Weyl–Wigner formulation of noncommutative quantum mechanicsCatarina Bastos, Orfeu Bertolami, Nuno Costa Dias, and João Nuno Prata Citation: J. Math. Phys. 49, 072101 (2008); doi: 10.1063/1.2944996 View online: http://dx.doi.org/10.1063/1.2944996 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v49/i7 Published by the AIP Publishing LLC. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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Weyl–Wigner formulation of noncommutative quantummechanics

Catarina Bastos,1,a� Orfeu Bertolami,1,b� Nuno Costa Dias,2,c� andJoão Nuno Prata2,d�

1Departamento de Física, Instituto Superior Técnico, Avenida Rovisco Pais 1,1049-001 Lisboa, Portugal2Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologias,Av. Campo Grande, 376, 1749-024 Lisboa, Portugal

�Received 11 February 2008; accepted 23 May 2008; published online 2 July 2008�

We address the phase-space formulation of a noncommutative extension of quan-tum mechanics in arbitrary dimension, displaying both spatial and momentum non-commutativities. By resorting to a covariant generalization of the Weyl–Wignertransform and to the Darboux map, we construct an isomorphism between theoperator and the phase-space representations of the extended Heisenberg algebra.This map provides a systematic approach to derive the entire structure of noncom-mutative quantum mechanics in phase space. We construct the extended star prod-uct and Moyal bracket and propose a general definition of noncommutative states.We study the dynamical and eigenvalue equations of the theory and prove that theentire formalism is independent of the particular choice of the Darboux map. Ourapproach unifies and generalizes all the previous proposals for the phase-spaceformulation of noncommutative quantum mechanics. For concreteness we rederivethese proposals by restricting our formalism to some two-dimensional spaces. ©2008 American Institute of Physics. �DOI: 10.1063/1.2944996�

I. INTRODUCTION

Noncommutative extensions of quantum mechanics have been recently widely discussed inthe literature.1–13 This interest has its roots in the noncommutative field theories and their connec-tion with quantum gravity and string theory. It is widely believed that the final theory of quantumgravity will determine the fundamental structure of space-time and does contain some sort ofnoncommutative structure. This will have of course a profound impact on the mathematical foun-dation of quantum mechanics and quantum field theory and will certainly have implications ontheir predictions. The quest to find deviations from the predictions of standard quantum mechanicsthat could be regarded as a signature of quantum gravity is one of the key motivations for therecent interest in noncommutative quantum mechanics �NCQM�, the nonrelativistic one-particlesector of noncommutative field theories.

Most of the models of NCQM considered in the literature are based on canonical extensionsof the Heisenberg algebra. In these models, time is required to be a commutative parameter. In ad-dimensional space with noncommuting position and momentum variables, the extended Heisen-berg algebra reads

a�Also at Centro de Física dos Plasmas, IST. Electronic mail: cbastos@fisica.ist.utl.pt.b�Also at Centro de Física dos Plasmas, IST. Electronic mail: orfeu@cosmos.ist.utl.pt.c�Also at Grupo de Física Matemática, UL, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal. Electronic mail:

ncdias@mail.telepac.pt.d�Also at Grupo de Física Matemática, UL, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal. Electronic mail:

joao.prata@mail.telepac.pt.

JOURNAL OF MATHEMATICAL PHYSICS 49, 072101 �2008�

49, 072101-10022-2488/2008/49�7�/072101/24/$23.00 © 2008 American Institute of Physics

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�qi, qj� = i�ij, �qi, pj� = i��ij, �pi, pj� = i�ij, i, j = 1, . . . ,d , �1�

where �ij and �ij are antisymmetric real constant �d�d� matrices and �ij is the identity matrix.Theoretical predictions for specific noncommutative systems have been compared to experimentaldata, leading to bounds on the noncommutative parameters:11,14

� � 4 � 10−40 m2, � � 1.76 � 10−61 kg2 m2 s−2. �2�

Moreover, a great deal of work has been devoted to structural and formal aspects of the quantumtheory based on the algebra �1�. The key property of the extended Heisenberg algebra is that it isrelated to the standard Heisenberg algebra:

