Symmetries in Discrete-Time Mechanics

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Annals of Physics � PH5617

annals of physics 252, 357�361 (1996)

Symmetries in Discrete-Time Mechanics

M. Khorrami*

Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 5531, Tehran 19395, Iran;Department of Physics, Tehran University, North Kargar Ave., Tehran, Iran; and

Institute for Advanced Studies in Basic Sciences,P.O. Box 45195-159, Gava Zang, Zanjan, Iran

Received February 2, 1996

Based on a general formulation for discrete-time quantum mechanics, introduced byM. Khorrami (Annals Phys. 224 (1995), 101), symmetries in discrete-time quantum mechanicsare investigated. It is shown that any classical continuous symmetry leads to a conserved quan-tity in classical mechanics, as well as quantum mechanics. The transformed wave function,however, has the correct evolution if and only if the symmetry is nonanomalous. � 1996

Academic Press, Inc.

INTRODUCTION

Discrete-space-time physics is an old tradition originated in solid-state physics.This has been the starting point for lattice physics. On the other hand, continuous-space-time field theory has still some problems, not only from the mathematical butalso from the computational point of view. The first kind of problem is, essentially,the lack of an exact definition for functional integration, and the problem ofultraviolet divergences in interacting field theories ([2], for example). The secondkind of problem is related to the fact that most of the numerical results of fieldtheory are in fact perturbative results. A promising answer to this kind of problemseems to be lattice field theories, specially lattice gauge theories [3]. These theories,however, are interesting not only as an approximation for continuous-space-timetheories, but also as independent models [4�8].

Another problem, very much related to the above problems, is the lack of aconsistent theory of quantum gravity. In the context of quantum gravity, therearises a natural scale for space-time, the Planck scale, and it seems plausible thatsomething new should happen at this scale. In string theories, this scale is the size,or the tension, of the string [9]. One can also use this scale as the size of a possiblespace-time lattice. In fact, there is no true reason for the continuous of space-time:all of the measurements of space-time (direct or indirect) have a certain resolution,which is very much larger than the Planck scale. On the other hand, there aretheories which force the time to be discrete ([10, 11], for example).

article no. 0136

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Copyright � 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

* E-mail: mamwad�rose.ipm.ac.ir.


Also, there have been attempts to discretize the time in quantum mechanics([12, 1], for example). In a previous paper [1], a general formulation of discrete-timequantum mechanics, based on the action principle, or the Feynman path-integralformalism, was introduced; and the unitary criteria and the equation of motionwere investigated.

This paper follows Ref. [1]. Here the symmetries in discrete-time mechanics areinvestigated. In Section I, the notion of symmetry in discrete-time classicalmechanics and a modification of Noether's theorem are introduced. In Section II,these concepts are generalized to quantum mechanics. It is then shown that anyclassical symmetry leads to a conservation law in quantum mechanics. Finally,Section III is devoted to the concept of anomaly. It is shown that the transformedwave function is a solution of the equation of motion, if and only if the symmetryis not anomalous.

I. SYMMETRY IN DISCRETE-TIME CLASSICAL MECHANICS

In discrete-time classical mechanics, the equation of motion is obtained (as in thecontinuous-time case) by extremizing the action [1]. As in the case of continuous-time classical mechanics, however, there are equivalent actions, that is, there areactions which differ from each other by only a boundary term. The action S ,,

S,(x, y)=S(x, y)+,(x)&,( y), (I.1)

is equivalent to S. Here S and S, are unit-time-interval actions [1], and forsimplicity we have dropped the (possible) explicit time dependence of the action. Itis easy to see that the equation of motion,

��xn

[S(xn+1, xn)+S(xn , xn&1)]=0, (I.2)

is the same for S and S,.A symmetry in classical mechanics is defined as a transformation which leaves the

action invariant or, to be more general, changes the action to an action equivalentto the previous one:

S(x~ , y~ )=S(x, y)+,(x)&,( y). (I.3)

Here too, the transformation x � x~ , as well as the action itself and the function ,,can explicitly depend on time. It is an easy task to generalize the conceptsintroduced here to the time dependent case. But to make everything simple andclear, the explicit time dependence is dropped.

Now, suppose that the symmetry transformation has an infinitesimal form, thatis, the transformation is not discrete. Writing

x~ =x+=G, ,==f (I.4)

358 M. KHORRAMI


it is seen that

G(x) }�S�x

+G( y) }�S�y

= f (x)& f ( y), (I.5)

or

G(xn) }�S(xn , xn&1)

�xn& f (xn)=&G(xn&1) }

�S(xn , xn&1)�xn&1

& f (xn&1). (I.6)

Using the equation of motion (I.2), one can define

pn : =�S(xn , xn&1)

�xn=&

�S(xn+1, xn)�xn

. (I.7)

So (I.6) is a conservation law:

pn } G(xn)& f (xn)=const. (I.8)

This is the discrete-time analogue of the Noether's theorem.

