Time dependent transformations in deformation quantizationNuno Costa Dias and João Nuno Prata Citation: Journal of Mathematical Physics 45, 887 (2004); doi: 10.1063/1.1641152 View online: http://dx.doi.org/10.1063/1.1641152 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/45/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Strict deformation quantization for a particle in a magnetic field J. Math. Phys. 46, 052105 (2005); 10.1063/1.1887922 The Bargmann transform and canonical transformations J. Math. Phys. 43, 2249 (2002); 10.1063/1.1468254 Theory of measurement and second quantization AIP Conf. Proc. 461, 91 (1999); 10.1063/1.57891 On time-dependent quasi-exactly solvable problems AIP Conf. Proc. 453, 257 (1998); 10.1063/1.57099 Strict quantization of coadjoint orbits J. Math. Phys. 39, 6372 (1998); 10.1063/1.532644

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Time dependent transformations in deformationquantization

Nuno Costa Diasa) and Joao Nuno Pratab)

Departamento de Matema´tica, Universidade Luso´fona de Humanidades e Tecnologias,Av. Campo Grande, 376, 1749-024 Lisboa, Portugal

~Received 20 August 2003; accepted 10 November 2003!

We study the action of time dependent canonical and coordinate transformations inphase space quantum mechanics. We extend the covariant formulation of the theoryby providing a formalism that is fully invariant under both standard and timedependent coordinate transformations. This result considerably enlarges the set ofpossible phase space representations of quantum mechanics and makes it possibleto construct a causal representation for the distributional sector of Wigner quantummechanics. ©2004 American Institute of Physics.@DOI: 10.1063/1.1641152#

I. INTRODUCTION

The phase space formulation of quantum mechanics was originally introduced by Weyl1 andWigner2 and further developed by Moyal.3 The theory lives on the classical phase space and itskey algebraic structures~the star-product and the Moyal bracket! are both\-deformations of thestandard algebraic structures of classical mechanics.4–11 Because of this its mathematical formal-ism is remarkably similar to that of classical statistical mechanics, a property that has beenperceived by many as a conceptual and technical advantage when addressing a wide range ofspecific problems.11–17 This relative success, together with the fact that the deformed algebraicstructures play a key part in some current developments in M-theory,18–21 led to an intenseresearch on applications of the deformation quantization approach as well as on the further devel-opment of its mathematical structure.

The Wigner theory uses the symmetric ordering prescription~the Weyl order! to find a par-ticular phase space representation of quantum mechanics. Different representations provide differ-ent points of view and may suggest new solutions for both technical and conceptual problems.They may even suggest new interpretations for the entire quantum theory, as in the case of the deBroglie Bohm formulation. For its importance, the topic of finding new, more general phase spacerepresentations of quantum mechanics has been studied in depth. Cohen22 introduced a generali-zation of the Weyl map, providing in an unified fashion all phase space representations thatcorrespond to different ordering prescriptions of operator quantum mechanics. The resultingtheory of quasidistributions includes as particular cases the Wigner and the de Broglie Bohm23,24

formulations. Vey25 and several others,8,11,26–30developed the covariant generalization of Wigner’stheory. The new formulation renders phase space quantum mechanics fully invariant under theaction of phase space coordinate transformations. By doing so it provides a general formula for the\-deformations of the Poisson bracket and makes it possible to apply deformation quantizationmethods to a larger set of dynamical systems including those displaying the structure of a curvedphase space manifold.

The aim of this article is to extend the covariant formulation further by admitting the possi-bility of time dependent coordinate transformations. We will study the action of these transforma-tions in phase space quantum mechanics and rewrite the covariant Wigner theory in a fullyinvariant form under their action. We derive the time dependent covariant form of the starproduct,

a!Electronic mail: ncdias@mail.telepac.ptb!Electronic mail: joao.prata@ulusofona.pt

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 45, NUMBER 3 MARCH 2004

8870022-2488/2004/45(3)/887/15/$22.00 © 2004 American Institute of Physics

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Moyal bracket, Moyal dynamical equation and stargenvalue equation as well as the covariantprobabilistic functionals, providing all key ingredients of the time dependent covariant formulationof Wigner quantum mechanics. This result enlarges the set of possible phase space representationsof quantum mechanics and provides a more general formula for the\-deformations of the Poissonbracket, which may now include an explicit time dependence.

The new set of representations provides new possible formulations for a generic quantummechanical problem. In some cases this may considerably simplify the technical resolution of theproblem. In Sec. VI a simple example illustrates how a suitable time dependent representationleads to a far simpler description of the dynamics of the quasidistribution. More relevant is the factthat the new formalism makes it possible to construct a phase space representation of quantummechanics where the quasidistribution displays a classical causal structure, i.e., the Wigner func-tion evolves according to the Liouville equation. A formulation displaying this set of propertiescannot be~easily! accomplished using the standard methods of quantum mechanics~not evenwithin the standard covariant Wigner formalism! and it proves that the de Broglie Bohm theory isnot the unique possible causal formulation of quantum mechanics. In the new causal representa-tion the quantum dynamical behavior is completely removed from the distributional sector of thetheory and is exclusively placed on the observables’ sector. In particular, if the quasidistribution ispositive defined at the initial time, it will remain so for all times. These properties reinforce theformal analogy between phase space quantum mechanics and classical statistical mechanics andmake the causal formulation especially suitable to study the semiclassical limit of quantum me-chanics.

This article is organized as follows: in Sec. II we review the main topics of the covariantformulation of the Wigner theory. In Sec. III we study the action of time dependent canonicaltransformations in standard operator quantum mechanics. Particular attention is devoted to thebehavior of the density matrix. In Sec. IV we derive the time dependent covariant formulation ofWigner quantum mechanics. In Sec. V a particular set of coordinates is used to obtain the causalphase space representation. In Sec. VI a simple example illustrates some of the former results andin Sec. VII we present the conclusions.