�Ri,Rj� = 0, �Ri,� j� = i��ij, ��i,� j� = 0, i, j = 1, . . . ,d , �3�

by a class of linear �noncanonical� transformations:

qi = qi�Rj,� j�, pi = pi�Rj,� j� . �4�

This is a well-known result. Every even-dimensional, nondegenerate, antisymmetric bilinear formcan be brought to a normal form according to Darboux’s theorem. We shall therefore denote theprevious linear noncanonical transformations by Darboux �D� maps. They are a sort of quantummechanical version of the Seiberg–Witten map.15 By resorting to one of these transformations, oneis able to find a representation of the noncommutative observables as operators acting on theconventional Hilbert space of ordinary quantum mechanics, i.e., to convert the noncommutativesystem into a modified commutative system which exhibits an explicit dependence on the non-commutative parameters as well as on the particular D map. The states of the system are thenwave functions of the ordinary Hilbert space and its dynamics is determined by the Schrödingerequation with a modified � ,�-dependent Hamiltonian. Although the entire formulation is depen-dent on the particular D map used to perform the noncommutative-commutative conversion, thisis not so for the physical predictions such as expectation values and probability distributions. Still,the fact that the formalism is not manifestly invariant under a modification of the D map is adisadvantage. Moreover, both the observables and the states, being written in terms of the Heisen-berg variables �3�, do not display a simple mathematical structure. This tends to obscure theirphysical meaning.

The deformation quantization method16–29 leads to a phase-space formulation of quantummechanics alternative to the more conventional path integral and operator formulations. This is apowerful quantization procedure that can be applied to classical systems defined on arbitrarynonflat Poisson or symplectic manifolds. In the flat case it leads to the well-known Weyl–Wignerformulation of quantum mechanics which is tantamount to a phase-space representation of theHeisenberg algebra �3�. This formulation of quantum mechanics is akin to classical statisticalmechanics. Observables are represented by phase-space functions and states by a quasidistributionknown as the Wigner function. The key algebraic structure of the theory is an associative andnoncommutative �-product which carries the information about the commutation relations �3�.

The most natural representation of the algebra �1� is also in phase space where the commu-tation relations �1� can be implemented in a Lie algebraic way, i.e., following from a noncommu-tative and associative extended �-product of phase-space functions: i��A ,B�=A�B−B�A. Thisformulation leads directly to a noncommutative extension of the Weyl–Wigner formulation ofquantum mechanics. Some work concerning the deformation quantization of NCQM has recentlyappeared in the literature. Various authors proposed formulas for the quasidistribution on the planewith only spatial noncommutativity30–32 or with phase-space noncommutativity.33

The former approaches point in an interesting direction. However, they have been developedfor a particular dimension �d=2� and for specific values of the noncommutative parameters and itis difficult to see how they relate to each other and to the operator representations of the noncom-mutative algebra. A global formalism from where these formulations could be derived as particularcases is still missing. One of the main issues is the absence of a unified point of view on the

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definition of state for phase-space NCQM. A clear relation to the operator formulation of NCQMis also quite relevant and is still lacking and so is a definitive proof that the formalism is inde-pendent of the particular choice of the D map.

In this paper we will study the phase-space formulation of NCQM and address these issues.Our main tool will be the covariant generalization of the Weyl–Wigner map.20 In its originalformulation the non-covariant Weyl–Wigner map is an isomorphism between the operator and thephase-space representations of the ordinary quantum mechanics based on the algebra �3� and itprovides the simplest approach to derive most of the mathematical structure of conventionalphase-space quantum mechanics.16–29 The covariant extension of this map was studied in Ref. 20in connection with a diffeomorphism invariant formulation of Weyl–Wigner quantum mechanics.

By resorting to the covariant generalization of the Weyl–Wigner transform and to the D map,we will construct an extended Weyl–Wigner map for NCQM. This map is an isomorphism be-tween the operator and phase-space representations of the extended Heisenberg algebra �1� and,therefore, it provides a systematic approach to derive the phase-space formulation of NCQM in itsmost general form, i.e., for arbitrary d-dimensions and any value of the noncommutative param-eters. By applying it to the density matrix, we get the noncommutative counterpart of the Wignerfunction, which we shall call the noncommutative Wigner function. It represents the state of thesystem. Moreover, this extended Weyl–Wigner transform can be applied to derive the fundamentalalgebraic structure of the theory. We shall use it to

�i� get the extended �-product and a Moyal bracket encompassing the noncommutativity of thetheory and

�ii� derive the dynamical and eigenvalue equations for NCQM.

