*h*¯

and the operator *G*<sup>2</sup> depends only on observables *A*<sup>2</sup> at time *t*2. If both sets *A*1, *A*<sup>2</sup> are

If the time evolution from state |*a*1, *<sup>t</sup>*1� to |*a*2, *<sup>t</sup>*2� can be thought of as occurring continuously

 *<sup>t</sup>*<sup>2</sup> *t*1

where *L*(*t*) is called the Lagrange operator. As a consequence of (23), it follows that if the dynamical variables which enter *L*(*t*) are altered during an arbitrary infinitesimal change

The operator equations of motion are implied in this result. The usual form for the Lagrange

not as usual commute. It is assumed the Hamiltonian *H* is a Hermitian operator. The action operator (24) is used to calculate the variation *δW*21. In order to vary the endpoints *t*1, *t*2,

*W*<sup>21</sup> =

*<sup>L</sup>*(*t*) = <sup>1</sup> 2 (*pi x*˙ *<sup>i</sup>* + *x*˙ *i*

The first term has been symmetrized to give a Hermitian *L*(*t*), as the operators *x*˙

*<sup>G</sup>*1|*a*1, *<sup>t</sup>*1�. (20)

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163

The Schwinger Action Principle and Its Applications to Quantum Mechanics

�*a*2, *<sup>t</sup>*2|*G*2, (21)

�*a*2, *<sup>t</sup>*2|*G*<sup>2</sup> − *<sup>G</sup>*1|*a*1, *<sup>t</sup>*1�. (22)

*L*(*t*) *dt*, (24)

*δW*<sup>21</sup> = 0. (25)

*pi*) − *<sup>H</sup>*(**x**, **<sup>p</sup>**, *<sup>t</sup>*). (26)

*<sup>i</sup>* and *pi* do

*<sup>δ</sup>W*<sup>21</sup> = *<sup>G</sup>*<sup>2</sup> − *<sup>G</sup>*1. (23)

*<sup>h</sup>*¯ *G*, where *G* is Hermitean. It is then possible to define a variation

*<sup>δ</sup>*|*a*1, *<sup>t</sup>*1� <sup>=</sup> <sup>−</sup> *<sup>i</sup>*

*<sup>δ</sup>*�*a*2, *<sup>t</sup>*2<sup>|</sup> <sup>=</sup> *<sup>i</sup>*

altered infinitesimally, then the change in the transformation function is given by

*h*¯

*<sup>δ</sup>*�*a*2, *<sup>t</sup>*2|*a*1, *<sup>t</sup>*1� <sup>=</sup> *<sup>i</sup>*

Comparing this with (18), it is concluded that

in time, then *W*<sup>21</sup> can be expressed as

between *t*<sup>1</sup> and *t*2, then it must be that

operator is

*t*1. Similarly, if observables *A*<sup>2</sup> are altered at *t*2, it is the case that

*U* = *I* + *<sup>i</sup>*

Identifying the *a* and *c* pictures in (11) gives,

$$
\delta \mathcal{W}\_{ba} = -\delta \mathcal{W}\_{ab}.\tag{14}
$$

The complex conjugate of (10) with *a* and *b* descriptions reversed implies

$$-\frac{i}{\hbar}\langle b|\delta W\_{ba}|a\rangle^\* = \frac{i}{\hbar}\langle a|\delta W\_{ab}|b\rangle. \tag{15}$$

This has the equivalent form

$$-\left\langle a|\delta \mathcal{W}\_{ba}^{\dagger}|b\right\rangle = \left\langle a|\delta \mathcal{W}\_{ab}|b\right\rangle. \tag{16}$$

Using (14), this yields the property

$$
\delta \mathcal{W}\_{ba}^{\dagger} = \delta \mathcal{W}\_{ab}.\tag{17}
$$

The basic properties of the transformation function and the definition (10) have produced all of these additional properties [5,6].

In the Heisenberg picture, the basis kets become time dependent. The transformation function relates states which are eigenstates of different complete sets of commuting observables at different times. Instead of using different letters, different subscripts 1, 2 can be used to denote different complete sets of commuting observables. In this event, (10) takes the form

$$
\delta \langle a\_{2'}' t\_2 | a\_{1'}' t\_1 \rangle = \frac{i}{\hbar} \langle a\_{2'}' t\_2 | \delta W\_{21} | a\_{1'}' t\_1 \rangle. \tag{18}
$$

The assumption at the heart of this approach is that the operator *δW*<sup>21</sup> in (18) is obtained from the variation of a single operator *W*21. This is referred to as the action operator.

To adapt the results of the previous notation to the case with subscripts, we should have

$$W\_{31} = W\_{32} + W\_{21\prime} \qquad W\_{11} = 0, \qquad W\_{21} = -W\_{12} = W\_{21}^{\dagger}.\tag{19}$$

At this point, a correspondence between the Schwinger action principle and the classical principle of stationary action can be made. Suppose the members of a complete set of commuting observables *<sup>A</sup>*<sup>1</sup> which have eigenvectors |*a*1, *<sup>t</sup>*� in the Heisenberg picture are deformed in some fashion at time *t*1. For example, take the alteration in the observables to correspond to a unitary transformation *<sup>A</sup>* → *<sup>U</sup>*†*AU* such that *<sup>U</sup>*† = *<sup>U</sup>*<sup>−</sup>1. To remain

eigenstates of the transformed operator, it must be that states transform as |*a*� → *U*†|*a*�. Thinking of the transformation as being infinitesimal in nature, the operator *U* can be written *U* = *I* + *<sup>i</sup> <sup>h</sup>*¯ *G*, where *G* is Hermitean. It is then possible to define a variation

$$
\delta|a\_1, t\_1\rangle = -\frac{i}{\hbar}G\_1|a\_1, t\_1\rangle. \tag{20}
$$

Here operator *G* is a Hermitian operator and depends only on the observables *A*<sup>1</sup> at the time *t*1. Similarly, if observables *A*<sup>2</sup> are altered at *t*2, it is the case that

$$
\delta \langle a\_2, t\_2 \vert = \frac{i}{\hbar} \langle a\_2, t\_2 \vert \mathcal{G}\_{2\prime} \tag{21}
$$

and the operator *G*<sup>2</sup> depends only on observables *A*<sup>2</sup> at time *t*2. If both sets *A*1, *A*<sup>2</sup> are altered infinitesimally, then the change in the transformation function is given by

$$
\delta \langle a\_2, t\_2 | a\_1, t\_1 \rangle = \frac{i}{\hbar} \langle a\_2, t\_2 | \mathcal{G}\_2 - \mathcal{G}\_1 | a\_1, t\_1 \rangle. \tag{22}
$$

Comparing this with (18), it is concluded that

4 Quantum Mechanics

Identifying the *a* and *c* pictures in (11) gives,

This has the equivalent form

Using (14), this yields the property

of these additional properties [5,6].

the form

results

In the case in which the *<sup>a</sup>* and *<sup>b</sup>* descriptions are identified then using *<sup>δ</sup>*�*a*|*a*′

The complex conjugate of (10) with *a* and *b* descriptions reversed implies

− �*a*|*δW*†

*δW*†

<sup>1</sup>, *<sup>t</sup>*1� <sup>=</sup> *<sup>i</sup> h*¯ �*a*′

The basic properties of the transformation function and the definition (10) have produced all

In the Heisenberg picture, the basis kets become time dependent. The transformation function relates states which are eigenstates of different complete sets of commuting observables at different times. Instead of using different letters, different subscripts 1, 2 can be used to denote different complete sets of commuting observables. In this event, (10) takes

The assumption at the heart of this approach is that the operator *δW*<sup>21</sup> in (18) is obtained from the variation of a single operator *W*21. This is referred to as the action operator.

To adapt the results of the previous notation to the case with subscripts, we should have

*<sup>W</sup>*<sup>31</sup> = *<sup>W</sup>*<sup>32</sup> + *<sup>W</sup>*21, *<sup>W</sup>*<sup>11</sup> = 0, *<sup>W</sup>*<sup>21</sup> = −*W*<sup>12</sup> = *<sup>W</sup>*†

At this point, a correspondence between the Schwinger action principle and the classical principle of stationary action can be made. Suppose the members of a complete set of commuting observables *<sup>A</sup>*<sup>1</sup> which have eigenvectors |*a*1, *<sup>t</sup>*� in the Heisenberg picture are deformed in some fashion at time *t*1. For example, take the alteration in the observables to correspond to a unitary transformation *<sup>A</sup>* → *<sup>U</sup>*†*AU* such that *<sup>U</sup>*† = *<sup>U</sup>*<sup>−</sup>1. To remain

2, *<sup>t</sup>*2|*δW*21|*a*′

�*b*|*δWba*|*a*�<sup>∗</sup> <sup>=</sup> *<sup>i</sup>*

*h*¯

− *i h*¯

*<sup>δ</sup>*�*a*′ 2, *<sup>t</sup>*2|*a*′ � = 0, there

*δWaa* = 0. (13)

*δWba* = −*δWab*. (14)

*ba*|*b*� = �*a*|*δWab*|*b*�. (16)

*ba* = *δWab*. (17)

<sup>1</sup>, *<sup>t</sup>*1�. (18)

<sup>21</sup>. (19)

�*a*|*δWab*|*b*�. (15)

$$
\delta \mathcal{W}\_{21} = \mathcal{G}\_2 - \mathcal{G}\_1. \tag{23}
$$

If the time evolution from state |*a*1, *<sup>t</sup>*1� to |*a*2, *<sup>t</sup>*2� can be thought of as occurring continuously in time, then *W*<sup>21</sup> can be expressed as

$$W\_{21} = \int\_{t\_1}^{t\_2} L(t) \, dt\_1 \tag{24}$$

where *L*(*t*) is called the Lagrange operator. As a consequence of (23), it follows that if the dynamical variables which enter *L*(*t*) are altered during an arbitrary infinitesimal change between *t*<sup>1</sup> and *t*2, then it must be that

$$
\delta W\_{21} = 0.\tag{25}
$$

The operator equations of motion are implied in this result. The usual form for the Lagrange operator is

$$L(t) = \frac{1}{2}(p\_i \dot{\mathbf{x}}^i + \dot{\mathbf{x}}^i p\_i) - H(\mathbf{x}, \mathbf{p}\_\prime t). \tag{26}$$

The first term has been symmetrized to give a Hermitian *L*(*t*), as the operators *x*˙ *<sup>i</sup>* and *pi* do not as usual commute. It is assumed the Hamiltonian *H* is a Hermitian operator. The action operator (24) is used to calculate the variation *δW*21. In order to vary the endpoints *t*1, *t*2, we follow Schwinger exactly and change the variable of integration from *t* to *τ* such that *t* = *t*(*τ*). This allows for the variation of the functional dependence of *t* to depend on *τ* with the variable of integration *τ* held fixed. Then *W*<sup>21</sup> takes the form,

$$\mathcal{W}\_{21} = \int\_{\tau\_1}^{\tau\_2} d\tau \left[ \frac{1}{2} (P\_i \frac{dy^i}{d\tau} + \frac{dy^i}{d\tau} P\_i) - \mathcal{H}(\mathbf{y}, \mathbf{P}, \tau) \right. \\ \left. \frac{dt}{d\tau} \right]\_{\prime} \tag{27}$$

where in (27),

$$
\mathbf{H}(\mathbf{y}, \mathbf{P}, \tau) = H(\mathbf{x}, \mathbf{p}\_\prime t). \tag{28}
$$

then *δW*<sup>21</sup> can be expressed in the form,

*dt* {*δpi*(*x*˙

*x*˙ *<sup>i</sup>* <sup>=</sup> *<sup>∂</sup><sup>H</sup> ∂pi*

*<sup>i</sup>* − *∂H ∂pi*

) − *δx<sup>i</sup>*

, *<sup>p</sup>*˙*<sup>i</sup>* <sup>=</sup> <sup>−</sup>*∂<sup>H</sup>*

Let *<sup>B</sup>* represent any observable and consider the matrix element �*a*|*B*|*a*′

�*a*¯|*B*¯|*a*¯ ′ (*p*˙*<sup>i</sup>* +

Taking the variations with endpoints fixed, it follows that *G*<sup>1</sup> = *G*<sup>2</sup> = 0. Consequently, the operator equations of motion which follow from equating *δW*<sup>21</sup> in (34) to zero are then

*∂x<sup>i</sup>*

The results produced in this way are exactly of the form of the classical Hamilton equations of motion, and the derivatives in the first two equations of (35) are with respect to operators.

are subjected to a unitary transformation *A* → *A*¯ = *UAU*†, where *U* is a unitary operator, then the eigenstates |*a*� are transformed into |*a*¯� = *U*|*a*� having the eigenvalue *a*. Define the

� <sup>=</sup> �*a*|*B*|*a*′

*h*¯

*h*¯

*Ga*) = *<sup>B</sup>* <sup>−</sup> *<sup>i</sup>*

*h*¯

Let *U* now be an infinitesimal unitary transformation which can be expressed in the form

*<sup>U</sup>* <sup>=</sup> *<sup>I</sup>* <sup>−</sup> *<sup>i</sup>*

the change produced by the canonical transformation whose generator is *Ga*, then

*Ga*)*B*(*I* +

*<sup>B</sup>*¯ = (*<sup>I</sup>* <sup>−</sup> *<sup>i</sup>*

*h*¯

*<sup>δ</sup>Ga* <sup>|</sup>*a*� <sup>=</sup> <sup>|</sup>*a*¯�−|*a*� <sup>=</sup> <sup>−</sup> *<sup>i</sup>*

In (38), *Ga* is a Hermitian quantity and can depend on observables *A*. Consequently, if *δGa* is

*i h*¯ , *dH*

*∂H <sup>∂</sup>x<sup>i</sup>* )+( *dH*

*dt* <sup>−</sup> *<sup>∂</sup><sup>H</sup>*

The Schwinger Action Principle and Its Applications to Quantum Mechanics

*dt* <sup>=</sup> *<sup>∂</sup><sup>H</sup>*

*B*¯ = *UBU*†. (36)

�. (37)

*Ga*. (38)

*Ga*|*a*�, (39)

[*Ga*, *B*]. (40)

*<sup>∂</sup><sup>t</sup>* )*δt*} <sup>+</sup> *<sup>G</sup>*<sup>2</sup> <sup>−</sup> *<sup>G</sup>*1. (34)

http://dx.doi.org/10.5772/53472

165

*<sup>∂</sup><sup>t</sup>* . (35)

�. If the variables *A*

 *<sup>t</sup>*<sup>2</sup> *t*1

**3. Commutation relations**

These operators have the property that

operator *B*¯ to be

and *B*¯ is given by

*δW*<sup>21</sup> =

Thus *y<sup>i</sup>* (*τ*) = *x<sup>i</sup>* (*t*) and *Pi*(*τ*) = *pi*(*t*) when the transformation *t* = *t*(*τ*) is implemented. Evaluating the infinitesimal variation of (27), it is found that

$$\delta \mathcal{W}\_{21} = \int\_{\tau\_1}^{\tau\_2} d\tau [\frac{1}{2} \delta P\_i \frac{dy^i}{d\tau} + \frac{1}{2} P\_i \delta(\frac{dy^i}{d\tau}) + \frac{1}{2} \delta(\frac{dy^i}{d\tau}) P\_i + \frac{1}{2} \frac{dy^i}{d\tau} \delta P\_i - \delta \tilde{H} \frac{dt}{d\tau} - \tilde{H} \delta(\frac{dt}{d\tau})]. \tag{29}$$