II. COVARIANT WIGNER QUANTUM MECHANICS

Let us consider anN dimensional dynamical system. Its classical formulation lives on thephase spaceT* M which, to make it simple, we assume to be flat. A global Darboux chart can thenbe naturally defined onT* M , for which the symplectic structure readsw5dqi∧dpi , where$qi ,pi ,i 51,...,N% is a set of canonical variables.

Upon quantization the set$qi% yields a complete set of commuting observables. Let then

A(qW ,pW ) be a generic operator acting on the physical Hilbert spaceH. The Weyl map

W(q,p)~A!5\NE dNyWe2 ipW •yW K qW 1\

2yW UAUqW 2

\

2yW L , ~1!

where we introduced the vector notationyW[(y1 ,...,yN) and uqW 6\/2yW & are eigenstates ofqW ,provides a Lie algebra isomorphism between the algebraA(H) of linear operators acting on theHilbert spaceH and the algebra of phase space functionsA(T* M ) endowed with a*-product andMoyal bracket@let A,BPA(T* M )]:

A* (q,p)B5Ae~ i\/2!]QkJ(q,p)kl ]W lB, @A,B#M (q,p)

52

\A sinS \

2]Q kJ(q,p)

kl ]W l DB, ~2!

where the derivatives]Q and]W act onA andB, respectively, andJ(q,p)kl is theklth element of the

symplectic matrix in the variables (qW ,pW ):

888 J. Math. Phys., Vol. 45, No. 3, March 2004 N. C. Dias and J. N. Prata

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J(q,p)5S 0N3N 21N3N

1N3N 0N3ND . ~3!

We also introduced the compact notation:Ok5pk ,k51,...,N; Ok5qk2N ,k5N11,...,2N;]/]Ok5]k and sum over repeated indices is understood.

We now consider a second set of fundamental operators (QW ,PW ) unitarily related to (qW ,pW ), i.e.,

qW 5UQW U21 and pW 5UPW U21 where U is some unitary operator. The new operators satisfy theHeisenberg commutation relations, yield a new Weyl mapW(Q,P) and induce a phase spacetransformation (qW ,pW )→(QW ,PW ) acting on a generic observable through the procedure~let U

5W(Q,P)(U)):

A~qW ,pW !5W(q,p)~A!→A8~QW ,PW !5W(Q,P)~A!5U* (Q,P)A~QW ,PW !* (Q,P)U21 . ~4!

The phase space implementation of the unitary transformation preserves the starproduct and theMoyal bracket but, as is well known, it does not act as a coordinate transformation~the exceptionsare the linear transformations!: let qW (QW ,PW )5W(Q,P)(q) andpW (QW ,PW )5W(Q,P)( p) and we find thatin generalA8(QW ,PW )ÞA(qW (QW ,PW ),pW (QW ,PW )). We conclude that the standard Wigner formulation isnon-covariant.

We now introduce the generalized Weyl map.8 Let the transformation (qW ,pW )→(QW ,PW ) be aphase space diffeomorphism defined, in general terms, byqW 5qW (QW ,PW ) and pW 5pW (QW ,PW ). In par-ticular, the transformation of the canonical variables (qW ,pW ) might be given by the unitary trans-formation above, but this is not required. The generalized Weyl map is then defined by8

W(Q,P)(q,p) ~A!5\NE dNxWE dNyW e2 ipW (QW ,PW )•yWd~xW2qW ~QW ,PW !!K xW1

\

2yW UAUxW2

\

2yW L , ~5!

where uxW6\/2yW & are eigenstates ofqW . The new map implements the transformation (qW ,pW )→(QW ,PW ) as a coordinate transformation in quantum phase space; letA8(QW ,PW )5W(Q,P)

(q,p) (A) andA(qW ,pW )5W(q,p)(A) and we haveA8(QW ,PW )5A(qW (QW ,PW ),pW (QW ,PW )), though in general it does notpreserve the functional form of the star-product and Moyal bracket. Instead it yields the moregeneral covariant star-product and Moyal bracket:8,11

A8~QW ,PW !* (Q,P)8 B8~QW ,PW !5A8~QW ,PW !e~ i\/2!¹Q i8J(Q,P)8 i j ¹W j8B8~QW ,PW !,~6!

@A8~QW ,PW !,B8~QW ,PW !#M(Q,P)8 5

2

\A8~QW ,PW !sinS \

2¹Q i8J(Q,P)8 i j ¹W j8DB8~QW ,PW !,

where the covariant derivative is given by~let O8 i5Pi ,i 51,...,N; O8 i5Qi 2N ,i 5N11,...,2N)

¹ i8A85] i8A8, ¹ i8¹ j8A85] i8] j8A82G i j8k]k8A8, ] i85]/]O8 i , i , j ,k51,...,2N, ~7!

and

J(Q,P)8 i j ~QW ,PW !5]O8 i

]Ok

]O8 j

]Ol J(q,p)kl , G jk8

i~QW ,PW !5]O8 i

]Ob

]2Ob

]O8 j]O8k ~8!

are the new symplectic structure and Poisson connection associated to the coordinates (QW ,PW ).Notice that in Eq.~8! we explicitly took into account the phase space flat structure.

When formulated in terms of these structures Wigner mechanics becomes invariant under theaction of general coordinate transformations:

889J. Math. Phys., Vol. 45, No. 3, March 2004 Time dependent transformations

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A8~QW ,PW !* (Q,P)8 B8~QW ,PW !5A~qW ~QW ,PW !,pW ~QW ,PW !!* (q,p)B~qW ~QW ,PW !,pW ~QW ,PW !! ;A,BPA(T* M ) ,~9!

the covariant generalization of the Moyal and stargenvalue equations reading

f W8 5@H8, f W8 #M(Q,P)8 ,

~10!A8* (Q,P)8 ga85ga8* (Q,P)8 A85aga8 ,

where f W8 (QW ,PW ;t)5 (1/(2p\)N) W(Q,P)(q,p) (uc(t)&^c(t)u) is the covariant Wigner function andga8 is

the left- and right-stargenfunction associated to the eigenvaluea.These equations transform covariantly under arbitrary phase space diffeomorphisms yielding,

in any coordinates, identical mathematical solutions and thus identical physical predictions:

