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Generalized Weyl–Wigner map and Vey quantum mechanics
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Generalized Weyl–Wigner map and Vey quantum mechanics Nuno Costa Dias and João Nuno Prata Citation: Journal of Mathematical Physics 42, 5565 (2001); doi: 10.1063/1.1415086 View online: http://dx.doi.org/10.1063/1.1415086 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/42/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Wigner functions and Weyl transforms for pedestrians Am. J. Phys. 76, 937 (2008); 10.1119/1.2957889 Weyl–Wigner formulation of noncommutative quantum mechanics J. Math. Phys. 49, 072101 (2008); 10.1063/1.2944996 The Wigner-Weyl transformation and the quantum path integral J. Math. Phys. 47, 042101 (2006); 10.1063/1.2184768 Decoherence, Wigner functions, and the classical limit of quantum mechanics in cavity QED AIP Conf. Proc. 461, 151 (1999); 10.1063/1.57851 On simulating Liouvillian flow from quantum mechanics via Wigner functions J. Math. Phys. 39, 4492 (1998); 10.1063/1.532521 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sun, 30 Nov 2014 18:00:29
Transcript of Generalized Weyl–Wigner map and Vey quantum mechanics
Generalized Weyl–Wigner map and Vey quantum mechanicsNuno Costa Dias and João Nuno Prata Citation: Journal of Mathematical Physics 42, 5565 (2001); doi: 10.1063/1.1415086 View online: http://dx.doi.org/10.1063/1.1415086 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/42/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Wigner functions and Weyl transforms for pedestrians Am. J. Phys. 76, 937 (2008); 10.1119/1.2957889 Weyl–Wigner formulation of noncommutative quantum mechanics J. Math. Phys. 49, 072101 (2008); 10.1063/1.2944996 The Wigner-Weyl transformation and the quantum path integral J. Math. Phys. 47, 042101 (2006); 10.1063/1.2184768 Decoherence, Wigner functions, and the classical limit of quantum mechanics in cavity QED AIP Conf. Proc. 461, 151 (1999); 10.1063/1.57851 On simulating Liouvillian flow from quantum mechanics via Wigner functions J. Math. Phys. 39, 4492 (1998); 10.1063/1.532521
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Generalized Weyl–Wigner map and Vey quantummechanics
Nuno Costa Diasa) and Joao Nuno Pratab)
Departamento de Matema´tica, Departamento de Engenharias, Universidade Luso´fona deHumanidades e Tecnologias, Av. Campo Grande, 376, 1749-024 Lisboa, Portugal
~Received 16 April 2001; accepted for publication 6 September 2001!
The Weyl–Wigner map yields the entire structure of Moyal quantum mechanicsdirectly from the standard operator formulation. The covariant generalization ofMoyal theory, also known as Vey quantum mechanics, was presented in the litera-ture many years ago. However, a derivation of the formalism directly from standardoperator quantum mechanics, clarifying the relation between the two formulations,is still missing. In this article we present a covariant generalization of the Weylorder prescription and of the Weyl–Wigner map and use them to derive Vey quan-tum mechanics directly from the standard operator formulation. The proceduredisplays some interesting features: it yields all the key ingredients and provides amore straightforward interpretation of the Vey theory including a direct implemen-tation of unitary operator transformations as phase space coordinate transforma-tions in the Vey idiom. These features are illustrated through a simpleexample. ©2001 American Institute of Physics.@DOI: 10.1063/1.1415086#
I. INTRODUCTION
The Weyl–Wigner map1,2 yields the Moyal formulation of quantum mechanics,3–8 alternativeto the more conventional standard operator9–12 and path integral formulations. The main featuresof Moyal quantum mechanics are that it is formulated in terms of phase space functions and thedynamics is based on a deformation of the Poisson bracket, named the Moyal bracket.3,13–15Thisformulation of quantum mechanics has been receiving increased attention namely in the context ofthe fields of the semiclassical limit of quantum mechanics,5,14–19 quantum chaos,20,21 hybriddynamics22,23 and also inM -theory.24–26
The Weyl–Wigner isomorphism between operators and phase space functions~symbols! pro-vides the entire structure of Moyal quantum mechanics directly from the standard operator for-mulation. Let us choose a set of fundamental operators (qi ,pi) and the corresponding set ofcanonical variables (qi ,pi),i 51,...,N, for an arbitraryN dimensional dynamical system. TheWeyl–Wigner mapW(q,p) :A→A(T* M ) attributes to a given operatorA in the quantum algebraA a unique element of the algebra of functions over the phase spaceT* M :
W(q,p)~A!5E dNyWe2 ipW •yW K qW 2\
2yW UAUqW 1
\
2yW L , ~1!
where we used the compact notation:
yW[~y1 ,...,yN!; dNyW[dy1¯dyN ; pW [~p1 ,...,pN! qW [~q1 ,...,qN!,
and the subscript (q,p) means that the corresponding object~the Weyl–Wigner map in this case!is defined in the variables (qW ,pW ). This specification seems redundant now but is important for thesequel. The Weyl–Wigner map is bijective and unequivocal. Moreover, it is an isomorphism
a!Electronic mail: [email protected]!Electronic mail: [email protected]
JOURNAL OF MATHEMATICAL PHYSICS VOLUME 42, NUMBER 12 DECEMBER 2001
55650022-2488/2001/42(12)/5565/15/$18.00 © 2001 American Institute of Physics
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between the quantum (A,•,@ ,#) and the ‘‘classical’’ (A,* ,@ ,#M) algebras. The quantum algebraAis based on the operator product• and the quantum commutator@,#, whereas the classical algebraicstructures are the ‘‘star’’ product* and the Moyal bracket@ ,#M . The Weyl–Wigner map is amorphism in the sense that
W(q,p)~A•B!5W(q,p)~A!* (q,p)W(q,p)~B!,~2!