Moving the operator *δ* through the derivative, this becomes

$$
\delta \delta W\_{21} = \int\_{t\_1}^{t\_2} d\tau \left\{ \frac{1}{2} (\delta P\_l \frac{d\mathbf{y}^i}{d\tau} + \frac{d\mathbf{y}^i}{d\tau} \delta P\_l - \frac{dP\_i}{d\tau} \delta \mathbf{y}^i - \delta \mathbf{y}^i \frac{dP\_i}{d\tau}) - \delta \tilde{H} \frac{dt}{d\tau} + \frac{d\tilde{H}}{d\tau} \delta t
$$

$$
+ \frac{d}{d\tau} [\frac{1}{2} (P\_i \delta \mathbf{x}^i + \delta \mathbf{x}^i P\_l) - \tilde{H} \delta t] \right\}. \tag{30}
$$

No assumptions with regard to the commutation properties of the variations with the dynamical variables have been made yet. It may be assumed that the variations are multiples of the identity operator, which commutes with everything. After returning to the variable *t* in the integral in *δW*21, the result is

$$
\delta \mathsf{W}\_{21} = \int\_{t\_1}^{t\_2} dt (\delta p\_i \dot{\mathbf{x}}^i - \dot{p}\_i \delta \mathbf{x}^i + \frac{dH}{dt} \delta t - \delta H) + \mathsf{G}\_2 - \mathsf{G}\_1. \tag{31}
$$

Here *G*<sup>1</sup> and *G*<sup>2</sup> denote the quantity

$$\mathbf{G} = p\_{\mathbf{i}} \delta \mathbf{x}^{\bar{\mathbf{i}}} - H \,\delta \mathbf{t},\tag{32}$$

when it is evaluated at the two endpoints *t* = *t*<sup>1</sup> and *t* = *t*2. If we define,

2

$$
\delta H = \delta \mathbf{x}^i \frac{\partial H}{\partial \mathbf{x}^i} + \delta p\_i \frac{\partial H}{\partial p\_i} + \delta t \frac{\partial H}{\partial t} \,' \tag{33}
$$

then *δW*<sup>21</sup> can be expressed in the form,

6 Quantum Mechanics

where in (27),

*δW*<sup>21</sup> =

(*τ*) = *x<sup>i</sup>*

 *τ*<sup>2</sup> *τ*1 *dτ*[ 1 2 *δPi dy<sup>i</sup> dτ* + 1 2 *Piδ*( *dy<sup>i</sup> <sup>d</sup><sup>τ</sup>* ) + <sup>1</sup> 2 *δ*( *dy<sup>i</sup> <sup>d</sup><sup>τ</sup>* )*Pi* <sup>+</sup>

*δW*<sup>21</sup> =

 *<sup>t</sup>*<sup>2</sup> *t*1

in the integral in *δW*21, the result is

Here *G*<sup>1</sup> and *G*<sup>2</sup> denote the quantity

*δW*<sup>21</sup> =

 *<sup>t</sup>*<sup>2</sup> *t*1

*dt*(*δpix*˙

when it is evaluated at the two endpoints *t* = *t*<sup>1</sup> and *t* = *t*2. If we define,

*<sup>δ</sup><sup>H</sup>* <sup>=</sup> *<sup>δ</sup>x<sup>i</sup> <sup>∂</sup><sup>H</sup>*

*dτ* { 1 2 (*δPi dy<sup>i</sup> dτ*

Thus *y<sup>i</sup>*

we follow Schwinger exactly and change the variable of integration from *t* to *τ* such that *t* = *t*(*τ*). This allows for the variation of the functional dependence of *t* to depend on *τ* with

> <sup>+</sup> *dy<sup>i</sup> dτ*

*Pi*) <sup>−</sup> *<sup>H</sup>*˜ (**y**, **<sup>P</sup>**, *<sup>τ</sup>*) *dt*

(*t*) and *Pi*(*τ*) = *pi*(*t*) when the transformation *t* = *t*(*τ*) is implemented.

1 2 *dy<sup>i</sup> dτ*

*<sup>δ</sup>y<sup>i</sup>* <sup>−</sup> *<sup>δ</sup>y<sup>i</sup> dPi*

*H*˜ (**y**, **P**, *τ*) = *H*(**x**, **p**, *t*). (28)

*<sup>δ</sup>Pi* <sup>−</sup> *<sup>δ</sup>H*˜ *dt*

*<sup>d</sup><sup>τ</sup>* ) <sup>−</sup> *<sup>δ</sup>H*˜ *dt*

*dτ* + *dH*˜ *dτ δt*

*Pi*) − *<sup>H</sup>*˜ *<sup>δ</sup>t*]}. (30)

*dt <sup>δ</sup><sup>t</sup>* <sup>−</sup> *<sup>δ</sup>H*) + *<sup>G</sup>*<sup>2</sup> <sup>−</sup> *<sup>G</sup>*1. (31)

*<sup>∂</sup><sup>t</sup>* , (33)

*<sup>G</sup>* = *pi <sup>δ</sup>x<sup>i</sup>* − *<sup>H</sup> <sup>δ</sup>t*, (32)

*<sup>d</sup><sup>τ</sup>* <sup>−</sup> *<sup>H</sup>*˜ *<sup>δ</sup>*(

*dt*

*<sup>d</sup><sup>τ</sup>* )]. (29)

*<sup>d</sup><sup>τ</sup>* ], (27)

the variable of integration *τ* held fixed. Then *W*<sup>21</sup> takes the form,

*dτ* [ 1 2 (*Pi dy<sup>i</sup> dτ*

 *τ*<sup>2</sup> *τ*1

Evaluating the infinitesimal variation of (27), it is found that

Moving the operator *δ* through the derivative, this becomes

+ *d dτ* [ 1 2

<sup>+</sup> *dy<sup>i</sup> dτ*

*<sup>δ</sup>Pi* <sup>−</sup> *dPi dτ*

(*Piδx<sup>i</sup>* + *δx<sup>i</sup>*

*<sup>i</sup>* − *<sup>p</sup>*˙*iδx<sup>i</sup>* +

*<sup>∂</sup>x<sup>i</sup>* <sup>+</sup> *<sup>δ</sup>pi*

*dH*

*∂H ∂pi* + *δt ∂H*

No assumptions with regard to the commutation properties of the variations with the dynamical variables have been made yet. It may be assumed that the variations are multiples of the identity operator, which commutes with everything. After returning to the variable *t*

*W*<sup>21</sup> =

$$\delta \mathcal{W}\_{21} = \int\_{t\_1}^{t\_2} dt \left\{ \delta p\_i(\dot{\mathbf{x}}^i - \frac{\partial H}{\partial p\_i}) - \delta \mathbf{x}^i (\dot{p}\_i + \frac{\partial H}{\partial \mathbf{x}^i}) + (\frac{dH}{dt} - \frac{\partial H}{\partial t}) \delta t \right\} + \mathcal{G}\_2 - \mathcal{G}\_1. \tag{34}$$

Taking the variations with endpoints fixed, it follows that *G*<sup>1</sup> = *G*<sup>2</sup> = 0. Consequently, the operator equations of motion which follow from equating *δW*<sup>21</sup> in (34) to zero are then

$$\dot{\mathbf{x}}^{i} = \frac{\partial H}{\partial p\_{i}}, \qquad \dot{p}\_{i} = -\frac{\partial H}{\partial \mathbf{x}^{i}}, \qquad \frac{dH}{dt} = \frac{\partial H}{\partial t}. \tag{35}$$

The results produced in this way are exactly of the form of the classical Hamilton equations of motion, and the derivatives in the first two equations of (35) are with respect to operators.

#### **3. Commutation relations**

Let *<sup>B</sup>* represent any observable and consider the matrix element �*a*|*B*|*a*′ �. If the variables *A* are subjected to a unitary transformation *A* → *A*¯ = *UAU*†, where *U* is a unitary operator, then the eigenstates |*a*� are transformed into |*a*¯� = *U*|*a*� having the eigenvalue *a*. Define the operator *B*¯ to be

$$
\vec{B} = \mathsf{U} \mathsf{B} \mathsf{U} \mathsf{I}^{\dagger}.\tag{36}
$$

These operators have the property that

$$
\langle \vec{a} | \vec{B} | \vec{a}' \rangle = \langle a | B | a' \rangle. \tag{37}
$$

Let *U* now be an infinitesimal unitary transformation which can be expressed in the form

$$
\mathcal{U} = I - \frac{\dot{l}}{\hbar} \mathcal{G}\_a. \tag{38}
$$

In (38), *Ga* is a Hermitian quantity and can depend on observables *A*. Consequently, if *δGa* is the change produced by the canonical transformation whose generator is *Ga*, then

$$
\delta\_{\mathcal{G}\_{\mathfrak{a}}}|a\rangle = |\vec{a}\rangle - |a\rangle = -\frac{i}{\hbar} \mathcal{G}\_{\mathfrak{a}}|a\rangle,\tag{39}
$$

and *B*¯ is given by

$$\vec{B} = (I - \frac{\dot{l}}{\hbar} \mathbf{G}\_a) B (I + \frac{\dot{l}}{\hbar} \mathbf{G}\_d) = B - \frac{\dot{l}}{\hbar} [\mathbf{G}\_{a\prime} B]. \tag{40}$$

The change in the matrix element �*a*|*B*|*a*′ � can then be considered to be entirely due to the change in the state vector, with *B* held fixed. To first order in the operator *Ga*, there results

$$
\delta\_{\mathbb{G}\_{\hbar}} \langle a|B|a'\rangle = \langle \vec{a}|B|\vec{a}'\rangle - \langle a|B|a'\rangle = \langle a|\frac{i}{\hbar}[\mathbb{G}\_{a\prime}B]|a'\rangle. \tag{41}
$$

Alternatively, this can be approached by taking the change in �*a*|*B*|*a*′ � to be due to a change in the operator *B*, but with the states held fixed. Define then the change in *B* as *δGaB* such that

$$
\delta\_{\mathcal{G}\_{\mathbf{u}}} \langle a|B|a'\rangle = \langle a|\delta\_{\mathcal{G}\_{\mathbf{u}}}B|a'\rangle. \tag{42}
$$

The general equations of motion have been produced [7].

Then for any operator *B*, equation (43) gives

In order to obtain the commutator [*x<sup>i</sup>*

which satisfies *F*† = *F*, and consequently

Then *G*¯ is calculated to be

*B* = *x<sup>j</sup>*

*<sup>δ</sup><sup>F</sup>* <sup>=</sup> <sup>1</sup> 2

Taking *<sup>δ</sup><sup>t</sup>* = 0 so that *Gp* can be defined as *Gp* = −*δpi <sup>x</sup><sup>i</sup>*

for any operator *B*. Taking *B* = *pj*, the bracket [*x<sup>i</sup>*

this transformation

relations

The canonical commutation relations are also a consequence of the action principle. To see this, first fix *δt* at the times *t*<sup>1</sup> and *t*<sup>2</sup> but permit *δx<sup>i</sup>* to vary. Then (32) gives the generator of

*G* = *piδx<sup>i</sup>*

Putting *B* = *x<sup>i</sup>* and *B* = *pj* respectively in (50) leads to the following pair of commutation

*j*

[*pi*, *<sup>B</sup>*] = <sup>−</sup> *<sup>i</sup>*

, *x<sup>j</sup>*

*<sup>F</sup>* <sup>=</sup> <sup>1</sup> 2

*<sup>δ</sup><sup>B</sup>* <sup>=</sup> <sup>−</sup> *<sup>i</sup> h*¯ *δpi* [*x<sup>i</sup>*

, using the independence of *δx<sup>i</sup>* and *δpi* there results the bracket

(*δpix<sup>i</sup>* + *piδx<sup>i</sup>* + *δx<sup>i</sup>*

*h*¯ *∂*

(*pix<sup>i</sup>* + *x<sup>i</sup>*

*pi* + *x<sup>i</sup>*

*<sup>δ</sup><sup>B</sup>* <sup>=</sup> *<sup>i</sup> h*¯

, *pj*] = *ih*¯ *δ<sup>i</sup>*

[*xi*

If *B* has a dependence on *x*, then *δB* = *B*[*x*(*t*)] − *B*[*x*(*t*) − *δx*(*t*)] implies

by the addition of a total time derivative can be used. Suppose that

. (49)

http://dx.doi.org/10.5772/53472

167

*δx<sup>i</sup>* [*pi*, *B*]. (50)

The Schwinger Action Principle and Its Applications to Quantum Mechanics

, [*pi*, *pj*] = 0. (51)

*<sup>∂</sup>x<sup>i</sup> <sup>B</sup>*. (52)

*pi*), (53)

, *B*], (56)

*<sup>j</sup>* is obtained from (56), and when

. (54)

], the freedom of altering the Lagrangian operator

*δpi*) = *δpix<sup>i</sup>* + *piδx<sup>i</sup>*

*<sup>G</sup>*¯ = *<sup>G</sup>* − *<sup>δ</sup><sup>F</sup>* = −*δpix<sup>i</sup>* − *<sup>H</sup> <sup>δ</sup>t*. (55)

, *pj*] = *ih*¯ *δ<sup>i</sup>*

, it follows that

If the result obtained for the change in �*a*|*B*|*a*′ � is not to depend on which of these approaches is taken, by comparing (41) and (42) it is concluded that

$$
\delta\_{G\_{\theta}} B = \frac{i}{\hbar} [G\_{\theta}, B]. \tag{43}
$$

As an example, let the change in the operators *A* at times *t*<sup>1</sup> and *t*<sup>2</sup> be due to a change in time *<sup>t</sup>* → *<sup>t</sup>* + *<sup>δ</sup><sup>t</sup>* with the *<sup>x</sup><sup>i</sup>* fixed at times *<sup>t</sup>*<sup>1</sup> and *<sup>t</sup>*<sup>2</sup> so that the generator for this tranformation is given by *G*(*t*) = −*H*(*t*)*δt* and (43) in this case is

$$
\delta\_{G\_\hbar} B = \frac{i}{\hbar} [B, H] \delta t\_\prime \tag{44}
$$

where *B* and *H* are evaluated at the same time. It is only the change in *B* resulting from a change in operators which is considered now. Specifically, if *B* = *B*[*A*(*t*), *t*], then *B*¯ = *B*[*A*(*t* + *δt*), *t*] and so

$$
\delta B = -\delta t (\frac{dB}{dt} - \frac{\delta B}{\delta t}).\tag{45}
$$

Using (45) in (44), a result in agreement with the Heisenberg picture equation of motion results,

$$\frac{dB}{dt} = \frac{\partial B}{\partial t} + \frac{i}{\hbar}[H, B]. \tag{46}$$

The Heisenberg equation of motion has resulted by this process. If *B* is taken to be the operator *B* = *x<sup>i</sup>* , then (46) becomes,

$$\dot{\mathbf{x}}^{i} = \frac{\dot{t}}{\hbar} [H, \mathbf{x}^{i}]. \tag{47}$$

If *B* = *pi* is used in (46), there results

$$
\dot{p}\_{\dot{l}} = \frac{\dot{l}}{\hbar} [H\_{\prime} p\_{\dot{l}}].\tag{48}
$$

The general equations of motion have been produced [7].