P~A8~QW ,PW ;t !5a!5E dNQW E dNPW ~detJ(Q,P)8 i j !21/2d* (Q,P)8 ~A8~QW ,PW !2a! f W8 ~QW ,PW ;t !, ~11!

whered* (Q,P)8 (A82a) is a particular solution of~10!, displaying the following explicit form:9,11

d* (Q,P)8 ~A8~QW ,PW !2a!51

2p E dk e* (Q,P)8ik(A8(QW ,PW )2a)

, ~12!

the *-exponential being given bye* (Q,P)8A8 5(n50

` (1/n!) Vn where V051 and Vn11

5Vn* (Q,P)8 A8.This concludes our review of the main topics of the covariant formulation of Wigner quantum

mechanics. The reader should refer to Refs. 8 and 11 for more detailed presentations of the theory.

III. TIME DEPENDENT CANONICAL TRANSFORMATIONS

The aim of this section is to succinctly review some aspects of time dependent canonical

transformations in standard operator quantum mechanics. LetAW 5AW (qW ,pW ,t) andBW 5BW (qW ,pW ,t) be anew set of fundamental operators@Ai ,Bj #5 i\d i j ,;t. Let T be the generator of the canonicaltransformation,

]

]tAW 5

1

i\@AW ,T# and

]

]tBW 5

1

i\@BW ,T#, ~13!

and to make it simple let us also impose the initial conditionsAW (0)5qW andBW (0)5pW . Then, theunitary transformation reads

AW 5V~ t !qW V21~ t !5AW ~qW ,pW ,t ! and BW 5V~ t !pW V21~ t !5BW ~qW ,pW ,t !, ~14!

where V(t)5exp((i/\)Tt). The former relations can be immediately inverted,qW 5V21(t)AW V(t)

andpW 5V21(t)BW V(t), and lead straightforwardly to the dynamical equation for a generic observ-able:

d

dtF~AW ,BW ,t !5

1

i\@ F,H1T#1

]

]tF~AW ,BW ,t !. ~15!

We now consider the density matrix of the system:r(t)5uc(t)&^c(t)u where uc(t)& is thecorresponding quantum state at the timet. In the q representation we have9

890 J. Math. Phys., Vol. 45, No. 3, March 2004 N. C. Dias and J. N. Prata

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r~ t !5E dNqW 8 dNqW 9^qW 9uc~ t !&^c~ t !uqW 8&uqW 9&^qW 8u

5E dNqW 8 dNqW 9 c~qW 9,t !c* ~qW 8,t !e2 ~ i /\!(qW 92qW 8)•pW D~qW 2qW 8!

5E dNqW 8 dNqW 9 c~qW 9,0!c* ~qW 8,0!e2 ~ i /\!(qW 92qW 8)•pW (qW ,pW ,2t)D~qW ~qW ,pW ,2t !2qW 8!5 r~qW ,pW ,t !,

~16!

whereD(qW 2qW 8)5D(q12q18) ¯ D(qN2qN8 ) and D(qi2qi8)5 (1/2p) *dkeik(qi2qi8). Moreover,

qW (qW ,pW ,t) andpW (qW ,pW ,t) are the Heisenberg time evolutions of the fundamental operatorsqW andpW .The action of the time dependent canonical transformation on the density matrix is now easily

implemented,r(t)5 r(qW (AW ,BW ,t),pW (AW ,BW ,t),t)5V21(t) r(AW ,BW ,t)V(t)5 r8(AW ,BW ,t), from where itfollows that in theA representation

]

]tr8~AW ,BW ,t !5

]V21~ t !

]trV~ t !1V21~ t !S ]

]tr~AW ,BW ,t ! D V~ t !1V21~ t !r

]V~ t !

]t

51

i\@ T,r8#1

1

i\V21@H,r #V5

1

i\@ T1H8,r8#, ~17!

whereH8(AW ,BW ,t)5H(qW (AW ,BW ,t),pW (AW ,BW ,t))5V21(t)H(AW ,BW )V(t).

IV. TIME DEPENDENT TRANSFORMATIONS IN PHASE SPACE QUANTUM MECHANICS

In this section we study the action of time dependent transformations in phase space quantummechanics. We consider an arbitraryN dimensional quantum system with HamiltonianH anddescribed by the wave functionc(t). As we have seen, the original Weyl transformW(q,p) yieldsthe standard Wigner formulation of the system. The time evolution of the Wigner functionf W(qW ,pW ,t)5 (1/(2p\)N) W(q,p)(uc(t)&^c(t)u) is dictated by the standard Moyal equation wherethe Moyal bracket and the starproduct are given by Eq.~2!. We then consider two different phasespace implementations of a time dependent operator transformation.

~1! Unitary time dependent transformations and the map W(A,B) . At the quantum operator

level we introduce the unitary transformation (qW ,pW )→(AW ,BW ) defined by Eqs.~13! and ~14!. The

new variables (AW ,BW ) satisfy the Heisenberg commutation relations and thus a new Weyl mapW(A,B) can be constructed. It displays the standard functional structure given by Eq.~1! and yieldsa starproduct* (A,B) and Moyal bracket@ ,#M (A,B)

also displaying the non-covariant functional formEq. ~2!. The time evolution of the new Wigner function,

f W8 ~AW ,BW ,t !51

~2p\!N W(A,B)~ r8~AW ,BW ,t !!5V21~ t !* (A,B) f W~AW ,BW ,t !* (A,B)V~ t !, ~18!

whereV(t)5W(A,B)(V), reads

]

]tf W8 ~AW ,BW ,t !5@H81T, f W8 #M (A,B)

, ~19!

and is just the (A,B)-Weyl transform of Eq.~17!. Notice that just like in the time independent casethe unitary transformation (qW ,pW )→(AW ,BW ) does not, in general, act as a coordinate transformation:f W8 (AW ,BW ,t)Þ f W(qW (AW ,BW ,t),pW (AW ,BW ,t),t).