W(q,p)S 1
i\@A,B# D5@W(q,p)~A!,W(q,p)~B!#M (q,p)
, ;A,BPA ,
and it yields the functional structure of the star product and Moyal bracket:
A* (q,p)B5Ae~ i\/2!J(q,p)B, @A,B#M (q,p)5
2
\A sinS \
2J(q,p)DB, A,BPA, ~3!
whereJ(q,p) is the ‘‘Poisson’’ operator:
J(q,p)[(i 51
N S ]Q
]qi
]W
]pi2
]Q
]pi
]W
]qiD ,
the derivatives]Q and ]W acting onA and B, respectively. Alternatively,J(q,p) can be written asJ(q,p)5]Q kJ(q,p)
kl ]W l , where J(q,p)kl is the klth element of the symplectic matrix in the variables
(qW ,pW ):
J(q,p)5S 0N3N 21N3N
1N3N 0N3ND . ~4!
We introduced the compact notation,Ok5pk ,k51,...,N; Ok5qk2N ,k5N11,...,2N; ]/]Ok
5]k , and sum over repeated indices is understood.From Eq.~3! it is trivial to obtain the following expansion in powers of\:
A* (q,p)B5A•B1i\
2A]Q kJ(q,p)
kl ]W lB11
2 S i\
2 D 2
A]Q k]Q sJ(q,p)kl J(q,p)
sn ]W l]WnB1¯ . ~5!
The Weyl–Wigner transform of the density matrix operator is the Wigner distribution function,2
f W(qW ,pW ;t)5 (1/2P\)W(q,p)(uc(t)&^c(t)u), which is the fundamental mathematical object ofMoyal quantum mechanics. Its time evolution is given by the dynamical equation
f W~qW ,pW ;t !5@H~qW ,pW !, f W~qW ,pW ;t !#M (q,p), H5W(q,p)~H !, ~6!
whereH is the quantum Hamiltonian. We see that the mathematical structure of Moyal quantummechanics is very similar to that of classical statistical mechanics. However, these similaritiesshould not be taken too seriously. The procedure by which physical relevant information is ob-tained is a lot more elaborate. In classical statistical mechanics the fundamental predictions are theprobabilities for finding the system in an arbitrary configuration (qW 0 ,pW 0), which are given by thevalues of a true probability distribution functionr(qW 5qW 0 ,pW 5pW 0).
On the contrary, in Moyal quantum mechanics the value off W(qW 5qW 0 ,pW 5pW 0) cannot be givensuch straightforward interpretation, given the fact thatf W(qW ,pW ) might take on negative values. Thefundamental physical predictions of Moyal quantum mechanics are obtained through a procedureanalogous to that of standard operator quantum mechanics. Given a general observableA(qW ,pW ) weshould solve the star-genvalue equation,8,27,28
A~qW ,pW !* (q,p)gan~qW ,pW !5aga
n~qW ,pW !, ~7!
5566 J. Math. Phys., Vol. 42, No. 12, December 2001 N. C. Dias and J. N. Prata
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wheren is a degeneracy index, to obtain the probability for a measurement ofA at the timetyielding the valuea:
P~A~qW ,pW ;t !5a!5(n
E dNqW E dNpW gan~qW ,pW ! f W~qW ,pW ;t !, ~8!
where we assumed that the degeneracy index is discrete. If this is not the case, then the sum innis replaced by a~set of! integral~s! in n. For the fundamental observablesqi andpi , i 51,...,N, Eq.~8! reduces to a more ‘‘classical type’’ result:
P~qi~ t !5q0!5E dNqW E dNpW f W~qW ,pW ;t !d~qi2q0!;
~9!
P~pi~ t !5p0!5E dNqW E dNpW f W~qW ,pW ;t !d~pi2p0!,
the same happening to the average value ofA(qW ,pW ):
^A~qW ,pW ;t !&5E dNqW E dNpW A~qW ,pW ! f W~qW ,pW ;t !. ~10!
An important subject in any dynamical theory is the study of its invariances. Just like standardoperator quantum mechanics the Moyal formulation is invariant under the action of general uni-tary transformations and, contrary to what happens in classical mechanics, is not invariant underthe action of a significant set of coordinate transformations. In fact, most unitary operator trans-formations are not implemented as phase space coordinate transformations in the Moyal idiom.
Many years ago Vey29 presented a generalization of the star product that renders Moyalquantum mechanics fully invariant under phase space coordinate transformations. The originalmotivation was not to enlarge the invariance properties of Moyal quantum mechanics, but toderive the general form of the Poisson algebra deformations for curved phase spaces. Vey’soriginal developments have been used in investigations aiming at two major directions: First, toprovide a consistent classical interpretation of Moyal dynamics,14,15 and second, in more math-ematically oriented research, to generalize the Moyal–Weyl–Wigner quantization procedure tononflat phase space manifolds. In Refs. 14 and 15 the problem of constructing an associative starproduct in a general sympletic manifold was considered. The question of existence of such aproduct was completely solved in Ref. 30. The same problem for Poisson–Lie groups was con-sider in Ref. 31. In Refs. 32–34 an alternative construction of star products for a general sympleticmanifold was proposed. This construction is given in pure geometrical terms and is thus quite anappealing framework to study the invariance properties of phase space quantum mechanics. InRef. 33 the action of unitary transformations is implemented, in geometrical terms, as sympleticmorphisms of the phase space manifold.