The canonical commutation relations are also a consequence of the action principle. To see this, first fix *δt* at the times *t*<sup>1</sup> and *t*<sup>2</sup> but permit *δx<sup>i</sup>* to vary. Then (32) gives the generator of this transformation

$$
\delta G = p\_i \delta \mathfrak{x}^i. \tag{49}
$$

Then for any operator *B*, equation (43) gives

8 Quantum Mechanics

that

The change in the matrix element �*a*|*B*|*a*′

*<sup>δ</sup>Ga* �*a*|*B*|*a*′

If the result obtained for the change in �*a*|*B*|*a*′

given by *G*(*t*) = −*H*(*t*)*δt* and (43) in this case is

, then (46) becomes,

If *B* = *pi* is used in (46), there results

*B*[*A*(*t* + *δt*), *t*] and so

results,

operator *B* = *x<sup>i</sup>*

is taken, by comparing (41) and (42) it is concluded that

� = �*a*¯|*B*|*a*¯

*<sup>δ</sup>Ga* �*a*|*B*|*a*′

*<sup>δ</sup>GaB* <sup>=</sup> *<sup>i</sup> h*¯

*<sup>δ</sup>GaB* <sup>=</sup> *<sup>i</sup> h*¯

*δB* = −*δt*(

*dB dt* <sup>=</sup> *<sup>∂</sup><sup>B</sup> <sup>∂</sup><sup>t</sup>* <sup>+</sup> *i h*¯

> *x*˙ *<sup>i</sup>* <sup>=</sup> *<sup>i</sup> h*¯ [*H*, *x<sup>i</sup>*

*<sup>p</sup>*˙*<sup>i</sup>* <sup>=</sup> *<sup>i</sup> h*¯

As an example, let the change in the operators *A* at times *t*<sup>1</sup> and *t*<sup>2</sup> be due to a change in time *<sup>t</sup>* → *<sup>t</sup>* + *<sup>δ</sup><sup>t</sup>* with the *<sup>x</sup><sup>i</sup>* fixed at times *<sup>t</sup>*<sup>1</sup> and *<sup>t</sup>*<sup>2</sup> so that the generator for this tranformation is

where *B* and *H* are evaluated at the same time. It is only the change in *B* resulting from a change in operators which is considered now. Specifically, if *B* = *B*[*A*(*t*), *t*], then *B*¯ =

> *dB dt* <sup>−</sup> *<sup>δ</sup><sup>B</sup> δt*

Using (45) in (44), a result in agreement with the Heisenberg picture equation of motion

The Heisenberg equation of motion has resulted by this process. If *B* is taken to be the

Alternatively, this can be approached by taking the change in �*a*|*B*|*a*′

� can then be considered to be entirely due to the

[*Ga*, *<sup>B</sup>*]|*a*′

�. (41)

� to be due to a change

�. (42)

� is not to depend on which of these approaches

[*Ga*, *B*]. (43)

[*B*, *H*]*δt*, (44)

). (45)

[*H*, *B*]. (46)

]. (47)

[*H*, *pi*]. (48)

change in the state vector, with *B* held fixed. To first order in the operator *Ga*, there results

�−�*a*|*B*|*a*′

in the operator *B*, but with the states held fixed. Define then the change in *B* as *δGaB* such

� <sup>=</sup> �*a*|*δGaB*|*a*′

� = �*a*| *i h*¯

′

$$
\delta \mathcal{B} = \frac{i}{\hbar} \delta \mathbf{x}^{i} \, [p\_{i\nu} \mathcal{B}]. \tag{50}
$$

Putting *B* = *x<sup>i</sup>* and *B* = *pj* respectively in (50) leads to the following pair of commutation relations

$$[\mathbf{x}^i, p\_j] = i\hbar \,\delta^i\_{\mathbf{j}\prime} \qquad [p\_{i\prime} p\_{\mathbf{j}}] = 0. \tag{51}$$

If *B* has a dependence on *x*, then *δB* = *B*[*x*(*t*)] − *B*[*x*(*t*) − *δx*(*t*)] implies

$$[p\_{\dot{\nu}}B] = -\frac{i}{\hbar} \frac{\partial}{\partial x^{i}} B.\tag{52}$$

In order to obtain the commutator [*x<sup>i</sup>* , *x<sup>j</sup>* ], the freedom of altering the Lagrangian operator by the addition of a total time derivative can be used. Suppose that

$$F = \frac{1}{2} (p\_i \mathbf{x}^i + \mathbf{x}^i p\_i)\_\prime \tag{53}$$

which satisfies *F*† = *F*, and consequently

$$
\delta F = \frac{1}{2} (\delta p\_i \mathbf{x}^i + p\_i \delta \mathbf{x}^i + \delta \mathbf{x}^i p\_i + \mathbf{x}^i \delta p\_i) = \delta p\_i \mathbf{x}^i + p\_i \delta \mathbf{x}^i. \tag{54}
$$

Then *G*¯ is calculated to be

$$
\tilde{G} = G - \delta F = -\delta p\_i \mathbf{x}^i - H \,\delta t.\tag{55}
$$

Taking *<sup>δ</sup><sup>t</sup>* = 0 so that *Gp* can be defined as *Gp* = −*δpi <sup>x</sup><sup>i</sup>* , it follows that

$$
\delta B = -\frac{i}{\hbar} \delta p\_i \left[ \mathbf{x}^i, B \right], \tag{56}
$$

for any operator *B*. Taking *B* = *pj*, the bracket [*x<sup>i</sup>* , *pj*] = *ih*¯ *δ<sup>i</sup> <sup>j</sup>* is obtained from (56), and when *B* = *x<sup>j</sup>* , using the independence of *δx<sup>i</sup>* and *δpi* there results the bracket

$$[\mathbf{x}^i, \mathbf{x}^j] = \mathbf{0}.\tag{57}$$

Since *<sup>δ</sup>*W<sup>21</sup> is well-ordered, it is possible to write

Comparing this with (63), it must be that

*<sup>∂</sup>xi*(*t*2) <sup>=</sup> *pi*(*t*2), *<sup>∂</sup>*W<sup>21</sup>

�*a*2, *<sup>t</sup>*2|*a*1, *<sup>t</sup>*1� =

quantized by replacing the classical fields *φ<sup>i</sup>*

two spacetime points of Σ, then

*<sup>∂</sup>*W<sup>21</sup>

then have a form of (62),

*<sup>δ</sup>*�*a*2, *<sup>t</sup>*2|*a*1, *<sup>t</sup>*1� <sup>=</sup> *<sup>i</sup>*

*∂t*<sup>2</sup>

 *dnx*<sup>2</sup>  *h*¯

<sup>=</sup> <sup>−</sup>*H*(*t*2), *<sup>∂</sup>*W<sup>21</sup>

�**x**2, *<sup>t</sup>*2|**x**1, *<sup>t</sup>*1� = exp(

Any other transformation function can be recovered from the propagator equation

amplitudes will be seen when the path integral approach is formulated.

[*ϕi*

eigenvalues of the observables. A quantum state is then denoted by

(*x*1), *ϕ<sup>j</sup>*

**4. Action principle adapted to case of quantum fields**

If we consider a matrix element �*x*2, *<sup>t</sup>*2|*x*1, *<sup>t</sup>*1�, which could be regarded as the propagator, we

The arbitrary integration constant is determined by requiring that �**x**2, *<sup>t</sup>*2|**x**1, *<sup>t</sup>*1� = *<sup>δ</sup>*(**x**2, **<sup>x</sup>**1).

which follows from the completeness relation. Another way of formulating transition

The Schwinger action principle, much like the Feynman path integral, concentrates on the transition amplitude between two quantum states. The action principle will be formulated here for a local field theory. Classically, a local field is a function which depends only on a single spacetime point, rather than on an extended region of spacetime. The theory can be

Let Σ denote a spacelike hypersurface. This means that any two points on Σ have a spacelike separation and consequently must be causally disconnected. As a consequence of this, the values of the field at different points of the surface Σ must be independent. If *x*<sup>1</sup> and *x*<sup>2</sup> are

This follows since it must be the case that a measurement at *x*<sup>1</sup> must not influence one at *x*2. A fundamental assumption of local field theory is that a complete set of commuting observables can be constructed based on the fields and their derivatives on the surface Σ. Let *<sup>τ</sup>* denote such a complete set of commuting observables on <sup>Σ</sup> such that *<sup>τ</sup>*′ represents the


(*x*) with field operators *ϕ<sup>i</sup>*

*i h*¯

�*a*2, *<sup>t</sup>*2|*δ*W21|*a*1, *<sup>t</sup>*1�. (64)

*∂t*<sup>1</sup>

W21). (66)

(*x*).

(*x*2)] = 0. (68)

, Σ�. (69)

= *H*(*t*1). (65)

http://dx.doi.org/10.5772/53472

169

*<sup>∂</sup>xi*(*t*1) <sup>=</sup> <sup>−</sup>*pi*(*t*1), *<sup>∂</sup>*W<sup>21</sup>

The Schwinger Action Principle and Its Applications to Quantum Mechanics

*<sup>d</sup>nx*<sup>1</sup> �*a*2, *<sup>t</sup>*2|**x**2, *<sup>t</sup>*2��**x**2, *<sup>t</sup>*2|**x**1, *<sup>t</sup>*1��**x**1*t*1|*a*1, *<sup>t</sup>*1�, (67)

It has been shown that the set of canonical commutation relations can be obtained from this action principle.

Consider a matrix element �*a*|*F*(*A*, *B*)|*b*� where *A* and *B* each represent a complete set of mutually compatible observables and *F*(*A*, *B*) is some function, and it is not assumed the observables from the two different sets commute with each other. If the commutator [*B*, *A*] is known, it is always possible to order the operators in *F*(*A*, *B*) so all *A* terms are to the left of *B*, which allows the matrix element to be evaluated. Let

$$\mathcal{F}(A,B) = F(A,B) \tag{58}$$

denote the operator where the commutation relation for [*A*, *B*] has been used to move all occurrences of *A* in *F*(*A*, *B*) to the left of all *B*, and (58) is said to be well-ordered

$$
\langle a|F(A,B)|b\rangle = \mathcal{F}(a,b)\langle a|b\rangle. \tag{59}
$$

The matrix element of *F*(*A*, *B*) is directly related to the transformation function �*a*|*b*�.

The idea of well-ordering operators can be used in the action principle. Define

$$
\delta \mathcal{W}\_{21} = \delta \mathcal{W}\_{21}.\tag{60}
$$

be the well-ordered form of *δW*21. Then

$$
\delta \langle a\_{2\prime} t\_2 | a\_{1\prime} t\_1 \rangle = \frac{i}{\hbar} \delta \mathcal{W}\_{21} \langle a\_{2\prime} t\_2 | a\_{1\prime} t\_1 \rangle,\tag{61}
$$

where *<sup>δ</sup>*W<sup>21</sup> denotes the replacement of all operators with their eigenvalues. Equation (61) can be integrated to yield

$$
\langle a\_{2}, t\_{2} | a\_{1}, t\_{1} \rangle = \exp(\frac{i}{\hbar} \mathcal{W}\_{21}).\tag{62}
$$

Using (23) and (32) in the action principle,

$$\delta\langle a\_2, t\_2 | a\_1, t\_1 \rangle = \frac{i}{\hbar} \delta x^i(t\_2) \langle a\_2, t\_2 | p(t\_2) | a\_1, t\_1 \rangle - \frac{i}{\hbar} \delta t\_2 \langle a\_2, t\_2 | H(t\_2) | a\_1, t\_1 \rangle$$

$$-\frac{i}{\hbar}\delta\mathbf{x}^{i}(t\_{1})\langle a\_{2},t\_{2}|p\_{i}(t\_{1})|a\_{1},t\_{1}\rangle+\frac{i}{\hbar}\delta t\_{1}\langle a\_{2},t\_{2}|H(t\_{1})|a\_{1},t\_{1}\rangle.\tag{63}$$

Since *<sup>δ</sup>*W<sup>21</sup> is well-ordered, it is possible to write

$$
\delta \langle a\_{2\prime} t\_2 | a\_{1\prime} t\_1 \rangle = \frac{i}{\hbar} \langle a\_{2\prime} t\_2 | \delta \mathcal{W}\_{21} | a\_{1\prime} t\_1 \rangle. \tag{64}
$$

Comparing this with (63), it must be that

10 Quantum Mechanics

action principle.

[*xi* , *xj*

of *B*, which allows the matrix element to be evaluated. Let

be the well-ordered form of *δW*21. Then

Using (23) and (32) in the action principle,

*<sup>δ</sup>*�*a*2, *<sup>t</sup>*2|*a*1, *<sup>t</sup>*1� <sup>=</sup> *<sup>i</sup>*

− *i h*¯ *δxi* *h*¯ *δxi*

can be integrated to yield

It has been shown that the set of canonical commutation relations can be obtained from this

Consider a matrix element �*a*|*F*(*A*, *B*)|*b*� where *A* and *B* each represent a complete set of mutually compatible observables and *F*(*A*, *B*) is some function, and it is not assumed the observables from the two different sets commute with each other. If the commutator [*B*, *A*] is known, it is always possible to order the operators in *F*(*A*, *B*) so all *A* terms are to the left

denote the operator where the commutation relation for [*A*, *B*] has been used to move all

occurrences of *A* in *F*(*A*, *B*) to the left of all *B*, and (58) is said to be well-ordered

The matrix element of *F*(*A*, *B*) is directly related to the transformation function �*a*|*b*�.

*h*¯

where *<sup>δ</sup>*W<sup>21</sup> denotes the replacement of all operators with their eigenvalues. Equation (61)

(*t*2)�*a*2, *<sup>t</sup>*2|*p*(*t*2)|*a*1, *<sup>t</sup>*1� − *<sup>i</sup>*

*i h*¯

*i h*¯ *h*¯

�*a*2, *<sup>t</sup>*2|*a*1, *<sup>t</sup>*1� = exp(

The idea of well-ordering operators can be used in the action principle. Define

*<sup>δ</sup>*�*a*2, *<sup>t</sup>*2|*a*1, *<sup>t</sup>*1� <sup>=</sup> *<sup>i</sup>*

(*t*1)�*a*2, *<sup>t</sup>*2|*pi*(*t*1)|*a*1, *<sup>t</sup>*1� +

] = 0. (57)

F(*A*, *B*) = *F*(*A*, *B*) (58)

�*a*|*F*(*A*, *B*)|*b*� = F(*a*, *b*)�*a*|*b*�. (59)

*<sup>δ</sup>*W<sup>21</sup> = *<sup>δ</sup>W*21. (60)

*<sup>δ</sup>*W21�*a*2, *<sup>t</sup>*2|*a*1, *<sup>t</sup>*1�, (61)

W21). (62)

*<sup>δ</sup>t*2�*a*2, *<sup>t</sup>*2|*H*(*t*2)|*a*1, *<sup>t</sup>*1�

*<sup>δ</sup>t*1�*a*2, *<sup>t</sup>*2|*H*(*t*1)|*a*1, *<sup>t</sup>*1�. (63)

$$\frac{\partial \mathcal{W}\_{21}}{\partial \mathbf{x}^{i}(t\_{2})} = p\_{i}(t\_{2}), \quad \frac{\partial \mathcal{W}\_{21}}{\partial t\_{2}} = -H(t\_{2}), \quad \frac{\partial \mathcal{W}\_{21}}{\partial \mathbf{x}^{i}(t\_{1})} = -p\_{i}(t\_{1}), \quad \frac{\partial \mathcal{W}\_{21}}{\partial t\_{1}} = H(t\_{1}).\tag{65}$$

If we consider a matrix element �*x*2, *<sup>t</sup>*2|*x*1, *<sup>t</sup>*1�, which could be regarded as the propagator, we then have a form of (62),

$$
\langle \mathbf{x}\_{2}, t\_{2} | \mathbf{x}\_{1}, t\_{1} \rangle = \exp(\frac{i}{\hbar} \mathcal{W}\_{21}). \tag{66}
$$

The arbitrary integration constant is determined by requiring that �**x**2, *<sup>t</sup>*2|**x**1, *<sup>t</sup>*1� = *<sup>δ</sup>*(**x**2, **<sup>x</sup>**1). Any other transformation function can be recovered from the propagator equation

$$
\langle \langle a\_2, t\_2 | a\_1, t\_1 \rangle \rangle = \int d^n \mathbf{x}\_2 \int d^n \mathbf{x}\_1 \,\langle a\_2, t\_2 | \mathbf{x}\_2, t\_2 \rangle \langle \mathbf{x}\_2, t\_2 | \mathbf{x}\_1, t\_1 \rangle \langle \mathbf{x}\_1 t\_1 | a\_1, t\_1 \rangle,\tag{67}
$$

which follows from the completeness relation. Another way of formulating transition amplitudes will be seen when the path integral approach is formulated.