891J. Math. Phys., Vol. 45, No. 3, March 2004 Time dependent transformations

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~2! Coordinate transformations and the map W(A,B)(q,p) . We follow the steps of the time inde-

pendent case and introduce a time dependent phase space diffeomorphism (qW ,pW )→(AWW ,BWW ) definedin generic terms byqW 5qW (AWW ,BWW ,t); pW 5pW (AWW ,BWW ,t). This coordinate transformation is not required tobe a symplectomorphism~i.e., to preserve the Poisson bracket! nor to preserve the Moyal bracketbetween the fundamental variables~i.e., to satisfy @qi(AWW ,BWW ,t),pj (AWW ,BWW ,t)#M (A,B)

5d i j and

@qi(AWW ,BWW ,t),qj (AWW ,BWW ,t)#M (A,B)5@pi(AWW ,BWW ,t),pj (AWW ,BWW ,t)#M (A,B)

50 for all i , j 51 ,. . .,N).

We then define the time dependent generalized Weyl transform in the variables (AWW ,BWW ):

W(A,B)(q,p) :A~H!→A~T* M !; F→F8~AWW ,BWW ,t !5W(A,B)

(q,p) ~ F !5W(q,p)~ F !uqW 5qW (AW ,BW ,t)∧pW 5pW (AW ,BW ,t) .~20!

The explicit form ofW(A,B)(q,p) is given by the trivial time dependent generalization of Eq.~5!:

W(A,B)(q,p) ~ F !5\NE dNxWE dNyW e2 ipW (AW ,BW ,t)•yWd~xW2qW ~AWW ,BWW ,t !!K xW1

\

2yW UFUxW2

\

2yW L , ~21!

from which follows the covarianttime dependent* -product and Moyal bracket:

W(A,B)(q,p) ~ FG!5F8~AWW ,BWW ,t !* (A,B)8 G8~AWW ,BWW ,t !5F~qW ~AWW ,BWW ,t !,pW ~AWW ,BWW ,t !!* (q,p)G~qW ~AWW ,BWW ,t !,pW ~AWW ,BWW ,t !!,

~22!

and @F8,G8#M(A,B)8 5 (1/i\) (F8* (A,B)8 G82G8* (A,B)8 F8) whereF8(AWW ,BWW ,t)5W(A,B)

(q,p) (F), F(qW ,pW ,t)

5W(q,p)(F) and likewise forG andG8. The two algebraic structures display the functional formgiven by Eqs.~6!–~8! with the obvious inclusion of an explicit time dependence.

The dynamical structure of the theory displays more significant corrections. Letf W8 (AWW ,BWW ,t)5 (1/(2p\)N) W(A,B)

(q,p) (uc(t)&^c(t)u) be the covariant Wigner function. It satisfiesf W8 (AWW ,BWW ,t)5 f W(qW (AWW ,BWW ,t),pW (AWW ,BWW ,t),t) and thus

]

]tf W8 ~AWW ,BWW ,t !5S ]

]t1

1]

]t2D f W~qW ~AWW ,BWW ,t1!,pW ~AWW ,BWW ,t1!,t2!U

t15t25t

5@H~qW ~AWW ,BWW ,t !,pW ~AWW ,BWW ,t !,t !, f W~qW ~AWW ,BWW ,t !,pW ~AWW ,BWW ,t !,t !#M (q,p)1

] f W

]qW•

]qW

]t1

1] f W

]pW

•

]pW

]t1U

t15t

5@H8~AWW ,BWW ,t !, f W8 ~AWW ,BWW ,t !#M(A,B)8 1S ] f W8

]Ai

]Ai

]qj

1] f W8

]Bi

]Bi

]qjD ]qj

]t

1S ] f W8

]Ai

]Ai

]pj

1] f W8

]Bi

]Bi

]pjD ]pj

]t5@H8~AWW ,BWW ,t !, f W8 ~AWW ,BWW ,t !#M

(A,B)8 1] f W8

]AiS ]Ai

]qj

]qj

]t

1]Ai

]pj

]pj

]tD 1

] f W8

]BiS ]Bi

]qj

]qj

]t1

]Bi

]pj

]pj

]tD 5@H8~AWW ,BWW ,t !, f W8 ~AWW ,BWW ,t !#M

(A,B)8 2] f W8

]AWW

•

]AWW

]t2

] f W8

]BWW•

]BWW

]t, ~23!

where in the last step we used the fact thatAWW5AWW (qW(AWW ,BWW ,t),pW(AWW ,BWW ,t),t) and likewise forBWW .Further, contraction over repeated indices is understood, i.e.,

892 J. Math. Phys., Vol. 45, No. 3, March 2004 N. C. Dias and J. N. Prata

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] f W

]qi

]qi

]t15

] f W

]qW•

]qW

]t15(

i 51

N] f W

]qi

]qi

]t1.

Equation~23! constitutes a generalization of the Moyal covariant equation~10! and renders thedynamics of the Wigner function fully invariant under the action of general time dependent phasespace diffeomorphisms.

Let us then consider several particular cases in more detail:~a! If the transformation (qW ,pW )→(AWW ,BWW ) is time independent, then Eq.~23! reduces to the

standard covariant Moyal equation~10!.~b! If, on the other hand, it is unitary, i.e., ifAWW (qW,pW,t)5V(t)* (q,p)qW* (q,p)V

21(t) andBWW (qW,pW,t)5V(t)* (q,p)pW* (q,p)V

21(t) satisfying]AWW /]t 5@AWW ,T#M (q,p)and ]BWW /]t 5@BWW ,T#M (q,p)

with

initial conditionsAWW (qW,pW,0)5qW andBWW (qW,pW,0)5pW , then in Eq.~23! we have

]AWW

]t5

]

]tAWW ~qW ,pW ,t !U

qW 5qW (AW ,BW ,t)∧pW 5pW (AW ,BW ,t)

5@AWW ~qW ~AWW ,BWW ,t !,pW ~AWW ,BWW ,t !,t !,T~qW ~AWW ,BWW ,t !,pW ~AWW ,BWW ,t !!#M (q,p)

5@AWW ,T8~AWW ,BWW ,t !#M(A,B)8 , ~24!

whereT8(AWW ,BWW ,t)5W(A,B)(q,p) (T). An equivalent result is valid for]BWW /]t. Substituting these results in

Eq. ~23! we get

]

]tf W8 ~AWW ,BWW ,t !5@H8~AWW ,BWW ,t !, f W8 ~AWW ,BWW ,t !#M

(A,B)8 1] f W8

]AWW•@T8,AWW #M

(A,B)8 1] f W8

]BWW•@T8,BWW #M

(A,B)8 .