To our knowledge, however, a complete study of the relation between Vey covariant quantummechanics and standard operator quantum mechanics casting Vey theory at the same level ofcompleteness as Moyal quantum mechanics has not yet been presented. Take, for instance, theWeyl–Wigner map. Although trivial, the covariant generalization of this map is still missing in theliterature.
In this article we emphasize the relation between standard operator quantum mechanics andMoyal quantum mechanics and attempt to derive the Vey formulation in a similar fashion. Thereare two main virtues in this approach: first, it clarifies the relation between the standard operatorand Vey quantum mechanics~providing, for instance, a new point of view for the analysis of theinvariance properties of covariant phase space quantum mechanics!. Second, it yields a~previ-ously missing! covariant generalization of some key ingredients of phase space quantum mechan-ics, such as the Weyl–Wigner map, the Weyl order prescription, the average and the marginalprobability distribution functionals and the star-genvalue equation.
5567J. Math. Phys., Vol. 42, No. 12, December 2001 Generalized Weyl–Wigner map
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This article is organized as follows: in Sec. II we start by reviewing some well knownproperties concerning the action of canonical and coordinate transformations in the Moyal formal-ism. We then present~Sec. III! a covariant generalization of the Weyl–Wigner map. The new mapmakes it possible to implement a general unitary transformation of standard operator quantummechanics as a coordinate transformation in the Moyal idiom. In Sec. IV we use the new map toderive the covariant star product and, as a by-product, the dynamical structure of Vey quantummechanics. In Sec. V we present a summary of the structure of covariant quantum mechanics,including the generalizations of the star-genvalue equation, the average value and the probabilityfunctionals. Finally, in Sec. VI some of the features of the formalism are illustrated through asimple example.
Before proceeding let us make an important remark: from the outset we shall restrict ourattention to the simpler case of dynamical systems displaying a phase space with the structure ofa flat manifold.
II. CANONICAL AND COORDINATE TRANSFORMATIONS
In classical mechanics all canonical transformations are phase space coordinate transforma-tions. Their action is of the form~we shall take the passive point of view!
T:T* M→T* M ;~qW ,pW !→~qW 5qW ~QW ,PW !,pW 5pW ~QW ,PW !!, ~11!
where (QW ,PW ) and (qW ,pW ) are two sets of canonical variables.T yields a transformation of a generalobservable given by
T:A~qW ,pW !→A8~QW ,PW !5A~qW ~QW ,PW !,pW ~QW ,PW !!, ~12!
and for two general observablesA8(QW ,PW )5T$A(qW ,pW )% andB8(QW ,PW )5T$B(qW ,pW )% we have
$A~qW ~QW ,PW !,pW ~QW ,PW !!,B~qW ~QW ,PW !,pW ~QW ,PW !!%(q,p)5$A8~QW ,PW !,B8~QW ,PW !% (Q,P) , ~13!
and thus the Hamiltonian equations of motion in the variables (qW ,pW ) and (QW ,PW ) are fully equiva-lent: they yield identical mathematical solutions and thus identical physical predictions.
This picture does not translate to Moyal quantum mechanics. To see this explicitly let us goback to standard operator quantum mechanics and consider, to make it simpler, the unitary trans-formation:
qW [qW ~QW ,PW !5UQW U21, pW [pW ~QW ,PW !5UPW U21, A~qW ,pW 
T:A~qW ,pW !5W(q,p)~A~qW ,pW !!→A8~QW ,PW !
5W(Q,P)~A8~QW ,PW !!
5U~QW ,PW !* (Q,P)A~QW ,PW !* (Q,P)U21~QW ,PW !, ~17!
which, in general, does not correspond to the action of a coordinate transformation~except ifT islinear! since
A8~QW ,PW !ÞA~qW ~QW ,PW !,pW ~QW ,PW !!, ~18!
even though the transformation is canonical. For two general observablesA(qW ,pW ) andB(qW ,pW ) wehave
@A8~QW ,PW !,B8~QW ,PW !#M (Q,P)5T~@A~qW ,pW !,B~qW ,pW !#M (q,p)
!, ~19!
and thus the Moyal dynamical equations in the variables (QW ,PW ) and (qW ,pW ),
A~qW ,pW ;t !5@A~qW ,pW ;t !,H~qW ,pW !#M (q,p)and A8~QW ,PW ;t !5@A8~QW ,PW ;t !,H8~QW ,PW !#M (Q,P)
, ~20!
yield two mathematical solutions, related by
A~ t !5F~qW ,pW ,t ! and A8~ t !5F8~QW ,PW ,t !5U* (Q,P)F~QW ,PW ,t !* (Q,P)U21, ~21!
which, in general, are not the same phase space function:F8(QW ,PW ,t)ÞF(qW (QW ,PW ),pW (QW ,PW ),t),albeit providing the same physical predictions: Eqs.~8!–~10!. We see that no physical meaningcan be attached to a single value of the observableA since this value is dependent of the particularrepresentation chosen. Take, for instance, the Wigner distribution function that may be positivedefined in one representation and become negative under a unitary transformation.