#### **4. Action principle adapted to case of quantum fields**

The Schwinger action principle, much like the Feynman path integral, concentrates on the transition amplitude between two quantum states. The action principle will be formulated here for a local field theory. Classically, a local field is a function which depends only on a single spacetime point, rather than on an extended region of spacetime. The theory can be quantized by replacing the classical fields *φ<sup>i</sup>* (*x*) with field operators *ϕ<sup>i</sup>* (*x*).

Let Σ denote a spacelike hypersurface. This means that any two points on Σ have a spacelike separation and consequently must be causally disconnected. As a consequence of this, the values of the field at different points of the surface Σ must be independent. If *x*<sup>1</sup> and *x*<sup>2</sup> are two spacetime points of Σ, then

$$[\![\phi^{\dot{l}}(\mathbf{x}\_1), \phi^{\dot{l}}(\mathbf{x}\_2)] = 0. \tag{68}$$

This follows since it must be the case that a measurement at *x*<sup>1</sup> must not influence one at *x*2. A fundamental assumption of local field theory is that a complete set of commuting observables can be constructed based on the fields and their derivatives on the surface Σ. Let *<sup>τ</sup>* denote such a complete set of commuting observables on <sup>Σ</sup> such that *<sup>τ</sup>*′ represents the eigenvalues of the observables. A quantum state is then denoted by

$$|\mathfrak{r}',\Sigma\rangle. \tag{69}$$

Since causality properties are important in field theory, the surface Σ is written explicitly in the ket. The Heisenberg picture has been adopted here. The states are then time-independent with the time dependence located in the operators. This is necessary for manifest covariance, since the time and space arguments of the field are not treated differently.

Suppose that Σ<sup>1</sup> and Σ<sup>2</sup> are two spacelike hypersurfaces such that all points of Σ<sup>2</sup> are to the future of Σ1. Let *τ*<sup>1</sup> be a complete set of commuting observables defined on Σ1, and *τ*<sup>2</sup> a complete set of observables defined on Σ2, such that these observables have the same eigenvalue spectrum. Then *τ*<sup>1</sup> and *τ*<sup>2</sup> should be related by a unitary transformation

$$
\pi\_2 = \mathcal{U}\_{12}\pi\_1 \mathcal{U}\_{12}^{-1}.\tag{70}
$$

it must be that *δS*<sup>12</sup> is Hermitean. If Σ<sup>3</sup> is a spacelike hypersurface, all of whose points lie to

2, <sup>Σ</sup>2��*τ*′

2, <sup>Σ</sup>2|*τ*′

The Schwinger Action Principle and Its Applications to Quantum Mechanics

3, <sup>Σ</sup>3|*τ*′

*δS*<sup>13</sup> = *δS*<sup>23</sup> + *δS*12. (79)

*<sup>δ</sup>S*<sup>12</sup> = *<sup>F</sup>*<sup>2</sup> − *<sup>F</sup>*1. (81)

*dσxnµFµ*(*x*), (82)

2, <sup>Σ</sup>2��*τ*′

<sup>1</sup>, <sup>Σ</sup>1�. (78)

<sup>1</sup>, <sup>Σ</sup>1�. (80)

<sup>1</sup>, <sup>Σ</sup>1� <sup>+</sup> �*τ*′

<sup>1</sup>, <sup>Σ</sup>1�. (77)

http://dx.doi.org/10.5772/53472

2, <sup>Σ</sup>2|*δS*12|*τ*′

<sup>1</sup>, <sup>Σ</sup>1�}

171

the future of those on Σ2, the basic law for composition of probability amplitudes is

*τ*′ 2 �*τ*′ 3, <sup>Σ</sup>3|*τ*′

2, <sup>Σ</sup>2��*τ*′

3, <sup>Σ</sup>3|*δS*<sup>23</sup> <sup>+</sup> *<sup>δ</sup>S*12|*τ*′

*δS*<sup>12</sup> = 0.

*<sup>δ</sup>S*<sup>21</sup> = −*δS*12.

If the operators in *τ*<sup>1</sup> and *τ*<sup>2</sup> undergo infinitesimal, unitary transformations on the hypersurfaces Σ<sup>1</sup> and Σ2, respectively, and only on these two hypersurfaces, then the change

*h*¯ �*τ*′

Here *F*<sup>1</sup> and *F*<sup>2</sup> are Hermitean operators constructed from a knowledge of the fields and their

A generator *F* of this type on a spacelike hypersurface Σ should be expressible in the form,

and *dσ<sup>x</sup>* is the area element on Σ, *n<sup>µ</sup>* is the outward unit normal to Σ, and *Fµ*(*x*) may be put

2, <sup>Σ</sup>2|*F*<sup>2</sup> <sup>−</sup> *<sup>F</sup>*1|*τ*′

<sup>1</sup>, <sup>Σ</sup>1� <sup>=</sup> *<sup>i</sup>*

derivatives on Σ<sup>1</sup> and Σ2. The result (80) is of the form (76) provided that

*F* = Σ

together based on a knowledge of the fields on the surface Σ.

2, <sup>Σ</sup>2|*τ*′

<sup>1</sup>, <sup>Σ</sup>1� <sup>=</sup> ∑

3, <sup>Σ</sup>3|*δS*23|*τ*′

<sup>=</sup> �*τ*′

�*τ*′ 3, <sup>Σ</sup>3|*τ*′

Varying both sides of the expression in (77),

*τ*′ 2 {�*τ*′

Comparing both ends of the result in (78), it follows that

<sup>1</sup>, <sup>Σ</sup>1� <sup>=</sup> ∑

In the limit <sup>Σ</sup><sup>2</sup> → <sup>Σ</sup>1, it must be that

In the limit in which <sup>Σ</sup><sup>3</sup> → <sup>Σ</sup>1, it follows that

in the transformation function has the form

*<sup>δ</sup>*�*τ*′ 2, <sup>Σ</sup>2|*τ*′

�*τ*′

3, <sup>Σ</sup>3|*δS*13|*τ*′

The eigenstates are related by

$$
\langle |\tau\_{2'}'\Sigma\_2\rangle = \mathcal{U}\_{12} |\tau\_{1'}'\Sigma\_1\rangle. \tag{71}
$$

In (71), *U*<sup>12</sup> is a unitary operator giving the evolution of the state in the spacetime between the two spacelike hypersurfaces. The transition amplitude is defined by

$$
\langle \pi\_{2'}' \Sigma\_2 | \pi\_{1'}' \Sigma\_1 \rangle = \langle \pi\_{1'}' \Sigma\_1 | \mathcal{U}\_{12}^{-1} | \pi\_{1'}' \Sigma\_1 \rangle. \tag{72}
$$

The unitary operator *U*<sup>12</sup> depends on a number of details of the quantum system, namely, the choice made for the commuting observables *τ*, and the spacelike hypersurfaces Σ<sup>1</sup> and Σ2. A change in any of these quantities will induce a change in the transformation function according to

$$
\delta \langle \tau\_{2'}' \Sigma\_2 | \tau\_{1'}' \Sigma\_1 \rangle = \langle \tau\_{1'}' \Sigma\_1 | \delta U\_{12}^{-1} | \tau\_{1'}' \Sigma\_1 \rangle. \tag{73}
$$

The unitary operator *U*<sup>12</sup> can be expressed in the form

$$\mathcal{U}\_{12} = \exp(-\frac{\dot{l}}{\hbar}\mathcal{S}\_{12}),\tag{74}$$

in which *S*† <sup>12</sup> <sup>=</sup> *<sup>S</sup>*<sup>12</sup> is a Hermitian operator. Moreover, beginning with *<sup>U</sup>*12*U*−<sup>1</sup> <sup>12</sup> <sup>=</sup> *<sup>I</sup>* and using (74), it is found that

$$
\delta \mathcal{U}\_{12}^{-1} = -\mathcal{U}\_{12}^{-1} \delta \mathcal{U}\_{12} \mathcal{U}\_{12}^{-1} = \frac{i}{\hbar} \mathcal{U}\_{12}^{-1} \delta \mathcal{S}\_{12}.\tag{75}
$$

The change in the transformation function can be written in terms of the operator in (74) as

$$
\delta \langle \tau\_{2'}' \Sigma\_2 | \tau\_{1'}' \Sigma\_1 \rangle = \frac{i}{\hbar} \langle \tau\_{2'}' \Sigma\_2 | \delta \mathcal{S}\_{12} | \tau\_{1'}' \Sigma\_1 \rangle. \tag{76}
$$

Equation (76) can be regarded as a definition of *δS*12. In order that *δS*<sup>12</sup> be consistent with the basic requirement that

$$
\langle \pi\_2', \Sigma\_2 | \pi\_1', \Sigma\_1 \rangle^\* = \langle \pi\_1', \Sigma\_1 | \pi\_2', \Sigma\_2 \rangle\_{\prime \prime}
$$

it must be that *δS*<sup>12</sup> is Hermitean. If Σ<sup>3</sup> is a spacelike hypersurface, all of whose points lie to the future of those on Σ2, the basic law for composition of probability amplitudes is

$$
\langle \langle \tau\_{3'}' \Sigma\_3 | \tau\_{1'}' \Sigma\_1 \rangle = \sum\_{\tau\_2'} \langle \tau\_{3'}' \Sigma\_3 | \tau\_{2'}' \Sigma\_2 \rangle \langle \tau\_{2'}' \Sigma\_2 | \tau\_{1'}' \Sigma\_1 \rangle. \tag{77}
$$

Varying both sides of the expression in (77),

12 Quantum Mechanics

The eigenstates are related by

according to

in which *S*†

(74), it is found that

the basic requirement that

Since causality properties are important in field theory, the surface Σ is written explicitly in the ket. The Heisenberg picture has been adopted here. The states are then time-independent with the time dependence located in the operators. This is necessary for manifest covariance,

Suppose that Σ<sup>1</sup> and Σ<sup>2</sup> are two spacelike hypersurfaces such that all points of Σ<sup>2</sup> are to the future of Σ1. Let *τ*<sup>1</sup> be a complete set of commuting observables defined on Σ1, and *τ*<sup>2</sup> a complete set of observables defined on Σ2, such that these observables have the same

*<sup>τ</sup>*<sup>2</sup> <sup>=</sup> *<sup>U</sup>*12*τ*1*U*−<sup>1</sup>

2, <sup>Σ</sup>2� <sup>=</sup> *<sup>U</sup>*12|*τ*′

<sup>1</sup>, <sup>Σ</sup>1� <sup>=</sup> �*τ*′

<sup>1</sup>, <sup>Σ</sup>1� <sup>=</sup> �*τ*′

*<sup>U</sup>*<sup>12</sup> <sup>=</sup> exp(<sup>−</sup> *<sup>i</sup>*

<sup>12</sup> <sup>=</sup> *<sup>S</sup>*<sup>12</sup> is a Hermitian operator. Moreover, beginning with *<sup>U</sup>*12*U*−<sup>1</sup>

<sup>12</sup> *<sup>δ</sup>U*12*U*−<sup>1</sup>

The change in the transformation function can be written in terms of the operator in (74) as

*h*¯ �*τ*′

Equation (76) can be regarded as a definition of *δS*12. In order that *δS*<sup>12</sup> be consistent with

<sup>1</sup>, <sup>Σ</sup>1�<sup>∗</sup> <sup>=</sup> �*τ*′

<sup>1</sup>, <sup>Σ</sup>1� <sup>=</sup> *<sup>i</sup>*

In (71), *U*<sup>12</sup> is a unitary operator giving the evolution of the state in the spacetime between

The unitary operator *U*<sup>12</sup> depends on a number of details of the quantum system, namely, the choice made for the commuting observables *τ*, and the spacelike hypersurfaces Σ<sup>1</sup> and Σ2. A change in any of these quantities will induce a change in the transformation function

<sup>1</sup>, <sup>Σ</sup>1|*U*−<sup>1</sup>

<sup>1</sup>, <sup>Σ</sup>1|*δU*−<sup>1</sup>

*h*¯

<sup>12</sup> <sup>=</sup> *<sup>i</sup> h*¯ *U*−<sup>1</sup>

2, <sup>Σ</sup>2|*δS*12|*τ*′

<sup>1</sup>, <sup>Σ</sup>1|*τ*′

2, <sup>Σ</sup>2�,

<sup>12</sup> <sup>|</sup>*τ*′

<sup>12</sup> <sup>|</sup>*τ*′

<sup>12</sup> . (70)

<sup>1</sup>, <sup>Σ</sup>1�. (71)

<sup>1</sup>, <sup>Σ</sup>1�. (72)

<sup>1</sup>, <sup>Σ</sup>1�. (73)

<sup>12</sup> <sup>=</sup> *<sup>I</sup>* and using

*S*12), (74)

<sup>12</sup> *<sup>δ</sup>S*12. (75)

<sup>1</sup>, <sup>Σ</sup>1�. (76)

eigenvalue spectrum. Then *τ*<sup>1</sup> and *τ*<sup>2</sup> should be related by a unitary transformation

since the time and space arguments of the field are not treated differently.


the two spacelike hypersurfaces. The transition amplitude is defined by

�*τ*′ 2, <sup>Σ</sup>2|*τ*′

*<sup>δ</sup>*�*τ*′ 2, <sup>Σ</sup>2|*τ*′

*<sup>δ</sup>U*−<sup>1</sup>

*<sup>δ</sup>*�*τ*′ 2, <sup>Σ</sup>2|*τ*′

> �*τ*′ 2, <sup>Σ</sup>2|*τ*′

<sup>12</sup> <sup>=</sup> <sup>−</sup>*U*−<sup>1</sup>

The unitary operator *U*<sup>12</sup> can be expressed in the form

$$
\langle \pi'\_{3'} \Sigma\_3 | \delta \Sigma\_{13} | \pi'\_1 \Sigma\_1 \rangle = \sum\_{\pi'\_2} \langle \langle \pi'\_3 \Sigma\_3 | \delta \Sigma\_{23} | \pi'\_2 \Sigma\_2 \rangle \langle \pi'\_2 \Sigma\_2 | \pi'\_1 \Sigma\_1 \rangle \\
+ \langle \pi'\_3 \Sigma\_3 | \pi'\_2 \Sigma\_2 \rangle \langle \pi'\_2 \Sigma\_2 | \delta \Sigma\_{12} | \pi'\_1 \Sigma\_1 \rangle \\
= \sum\_{\pi'\_2} \langle \pi'\_3 \Sigma\_3 | \pi'\_1 \Sigma\_1 \rangle \langle \pi'\_2 \Sigma\_2 | \pi'\_1 \Sigma\_1 \rangle \\
= \sum\_{\pi'\_2} \langle \pi'\_3 \Sigma\_3 | \pi'\_1 \Sigma\_1 \rangle \langle \pi'\_2 | \delta \Sigma\_{12} | \pi'\_1 \Sigma\_1 \rangle \\
= \sum\_{\pi'\_2} \langle \pi'\_3 \Sigma\_3 | \pi'\_1 \Sigma\_1 \rangle \langle \pi'\_3 \Sigma\_2 | \pi'\_2 \Sigma\_1 \rangle = \sum\_{\pi'\_2} \langle \pi'\_3 \Sigma\_3 | \pi'\_1 \Sigma\_1 \rangle \langle \pi'\_3 \Sigma\_2 | \pi'\_2 \Sigma\_1 \rangle = \sum\_{\pi'\_2} \langle \pi'\_3 \Sigma\_3 | \pi'\_1 \Sigma\_1 \rangle \langle \pi'\_3 \Sigma\_2 | \pi'\_2 \Sigma\_1 \rangle = \langle \pi'\_2 \Sigma\_3 | \pi'\_1 \Sigma\_1 \rangle \langle \pi'\_3 \Sigma\_2 | \pi'\_2 \rangle = \langle \pi'\_4 \Sigma\_4 | \pi'\_2 \Sigma\_4 \rangle = \langle \pi'\_4 \Sigma\_4 | \pi'\_1 \Sigma\_1 \rangle
$$

$$
\hat{\mathbf{y}} = \langle \pi\_{3\prime}^{\prime} \Sigma\_3 | \delta \mathbf{S\_{23}} + \delta \mathbf{S\_{12}} | \pi\_{1\prime}^{\prime} \Sigma\_1 \rangle. \tag{78}
$$

Comparing both ends of the result in (78), it follows that

$$
\delta \mathbf{S}\_{13} = \delta \mathbf{S}\_{23} + \delta \mathbf{S}\_{12}.\tag{79}
$$

In the limit <sup>Σ</sup><sup>2</sup> → <sup>Σ</sup>1, it must be that

$$
\delta S\_{12} = 0.
$$

In the limit in which <sup>Σ</sup><sup>3</sup> → <sup>Σ</sup>1, it follows that

$$
\delta S\_{21} = -\delta S\_{12}.
$$

If the operators in *τ*<sup>1</sup> and *τ*<sup>2</sup> undergo infinitesimal, unitary transformations on the hypersurfaces Σ<sup>1</sup> and Σ2, respectively, and only on these two hypersurfaces, then the change in the transformation function has the form

$$
\delta \langle \tau\_{2\prime}^{\prime} \Sigma\_2 | \tau\_{1\prime}^{\prime} \Sigma\_1 \rangle = \frac{i}{\hbar} \left\langle \tau\_{2\prime}^{\prime} \Sigma\_2 | \mathcal{F}\_2 - \mathcal{F}\_1 | \tau\_{1\prime}^{\prime} \Sigma\_1 \right\rangle. \tag{80}
$$

Here *F*<sup>1</sup> and *F*<sup>2</sup> are Hermitean operators constructed from a knowledge of the fields and their derivatives on Σ<sup>1</sup> and Σ2. The result (80) is of the form (76) provided that

$$
\delta S\_{12} = F\_2 - F\_1. \tag{81}
$$

A generator *F* of this type on a spacelike hypersurface Σ should be expressible in the form,

$$F = \int\_{\Sigma} d\sigma\_{\mathfrak{X}} n^{\mu} F\_{\mu}(\mathfrak{x})\_{\prime} \tag{82}$$

and *dσ<sup>x</sup>* is the area element on Σ, *n<sup>µ</sup>* is the outward unit normal to Σ, and *Fµ*(*x*) may be put together based on a knowledge of the fields on the surface Σ.