~25!

~c! Finally, we consider the case where the transformation (qW ,pW )→(AWW ,BWW ) is a symplectomor-phism. LetT be the generator. ThenAWW (qW,pW,t) satisfies]AWW /]t 5$AWW ,T%(q,p)5$AWW ,T% (A,B) and like-wise for BWW . Hence, Eq.~23! reduces to

]

]tf W8 ~AWW ,BWW ,t !5@H8~AWW ,BWW ,t !, f W8 ~AWW ,BWW ,t !#M

(A,B)8 2] f W8

]AWW•$AWW ,T%(A,B)2

] f W8

]BWW•$BWW ,T%(A,B)

5@H8~AWW ,BWW ,t !, f W8 ~AWW ,BWW ,t !#M(A,B)8 1$T, f W8 % (A,B) . ~26!

To finish this section let us study the(A,B)(q,p) -representation of a general stargenfunction. LetF

be a generic operator andF8(AWW ,BWW ,t)5W(A,B)(q,p) (F). The *-genvalue equation in the

(A,B)(q,p) -representation is then written

F8~AWW ,BWW ,t !* (A,B)8 ga8~AWW ,BWW ,t !5ga8~AWW ,BWW ,t !* (A,B)8 F8~AWW ,BWW ,t !5aga8~AWW ,BWW ,t ! ~27!

and displays the solution

ga8~AWW ,BWW ,t !5d* (A,B)8 @F8~AWW ,BWW ,t !2a#51

2p E dke* (A,B)8ik[F8(AW ,BW ,t)2a]

51

2p E dke* (q,p)

ik[F(qW (AW ,BW ,t),pW (AW ,BW ,t),t)2a] , ~28!

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whereF(qW ,pW ,t)5W(q,p)(F). Further, if ga(qW ,pW ,t) is such thatF* (q,p)ga5ga* (q,p)F5aga thenga8(A

W ,BW ,t)5ga(qW (AW ,BW ,t),pW (AW ,BW ,t),t), a result that follows immediately from Eqs.~22! and~28!.Therefore the stargenvalue equation transforms covariantly under the action of general time de-pendent coordinate transformations.

As an illustrative example let us consider the one-dimensional simple caseF5q⇒W(A,B)

(q,p) (q)5q(A,B,t). The solution of the*-genvalue equation~27! is then

ga8~A,B,t !51

2p E dke* (A,B)8ik[q(A,B,t)2a]

51

2p E dke* (q,p)

ik[q(A,B,t)2a]51

2p E dkeik[q(A,B,t)2a]

5d@q~A,B,t !2a#, ~29!

and displays the time evolution

]

]tga8~A,B,t !5

]d

]q@q~A,B,t !2a#

]q

]t~A,B,t !. ~30!

Furthermore, if the transformation (q,p)→(A,B) is symplectic with generatorT, then ]q/]t5$q,2T% (A,B) and

]

]tga8~A,B,t !52$ga8 ,T% (A,B) . ~31!

V. THE CAUSAL REPRESENTATION

As an application of the formalism let us consider a finite dimensional dynamical systemdescribed by a generic HamiltonianH and use the generalized time dependent Weyl map to derivein a systematic way~1! the Schro¨dinger and~2! the Heisenberg phase space pictures and~3! a newphase space representation where the Wigner function displays a fully classical time evolution.

To begin with, we introduce the time dependent unitary transformation generated byT5

2H. A new set of fundamental operators is given byAW 5AW (qW ,pW ,t) andBW 5BW (qW ,pW ,t), solutions of

Eq. ~13! and satisfying the initial conditionsAW (0)5qW andBW (0)5pW . Given the relation betweenHand T they also satisfy

AW 5AW ~qW ,pW ,t !5qW ~qW ,pW ,2t ! and BW 5BW ~qW ,pW ,t !5pW ~qW ,pW ,2t !, ~32!

whereqW (qW ,pW ,t) and pW (qW ,pW ,t) are the Heisenberg time evolution of the fundamental operatorsqW

andpW . From ~32! we define a new set of phase space coordinatesAW 5AW M(qW ,pW ,t)5W(q,p)(A) andBW 5BW M(qW ,pW ,t)5W(q,p)(B) satisfying the Moyal equations,

]AW M

]t5@AW M ,T#M (q,p)

,]BW M

]t5@BW M ,T#M (q,p)

, T5W(q,p)~ T!, ~33!

and the initial conditionsAW M(qW ,pW ,0)5qW and BW M(qW ,pW ,0)5pW . The subscriptM indicates that thefunctions AM and BM obey the Moyal equations. Intuitively, they can be seen as the quantumphase space histories of the system.

A second set of~purely classical! phase space coordinates will also be required:QW ,PW are givenby QW 5QW (AW ,BW ,t) andPW 5PW (AW ,BW ,t) and satisfy the classical Hamiltonian equations,

]QW

]t5$QW ,2T~AW ,BW !%(A,B) ,

]PW

]t5$PW ,2T~AW ,BW !%(A,B) , T~AW ,BW !5W(A,B)~ T!, ~34!