On the other hand, most coordinate transformations act noncanonically in the Moyal formal-ism ~the exceptions, once again, are linear transformations!: consider the transformationT in Eq.~16! and the two general phase space functionsG(qW ,pW ) andF(qW ,pW ). In general we have
@G~qW ~QW ,PW !,pW ~QW ,PW !!,F~qW ~QW ,PW !,pW ~QW ,PW !!#M (q,p)
Þ@G~qW ~QW ,PW !,pW ~QW ,PW !!,F~qW ~QW ,PW !,pW ~QW ,PW !!#M (Q,P), ~22!
which is a consequence of the fact that the star product~sometimes expressed in terms of thesymmetric bracket13! is also not invariant under a general coordinate transformation:
G~qW ~QW ,PW !,pW ~QW ,PW !!* (q,p)F~qW ~QW ,PW !,pW ~QW ,PW !!
ÞG~qW ~QW ,PW !,pW ~QW ,PW !!* (Q,P)F~qW ~QW ,PW !,pW ~QW ,PW !!. ~23!
These features of the Weyl-Wigner map and consequently of the star product are well known andhave been extensively studied in the past~see, for instance Refs. 14 and 15!. Namely, it wasproved that the set of observables invariant under general unitary transformations is the set of firstorder polynomials in the fundamental variables and the coordinate transformations that preservethe Moyal star product are the linear transformations. These properties restrict the validity of thedeformation quantization procedure to those phase space manifolds where a global Darboux chartcan be naturally defined and thus completely exclude the possibility of extending this quantizationprocedure to nonflat phase space manifolds. In this context they motivated the original develop-ments by Vey29 and subsequently by Bayenet al.,14,15 aiming at producing a more robust, coor-
5569J. Math. Phys., Vol. 42, No. 12, December 2001 Generalized Weyl–Wigner map
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dinate invariant formulation of the deformed structures. This work culminated with the results ofFedosov33,34 where a pure geometrical implementation of the star product was proposed.
Our analysis of the next sections differs in the approach but not in the final goals. Instead ofworking directly in the context of Wigner quantum mechanics~like in the previous references! wefocus on the relation between standard operator and phase space quantum mechanics. From theanalysis leading to Eq.~18! and Eqs.~22! and ~23! we see that the behavior of Moyal quantummechanics under the action of canonical and coordinate transformations can be seen as a conse-quence of the definition of the Weyl–Wigner map~15!. This motivates the purpose of the nextsection where we will present a generalization of the Weyl–Wigner map and use it to prove thatunitary operator transformations can be implemented as coordinate transformations in the Moyalformalism. Furthermore, we will see in Sec. IV that the new map provides a derivation of Veycovariant quantum mechanics directly from standard operator quantum mechanics.
III. GENERALIZED WEYL–WIGNER MAP
We start by introducing a new Weyl–Wigner map in the variables (QW ,PW ) ~which are notrequired to be canonical! that copies the Weyl–Wigner map in the variables (qW ,pW ):
Definition: Generalized Weyl–Wigner map:Let W(q,p) be the standard Weyl–Wigner map@inthe variables (qW ,pW )# from the algebra of linear operatorsA(H) acting on the physical HilbertspaceH to the algebra of observables in the phase spaceT* M . For the variables (QW ,PW ), assumedin one-to-one correspondence with the canonical variables (qW ,pW ), we define a newgeneralizedWeyl–Wigner map:
W(Q,P)8 :A~H!→A~T* M !; W(Q,P)8 ~A!5W(q,p)~A!, ;APA(H) . ~24!
For each new choice of the canonical variables (qW ,pW ) we obtain a new Weyl–Wigner map in thevariables (QW ,PW ). If the transformation from (qW ,pW ) to (QW ,PW ) is a polynomial of first degree, thenW(Q,P)8 5W(Q,P) . Otherwise the two maps differ.
The aim of this section is to obtain the explicit expression forW(Q,P)8 . We start by deriving thegeneralizations of the Weyl order and Weyl symbol prescriptions.
A generic dynamical operatorA can be cast in a fully symmetrized form according to Weyl’sprescription:
AW(q,p)5A~qW ,pW !5E dNxWdNyWa~xW ,yW !eixW•qW 1 iyW•pW , ~25!
where the subscriptW(q,p) means thatA is displayed as a fully symmetrized functional of the
variables (qW ,pW ), andxW•qW 1yW•pW [( j 51N (xj qj1yj pj ). If A is Hermitian, then the numerical~usu-
ally singular! function a(xW ,yW ) is subject to the constrainta* (xW ,yW )5a(2xW ,2yW ). The Weyl sym-bol, W(q,p)(A), associated with the operatorA in Eq. ~25! is the c-function of 2N phase spacevariables (qW ,pW ) given by
A~qW ,pW 
pW [E dNxWdNyWrp~xW ,yW !eixW•QW 1 iyW•PW .
The new variables are not necessarily canonical: in general,@Qi ,Pj #Þ i\d i j . The (QW ,PW )-Weylsymbols associated with~27! are
qW [W(Q,P)~qW !5E dNzWdNwW rq~zW,wW !eizW•QW 1 iwW •PW ,
~28!
pW [W(Q,P)~pW !5E dNxWdNyWrp~xW ,yW !eixW•QW 1 iyW•PW .