II. SYMMETRY IN DISCRETE-TIME QUANTUM MECHANICS

Consider an action which has the classical symmetry of the previous section, thatis, satisfies (I.6). One can write this equation for xn � x̂A , and xn&1 � x. Here x̂A

is the Heisenberg position operator in unit-step evolution [1],

x̂A : =U� - x̂U� , (II.1)

where U� is the unit-step evolution operator [1]. The equation (I.6) then becomes

G(x̂A) }�S(x̂A , x)

�x̂A +G(x) }�S(x̂A , x)

�x= f (x̂A)& f (x). (II.2)

There is no ambiguity in this equation, because x̂ A commutes with the c number x.One can now multiply this equation by |x), and arrive at

G(x̂A) b� }�S(x̂A , x̂)

�x̂A b� + b��S(x̂A , x̂)

�x̂b� } G(x̂)= f (x̂A)& f (x̂), (II.3)

where in the left-hand side of this equation, the time ordering convention of Ref. 1has been used. Using the quantum equations of motion [1],

b��S(x̂, x̂a)

�x̂b� = p̂, b�

�S(x̂A , x̂)�x̂

b� =&p̂, (II.4)

359SYMMETRIES IN DISCRETE-TIME MECHANICS


(II.3) becomes

G(x̂A) } p̂A& p̂ } G(x̂)= f (x̂A)& f (x̂). (II.5)

This is still not a conservation law. But taking the Hermitian conjugate of (II.5),and adding the result to (II.5), the desired conservation equation is obtained:

12 [ p̂ } G(x̂)+G(x̂) } p̂]& f (x̂)=const. (II.6)

Note that the left-hand side is Hermitian and is, indeed, a physical quantity. In fact,it is a specially symmetrized form of the classical conserved quantity.

III. ANOMALY

A classical symmetry is said to be anomalous, if the symmetry transformationleaves the action invariant, but changes the measure of the path integral. In theprevious section we saw that in discrete-time quantum mechanics, any classicalsymmetry leads to a conservation law. There is, however, a difference betweenanomalous and non anomalous symmetries. To see this, consider the evolutionequation of the wave function [1]:

�n+1(x)=| dy U(x, y) �n( y)

=| dy e (i��) S(x, y) �n( y). (III.1)

This leads to

�n+1(x~ )=| dy e (i��) S(x~ , y)�n( y~ )

=| dy~ e (i��) S(x~ , y~ )�n( y~ )

=| dy~ e (i��) S(x, y)+(i��) ,(x)&(i��) ,( y)�n( y~ ),

or

e&(i��) ,(x) �n+1(x~ )=| dy~ e (i��) S(x, y)[e&(i��) ,( y) �n( y~ )]. (III.2)

Now define

�� (x) :=e&(i��) ,(x) �(x~ ). (III.3)

360 M. KHORRAMI


Equation (III.2) implies that �� satisfies the evolution equation (III.1), if and onlyif

dy~ =dy, or det \�y~�y+=1, (III.4)

that is, if and only if the integration measure is not changed by the transformation.This means that the transformed wave function has the correct evolution, if andonly if the symmetry is not anomalous.

Now suppose that the symmetry transformation is a continuous one (which leadsto a conserved quantity). (III.4) is then equivalent to

{ } G=0, (III.5)

which means that

p̂ } g(x̂)=G(x̂) } p̂. (III.6)

In this case, the conserved quantity is exactly the classical one (without need of anysymmetrization).

To conclude, it was shown that, in the context of discrete-time mechanics, anyclassical continuous symmetry leads to a conservation law, in classical, as well as,quantum mechanics. The transformed wave function, however, does not evolvecorrectly unless the symmetry is not anomalous.

REFERENCES

1. M. Khorrami, Ann. Phys. 224 (1995), 101.2. J. Collins, ``Renormalisation,'' Cambridge Univ. Press, Cambridge, UK, 1984.3. C. Rebbi (Ed.), ``Lattice Gauge Theories and Monte Carlo Simulations,'' Chap. 2, World Scientific,

Singapore.4. F. J. Wegner, J. Math. Phys. 12 (1971), 2559.5. K. G. Wilson, Phys. Rev. D 10 (1974), 2445.6. J. B. Kogut, Rev. Mod. Phys. 51 (1979), 659.7. M. Khorrami, Int. J. Theor. Phys. 33 (1994), 2297.8. M. Khorrami, Int. J. Theor. Phys. 35 (1996), 557.9. M. B. Green, J. H. Schwartz, and E. Witten, ``Superstring Theory,'' Cambridge Univ. Press,

Cambridge, UK, 1987.10. G. 't Hooft, Class. Quantum Grav. 10 (1993), 1023.11. A. P. Balachandran and L. Chandar, Discrete time quantum physics, SU-4240-579.12. R. Friedberg and T. D. Lee, Nucl. Phys. B 225 (1983), 1.

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Symmetries in Discrete-Time Mechanics