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and the initial conditionsQW (AW ,BW ,0)5AW , PW (AW ,BW ,0)5BW . Notice thatT(AW ,BW ) displays the samefunctional form as T(qW ,pW )5W(q,p)(T). Solving the algebraic equationsQW 5QW (AW ,BW ,t), PW

5PW (AW ,BW ,t) with respect toAW ,BW we getAW 5AW C(QW ,PW ,t) andBW 5BW C(QW ,PW ,t) where the subscriptCindicates thatAW C andBW C are the solutions of the classical Hamilton’s equations,

]AW C

]t5$AW C ,T~QW ,PW !% (Q,P) ,

]BW C

]t5$BW C ,T~QW ,PW !% (Q,P) , ~35!

whereT(QW ,PW )5T(AW C(QW ,PW ,t),BW C(QW ,PW ,t)) displays the same functional form asT(AW ,BW ) ~noticethat T is the generator of the canonical transformation!. Also notice that in generalqW (AW ,BW ,t)ÞQW (AW ,BW ,t) andpW (AW ,BW ,t)ÞPW (AW ,BW ,t) ~the exceptions happen forT quadratic in the phase spacevariables!. While the variables (qW ,pW ) describe thequantumphase space time evolution, the vari-ables (QW ,PW ) describe theclassicalphase space trajectories. The transformation (AW ,BW )→(QW ,PW ) isa phase space symplectomorphism exclusively defined at the classical level, i.e., it is not@andunlike (qW ,pW )→(AW ,BW ) it could not be# inherited from a quantum operator transformation.

Finally, from Eqs.~16! and~32! the density matrixr(t)5uc(t)&^c(t)u admits the expansions

r~ t !5E dNqW 8dNqW 9c~qW 9,0!c* ~qW 8,0!e2~ i /\!(qW 92qW 8)•pW (qW ,pW ,2t)D@qW ~qW ,pW ,2t !2qW 8#5 r~qW ,pW ,t !,

~36!

r~ t !5E dNqW 8dNqW 9c~qW 9,0!c* ~qW 8,0!e2~ i /\!(qW 92qW 8)•BW D~AW 2qW 8!5 r~AW ,BW ,0!.

With these preliminaries settled down we address the derivation of the three phase spacepictures:

~1! Schrodinger picture and the map W(q,p) . From the first expansion for the density matrix~36! we immediately get

f W~qW ,pW ,t !51

~2p\!N W(q,p)@ r~ t !#

51

~2p\!N E dNqW 8dNqW 9c~qW 9,0!c* ~qW 8,0!e* (q,p)~2 i /\!(qW 92qW 8)•pW (qW ,pW ,2t)

* (q,p)

3d* (q,p)@qW ~qW ,pW ,2t !2qW 8# ~37!

and the time evolution of the Wigner function obeys the standard Moyal equation: (]/]t) f W

5@H, f W#M (q,p). We also haveW(q,p)(uqW 0&^qW 0u)5d* (qW 2qW 0)5d(qW 2qW 0) and so (]/]t) d(qW 2qW 0)

50 and likewise forpW , i.e., the dynamics is cast in the Schro¨dinger picture.~2! Heisenberg picture and the map W(A,B) . From the second expansion in~36! we get

1

~2p\!N W(A,B)@ r~ t !#51

~2p\!N W(A,B)@ r~AW ,BW ,0!#

51

~2p\!N E dNqW 8dNqW 9c~qW 9,0!c* ~qW 8,0!e* (A,B)2 ~ i /\!(qW 92qW 8)•BW

* (A,B)

3dN

* (A,B)~AW 2qW 8!. ~38!

Let us check explicit that the previous formula yields the standard~in this case time independent!expression of the Wigner function. The same derivation would also apply to Eq.~37!. We start byconsidering the term evolving starproducts in more detail:

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e* (A,B)2 i /\(qW 92qW 8)•BW

* (A,B)dN

* (A,B)~AWW 2qW 8!5e2 ~ i /\!(qW 92qW 8)•BW * (A,B)dN~AWW 2qW 8!

51

~2p!N (n50

1`1

n!S 2

i\

2D n

e2 ~ i /\!(qW 92qW 8)•BW F2i

\~qW 9

2qW 8!•]

]AWWG nE dNkW eikW•(AW 2qW 8)

51

~2p!Ne2 ~ i /\!(qW 92qW 8)•BW E dNkW (

n50

1`1

n!S 2

i

2kW •~qW 9

2qW 8!D n

eikW•(AW 2qW 8)

5e2 ~ i /\!(qW 92qW 8)•BW1

~2p!N E dNkW eikW•(AW 2qW 81 qW 8/2 2 qW 9/2)

5e2 i /\(qW 92qW 8)•BW dNS AWW 2qW 8

22

qW 9

2D . ~39!

Substituting this expression in~38! we get

1

~2p\!N W(A,B)@ r~ t !#51

~2p\!N E dNqW 8dNqW 9c~qW 9,0!c* ~qW 8,0!2NdN~2AWW 2qW 8

2qW 9!e2 ~2i /\!(AW 2qW 8)•BW 51

~p\!N E dNqW 8c~2AWW

2qW 8,0!c* ~qW 8,0!e2 ~2i /\!(AW 2qW 8)•BW 51

~p\!N E dNyWc~AWW 1yW ,0!c* ~AWW

2yW ,0!e2 ~2i /\!yW•BW 5 f W~AWW ,BWW ,0!, ~40!

where in the last step we madeyW5AWW 2qW 8. We indeed recovered the standard definition of theWigner function. Moreover we see that (]/]t) f W50, as it should and in perfect agreement withEq. ~19! taking into account thatT52H. On the other hand, we also have

gqW 0~AWW ,BWW ,t !5W(A,B)~ uqW 0&^qW 0u!5d

* (A,B)

N @qW ~AWW ,BWW ,t !2qW 0#51

~2p!N E dNkW e* (A,B)

ikW•(qW (AW ,BW ,t)2qW 0) ,

~41!

whereqW 5qW (AWW ,BWW ,t) is the solution with respect toqW of the algebraic equationsAWW5AWW M(qW,pW,t),BWW5BWW M(qW,pW,t) defined in Eq.~33!. It also follows from Eq.~33! that qW (AWW ,BWW ,t) satisfies]qW /]t5@qW ,2T#M (A,B)