Let us now consider again the operatorA given by Eq.~25!. In the (QW ,PW ) representationA is
written asA5A(qW (QW ,PW ),pW (QW ,PW ))5A8(QW ,PW ) and the explicit functional form ofA8 is given bythe generalized Weyl prescription:
AW8(Q,P)5A8~QW ,PW !5E dNxWdNyWa~xW ,yW !eixW•qW (QW ,PW )1 iyW•pW (QW ,PW ), ~29!
whereqW (QW ,PW ) andpW (QW ,PW ) are given~in a fully symmetrized order! by Eq.~27! and the subscript
W8(Q,P) means thatA is displayed as a functional of the variables (QW ,PW ) in a fully symmetrized
order in the variables (qW ,pW ). Notice that the numerical functiona(xW ,yW ) is the same as in Eq.~25!.The standard (qW ,pW )-Weyl symbol associated withA is given by Eq.~26! and thus, using thedefinition ~24!, it is straightfoward to conclude that the generalized Weyl symbol associated withA is given by
A8~QW ,PW 
W(Q,P)8 ~A!5E dNxWE dNyWe2 ipW (QW ,PW )•yWd~xW2qW ~QW ,PW !!K xW2\
2yW UQAUxW1
\
2yW L
Q
~34!
satisfying definition~24!.To illustrate the features of the new map let us consider the unitary transformation~14!. The
generalized map yields a different phase space version of this transformation. The analog of Eq.~17! is given by
T:A~qW ,pW !5W(q,p)~A~qW ,pW !!→A8~QW ,PW !5W(Q,P)8 ~A8~QW ,PW !!, ~35!
and we get
W(Q,P)8 ~A8~QW ,PW !!5W(Q,P)8 ~UA~QW ,PW !U21!
5E dNxWE dNyWe2 ipW (QW ,PW )•yWd~xW2qW ~QW ,PW !!
3 K xW2\
2yW UU21~UA~QW ,PW !U21!UUxW1
\
2yW L
5A~qW ~QW ,PW !,pW ~QW ,PW !!, ~36!
where qW (QW ,PW ) and pW (QW ,PW ) are given by Eq.~16! and uxW6 \2yW & are eigenstates ofQW and thus
UuxW6 \2yW & are eigenstates ofqW , displayed in theQW representation, and with associated eigenvalues
xW6 \2yW . As expected, the previous result means that the unitary transformation is mapped by the
generalized Weyl–Wigner map to a phase space coordinate transformation.
IV. COVARIANT STAR PRODUCT
The new Weyl–Wigner map yields a new star product through the definition
W(Q,P)8 ~A•B!5W(Q,P)8 ~A!* (Q,P)8 W(Q,P)8 ~B!; ;A,BPA(H) , ~37!
and one immediately recognizes that the new product satisfies
W(Q,P)8 ~A!* (Q,P)8 W(Q,P)8 ~B!5W(Q,P)8 ~AB!5W(q,p)~AB!5W(q,p)~A!* (q,p)W(q,p)~B!. ~38!
If W(q,p)(A)5A(qW ,pW ), thenW(Q,P)8 (A)5A(qW (QW ,PW ),pW (QW ,PW ))5A8(QW ,PW ) and thus
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 ) .~39!
The former result immediately implies that the new product is also a noncommutative, associativeproduct for the algebra of functions over the classical phase space. Moreover, using the newproduct we can define a new bracket~named generalized Moyal bracket! alternative to the stan-dard Moyal bracket:
@A,B#M(Q,P)8 5
1
i\~A* (Q,P)8 B2B* (Q,P)8 A!. ~40!
It follows from Eq. ~39! that this is also a Lie bracket.
5572 J. Math. Phys., Vol. 42, No. 12, December 2001 N. C. Dias and J. N. Prata
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Neither the new star product nor the new Moyal bracket display the same functional structureas the standard ones* (Q,P) and@ ,#M (Q,P)
. The aim of the rest of this section is to derive the explicitform of the new product and in the sequel of the new bracket from the generalized Weyl–Wignermap. Let us start by introducing the notation:
Ok5 pk , k51,...,N; O8k5 Pk , k51,...,N;
Ok5qk2N , k5N11,...,2N; O8k5Qk2N , k5N11,...,2N;
and the symbolsOk[W(q,p)(Ok), O8k[W(Q,P)(O8k). Consider the two following operators dis-
played in the generalized Weyl order~sum over repeated indices is understood!:
AW8(Q,P)5A8~QW ,PW !5E d2NaW a~aW !eiakOk(O8s),
~41!
BW8(Q,P)5B8~QW ,PW !5E d2NbW b~bW !eibl Ol (O8r ),
whereaW 5(a1 ,...,a2N), bW 5(b1 ,...,b2N). Let us then calculateW(Q,P)8 (A•B) explicitly and ex-press the result in terms of the symbolsA8(QW ,PW )5W(Q,P)8 (A) andB8(QW ,PW )5W(Q,P)8 (B):
W(Q,P)8 ~A•B!5E d2NaW d2NbW a~aW !b~bW !W(Q,P)8 $eiakOk(O8s)•eibl O
l (O8r )%
5E d2NaW d2NbW a~aW !b~bW !W(q,p)$eiakOk(O8s)
•eibl Ol (O8r )%
5E d2NaW d2NbW a~aW !b~bW !@eiakOk(O8s)#* (q,p)@eiblOl (O8r )#. ~42!
