arXiv:quant-ph/9812015v1 7 Dec 1998 H rcesaddsusterrltosi oteeultm P equal-time commutators. the non-equal-time constru to and explicit relationship is equal- an their give this discuss we that and article seems brackets this It In a classical [2]. constructed. its specialists explicitly where reference are any – know e brackets not in do Howev discussed we is [1], commutators mechanics. theory non-equal-time quantum the br in to Poisson commutators ization the equal-time between the correspondence and well-known a is There in ftetime the of tions ontdpn napriua yais ..aHmloin W Hamiltonian. a i.e. variables dynamics, The brackets. particular Poisson a equal-time on actually depend not do eitouetenotation the introduce we prtr nteSchr¨odinger picture. the in operators oethat Note nparticular, In ( ,p x, e sfis ics ytmwt n ereo reo,descr freedom, of degree one with system a discuss first us Let e sdsustetm eedneo h aoia variable canonical the of dependence time the discuss us Let h qa-adnneultm omttr,i discussed. is Poi commutators, equal-time non-equal-time and the equal- to the relationship Their brackets. Poisson .Teodnr oso rcesaedfie by defined are brackets Poisson ordinary The ). h tnaddfiiino h oso rcesi generaliz is brackets Poisson the of definition standard The x hoeia hsc iiin ujrBoˇskovi´c Rudjer Division, Institut Physics Theoretical and t p n h nta conditions initial the and o-qa-iePisnbrackets Poisson Non-equal-time n()ad()hv oepii iedpnec.TePoisson The time-dependence. explicit no have (2) and (1) in ...11,H-00 arb Croatia Zagreb, HR-10001 1016, P.O.B. { A { ( ,p x, ,p x, } ) x B , 1 = G eray1 2008 1, February [email protected] ( roeNikoli´cHrvoje ( t = ) ,p x, , Abstract ) x } { ( ,x x, ,x t, = x 1 ( τ ∂A ∂x ( } ), τ = ) p , p ∂B ∂p x ( { τ ( ,p p, τ and ,where ), − )) to ftenneultm Poisson non-equal-time the of ction } ∂A , ∂p p 0 = isnbakt,a ela othe to as well as brackets, oisson r lsia nlg fquantum of analogs classical are eytxbo nqatmfield quantum on textbook very ato ai nweg among knowledge tacit of part snbakt,a ela to as well as brackets, sson ao o-qa-iePoisson non-equal-time – nalog ∂B ∂x . τ cesi lsia mechanics classical in ackets dt h non-equal-time the to ed .Te r eeal func- generally are They s. . ssome is hl e htteeare these that see shall e r lhuhtegeneral- the although er, bdb Hamiltonian a by ibed fixed e, ntn.Thus instant. brackets (1) (2) (3) and similarly for pG(t), where the label G indicates that this is the general solution of the equations of motion. If the equations of motion are linear, then xG(t) and pG(t) are linear in x(τ) and p(τ). We define the derivative

t +ε t +ε ∂F (x(ti)) j δF (x(ti)) j ′ ′ ≡ lim ds = lim ds F (x(ti)) δ(ti − s)= δti,tj F (x(ti)) (4) ε→0+ − ε→0+ − ∂x(tj ) Ztj ε δx(s) Ztj ε and the non-equal-time Poisson bracket ∂A ∂B ∂A ∂B {A(x (t),p (t)), B(x (t′),p (t′))} = − . (5) G G G G τ ∂x(τ) ∂p(τ) ∂p(τ) ∂x(τ)

The right-hand side of (5) can also be calculated if xG(t) is replaced by x(t) or pG(t) by p(t). For example,

′ {x(t),p(t )}τ = δt,t′ δt,τ , ′ ′ {x(t), x(t )}τ = {p(t),p(t )}τ =0 . (6)

The ordinary Poisson bracket (1) can be obtained from (5) using the relation

{A(xG(τ),pG(τ)), B(xG(τ),pG(τ))}τ = {A(x, p), B(x, p)} (7)

and the identifications x(τ) ≡ x and p(τ) ≡ p. The transition from the non-equal-time Poisson brackets in classical mechanics to the non-equal-time commutators in the Heisenberg representation of quantum physics is given by

′ ′ ′ ′ {A(xG(t),pG(t)), B(xG(t ),pG(t ))}τ=0 −→ −i[A(ˆx(t), pˆ(t)), B(ˆx(t ), pˆ(t ))] . (8)