. Therefore,

]

]tgqW 0

~AWW ,BWW ,t !5@gqW 0~AWW ,BWW ,t !,H#M (A,B)

, ~42!

and we obtained the phase space Heisenberg picture.~3! Causal picture and the map W(Q,P)

(A,B) . We finally consider the action of the mapW(Q,P)(A,B) on

the second expansion for the density matrix~36!. From Eq. ~40! it follows that ~notice thatW(Q,P)

(A,B) 5W(A,B))

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f W8 ~QW ,PW ,t !51

~2p\!N W(Q,P)(A,B) @ r~ t !#

51

~2p\!N E dNqW 8dNqW 9c~qW 9,0!c* ~qW 8,0!e* 8(Q,P)

2 ~ i /\!(qW 92qW 8)•BW C(QW ,PW ,t)* (Q,P)8 d

* 8(Q,P)N

@AW C~QW ,PW ,t !2qW 8#

51

~2p\!N E dNqW 8dNqW 9c~qW 9,0!c* ~qW 8,0!e* (A,B)2 ~ i /\!(qW 92qW 8)•BW C(QW ,PW ,t)

* (A,B)d* (A,B)N

@AW C~QW ,PW ,t !2qW 8#

5 f W~AW C~QW ,PW ,t !,BW C~QW ,PW ,t !,0!5 f W~QW ~QW ,PW ,2t !,PW ~QW ,PW ,2t !,0!, ~43!

whereQW (QW ,PW ,t)5AW C(QW ,PW ,2t) andPW (QW ,PW ,t)5BW C(QW ,PW ,2t) are the classical time evolution ofthe canonical variables (QW ,PW ) @cf. ~35!#

QW 5$QW ,2T~QW ,PW !%(Q,P) and PW 5$PW ,2T~QW ,PW !%(Q,P) ~44!

associated with the HamiltonianH(QW ,PW )52T(QW ,PW )5W(A,B)(H)uAW 5QW ∧BW 5PW . Hence, the timeevolution of the Wigner function is given by

]

]tf W8 ~QW ,PW ,t !5$ f W8 ,T% (Q,P)5$H, f W8 %(Q,P) . ~45!

On the other hand, for the*-genfunctionsW(Q,P)(A,B) (uqW 0&^qW 0u) we have from Eq.~41!

gqW 08 ~QW ,PW ,t !5d

* (Q,P)8N

~qW 2qW 0!5d* (A,B)

N ~qW ~AW ,BW ,t !2qW 0!uAW 5AW C(QW ,PW ,t)∧BW 5BW C(QW ,PW ,t)

5gqW 0~AW C~QW ,PW ,t !,BW C~QW ,PW ,t !,t !, ~46!

and so@cf. ~35! and ~42!#

]

]tgqW 08 ~QW ,PW ,t !5H ]

]t11

]

]t2J gqW 0

~AW C~QW ,PW ,t2!,BW C~QW ,PW ,t2!,t1!Ut15t25t

5@gqW 08 ,H#M

(Q,P)8 2$gqW 08 ,H%(Q,P) . ~47!

In this representation both the Wigner function and the stargenfunctions evolve in time, theWigner function displaying a fully classical evolution. In particular, if the Wigner function ispositive defined at the initial time, it will remain so for all times. We conclude that the source ofthe quantum behavior has been completely removed from the distributional sector of the theoryand is now exclusively placed on the observables~stargenfunctions! sector.

VI. EXAMPLE

To illustrate our previous results let us consider a two particle system described by theHamiltonian,

H5p1

2

2M1

p22

2m1kq1p2

2, ~48!

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where (q1 ,p1) are the canonical variables of the particle of massM and (q2 ,p2) are those of theparticle of massm andk is a coupling constant.

The Weyl mapW(q,p) yields the (q,p)-Hamiltonian symbol,

H5W(q,p)~H !5p1

2

2M1

p22

2m1kq1p2

2, ~49!

and the Moyal equationsz5@z,H#M (q,p)for the fundamental variablesz5q1 ,q2 ,p1 or p2 . These

display the solutions

q1~ t !5q1~0!1p1~0!

Mt2

k

2Mp2~0!2t2,

p1~ t !5p1~0!2kp2~0!2t,~50!

q2~ t !5q2~0!1H p2~0!

m12kq1~0!p2~0!J t1

k

Mp1~0!p2~0!t22

k2

3Mp2~0!3t3,

p2~ t !5p2~0!,

which coincide exactly with the classical time evolution, i.e., with the solutions of the classicalHamiltonian equations for the classical Hamiltonian~49!. This property is not shared by theWigner function, its time evolution satisfying the equation

] f W

]t5@H, f W#M (q,p)

⇔ ] f W

]t5$H, f W% (q,p)1

\2

24F2H 2kp2 ,]2f W

]q2]p1J

(q,p)

2H 2kq1 ,]2f W

]q22 J

(q,p)G ,

~51!

which is obviously not of the form of the Liouville equation. Consequently, the Wigner functiondoes not satisfyf W(qW ,pW ,t)5 f W(qW (2t),pW (2t),0) with qW (t),pW (t) given by Eq. ~50!, and qW5(q1 ,q2), pW 5(p1 ,p2), i.e., it does not display a classical causal structure.

We now introduce a new set of fundamental operators,

A15q12p1

Mt2

k

2Mp2

2t2,

B15 p11kp22t,

~52!

A25q22H p2

m12kq1p2J t1

k

Mp1p2t21

k2

3Mp2

3t3,

B25 p2 ,

satisfying the Heisenberg algebra@A1 ,B1#5@A2 ,B2#5 i\, all other commutators being zero. Thetransformation~52! is unitary and generated byT52H. Applying the Weyl mapW(q,p) to Eq.~52! and comparing the result with~50! we get

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A1~qW ,pW ,t !5q1~2t !B1~qW ,pW ,t !5p1~2t !A2~qW ,pW ,t !5q2~2t !B2~qW ,pW ,t !5p2~2t !