Using the explicit expression of the (qW ,pW ) star product in Eq.~5!, we obtain the following expan-sion in powers of\:
W(Q,P)8 ~A•B!5 (k50
`1
k! S i\
2 D kE d2NaW d2NbW a~aW !b~bW !@eiatOt(O8s)#Jk@eiblO
l (O8r )#, ~43!
where
Jk5]Q i 1¯]Q i k
J(q,p)i 1 j 1
¯J(q,p)i kj k ]W j 1
¯]W j k. ~44!
At this point we recall that the phase space is assumed to have the structure of a flat manifold andintroduce the 2N32N ‘‘Euclidean’’ metric,
~gi j !5S a1N3N 0N3N
0N3N b1N3ND , ~45!
and the associated covariant derivative¹ i ,
¹ iA5] iA,~46!
¹ i¹ jA5] i] jA2G i jk ]kA, i , j ,k51,...,2N,
where the Christoffel symbolsG i jk are fully determined by the metric:
5573J. Math. Phys., Vol. 42, No. 12, December 2001 Generalized Weyl–Wigner map
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G jki 5 1
2 gil ~]kgl j 1] jglk2] lgjk!, i , j ,k51,...,2N. ~47!
Obviously, in the coordinates (qW ,pW ) we haveG jki 50, ; i , j ,k51,...,2N and thus¹ i5] i . ~a andb
are arbitrary constants introduced to ensure the correct dimensions.!Under the general coordinate transformationOi→Oi(O8s) ( i 51,...,2N) the symplectic ma-
trix and the covariant derivative transform according to
J(Q,P)8 i j 5]O8 i
]Ok
]O8 j
]Ol J(q,p)kl 5$O8 i ,O8 j%(q,p)5O8 iJ(q,p)O8 j , ~48!
G jk8i5Gbc
m ]O8 i
]Om
]Ob
]O8 j
]Oc
]O8k 1]O8 i
]Ob
]2Ob
]O8 j]O8k 5]O8 i
]Ob
]2Ob
]O8 j]O8k . ~49!
Moreover, the terms@eiatOt(O8s)#Jk@eiblO
l (O8r )# are scalars and thus are left invariant:
@eiatOt(O8s)#]Q i 1
¯ ]Q i kJ(q,p)
i 1 j 1¯ J(q,p)
i kj k ]W j 1¯ ]W j k
@eiblOl (O8r )#
5@eiatOt(O8s)#¹Q i 1
8 ¯ ¹Q i k8 J(Q,P)
8 i 1 j 1¯ J(Q,P)
8 i kj k ¹W j 18 ¯ ¹W j k
8 @eiblOl (O8r )#, ~50!
where the new covariant derivative¹8 is given by
¹ i8A5] i8A,~51!
¹ i8¹ j8A5] i8] j8A2G i j8k]k8A, ] i85]/]O8 i ; i , j ,k51,...,2N.
Substituting the result~50! in Eq. ~43! and taking into account~41!, it is trivial to obtainW(Q,P)8 (A•B)5W(Q,P)8 (A)* (Q,P)8 W(Q,P)8 (B) where the new star product is given by
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 !, ~52!
and we recovered the covariant formulation of the star product first introduced by Vey.29 Thecovariant formulation ensured the invariant nature of the numerical value for the star-product oftwo observables in any coordinate system. However, in general, the functional form of the productis altered under an arbitrary coordinate transformation.
We should point out that our construction of the covariant star product being based upon asympletic and a metric structure over the phase space manifold slightly differs from the construc-tion of Bayenet al., Fedosov and others, where the covariant star product is built upon a sympleticstructurew and a Poisson connection¹, satisfying ¹w50. This difference, however, is onlyapparent since our metric uniquely determines the Poisson connection and vice versa.
Finally, we can easily obtain the functional form of the new bracket:@A8,B8#M(Q,P)8
5 (1/i\) W(Q,P)8 (@A,B#):
@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 !, ~53!
and if (QW ,PW ) is a set of canonical variables this is equally a deformation of the Poisson bracket:
@A8,B8#M(Q,P)8 [
1
i\~A8* (Q,P)8 B82B8* (Q,P)8 A8!5$A8,B8%(Q,P)1O~\2!. ~54!
5574 J. Math. Phys., Vol. 42, No. 12, December 2001 N. C. Dias and J. N. Prata
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V. COVARIANT VEY QUANTUM MECHANICS AND DISCUSSION
The covariant formulation of Moyal quantum mechanics lives on the classical phase spaceT* M with the structure of the tangent bundle of the configuration space, where a symplecticstructureJi j and a metric structuregi j ~or alternatively a Poisson connection! can be naturallydefined.
The fundamental mathematical objects of the theory are the Wigner distribution functionf W8 (QW ,PW ;t) and the observablesA8(QW ,PW ). They all are scalar functions over (T* M ,Ji j ,gi j ), andare related with the mathematical objects of standard operator quantum mechanics through thegeneralized Weyl–Wigner map:f W8 (QW ,PW ;t)5 (1/2p\) W(Q,P)8 (uc(t)&^c(t)u) and A8(QW ,PW ;t)5W(Q,P)8 (A). The time evolution of the Wigner function is given by the dynamical equation:
f W8 5@H8, f W8 #M(Q,P)8 , ~55!
which transforms covariantly under arbitrary phase space difeomorphisms yielding, in any coor-dinates, identical mathematical solutions and thus identical physical predictions.