The points of Σ are all spacelike separated, hence independent and so the result follows by adding up all of these independent contributions. Applying (82) to Σ<sup>1</sup> and Σ<sup>2</sup> assuming that *Fµ*(*x*) is defined throughout the spacetime region bounded by these two surfaces, *δS*<sup>12</sup> can be expressed as

$$\delta\_{12}\mathbf{S} = \int\_{\Sigma\_2} d\sigma\_\mathbf{x} \, n^\mu F\_\mu(\mathbf{x}) - \int\_{\Sigma\_1} d\sigma\_\mathbf{x} \, n^\mu F\_\mu(\mathbf{x}) = \int\_{\Omega\_{12}} d\sigma\_\mathbf{x} \, \nabla^\mu F\_\mu(\mathbf{x}). \tag{83}$$

In (83), Ω<sup>12</sup> is the spacetime region bounded by Σ<sup>1</sup> and Σ2, and *dvx* is the invariant volume. This assumes that the operators are changes only on Σ<sup>1</sup> and Σ2. However, suppose the operators are changed in the spacetime region between Σ<sup>1</sup> and Σ2. Assume once more that *δS*<sup>12</sup> can be expressed as a volume integral as,

$$
\delta \mathcal{S}\_{12} = \int\_{\Omega\_{12}} dv\_{\mathcal{X}} \,\delta \mathcal{L}(\mathbf{x}) \tag{84}
$$

space-time also has a complementary momentum-energy implication, illustrated by the source idea [5,8]. Not only for the special balance of energy and momentum involved in the emission or absorption of a single particle is the source defined and meaningful. Given a sufficient excess of energy over momentum, or an excess of mass, several particles can be emitted or absorbed. For example, consider the emission of a pair of charged particles by an extended photon source. This process is represented as the conversion of a virtual photon into a pair of real particles. In ordinary scattering, particle-particle scattering, the particles persist while exchanging a space-like virtual photon. Another is an annihilation of the particle-antiparticle pair, producing a time-like virtual photon, which quickly decays

The Schwinger Action Principle and Its Applications to Quantum Mechanics

*JI*(*x*)*ϕ<sup>I</sup>*

<sup>2</sup>, <sup>Σ</sup>2|*τ*′

<sup>1</sup>, <sup>Σ</sup>1�*J*. If the Schwinger action principle is applied to this modified theory

2, <sup>Σ</sup>2|*δSJ*|*τ*′

*δφI*(*x*) <sup>+</sup> *JI*(*x*) = 0. (89)

and try to establish a connection between the action principle introduced here and a path

may be regarded as a functional of the source term *JI*(*x*). This will be denoted explicitly by

*h*¯ �*τ*′

The classical field *<sup>ϕ</sup>I*(*x*) has been replaced by the operator *<sup>φ</sup>I*(*x*). Moreover, in addition to its previous meaning, *δ* now includes a possible change in the source. By considering the variation to be with respect to the dynamical variables which are held fixed on Σ<sup>1</sup> and Σ2,

Assume that the variation in (88) is one in which the dynamical variables are held fixed and

*<sup>δ</sup>JI*(*x*)�*τ*′

*h*¯ �*τ*′ 2, <sup>Σ</sup>2|*φ<sup>i</sup>*

= *i h*¯ �2|*φ<sup>i</sup>*

2, <sup>Σ</sup>2|*φ<sup>I</sup>*

(*x*)|*τ*′

(*x*)|*τ*′

<sup>1</sup>, <sup>Σ</sup>1�*<sup>J</sup>* <sup>=</sup> *<sup>i</sup>*

*δS*[*φ*]

only the source is altered. Since the source enters in the simple way given in (87),

*h*¯ 

<sup>1</sup>, <sup>Σ</sup>1� *<sup>δ</sup>Ji*(*x*) <sup>=</sup> *<sup>i</sup>*

> *δ*�2|1� *δJi*

<sup>1</sup>, <sup>Σ</sup>1� <sup>=</sup> *<sup>i</sup>*

The result in (90) may be rewritten in the equivariant form,

*<sup>δ</sup>*�*τ*′ <sup>2</sup>, <sup>Σ</sup>2|*τ*′

To simplify this, it can be written in the alternate form,

(*x*) *<sup>d</sup>nx*, (87)

http://dx.doi.org/10.5772/53472

173

<sup>1</sup>, <sup>Σ</sup>1� discussed in the previous section

<sup>1</sup>, <sup>Σ</sup>1�*J*. (88)

<sup>1</sup>, <sup>Σ</sup>1�*<sup>J</sup> <sup>d</sup>nx*. (90)

<sup>1</sup>, <sup>Σ</sup>1�[*J*]. (91)


*SJ*[*ϕ*] = *<sup>S</sup>*[*ϕ*] +

back into particles.

writing �*τ*′

<sup>2</sup>, <sup>Σ</sup>2|*τ*′

Let us choose a simple scalar theory

integral picture [9]. The transition amplitude �*τ*′

the operator field equations are obtained as

*<sup>δ</sup>*�*τ*′ 2, <sup>Σ</sup>2|*τ*′

*<sup>δ</sup>*�*τ*′ 2, <sup>Σ</sup>2|*τ*′

for some *δ*L(*x*). Combining these two types of variation yields

$$
\delta S\_{12} = F\_2 - F\_1 + \int\_{\Omega\_{12}} dv\_X \, \delta \mathcal{L}(\mathbf{x}) = \int\_{\Omega\_{12}} dv\_X [\delta \mathcal{L}(\mathbf{x}) + \nabla^\mu F\_\mu(\mathbf{x})].\tag{85}
$$

It is an important result of (85) that altering *δ*L by the addition of the divergence of a vector field will result in a unitary transformation of the states on Σ<sup>1</sup> and Σ2.

To summarize, the fundamental assumption of the Schwinger action principle is that *δS*<sup>12</sup> may be obtained from a variation of

$$S\_{12} = \int\_{\Omega\_{12}} dv\_X \, \mathcal{L}(\mathbf{x}) ,\tag{86}$$

where L(*x*) is a Lagrangian density. The density depends on the fields and their derivatives at a single spacetime point. Since *δS*<sup>12</sup> is required to be Hermitian, *S*<sup>12</sup> must be Hermitian and similarly, so must the Lagrangian density.

#### **5. Correspondence with Feynman path integrals**

Suppose a classical theory described by the action *S*[*ϕ*] is altered by coupling the field to an external source *Ji* = *JI*(*x*). By external it is meant that it has no dependence on the field *<sup>ϕ</sup><sup>i</sup>* , and *<sup>i</sup>* stands for (*I*, *<sup>x</sup>*). For example, *<sup>F</sup>*,*i*[*ϕ*]*σ<sup>i</sup>* is an abbreviation for

$$F\_{,i}[\varphi]\sigma^{i} = \int d^{n}\mathbf{x}' \frac{\delta F[\varphi(\mathbf{x})]}{\delta \varphi^{I}(\mathbf{x}')} \sigma^{i}(\mathbf{x}').$$

The idea of introducing external sources originates with Schwinger. As he states, causality and space-time uniformity are the creative principles of source theory. Uniformity in space-time also has a complementary momentum-energy implication, illustrated by the source idea [5,8]. Not only for the special balance of energy and momentum involved in the emission or absorption of a single particle is the source defined and meaningful. Given a sufficient excess of energy over momentum, or an excess of mass, several particles can be emitted or absorbed. For example, consider the emission of a pair of charged particles by an extended photon source. This process is represented as the conversion of a virtual photon into a pair of real particles. In ordinary scattering, particle-particle scattering, the particles persist while exchanging a space-like virtual photon. Another is an annihilation of the particle-antiparticle pair, producing a time-like virtual photon, which quickly decays back into particles.

Let us choose a simple scalar theory

14 Quantum Mechanics

be expressed as

*δ*12*S* =

 Σ2

*δS*<sup>12</sup> can be expressed as a volume integral as,

*<sup>δ</sup>S*<sup>12</sup> = *<sup>F</sup>*<sup>2</sup> − *<sup>F</sup>*<sup>1</sup> +

and similarly, so must the Lagrangian density.

**5. Correspondence with Feynman path integrals**

and *<sup>i</sup>* stands for (*I*, *<sup>x</sup>*). For example, *<sup>F</sup>*,*i*[*ϕ*]*σ<sup>i</sup>* is an abbreviation for

*<sup>F</sup>*,*i*[*ϕ*]*σ<sup>i</sup>* <sup>=</sup>

may be obtained from a variation of

*<sup>d</sup>σ<sup>x</sup> <sup>n</sup>µFµ*(*x*) −

for some *δ*L(*x*). Combining these two types of variation yields

 Ω<sup>12</sup>

field will result in a unitary transformation of the states on Σ<sup>1</sup> and Σ2.

*S*<sup>12</sup> = Ω<sup>12</sup>

The points of Σ are all spacelike separated, hence independent and so the result follows by adding up all of these independent contributions. Applying (82) to Σ<sup>1</sup> and Σ<sup>2</sup> assuming that *Fµ*(*x*) is defined throughout the spacetime region bounded by these two surfaces, *δS*<sup>12</sup> can

In (83), Ω<sup>12</sup> is the spacetime region bounded by Σ<sup>1</sup> and Σ2, and *dvx* is the invariant volume. This assumes that the operators are changes only on Σ<sup>1</sup> and Σ2. However, suppose the operators are changed in the spacetime region between Σ<sup>1</sup> and Σ2. Assume once more that

*dσ<sup>x</sup> nµFµ*(*x*) =

 Ω<sup>12</sup>

*dvx* ∇*µFµ*(*x*). (83)

*dvx δ*L(*x*) (84)

*dvx* L(*x*), (86)

,

*dvx*[*δ*L(*x*) + ∇*µFµ*(*x*)]. (85)

 Σ1

*δS*<sup>12</sup> =

 Ω<sup>12</sup>

*dvx δ*L(*x*) =

It is an important result of (85) that altering *δ*L by the addition of the divergence of a vector

To summarize, the fundamental assumption of the Schwinger action principle is that *δS*<sup>12</sup>

where L(*x*) is a Lagrangian density. The density depends on the fields and their derivatives at a single spacetime point. Since *δS*<sup>12</sup> is required to be Hermitian, *S*<sup>12</sup> must be Hermitian

Suppose a classical theory described by the action *S*[*ϕ*] is altered by coupling the field to an external source *Ji* = *JI*(*x*). By external it is meant that it has no dependence on the field *<sup>ϕ</sup><sup>i</sup>*

The idea of introducing external sources originates with Schwinger. As he states, causality and space-time uniformity are the creative principles of source theory. Uniformity in

*<sup>d</sup>nx*′ *<sup>δ</sup>F*[*ϕ*(*x*)] *δϕI*(*x*′) *<sup>σ</sup><sup>i</sup>*

(*x*′ ).

 Ω<sup>12</sup>

$$\mathcal{S}\_I[\varphi] = \mathcal{S}[\varphi] + \int f\_I(\mathbf{x}) \varphi^I(\mathbf{x}) \, d^n \mathbf{x},\tag{87}$$

and try to establish a connection between the action principle introduced here and a path integral picture [9]. The transition amplitude �*τ*′ <sup>2</sup>, <sup>Σ</sup>2|*τ*′ <sup>1</sup>, <sup>Σ</sup>1� discussed in the previous section may be regarded as a functional of the source term *JI*(*x*). This will be denoted explicitly by writing �*τ*′ <sup>2</sup>, <sup>Σ</sup>2|*τ*′ <sup>1</sup>, <sup>Σ</sup>1�*J*. If the Schwinger action principle is applied to this modified theory

$$
\delta \langle \pi\_2', \Sigma\_2 | \pi\_1', \Sigma\_1 \rangle\_I = \frac{i}{\hbar} \langle \pi\_2', \Sigma\_2 | \delta \mathcal{S}\_I | \pi\_1', \Sigma\_1 \rangle\_I. \tag{88}
$$

The classical field *<sup>ϕ</sup>I*(*x*) has been replaced by the operator *<sup>φ</sup>I*(*x*). Moreover, in addition to its previous meaning, *δ* now includes a possible change in the source. By considering the variation to be with respect to the dynamical variables which are held fixed on Σ<sup>1</sup> and Σ2, the operator field equations are obtained as

$$\frac{\delta \mathcal{S}[\phi]}{\delta \phi^I(\mathbf{x})} + f\_I(\mathbf{x}) = 0. \tag{89}$$

Assume that the variation in (88) is one in which the dynamical variables are held fixed and only the source is altered. Since the source enters in the simple way given in (87),

$$
\delta \langle \pi\_{2'}' \Sigma\_2 | \pi\_{1'}' \Sigma\_1 \rangle = \frac{i}{\hbar} \int \delta I\_I(\mathbf{x}) \langle \pi\_{2'}' \Sigma\_2 | \phi^I(\mathbf{x}) | \pi\_{1'}' \Sigma\_1 \rangle\_I d^n \mathbf{x}. \tag{90}
$$

The result in (90) may be rewritten in the equivariant form,

$$\frac{\delta \langle \pi\_{2'}' \Sigma\_2 | \pi\_{1'}' \Sigma\_1 \rangle}{\delta f\_l(\mathbf{x})} = \frac{i}{\hbar} \langle \pi\_{2'}' \Sigma\_2 | \phi^i(\mathbf{x}) | \pi\_{1'}' \Sigma\_1 \rangle [f]. \tag{91}$$

To simplify this, it can be written in the alternate form,

$$\frac{\delta \langle 2 | 1 \rangle}{\delta f\_i} = \frac{i}{\hbar} \langle 2 | \phi^i | 1 \rangle [J]. \tag{92}$$

An abbreviated notation for the initial and final state has been introduced. This result can be varied with respect to the source which gives

$$
\delta \frac{\delta \langle 2 | 1 \rangle [J]}{\delta f\_i} = \frac{i}{\hbar} \delta \langle 2 | \phi^i | 1 \rangle [J]. \tag{93}
$$

Differentiating (91) and using the result (97), we get

This can be generalized, and omitting details,

�2|1�[*J*] =

The series may be formally summed to yield

previous result

with a comma,

If *φ<sup>i</sup>* is replaced by *<sup>h</sup>*¯

*i δ*

(102), the following differential equation arises

*S*,*i*[ *h*¯ *i δ δJi*

The operator equation of motion (89) implies

*δ*2�2|1�[*J*] *δJiδJj*

*δn*�2|1�[*J*] *<sup>δ</sup>Ji*<sup>1</sup> ··· *<sup>δ</sup>Jin*

> ∞ ∑ *n*=0

*S*[*φ*] =

*S*,*i*[*φ*] =

*S*,*i*[ *h*¯ *i δ δJi*

1 *n*! ( *i h*¯

�2|1�[*J*] = �2|*T*(exp(

∞ ∑ *n*=0

Similarly, the derivative of *S* with respect to the field has the expansion [7],

∞ ∑ *n*=0

and *Ji* is set to zero everywhere on the right-hand side except in the exponential.