If A and B are linear functions and if the equations of motion are linear, then relation (8) is an equality. In a more general case, this does not need to be an equality because of the ordering ambiguities of quantum operators. Let us illustrate all this on a simple example. We consider the Hamiltonian

p2 x2 H(x, p)= + . (9) 2 2 The general solution of the corresponding equations of motion is

−it ∗ it xG(t)= ae + a e , (10)

which we write in the form

xG(t)= x(τ) cos(t − τ)+ p(τ) sin(t − τ) . (11)

(There is a simple relationship between the (a, a∗) and (x(τ),p(τ)) coefficients.) The corre- sponding canonical momentum is

pG(t) =x ˙ G(t)= −x(τ) sin(t − τ)+ p(τ) cos(t − τ) . (12)

2 From (5) we obtain

′ ′ ′ {xG(t),pG(t )}τ = cos(t − τ) cos(t − τ) + sin(t − τ) sin(t − τ) (13) and {xG(t),pG(t)}τ =1 , (14) as a special case of (13). On the other hand, in quantum mechanics we work with the opera- tors in the Heisenberg picturex ˆ(t),p ˆ(t) and the corresponding operators in the Schr¨odinger picturex ˆ =x ˆ(0),p ˆ =p ˆ(0), which satisfy

xˆ(t)=ˆx cos t +ˆp sin t , pˆ(t)= −xˆ sin t +ˆp cos t . (15)

From the equal-time commutation relations

[ˆx, pˆ]= i , [ˆx, xˆ]=[ˆp, pˆ]=0 (16) we find the non-equal-time commutation relation

[ˆx(t), pˆ(t′)] = i(cos t cos t′ + sin t sin t′) . (17)

The equality (8) is obtained by putting τ = 0 in (13). The generalization of the non-equal-time Poisson brackets to a discrete set of degrees of freedom is trivial. Let us shortly discuss the generalization to field theory. The ordinary (i.e., equal-time) Poisson bracket is

δA δB δA δB {A(φ(x), π(x)), B(φ(x′), π(x′))} = d3y − . (18) Z " δφ(y) δπ(y) δπ(y) δφ(y)# In particular,

{φ(x), π(x′)} = δ3(x − x′) , {φ(x),φ(x′)} = {π(x), π(x′)} =0 . (19)

Now we introduce the space-time point x =(x0, x). The generalization of (4) is

+ ∂F (φ(x)) ∂F (φ(x0, x)) y0 ε δF (φ(x0, x)) = ≡ lim ds ε→0+ − ∂φ(y) ∂φ(y0, y) Zy0 ε δφ(s, y) y +ε 0 ′ 3 = lim ds F (φ(x0, x)) δ (x − y)δ(x0 − s) ε→0+ − Zy0 ε 3 ′ = δ (x − y)δx0,y0 F (φ(x0, x)) (20) and the generalization of (5) is

′ ′ 3 ∂A ∂B ∂A ∂B {A(φG(x), πG(x)), B(φG(x ), πG(x ))}τ = d y − . Z "∂φ(τ, y) ∂π(τ, y) ∂π(τ, y) ∂φ(τ, y)# (21)

3 As in the case of one degree of freedom, we have the correspondence between classical and quantum field theory in the form

′ ′ ′ ′ {A(φG(x), πG(x)), B(φG(x ), πG(x ))}τ=0 −→ −i[A(φˆ(x), πˆ(x)), B(φˆ(x ), πˆ(x ))] , (22) which is an equality for a linear case. In this article we have constructed the non-equal-time Poisson brackets by a generaliza- tion of the standard definition of the Poisson brackets. It is interesting to note that in [3] the non-equal-time Poisson brackets for field theory are constructed in a completely different way, directly from Lagrangians in a manifestly covariant way. However, this construction does not coincide with the conventional construction (such as ours), because the antisymmetry of the Poisson brackets is not generally provided in the approach of [3]. However, this construction still does coincide with the conventional approach for a large class of Lagrangians.

Acknowledgement

This work was supported by the Ministry of Science and Technology of the Republic of Croatia under Contract No. 00980102.

References

[1] See, for example, J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields (McGraw- Hill, New York, 1965).

[2] D. Marolf, private communication.

[3] R. E. Peierls, “The commutation laws of relativistic field theory,” Proc. Roy. Soc. (Lon- don) 214 (A 1117), 143-157 (1952).

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