Finally, the covariant form of Eqs.~7! and ~8! yield the physical relevant predictions. TheMoyal star-genvalue equation can be obtained, through the Weyl–Wigner map, from the standardoperator eigenvalue equation:Auca
n&^canu5auca
n&^canu ~where n is the degeneracy index!. We
follow the same procedure but use the generalized Weyl–Wigner map and obtain
A8~QW ,PW !* (Q,P)8 ga8n~QW ,PW !5aga8
n~QW ,PW !. ~56!
It is trivial to check thatga8n(QW ,PW )5ga
n(qW (QW ,PW ),pW (QW ,PW )) wheregan(qW ,pW ) is the solution of the
original star-genvalue equation. Futhermore, the probabilistic functionals are just the coordinatetransform of the original ones:
P~A8~QW ,PW ;t !5a!5(nE dNQW E dNPW ~detJ(Q,P)8 i j !21/2ga8
n~QW ,PW ! f W8 ~QW ,PW ;t !, ~57!
and finally the average value prediction is also trivially covariantized:
^A8~QW ,PW ;t !&5E dNQW E dNPW ~detJ(Q,P)8 i j !21/2A8~QW ,PW ! f W8 ~QW ,PW ;t !. ~58!
The covariant formulation of phase space quantum mechanics is a familiar field of mathemati-cal physics. The invariance properties of the theory are well studied even for the case of nonflatsympletic manifolds. The main point of this work was not to derive these results once again but tofocus on the relation between standard operator quantum mechanics and phase space quantummechanics. Such a relation given by the generalized Weyl–Wigner map provided a new point ofdeparture to derive the covariant generalization of phase space quantum mechanics and clarifiesthe nature of the invariance properties of the theory.
Let us then summarize our results: we presented an original generalized~covariant! formula-tion of the Weyl-order operator prescription and of the Weyl–Wigner map yielding the entirestructure of Vey quantum mechanics directly from standard operator quantum mechanics. All keyingredients of the theory were derived in this fashion, thus casting Vey quantum mechanics at thesame level of completeness as Moyal quantum mechanics. Furthermore, we studied the action ofstandard operator transformations in the Moyal formalism and concluded that through the gener-alized Weyl–Wigner map these transformations can be implemented as phase space coordinatetransformations in the Moyal idiom. It is now easy to realize that the group of ‘‘canonical’’transformations of Vey quantum mechanics~those that preserve the bracket structure! is the sub-group of the symplectic transformations which are also isometries, i.e., the coordinate transforma-tions Oi→Oi(O8s) such that
5575J. Math. Phys., Vol. 42, No. 12, December 2001 Generalized Weyl–Wigner map
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J(Q,P)8 i j 5]O8 i
]Ok
]O8 j
]Ol J(q,p)kl 5J(q,p)
i j ,
~59!
G jk8i5
]O8 i
]Ob
]2Ob
]O8 j]O8k 50,
and notice that in the limit\→0 the requirement on the isometry character of the canonicaltransformations disappears, as it should, and we recover the standard sympletic group of classicalmechanics.
Finally, let us make a small remark concerning possible future applications of the generalizedWeyl–Wigner map. Recently there has been a considerable interest in developing a fully classicalinterpretation for Moyal dynamics.14,15,19This is a difficult task for several reasons of which themost important are that Moyal dynamics is nonlocal and there is no general criterion to understandwhat is meant by ‘‘classical.’’ Very recently, however, the authors presented a proposal for aclassicality criterion and proved that it is fully compatible with the Weyl–Wigner map.18,19One ofthe problems that was left unsolved was how to understand the highly nonclassical behavior of anallegedly Moyal classical dynamics, under the action of canonical transformations. This problemhas been previously considered in Refs. 14 and 15 using an approach based on a privileged set ofobservables.
We may now expect that if the results of Ref. 19~concerning the compatibility between theclassicality criteria and the Weyl–Wigner map! turn out to be extendable to the covariant Weyl–Wigner map, then they will most likely open the path for a consistent classical interpretation of theaction of canonical transformations in Moyal dynamics. This will be the subject of a future work.35
VI. EXAMPLE
To illustrate some of the features of the formalism let us consider the system of two interact-ing particles described by the Hamiltonian
H5p2
2M1
y2
2m1kqy2, ~60!
where (q,p) are the fundamental variables of the particle of massM , (x,y) are the ones of theparticle of massm andk is a coupling constant.
A. Standard description in the original variables
In the Heisenberg picture the time evolution of the former system is given by
q~ t !5q1p
Mt2
k
2My2t2,
p~ t !5 p2ky2t,~61!
x~ t !5 x1H y
m12kqyJ t1
k
Mpyt22
k2
3My3t3,
y~ t !5 y.
The standard Weyl–Wigner transform yields the Moyal description of the system. The Hamil-tonian is trivially obtained,
H5W(q,x,p,y)~H !5p2
2M1
y2
2m1kqy2, ~62!
5576 J. Math. Phys., Vol. 42, No. 12, December 2001 N. C. Dias and J. N. Prata
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and the Moyal time evolution of the system is given by the dynamical equations
A5@A,H#M (q,x,p,y), ~63!
which yield the following solutions~to make it simpler let us concentrate on the dynamics of theparticle of massM !:
q~ t !5q~0!1p~0!