1 *n*!

> 1 *n*!

]�2|1�[*J*] = �2|*T*(*S*,*i*[*φ*] exp(

= ( *<sup>i</sup> h*¯

= ( *<sup>i</sup> h*¯

The amplitude �2|1�[*J*] may be defined by a Taylor expansion about *Ji* = 0, so using the

*i <sup>h</sup>*¯ *Jiφ<sup>i</sup>*

*<sup>δ</sup>Ji* in this expression and then operate on �2|1�[*J*] with *<sup>S</sup>*,*i*[ *<sup>h</sup>*¯

*i <sup>h</sup>*¯ *Jiφ<sup>i</sup>*

The action *S*[*φ*] can be expanded in a Taylor series about *φ<sup>i</sup>* = 0, indicating differentiation

)�2|*T*(*φ<sup>i</sup>*

*φj*

)|1�[*J*]. (98)

http://dx.doi.org/10.5772/53472

175

)*n*�2|*T*(*φi*<sup>1</sup> ··· *φin* )|1�[*J*]. (99)

The Schwinger Action Principle and Its Applications to Quantum Mechanics

)*<sup>n</sup> Ji*<sup>1</sup> ··· *Jin* �2|*T*(*φi*<sup>1</sup> ··· *<sup>φ</sup>in* )|1�[*<sup>J</sup>* <sup>=</sup> <sup>0</sup>]. (100)

*<sup>S</sup>*,*i*1···*in* [*<sup>φ</sup>* <sup>=</sup> <sup>0</sup>]*φi*<sup>1</sup> ··· *<sup>φ</sup>in* . (102)

*<sup>S</sup>*,*ii*1···*in* [*<sup>φ</sup>* <sup>=</sup> <sup>0</sup>]*φi*<sup>1</sup> ··· *<sup>φ</sup>in* . (103)

]�2|1�[*J*] = −*Ji*�2|1�[*J*]. (105)

*i δ δJi*

))|1�[*J* = 0]. (104)

], and use

))|1�[*J* = 0] (101)

To evaluate (93), a spatial hypersurface <sup>Σ</sup>′ is introduced which resides to the future of <sup>Σ</sup><sup>1</sup> and the past of Σ<sup>2</sup> and contains the spacetime point on which the *φ<sup>i</sup>* (*x*) depend. Any source variation can be represented as the sum of a variation which vanishes to the future of <sup>Σ</sup>′ , but is non-zero to the past, and one which vanishes to the past of <sup>Σ</sup>′ but is nonzero to the future. Consider the case where *δJi* vanishes to the future. In this event, any amplitude of the form �2|*φ<sup>i</sup>* |*τ*′ �[*J*], where <sup>|</sup>*τ*′ � represents a state on <sup>Σ</sup>′ , can not be affected by the variation of the source since *<sup>δ</sup>Ji* will vanish to the future of <sup>Σ</sup>′ . By using the completeness relation

$$
\langle \mathfrak{Q} | \phi^i | 1 \rangle [J] = \sum\_{\pi'} \langle \mathfrak{Q} | \phi^i | \pi' \rangle \langle \pi' | 1 \rangle [J]\_{\prime \prime}
$$

it follows that the right-hand side of (93) may be reexpressed with the use of

$$
\delta\left<2|\phi^i|1\right>[J] = \sum\_{\pi'} \langle 2|\phi^i|\pi'\rangle \delta\left<\pi'|1\right>[J].\tag{94}
$$

The Schwinger action principle then implies that

$$
\delta \langle \pi' | 1 \rangle [f] = \frac{i}{\hbar} \delta f\_k \langle \pi' | \phi^k | 1 \rangle [f]. \tag{95}
$$

Substituting (95) into (94) leads to the conclusion that

$$
\delta \langle 2 | \phi^{\dot{i}} | 1 \rangle [I] = \frac{\dot{i}}{\hbar} \delta f\_{\dot{j}} \sum\_{\tau'} \langle 2 | \phi^{\dot{i}} | \tau' \rangle \langle \tau' | \phi^{\dot{j}} | 1 \rangle [I] = \frac{\dot{i}}{\hbar} \delta f\_{\dot{j}} \langle 2 | \phi^{\dot{i}} \phi^{\dot{j}} | 1 \rangle [I]. \tag{96}
$$

Since it can be said that *<sup>δ</sup>Jj* vanishes to the future of <sup>Σ</sup>′ , which contains the spacetime point of *φ<sup>i</sup>* , the spacetime point of *φ<sup>j</sup>* must lie to the past of the former.

Consider the case in which *<sup>δ</sup>Jj* vanishes to the past of <sup>Σ</sup>′ . A similar argument yields the same conclusion as (96), but with *φ<sup>j</sup>* to the left of *φ<sup>i</sup>* . Combining this set of results produces the following conclusion

$$\frac{\delta \langle 2 | \phi^i | 1 \rangle}{\delta J\_j} = \frac{i}{\hbar} \langle 2 | T(\phi^i \phi^j) | 1 \rangle [J]. \tag{97}$$

In (97), *T* is the chronological, or time, ordering operator, which orders any product of fields in the sequence of increasing time, with those furthest to the past to the very right.

Differentiating (91) and using the result (97), we get

$$\frac{\delta^2 \langle 2 | 1 \rangle [J]}{\delta J\_i \delta J\_j} = (\frac{i}{\hbar}) \langle 2 | T(\phi^i \phi^j) | 1 \rangle [J]. \tag{98}$$

This can be generalized, and omitting details,

16 Quantum Mechanics

�2|*φ<sup>i</sup>* |*τ*′

of *φ<sup>i</sup>*

following conclusion

�[*J*], where <sup>|</sup>*τ*′

varied with respect to the source which gives

source since *<sup>δ</sup>Ji* will vanish to the future of <sup>Σ</sup>′

The Schwinger action principle then implies that

Substituting (95) into (94) leads to the conclusion that

<sup>|</sup>1�[*J*] = *<sup>i</sup>*

Since it can be said that *<sup>δ</sup>Jj* vanishes to the future of <sup>Σ</sup>′

Consider the case in which *<sup>δ</sup>Jj* vanishes to the past of <sup>Σ</sup>′

conclusion as (96), but with *φ<sup>j</sup>* to the left of *φ<sup>i</sup>*

*h*¯ *<sup>δ</sup>Jj* ∑ *τ*′ �2|*φ<sup>i</sup>* |*τ*′ ��*τ*′ |*φj*

, the spacetime point of *φ<sup>j</sup>* must lie to the past of the former.

*δ*�2|*φ<sup>i</sup>* |1�

*δJj*

= *i h*¯

in the sequence of increasing time, with those furthest to the past to the very right.

�2|*T*(*φ<sup>i</sup> φj*

In (97), *T* is the chronological, or time, ordering operator, which orders any product of fields

*δ*�2|*φ<sup>i</sup>*

*δ*

and the past of Σ<sup>2</sup> and contains the spacetime point on which the *φ<sup>i</sup>*

� represents a state on <sup>Σ</sup>′


it follows that the right-hand side of (93) may be reexpressed with the use of


<sup>|</sup>1�[*J*] = *<sup>i</sup>*

�2|*φ<sup>i</sup>*

*δ*�2|*φ<sup>i</sup>*

*<sup>δ</sup>*�*τ*′

*δ*�2|1�[*J*] *δJi*

An abbreviated notation for the initial and final state has been introduced. This result can be

= *i h*¯ *δ*�2|*φ<sup>i</sup>*

To evaluate (93), a spatial hypersurface <sup>Σ</sup>′ is introduced which resides to the future of <sup>Σ</sup><sup>1</sup>

is non-zero to the past, and one which vanishes to the past of <sup>Σ</sup>′ but is nonzero to the future. Consider the case where *δJi* vanishes to the future. In this event, any amplitude of the form

variation can be represented as the sum of a variation which vanishes to the future of <sup>Σ</sup>′

*τ*′

*τ*′ �2|*φ<sup>i</sup>* |*τ*′ �*δ*�*τ*′

*h*¯ *<sup>δ</sup>Jk*�*τ*′

�2|*φ<sup>i</sup>* |*τ*′ ��*τ*′


, can not be affected by the variation of the



. By using the completeness relation


<sup>|</sup>1�[*J*] = *<sup>i</sup>*

*h*¯

*<sup>δ</sup>Jj* �2|*φ<sup>i</sup>*

*φj*

. Combining this set of results produces the

, which contains the spacetime point

. A similar argument yields the same

)|1�[*J*]. (97)


(*x*) depend. Any source

, but

$$\frac{\delta^n \langle 2 | 1 \rangle [I]}{\delta J\_{i\_1} \cdots \delta J\_{i\_n}} = (\frac{i}{\hbar})^n \langle 2 | T(\phi^{i\_1} \cdots \phi^{i\_n}) | 1 \rangle [I]. \tag{99}$$

The amplitude �2|1�[*J*] may be defined by a Taylor expansion about *Ji* = 0, so using the previous result

$$\langle 2|1\rangle [f] = \sum\_{n=0}^{\infty} \frac{1}{n!} (\frac{i}{\hbar})^n J\_{i\_1} \cdots J\_{i\_n} \langle 2|T(\phi^{i\_1} \cdots \phi^{i\_n})|1\rangle [f=0]. \tag{100}$$

The series may be formally summed to yield

$$\langle \langle 2 \vert 1 \rangle [f] = \langle 2 \vert T(\exp(\frac{i}{\hbar} J\_i \phi^i)) \vert 1 \rangle [f=0] \tag{101}$$

and *Ji* is set to zero everywhere on the right-hand side except in the exponential.

The action *S*[*φ*] can be expanded in a Taylor series about *φ<sup>i</sup>* = 0, indicating differentiation with a comma,

$$\mathcal{S}[\phi] = \sum\_{n=0}^{\infty} \frac{1}{n!} \mathcal{S}\_{j\_1 \dots j\_n}[\phi = 0] \phi^{i\_1} \cdots \phi^{i\_n}.\tag{102}$$

Similarly, the derivative of *S* with respect to the field has the expansion [7],

$$\mathcal{S}\_{,i}[\phi] = \sum\_{n=0}^{\infty} \frac{1}{n!} \mathcal{S}\_{,\bar{i}\bar{i}\_1 \cdots \bar{i}\_n}[\phi = 0] \phi^{i\_1} \cdots \phi^{i\_n}.\tag{103}$$

If *φ<sup>i</sup>* is replaced by *<sup>h</sup>*¯ *i δ <sup>δ</sup>Ji* in this expression and then operate on �2|1�[*J*] with *<sup>S</sup>*,*i*[ *<sup>h</sup>*¯ *i δ δJi* ], and use (102), the following differential equation arises

$$S\_{\boldsymbol{\beta}}[\frac{\hbar}{i}\frac{\delta}{\delta f\_{\boldsymbol{i}}}]\langle 2|1\rangle[\boldsymbol{I}] = \langle 2|T(S\_{\boldsymbol{\beta}}[\boldsymbol{\phi}]\exp(\frac{\boldsymbol{i}}{\hbar}I\_{\boldsymbol{i}}\boldsymbol{\phi}^{i}))|1\rangle[\boldsymbol{I}=\boldsymbol{0}].\tag{104}$$

The operator equation of motion (89) implies

$$\mathcal{S}\_{,i}[\frac{\hbar}{i}\frac{\delta}{\delta f\_i}]\langle 2|1\rangle\langle f| = -J\_i\langle 2|1\rangle\langle f|.\tag{105}$$

This results in a differential equation for the transition amplitude. In order to solve equation (105), the functional analogue of a Fourier transform may be used

$$
\langle 2|1 \rangle = \int (\prod\_I d\varphi^I(\mathbf{x})) F[\varphi] \exp(\frac{i}{\hbar} \int J\_I(\mathbf{x'}) \varphi^I(\mathbf{x'}) \, d^n \mathbf{x'}).\tag{106}
$$

**6. QED - A physical example and summary**

− *∂*2*Aµ*(*x*) + *∂µ∂A*(*x*) = *J*

experimental consequences.

of the particle field equation

This gives the partial action

*W* = *dx*[*J*

eliminate the vector potential

) is the Green function

*Af*

and the gauge condition determines *λ*(*x*).

by a propagation function *<sup>G</sup>*+(*x*, *<sup>x</sup>*′

*<sup>µ</sup>*(*x*) =

*j µ* *dx*′

*cons*(*x*) = *j*

*<sup>W</sup>*<sup>21</sup> <sup>=</sup> <sup>1</sup> 2 

*<sup>µ</sup>A<sup>µ</sup>* <sup>−</sup> <sup>1</sup> 4

and *G<sup>A</sup>*

+(*x*, *<sup>x</sup>*′

A given elementary interaction implies a system of coupled field equations. Thus for the

[*γ*(−*i∂* − *eqA*(*x*)) + *m*]*ψ*(*x*) = *ηA*(*x*),

*ψ*(*x*)*γ*0*γµeqψ*(*x*) −

Since this is a nonlinear system, the construction of the fields in terms of the sources will be given by doubly infinite power series. The succesive terms of this series *Wn<sup>ν</sup>* with *n* paricles and *ν* photon sources represent increasingly complicated physical processes. One of the simplest terms in the interaction skeleton will be discussed below to the point of obtaining

There are two asymmetrical ways to eliminate the fields. First, introduce the formal solution

*dx*′ *G<sup>A</sup>* +(*x*, *<sup>x</sup>*′ )*ηA*(*x*),

*dxdx*′ *<sup>η</sup>A*(*x*)*γ*0*G<sup>A</sup>*

) = *<sup>δ</sup>*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*′

).

+(*x*, *<sup>x</sup>*′

*cons*] + *∂µλ*(*x*).

)*∂*′ *ν*(*x*′ ), )*ηA*(*x*′ ).