⇔5q1~AW ,BW ,t !5A11

B1

Mt2

k

2MB2

2t2

p1~AW ,BW ,t !5B12kB22t

q2~AW ,BW ,t !5A21H B2

m12kA1B2J t1

k

MB1B2t22

k2

3MB2

3t3

p2~AW ,BW ,t !5B2 .

~53!

The density matrix satisfiesr(qW ,pW ,t)5 r(AW (qW ,pW ,t),BW (qW ,pW ,t),0) @cf. ~36!# and thus the Wignerfunction f W(AW ,BW ,t)5W(A,B)( r)5 f W(AW ,BW ,0) is static. On the other hand, in this representation,the fundamental stargenfunctions do evolve in time. For instance,~let ux& be the general eigenketof q1 with associated eigenvaluex),

gx~AW ,BW ,t !5W(A,B)~ ux&^xu!5d* (A,B)@q1~AW ,BW ,t !2x#5

1

2p E dk e* (A,B)

ik(q1(AW ,BW ,t)2x)

51

2p E dk eik(q1(AW ,BW ,t)2x)5d@q1~AW ,BW ,t !2x# ~54!

satisfies

]

]tgx~AW ,BW ,t !5@gx~AW ,BW ,t !,H#M (A,B)

5$gx~AW ,BW ,t !,H%(A,B) . ~55!

Hence, the Weyl transformW(A,B) casts the phase space dynamics in the Heisenberg picture.Accordingly, the time dependence is exclusively displayed by the observable~stargenfunction!sector of the theory.

We now consider the action of the generalized Weyl mapW(q,p)(A,B) . The associated time depen-

dent covariant starproduct* (q,p)8 and Moyal bracket@ , #M (q,p)are characterized by@using the time

dependent version of Eqs.~6!–~8! and making O815p1 , O825p2 , O835q1 , O845q2 , O1

5B1 , O25B2 , O35A1 , O45A2 and i , j 51,...,4]

J(q,p)8 i j 5J(q,p)i j ,

~56!

G228152kt, G228

35k

Mt2, G128

45G21845

k

Mt2, G228

452k2

Mp2t3, G328

45G2384522kt,

all other Christoffel symbols being zero. Notice that the connection is time dependent.The new Wigner function@cf. ~53!#

f W8 ~qW ,pW ,t !5W(q,p)(A,B)~ r !5 f W~AW ~qW ,pW ,t !,BW ~qW ,pW ,t !,0!5 f W~qW ~qW ,pW ,2t !,pW ~qW ,pW ,2t !,0! ~57!

satisfies the Liouville equation

] f W8

]t5

] f W

]AW•

]AW

]t1

] f W

]BW•

]BW

]t5

] f W

]qW•$H,qW %(q,p)1

] f W

]pW•$H,pW %(q,p)5$H, f W%(q,p)5$H, f W8 % (q,p) .

~58!

The quantum behavior is displayed by the stargenfunction sector alone. However, for this system,we also have~let z5q1 ,p1∨p2 anduz0& be a generic eigenket ofz with associated eigenvaluez0)

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W(q,p)(A,B)~ uz0&^z0u!5d* 8(q,p)~z2z0!5d* (A,B)~z~AW ,BW ,t !2z0!uAW 5AW (qW ,pW ,t)∧BW 5BW (qW ,pW ,t)

5d~z~AW ,BW ,t !2z0!uAW 5AW (qW ,pW ,t)∧BW 5BW (qW ,pW ,t)5d~z2z0!, ~59!

where in the third step we used the fact thate* (A,B)

ik(z(AW ,BW ,t)2z0)5eik(z(AW ,BW ,t)2z0). Hence, the former

three fundamental stargenfunctions display a classical structure and satisfy$d* 8(q,p)(z2z0),H%50. We conclude that for this system, in this representation, the nontrivial~quantum! behavior isdisplayed by the stargenfunctionz5q2 alone.

A final remark is in order: in this example we were not required to use the most generalformalism of Sec. V~3! ~causal picture and the mapW(Q,P)

(A,B) ) to derive the causal phase spacerepresentation of the system. This is so because the dynamical structure of the system is excep-tionally simple: the quantum and the classical trajectories@which in the most general case have tobe described by two different sets of coordinates, (qW ,pW ) and (QW ,PW ), respectively# are identical.Indeed, Eq.~50! solves both the Moyal and the Hamiltonian equations of motion and thus we werenot required to introduce a second set of ‘‘classical’’ coordinates (QW ,PW ).

VII. CONCLUSIONS

Using a time dependent extension of the generalized Weyl map we enlarged the set of possiblephase space representations of quantum mechanics, derived a more general formula for the\-deformations of the Poisson bracket and proved that there is a phase space representation wherethe quantum quasidistribution displays a causal dynamical structure. In this formulation the quan-tum behavior is displayed by the*-genfunctions~observables! sector alone. Such a property maylead to interesting applications in the field of the semiclassical limit of quantum mechanics giventhe fact that in the causal representation the quantum behavior has become, in fact, independentfrom the state of the system.

The comparison with the de Broglie Bohm formulation seems inevitable. In the de BroglieBohm theory the source of quantum behavior is the quantum potential together with a modificationof the momentum*-genvalue equation. The theory admits an interpretation in terms of causal~butnot classical! trajectories. In the Wigner causal formulation the effect of the quantum potential hasbeen replaced by further corrections in the observables~* -genfunctions! sector of the theory andthe achievement was that the particle trajectories became fully classical. It is quite remarkable thatthis shift of the source of the quantum behavior~from the distributional to the observables sector!could be fully performed and it may lead to an alternative causal interpretation for quantummechanics.

ACKNOWLEDGMENTS

This work was partially supported by the grants POCTI/MAT/45306/2002 and POCTI/FNU/49543/2002.

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900 J. Math. Phys., Vol. 45, No. 3, March 2004 N. C. Dias and J. N. Prata

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