Mt2
k
2My~0!2t2,
~64!p~ t !5p~0!2ky~0!2t.
Notice that the same predictions can be obtained by applying the Weyl–Wigner map to thesolutions~61!.
B. Canonical transformation and the standard description in the new variables
Let us now consider the canonical transformation:
q5q, x5 ln Q,~65!
p5 p, y5 12~QP1 PQ!.
In the new variablesH takes the form
H5p2
2M1
~QP1 PQ!2
8m1
k
4q~QP1 PQ!2, ~66!
and it yields the Heisenberg picture time evolution~still just for the particle of massM !:
q~ t !5q1p
Mt2
k
8M~QP1 PQ!2t2
~67!
p~ t !5 p2k
4~QP1 PQ!2t
To obtain the Moyal formulation of the system in the variables (q,Q,p,P) the first step is to usethe standard Weyl–Wigner map to get the phase space representation of the transformation~65!:
q5W(q,Q,p,P)~ q!5q, x5W(q,Q,p,P)~ x~Q,P!!5 ln Q,~68!
p5W(q,Q,p,P)~ p!5p, y5W(q,Q,p,P)~ y~Q,P!!5QP.
We then express the Hamiltonian using the standard Weyl order prescription:
HW(q,Q,p,P)5p2
2M1
~ P2Q2!S
2m1kq~Q2P2!S1
k
4\2q1
\2
8m, ~69!
where we used the fact that14(QP1 PQ)W(q,Q,p,P)
2 5( P2Q2)S1\2/4 ~the subscripted ‘‘S’’ standingfor the full symmetrization of the operator!, and so1
4(QP1 PQ)2 can be expressed in the standardWeyl order@Eq. ~25!# by makinga(a,b,c,d)5d(a)d(b)$d9(c)d9(d)1\/4d(c)d(d)%, wherea,b, c, d are the integration variables associated to the fundamental operatoresq,p,Q,P, respec-tively. The standard Weyl–Wigner transform ofH is thus
5577J. Math. Phys., Vol. 42, No. 12, December 2001 Generalized Weyl–Wigner map
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H5W(q,Q,p,P)~H !5p2
2M1
P2Q2
2m1kqQ2P21
k
4\2q1
\2
8m. ~70!
Finally, the Moyal dynamical equations in the variables (q,Q,p,P) yield
q~ t !5q~0!1p~0!
Mt2
k
2MQ~0!2P~0!2t22
k\2
8Mt2
p~ t !5p~0!2kQ~0!2P~0!2t2k\2
4t2. ~71!
Notice that equivalent results can be obtained by applying the Weyl–Wigner transform to the timeevolution operator equations~67!. Furthermore, we realize that the transformation~65! does notact as a coordinate transformation in the Moyal formalism. Taking, for instance, the Hamiltonian,we haveH(q,p,x(Q,P),y(Q,P))ÞW(q,Q,p,P)(H), the left-hand side being given by Eq.~62! andthe right-hand side by Eq.~70!. Consequently, the two phase space orbits@Eqs.~64! and~71!# arenot the coordinate transformation of each other.
C. Covariant formulation
Finally, let us use the generalized Weyl–Wigner map. In the generalized Weyl order prescrip-tion Eq. ~29!, we makea(a,b,c,d)52d(a)d(b)d(c)d9(d), where, this time,a,b,c,d are theintegration variables associated to the fundamental operatorsq,p,x(Q,P),y(Q,P), and we obtain
14 ~QP1 PQ!W8(q,Q,p,P)
25 1
4 ~QPQP1QP2Q1 PQ2P1 PQPQ!, ~72!
and thus
H8~q,Q,p,P!5W(q,Q,p,P)8 ~H !5p2
2M1
Q2P2
2m1kqQ2P2, ~73!
which, as expected, is the coordinate transformation of the observableH given by Eq.~62!. Thenew star product and the new bracket are given by Eqs.~52! and ~53!, respectively, whereJ(Q,P)8 i j 5J(q,p)
i j and the covariant derivatives are associated to the Christoffel symbols:
G1181521/Q; G118
25P/Q2; G12825G218
251/Q, ~74!
all the others being zero~we used the notation:O15x; O25y; O815Q; O825P; O835O3
5q; O845O45p!. Notice that using the new product we can go back and obtain the HamiltonianH8 through a slightly different procedure, by making
W(q,Q,p,P)8 $ 12 ~QP1 PQ! 1
2 ~QP1 PQ!%
5W(q,Q,p,P)8 $ 12 ~QP1 PQ!%* (q,Q,p,P)8 W(q,Q,p,P)8 $ 1
2 ~QP1 PQ!%
5QP* (q,Q,p,P)8 QP5Q2P2, ~75!
where in the last step we used the explicit expression for the new star product~52! and~74!. It isnow easy to check that the generalized Moyal dynamical equations,O8 i5@O8 i ,H8#M
(q,Q,p,P)8 , yield
solutions that are just the coordinate transformation of the original ones~64!. Equivalent resultscan obviously be obtained by applying the generalized Weyl–Wigner map to the operator timeevolution equations~67!.
5578 J. Math. Phys., Vol. 42, No. 12, December 2001 N. C. Dias and J. N. Prata
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ACKNOWLEDGMENTS
This work was partially supported by the grants ESO/PRO/1258/98 and CERN/P/Fis/15190/1999.
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