+(*x*, *<sup>x</sup>*′

*dx*′ *<sup>f</sup> <sup>µ</sup>*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*′

The Schwinger Action Principle and Its Applications to Quantum Mechanics

)*ψ*(*x*′

)*γ*0*ieqηA*(*x*′

http://dx.doi.org/10.5772/53472

). (110) 177

photon and the charged spin 1/2 particle as in quantum electrodynamics [5],

*<sup>µ</sup>*(*x*) + <sup>1</sup> 2

*<sup>ψ</sup>A*(*x*) =

[*γ*(−*i∂* − *eqA*(*x*)) + *m*]*G<sup>A</sup>*

*<sup>F</sup>µνFµν*] + <sup>1</sup>

*<sup>D</sup>*+(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*′

*<sup>µ</sup>*(*x*) − 

2 

The stationarity requirement on variations of *A<sup>µ</sup>* recovers the Maxwell equation above. If we

)[*J*

Finally, the first few successive *W*2*<sup>ν</sup>* are written out, noting each particle source is multiplied

) to form the field *ψ*,

*<sup>µ</sup>*(*x*) + *j µ*

*dx*′ *<sup>f</sup> <sup>µ</sup>*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*′

*d*4*xψ*(*x*)*γ*0*eqγA*(*x*)*ψ*(2),

The integration in (106) extends over all fields which correspond to the choice of states described by |1� and |2�. The functional *F*[*ϕ*] which is to be thought of as the Fourier transform of the transformation function, is to be determined by requiring (106) satisfy (105)

$$0 = \int \left(\prod\_{I,\mathbf{x}} d\varphi^I(\mathbf{x})\right) \{S\_{,i}[\varphi] + f\_i\} F[\varphi] \exp(\frac{i}{\hbar} \int J\_I(\mathbf{x}') \varphi^I(\mathbf{x}') \,d^n \mathbf{x}')$$

$$=\int (\prod\_I d\boldsymbol{\varrho}^I(\mathbf{x})) \{ S\_{,i}[\boldsymbol{\varrho}] F[\boldsymbol{\varrho}] + \frac{\hbar}{i} F[\boldsymbol{\varrho}] \frac{\delta}{\delta \boldsymbol{\varrho}^i} \} \exp(\frac{i}{\hbar} \int J\_I(\mathbf{x'}) \boldsymbol{\varrho}^I(\mathbf{x'}) \boldsymbol{d}^n \mathbf{x'}).$$

Upon carrying out an integration by parts on the second term here

$$0 = \int \left(\prod\_{i} d\varphi^{i}\right) \{\mathcal{S}\_{,i}[\varphi]F[\varphi] - \frac{\hbar}{i}F\_{,i}[\varphi]\} \exp(\frac{i}{\hbar} \int f\_{I}(\mathbf{x'})\rho^{I}(\mathbf{x'}) \,d^{n}\mathbf{x'})$$

$$+\frac{\hbar}{i}F[\varphi]\exp(\frac{i}{\hbar}\int J\_I(\mathbf{x'})\varphi^I(\mathbf{x'})\,d^\eta\mathbf{x'})|\_{\varphi\_1}^{\varphi\_2}.\tag{107}$$

Assuming the surface term at the end vanishes, it follows from (107) that

$$F[\varphi] = \mathcal{N} \exp(\frac{\dot{l}}{\hbar} \mathcal{S}[\varphi]),\tag{108}$$

where N is any field-independent constant. The condition for the surface term to vanish is that the action *S*[*ϕ*] be the same on both surfaces Σ<sup>1</sup> and Σ2. This condition is usually fulfilled in field theory by assuming that the fields are in the vacuum state on the initial and final hypersurface.

The transformation function can then be summarized as

$$\langle 2|1 \rangle [f] = \mathcal{N} \int \left( \prod\_i d\varphi^i \right) \exp(\frac{i}{\hbar} \{ S[\varphi] + J\_i \varphi^i \}). \tag{109}$$

This is one form of the Feynman path integral, or functional integral, which represents the transformation function. This technique turns out to be very effective with further modifications applied to the quantization of gauge theories. These theories have been particularly successful in understanding the weak and strong interactions [10].

### **6. QED - A physical example and summary**

18 Quantum Mechanics

This results in a differential equation for the transition amplitude. In order to solve equation

The integration in (106) extends over all fields which correspond to the choice of states described by |1� and |2�. The functional *F*[*ϕ*] which is to be thought of as the Fourier transform of the transformation function, is to be determined by requiring (106) satisfy (105)

*i h*¯ *JI*(*x*′ )*ϕI* (*x*′ ) *<sup>d</sup>nx*′

> *i h*¯ *JI*(*x*′ )*ϕI* (*x*′ ) *<sup>d</sup>nx*′ )

} exp( *i h*¯ *JI*(*x*′ )*ϕI* (*x*′ ) *<sup>d</sup>nx*′ ).

> *i h*¯ *JI*(*x*′ )*ϕI* (*x*′ ) *<sup>d</sup>nx*′ )

{*S*[*ϕ*] + *Jiϕ<sup>i</sup>*

). (106)

*<sup>ϕ</sup>*<sup>1</sup> . (107)

}). (109)

*<sup>h</sup>*¯ *<sup>S</sup>*[*ϕ*]), (108)

(*x*))*F*[*ϕ*] exp(

(*x*)){*S*,*i*[*ϕ*] + *Ji*}*F*[*ϕ*] exp(

*i <sup>F</sup>*[*ϕ*] *<sup>δ</sup> δϕi*

*i*

*i h*¯ *JI*(*x*′ )*ϕI* (*x*′ ) *<sup>d</sup>nx*′ )| *ϕ*2

*F*[*ϕ*] = N exp(

where N is any field-independent constant. The condition for the surface term to vanish is that the action *S*[*ϕ*] be the same on both surfaces Σ<sup>1</sup> and Σ2. This condition is usually fulfilled in field theory by assuming that the fields are in the vacuum state on the initial and

This is one form of the Feynman path integral, or functional integral, which represents the transformation function. This technique turns out to be very effective with further modifications applied to the quantization of gauge theories. These theories have been

*<sup>F</sup>*,*i*[*ϕ*]} exp(

*i*

(105), the functional analogue of a Fourier transform may be used

(*x*)){*S*,*i*[*ϕ*]*F*[*ϕ*] + *<sup>h</sup>*¯

Upon carrying out an integration by parts on the second term here

+ *h*¯ *i*

The transformation function can then be summarized as

�2|1�[*J*] = N

){*S*,*i*[*ϕ*]*F*[*ϕ*] <sup>−</sup> *<sup>h</sup>*¯

*F*[*ϕ*] exp(

Assuming the surface term at the end vanishes, it follows from (107) that

 (∏ *i dϕi* ) exp( *i h*¯

particularly successful in understanding the weak and strong interactions [10].

�2|1� =

0 = (∏ *I*,*x dϕI*

= (∏ *I dϕI*

final hypersurface.

0 = (∏ *i dϕi*

 (∏ *I dϕI* A given elementary interaction implies a system of coupled field equations. Thus for the photon and the charged spin 1/2 particle as in quantum electrodynamics [5],

$$[\gamma(-i\partial - eqA(\mathfrak{x})) + m]\psi(\mathfrak{x}) = \eta^A(\mathfrak{x})\_{\prime\prime}$$

$$-\partial^2 A^\mu(\mathbf{x}) + \partial^\mu \partial A(\mathbf{x}) = J^\mu(\mathbf{x}) + \frac{1}{2} \psi(\mathbf{x}) \gamma^0 \gamma^\mu \epsilon \eta \psi(\mathbf{x}) - \int d\mathbf{x}' f^\mu(\mathbf{x} - \mathbf{x}') \psi(\mathbf{x}') \gamma^0 i \epsilon \eta \eta^A(\mathbf{x}'). \tag{110}$$

Since this is a nonlinear system, the construction of the fields in terms of the sources will be given by doubly infinite power series. The succesive terms of this series *Wn<sup>ν</sup>* with *n* paricles and *ν* photon sources represent increasingly complicated physical processes. One of the simplest terms in the interaction skeleton will be discussed below to the point of obtaining experimental consequences.

There are two asymmetrical ways to eliminate the fields. First, introduce the formal solution of the particle field equation

$$\psi^A(\mathbf{x}) = \int d\mathbf{x}' G\_+^A(\mathbf{x}, \mathbf{x}') \eta^A(\mathbf{x})\_A$$

and *G<sup>A</sup>* +(*x*, *<sup>x</sup>*′ ) is the Green function

$$
\delta \left[ \gamma (-i\partial - eqA(\mathbf{x})) + m \right] G^A\_+(\mathbf{x}, \mathbf{x}') = \delta(\mathbf{x} - \mathbf{x}') .
$$

This gives the partial action

$$\mathcal{W} = \int d\mathbf{x} \left[ \mathcal{J}^{\mu} A\_{\mu} - \frac{1}{4} F^{\mu \nu} F\_{\mu \nu} \right] + \frac{1}{2} \int d\mathbf{x} d\mathbf{x}' \, \eta^{A}(\mathbf{x}) \gamma^{0} G\_{+}^{A}(\mathbf{x}, \mathbf{x}') \eta^{A}(\mathbf{x}').$$

The stationarity requirement on variations of *A<sup>µ</sup>* recovers the Maxwell equation above. If we eliminate the vector potential

$$A^f\_{\mu}(\mathbf{x}) = \int d\mathbf{x}' D\_+(\mathbf{x} - \mathbf{x}') [J^\mu(\mathbf{x}) + j^\mu\_{cons}] + \partial\_\mu \lambda(\mathbf{x}).$$

$$j\_{\rm cons}^{\mu}(\mathbf{x}) = j^{\mu}(\mathbf{x}) - \int d\mathbf{x}' f^{\mu}(\mathbf{x} - \mathbf{x}') \partial\_{\nu}'(\mathbf{x}'),$$

and the gauge condition determines *λ*(*x*).

Finally, the first few successive *W*2*<sup>ν</sup>* are written out, noting each particle source is multiplied by a propagation function *<sup>G</sup>*+(*x*, *<sup>x</sup>*′ ) to form the field *ψ*,

$$W\_{21} = \frac{1}{2} \int d^4x \psi(x)\gamma^0 \epsilon q \gamma A(x)\psi(2)\gamma$$

$$W\_{22} = \frac{1}{2} \int d\mathbf{x} d\mathbf{x}' \psi(\mathbf{x}) \gamma^0 \epsilon q \gamma A(\mathbf{x}) G\_+(\mathbf{x} - \mathbf{x}') \epsilon q \gamma A(\mathbf{x}') \psi(\mathbf{x}').$$

As a brief introduction to how this formalism can lead to important physical results, let us look at a specific term like *W*21, the interaction energy of an electron with a static electromagnetic field *Aext µ*

$$E = \int d^3x \, j\_{\mu} A^{\mu}\_{\text{ext}} = e \int d^3x \bar{\psi}\_{p'} (\gamma\_{\mu} + \Gamma^R\_{\mu}(p', p) + \frac{i}{4\pi} \Pi^R\_{\mu\nu} i D^{\nu\sigma} \gamma\_{\sigma}) \psi\_p A^{\mu}\_{\text{ext}}.\tag{111}$$

These terms include the bare electron-photon term, the electron-photon correction terms and then the photon vacuum-polarization correction term, *R* means a renormalized quantity and *γµ* denote Dirac matrices. The self-energy correction is left out, because for free particles, it contributes only to charge and mass renormalization. The polarization tensor Π*µν*(*q*2) is given by

$$
\Pi\_{\mu\nu}(q^2) = (q^2 \mathcal{g}\_{\mu\nu} - q\_{\mu}\mathcal{q}\_{\nu})\Pi(q^2),
\tag{112}
$$

structure constant. The Gordon decomposition allows this to be written

*α* 3*π*

*<sup>µ</sup>* <sup>=</sup> <sup>−</sup>*<sup>i</sup>*

<sup>2</sup>*<sup>m</sup> <sup>ψ</sup>*¯ *<sup>p</sup>*′(*x*)(*∂µ* <sup>−</sup> ←−*<sup>∂</sup>* )*ψp*[<sup>1</sup> <sup>−</sup> *<sup>α</sup>*

*α* <sup>2</sup>*<sup>π</sup>* ) <sup>1</sup> 4*m* 

*α* <sup>2</sup>*<sup>π</sup>* )<sup>2</sup> 

<sup>2</sup>*mc* (<sup>1</sup> <sup>+</sup>

*α*

*α* <sup>2</sup>*<sup>π</sup>* ) <sup>1</sup>

*q*2 *<sup>m</sup>*<sup>2</sup> (ln(

←−

*m <sup>µ</sup>* ) <sup>−</sup> <sup>3</sup>

The momentum factors can be transformed into gradients in configuration space, thus *<sup>q</sup><sup>µ</sup>* →

<sup>8</sup> <sup>−</sup> <sup>1</sup> 5

3*π* 1 *<sup>m</sup>*<sup>2</sup> (ln(

<sup>2</sup>*<sup>m</sup> <sup>ψ</sup>*¯ *<sup>p</sup>*(*x*)*σµνψp*(*x*)*∂νA<sup>µ</sup>*

<sup>2</sup> [*γµ*, *γν*], the second part is

The first term contains the convection current of the electron which interacts with the potential. In the special case of a purely magnetic field the second part can be identified as the dipole energy. By introducing the electromagnetic field strength tensor *Fµν* = *∂µA<sup>ν</sup>* − *∂νA<sup>µ</sup>*

When *<sup>F</sup>µν* represents a pure magnetic field, *<sup>F</sup>*<sup>12</sup> = −*B*3, *<sup>σ</sup>*<sup>12</sup> = <sup>Σ</sup><sup>3</sup> with cyclic permutations

*α* <sup>2</sup>*<sup>π</sup>* )2�

The magnetic moment is thus proportional to the spin expectation value of the electron. The

The first point to note is that the value of the *g*-factor obtained including quantum mechanics differs from the classical value of 2. The result in (122) was first calculated by Schwinger and it has been measured to remarkable accuracy many times. A modern experimental value for

*<sup>d</sup>*3*<sup>x</sup> <sup>ψ</sup>*¯(*x*)

<sup>Σ</sup>*ψ*(*x*) ·

*<sup>S</sup>*� <sup>=</sup> *<sup>g</sup>µB*�

)] + (1 +

The Schwinger Action Principle and Its Applications to Quantum Mechanics

*m <sup>µ</sup>* ) <sup>−</sup> <sup>3</sup>

*α* <sup>2</sup>*<sup>π</sup>* ) *<sup>i</sup>* 2*m*

<sup>8</sup> <sup>−</sup> <sup>1</sup> 5 )]*A<sup>µ</sup> ext*

*d*3*xψ*¯(*x*)*σµνψ*(*x*)*Fµν*. (119)

*<sup>B</sup>* <sup>=</sup> −�*<sup>µ</sup>*� ·

<sup>2</sup>*<sup>π</sup>* ) = <sup>2</sup>(<sup>1</sup> <sup>+</sup> 0.00116141). (122)

*gexp* = 2(1 + 0.001159652193), (123)

*ext*}. (118)

*B*. (120)

*S*�. (121)

*∂ <sup>µ</sup> p<sup>µ</sup>* = *i∂µ* act on the spinor field to the left and

*σµνqν*}*ψpA<sup>µ</sup>*

http://dx.doi.org/10.5772/53472

*ext*. (117) 179

)*µ*[1 +

− (1 +

*Wmag* = *e*(1 +

<sup>4</sup>*<sup>m</sup>* (<sup>1</sup> <sup>+</sup>

�*<sup>µ</sup>*� <sup>=</sup> *eh*¯

*g* = 2(1 +

*E* = *e*  *<sup>d</sup>*3*xψ*¯ *<sup>p</sup>*′ { <sup>1</sup>

*<sup>i</sup>∂µ* acts on the photon field and *<sup>p</sup>*′

*E* = *e* 

right respectively. Then (117) becomes

and using the antisymmetry of *σµν* = *<sup>i</sup>*

and the interaction energy becomes

The magnetic moment is given by

the *g*-factor is

*Wmag* <sup>=</sup> <sup>−</sup> *<sup>e</sup>*

proportionality factor is the so called *g*-factor

<sup>2</sup>*<sup>m</sup>* (*<sup>p</sup>* <sup>+</sup> *<sup>p</sup>*′

*<sup>d</sup>*3*x*{ *<sup>i</sup>*

where Π(*q*2) is the polarization function. A simple result is obtained in the limit of low momentum transfer, *q*<sup>2</sup> → 0, which is also of special physical significance and the case of interest here. The renormalized polarization function is

$$\Pi^R(q^2) = -\frac{e^2}{\pi} \frac{q^2}{m^2} (\frac{1}{15} + \frac{1}{140} \frac{q^2}{m^2} + \dotsb). \tag{113}$$

The regularized vertex function is

$$
\Gamma^{\mathbb{R}}\_{\mu}(p',p) = \gamma\_{\mu} \mathcal{F}\_1(q^2) + \frac{i}{2m} \sigma\_{\mu\nu} q^{\nu} \mathcal{F}\_2(q^2). \tag{114}
$$

The functions *F*1(*q*2) and *F*2(*q*2) are called form factors. The electron gets an apparent internal structure by the interaction with the virtual radiation field which alters it from a pure Dirac particle. In the limit, *q*<sup>2</sup> → 0, these functions can be calculated to be

$$F\_1(q^2) = \frac{\mathfrak{a}}{3\pi} \frac{q^2}{m^2} (\ln(\frac{m}{\mu}) - \frac{3}{8}), \qquad F\_2(q^2) = \frac{\mathfrak{a}}{2\pi}.\tag{115}$$

Substituting all of these factors and *Dµν <sup>F</sup>* (*q*2) = <sup>−</sup>4*πgµν*/*q*<sup>2</sup> into (111) yields for small values of *q*2,

$$E = e \int d^3 \mathbf{x} \,\bar{\psi}\_{p'} \left\{ \gamma\_\mu [1 + \frac{\mathfrak{a}}{3\pi} \frac{q^2}{m^2} (\ln(\frac{m}{\mu}) - \frac{3}{8} - \frac{1}{5}) + \frac{\mathfrak{a}}{2\pi} \frac{\mathrm{i}}{2m} \sigma\_{\mu\nu} q^\nu \right\} \psi\_p A^\mu\_{\mathrm{ext}}.\tag{116}$$

Note *µ* appears in (115) as an elementary attempt to regularize a photon propagator in one of the terms and does not interfere further with the application at this level and *α* is the fine structure constant. The Gordon decomposition allows this to be written

$$E = e \int d^3 x \bar{\psi}\_{p'} \{ \frac{1}{2m} (p + p')\_{\mu} [1 + \frac{a}{3\pi} \frac{q^2}{m^2} (\ln(\frac{m}{\mu}) - \frac{3}{8} - \frac{1}{5})] + (1 + \frac{a}{2\pi}) \frac{i}{2m} \sigma\_{\mu\nu} \eta^{\nu} \} \psi\_p A^{\mu}\_{ext.} \tag{117}$$

The momentum factors can be transformed into gradients in configuration space, thus *<sup>q</sup><sup>µ</sup>* → *<sup>i</sup>∂µ* acts on the photon field and *<sup>p</sup>*′ *<sup>µ</sup>* <sup>=</sup> <sup>−</sup>*<sup>i</sup>* ←−*∂ <sup>µ</sup> p<sup>µ</sup>* = *i∂µ* act on the spinor field to the left and right respectively. Then (117) becomes

$$E = e \int d^3 \mathbf{x} \{ \frac{i}{2m} \bar{\psi}\_{p'} (\mathbf{x}) (\partial\_{\mu} - \stackrel{\leftarrow}{\partial}) \psi\_p [1 - \frac{\mu}{3\pi} \frac{1}{m^2} (\ln(\frac{m}{\mu}) - \frac{3}{8} - \frac{1}{5})] A\_{ext}^{\mu}$$

$$-\left(1+\frac{\alpha}{2\pi}\right)\frac{1}{2m}\bar{\psi}\_p(\mathbf{x})\sigma\_{\mu\nu}\psi\_p(\mathbf{x})\partial^\nu A\_{ext}^\mu \{\}.\tag{118}$$

The first term contains the convection current of the electron which interacts with the potential. In the special case of a purely magnetic field the second part can be identified as the dipole energy. By introducing the electromagnetic field strength tensor *Fµν* = *∂µA<sup>ν</sup>* − *∂νA<sup>µ</sup>* and using the antisymmetry of *σµν* = *<sup>i</sup>* <sup>2</sup> [*γµ*, *γν*], the second part is

$$\mathcal{W}\_{mag} = e(1 + \frac{\alpha}{2\pi})\frac{1}{4m} \int d^3x \bar{\psi}(\mathbf{x}) \sigma\_{\mu\nu} \psi(\mathbf{x}) F^{\mu\nu}.\tag{119}$$

When *<sup>F</sup>µν* represents a pure magnetic field, *<sup>F</sup>*<sup>12</sup> = −*B*3, *<sup>σ</sup>*<sup>12</sup> = <sup>Σ</sup><sup>3</sup> with cyclic permutations and the interaction energy becomes

$$\mathcal{W}\_{\text{mag}} = -\frac{e}{4m}(1 + \frac{\mathfrak{a}}{2\pi})2 \int d^3 x \,\bar{\psi}(\mathbf{x}) \vec{\Sigma} \psi(\mathbf{x}) \cdot \vec{B} = -\langle \vec{\mu} \rangle \cdot \vec{B}.\tag{120}$$

The magnetic moment is given by

20 Quantum Mechanics

electromagnetic field *Aext*

*E* = 

given by

of *q*2,

*<sup>W</sup>*<sup>22</sup> <sup>=</sup> <sup>1</sup> 2 *dxdx*′

*µ*

*ext* = *e* 

interest here. The renormalized polarization function is

Γ*R <sup>µ</sup>* (*p*′

*<sup>F</sup>*1(*q*2) = *<sup>α</sup>*

*<sup>d</sup>*3*<sup>x</sup> <sup>ψ</sup>*¯ *<sup>p</sup>*′ {*γµ*[<sup>1</sup> +

Substituting all of these factors and *Dµν*

*E* = *e* 

The regularized vertex function is

<sup>Π</sup>*R*(*q*2) = <sup>−</sup>*e*<sup>2</sup>

particle. In the limit, *q*<sup>2</sup> → 0, these functions can be calculated to be

3*π*

*q*2 *<sup>m</sup>*<sup>2</sup> (ln(

*α* 3*π*

*q*2 *<sup>m</sup>*<sup>2</sup> (ln(

*π q*2 *<sup>m</sup>*<sup>2</sup> ( <sup>1</sup> <sup>15</sup> <sup>+</sup>

, *<sup>p</sup>*) = *γµF*1(*q*2) + *<sup>i</sup>*

The functions *F*1(*q*2) and *F*2(*q*2) are called form factors. The electron gets an apparent internal structure by the interaction with the virtual radiation field which alters it from a pure Dirac

> *m <sup>µ</sup>* ) <sup>−</sup> <sup>3</sup>

Note *µ* appears in (115) as an elementary attempt to regularize a photon propagator in one of the terms and does not interfere further with the application at this level and *α* is the fine

<sup>8</sup> <sup>−</sup> <sup>1</sup> 5 ) + *<sup>α</sup>* 2*π i* 2*m*

*m <sup>µ</sup>* ) <sup>−</sup> <sup>3</sup> 8

*<sup>d</sup>*3*x jµA<sup>µ</sup>*

*<sup>ψ</sup>*(*x*)*γ*0*eqγA*(*x*)*G*+(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*′

As a brief introduction to how this formalism can lead to important physical results, let us look at a specific term like *W*21, the interaction energy of an electron with a static

These terms include the bare electron-photon term, the electron-photon correction terms and then the photon vacuum-polarization correction term, *R* means a renormalized quantity and *γµ* denote Dirac matrices. The self-energy correction is left out, because for free particles, it contributes only to charge and mass renormalization. The polarization tensor Π*µν*(*q*2) is

where Π(*q*2) is the polarization function. A simple result is obtained in the limit of low momentum transfer, *q*<sup>2</sup> → 0, which is also of special physical significance and the case of

> 1 140

2*m*

*q*2

), *<sup>F</sup>*2(*q*2) = *<sup>α</sup>*

*<sup>F</sup>* (*q*2) = <sup>−</sup>4*πgµν*/*q*<sup>2</sup> into (111) yields for small values

*<sup>µ</sup>* (*p*′

, *<sup>p</sup>*) + *<sup>i</sup>* 4*π* Π*R*

<sup>Π</sup>*µν*(*q*2)=(*q*2*gµν* − *<sup>q</sup>µqν*)Π(*q*2), (112)

*d*3*xψ*¯ *<sup>p</sup>*′(*γµ* + Γ*<sup>R</sup>*

)*eqγA*(*x*′

)*ψ*(*x*′ ).

*µνiDνσγσ*)*ψpA<sup>µ</sup>*

*<sup>m</sup>*<sup>2</sup> <sup>+</sup> ···). (113)

*σµνqνF*2(*q*2). (114)

<sup>2</sup>*<sup>π</sup>* . (115)

*ext*. (116)

*σµνqν*}*ψpA<sup>µ</sup>*

*ext*. (111)

$$
\langle \vec{\mu} \rangle = \frac{e\hbar}{2mc} (1 + \frac{\alpha}{2\pi}) 2 \langle \vec{S} \rangle = g\mu\_B \langle \vec{S} \rangle. \tag{121}
$$

The magnetic moment is thus proportional to the spin expectation value of the electron. The proportionality factor is the so called *g*-factor

$$\mathbf{g} = \mathbf{2}(1 + \frac{\boldsymbol{\mathfrak{a}}}{2\pi}) = \mathbf{2}(1 + 0.00116141). \tag{122}$$

The first point to note is that the value of the *g*-factor obtained including quantum mechanics differs from the classical value of 2. The result in (122) was first calculated by Schwinger and it has been measured to remarkable accuracy many times. A modern experimental value for the *g*-factor is

$$g\_{\text{exp}} = 2(1 + 0.001159652193),\tag{123}$$

and only the last digit is uncertain.

Of course Schwinger's calculation has been carried out to much further accuracy and is the subject of continuing work. At around order *α*4, further corrections must be included, such as such effects as virtual hadron creation. The pure-QED contributions are represented by coefficients *Ci* as a power series in powers of *α*/*π*, which acts as a natural expansion parameter for the calculation

$$g\_{tho} = 2[1 + \mathbb{C}\_1(\frac{\mathfrak{a}}{\pi}) + \mathbb{C}\_2(\frac{\mathfrak{a}}{\pi})^2 + \mathbb{C}\_3(\frac{\mathfrak{a}}{\pi})^3 + \cdots]. \tag{124}$$

[8] W. K. Burton, Equivalence of the Lagrangian formulations of quantum field theory due

The Schwinger Action Principle and Its Applications to Quantum Mechanics

http://dx.doi.org/10.5772/53472

181

[9] P. Bracken, Quantum Mechanics in Terms of an Action Principle, Can. J. Phys. 75,

[10] K. Fujikawa, Path integral measure for gauge theories with fermions, Phys. Rev. D 21,

to Feynman and Schwinger, Nuovo Cimento, 1, 355-357 (1955).

261-271 (1991).

2848-2858 (1980).

Not all of the assumptions made in classical physics apply in quantum physics. In particular, the assumption that it is possible, at least in principle to perform a measurement on a given system in a way in which the interaction between the measured and measuring device can be made as small as desired. In the absence of concepts which follow from observation, principles such as the action principle discussed here are extremely important in providing a direction in which to proceed to formulate a picture of reality which is valid at the microscopic level, given that many assumptions at the normal level of perception no longer apply. These ideas such as the action principle touched on here have led to wide ranging conclusions about the quantum world and resulted in a way to produce useful tools for calculation and results such as transition amplitudes, interaction energies and the result concerning the *g* factor given here.

### **Author details**

Paul Bracken

<sup>⋆</sup> Address all correspondence to: bracken@panam.edu

Department of Mathematics, University of Texas, Edinburg, TX, USA

#### **References**


[8] W. K. Burton, Equivalence of the Lagrangian formulations of quantum field theory due to Feynman and Schwinger, Nuovo Cimento, 1, 355-357 (1955).

22 Quantum Mechanics

and only the last digit is uncertain.

parameter for the calculation

concerning the *g* factor given here.

<sup>⋆</sup> Address all correspondence to: bracken@panam.edu

Department of Mathematics, University of Texas, Edinburg, TX, USA

**Author details**

Paul Bracken

**References**

Mineola, NY, 2003.

Press, Cambridge, 2007.

and Co, Evanston, 1961.

*gtheo* = 2[1 + *C*1(

Of course Schwinger's calculation has been carried out to much further accuracy and is the subject of continuing work. At around order *α*4, further corrections must be included, such as such effects as virtual hadron creation. The pure-QED contributions are represented by coefficients *Ci* as a power series in powers of *α*/*π*, which acts as a natural expansion

> *α <sup>π</sup>* )<sup>2</sup> <sup>+</sup> *<sup>C</sup>*3(

Not all of the assumptions made in classical physics apply in quantum physics. In particular, the assumption that it is possible, at least in principle to perform a measurement on a given system in a way in which the interaction between the measured and measuring device can be made as small as desired. In the absence of concepts which follow from observation, principles such as the action principle discussed here are extremely important in providing a direction in which to proceed to formulate a picture of reality which is valid at the microscopic level, given that many assumptions at the normal level of perception no longer apply. These ideas such as the action principle touched on here have led to wide ranging conclusions about the quantum world and resulted in a way to produce useful tools for calculation and results such as transition amplitudes, interaction energies and the result

[1] A. L. Fetter and J. D. Walecka, Theoretical Mechanics of particles and Continua, Dover,

[2] J. Schwinger, Quantum Mechanics, Springer Verlag, Berlin Heidelberg, 2001.

[3] J. Schwinger, Selected Papers on Quantum Electrodynamics, Dover, NY, 1958.

[4] J. Schwinger, The theory of quantized fields. I, Phys. Rev. 82, 914-927 (1951).

[5] J. Schwinger, Particles Sources and Fields, Vols I-III, Addison-Wesley, USA, 1981.

[6] D. Toms, The Schwinger Action Principle and Effective Action, Cambridge University

[7] S. S. Schweber, An Introduction to Relativistic Quantum Field Theory, Row, Peterson

*α*

*<sup>π</sup>* )<sup>3</sup> <sup>+</sup> ··· ]. (124)

*α <sup>π</sup>* ) + *<sup>C</sup>*2(


**Chapter 9**

<sup>→</sup> )*δV* is adopted as a

**Generalized Path Integral Technique: Nanoparticles**

**Incident on a Slit Grating, Matter Wave Interference**

One of the crises of contemporary mathematics belongs in part to the subject of the infinite and infinitesimals [1]. It originates from the barest necessity to develop a rigorous language for description of observable physical phenomena. It was a time when foundations of inte‐ gral and differential calculi were developed. A theoretical foundation for facilitation of un‐ derstanding of classical mechanics is provided by the concepts of absolute time and space originally formulated by Sir Isaac Newton [2]. Space is distinct from body. And time passes uniformly without regard to whether anything happens in the world. For this reason New‐ ton spoke of absolute space and absolute time as of a "container" for all possible objects and events. The space-time container is absolutely empty until prescribed metric and a reference frame are introduced. Infinitesimals are main tools of differential calculus [3, 4] within chos‐

Infinitesimal increment being a cornerstone of theoretical physics has one receptee default belief, that increment *δV* tending to zero contains a lot of events to be under consideration.

smooth differentiable function with respect to its argument. From experience we know that for reproducing the probability one needs to accumulate enormous amount of events occur‐ ring within this volume. On the other hand we know, that as *δV* tends to zero we lose infor‐ mation about amount of the events. What is more, the information becomes uncertain. It means the infinitesimal increment being applied in physics faces with a conflict of depth of understanding physical processes on such minuscule scales. This trouble is avoided in quan‐ tum mechanics by proclamation that infinitesimal increments are operators, whereas ob‐

> © 2013 Sbitnev; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Sbitnev; licensee InTech. This is a paper 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.

distribution, and reproduction in any medium, provided the original work is properly cited.

Probability of detection of a particle within this infinitesimal volume *ρ*(*r*

Valeriy I. Sbitnev

**1. Introduction**

en reference frames.

servables are averaged on an ensemble.

http://dx.doi.org/10.5772/53471

Additional information is available at the end of the chapter