Finally, we will show that the noncommutative Weyl–Wigner transform, and therefore theentire formulation of NCQM, does not depend on the particular choice for the D map.

Our approach provides a unified framework for the phase-space formulation of NCQM. Theentire structure of the theory follows directly from the extended Weyl–Wigner map. Previousapproaches to phase-space NCQM are generalized and unified. We shall rederive some of theseformulations by restricting our formalism to their particular physical situations.

All the previous approaches focus mainly on two-dimensional systems with noncommutativityin the spatial sector only. The reason is that they do not allow for a straightforward simultaneousgeneralization to higher-dimensional systems and noncommutativity in the momentum sector.Moreover, the derivations are so different and involved that the passage from the operator formu-lation to the phase-space formulation is unclear, especially when one deals with the observables.The expressions obtained in the various approaches for the noncommutative Wigner functionslook very different �see, e.g., Refs. 30 and 31�. It would thus be very useful to investigate howthey relate to each other and whether they are equivalent. It should also be stressed that theformalisms either explicitly depend on a particular Darboux transformation30 or are not covariantunder phase-space coordinate transformations.30,31

Let us briefly summarize the main results of our approach:

�i� It generalizes the previous approaches. It is applicable to systems of arbitrary dimensionand admits noncommutativity in the momentum sector. This is important as three- andfour-dimensional systems are more realistic. Moreover, the noncommutativity in the mo-mentum sector should not be discarded, as there are situations where the momentum non-commutative corrections may be larger11 and hence more susceptible to experimental de-tection.

�ii� It unifies all previous approaches. For particular physical situations �i.e., two-dimensionalsystems, with only spatial noncommmutativity�, we are able to show �see Sec. IV� that allthe other proposals found in the literature are particular cases of our general formula �Eq.�71� of Definition 3.8�. This is an important point also because the formulas of Refs. 30 and31 do not seem to coincide.

�iii� It provides a single principle from where the entire theory can be derived. Our extendedWeyl–Wigner map �Eq. �42�� is applicable both to observables as well as density matrices.

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Consequently, by a simple application of this map to any quantum equation, one can obtainimmediately a phase-space counterpart. In particular, using this method, we can proveTheorems 3.9 and 3.14 and Lemmas 3.10–3.13.

�iv� It provides a formulation which is free from ad hoc prescriptions. the formalism is mani-festly independent of the Darboux map and is covariant, i.e., the generalized Weyl–Wignermap �Eq. �20� in Sec. II�, and thus the entire structure of the theory transforms covariantlyunder general diffeomorphisms in the noncommutative phase space.

�v� It proves several new results concerning the structure of the theory. To the best of ourknowledge Lemmas 3.10–3.13 had not been derived before for noncommutative systems.Lemmas 3.4 and 3.12 had not been proved for higher-dimensional systems with momentumnoncommutativity. It also corrects some results previously presented in the literature33 �seeSec. IV�.

This paper is organized as follows. In Sec. II, we discuss the general features of the Weyl–Wigner map and its covariant generalization. We also make a brief review of the deformationquantization method. In Sec. III, we construct the extended star product, the extended Moyalbracket. and the noncommutative Wigner function �NCWF� and extensively discuss their proper-ties. We also prove that the extended Weyl–Wigner map is independent of the particular choice ofthe D transformation. In Sec. IV, we specialize our findings to two dimensions and compare themto results already known in the literature. Section V summarizes our conclusions.

II. THE GENERALIZED WEYL–WIGNER MAP

In this section we aim to review the main structures of the covariant formulation of deforma-tion quantization with emphasis on the generalized Weyl–Wigner transform. Our analysis is re-stricted to the case of flat phase spaces. The formalism presented here will help make the forth-coming sections simpler and more self-contained. The reader is referred to Refs. 16–24 for a moredetailed presentation and to Refs. 25–29 for the generalization of the formalism to the nonflat case.

The general interest of the covariant version of the deformation quantization procedure is thatit leads to a larger set of quantum phase-space representations. We shall see in Sec. III that froman operational point of view, NCQM in phase space is a covariant phase-space quantum mechan-ics with a particular choice of noncanonical coordinates.

Let us then settle down the preliminaries: we consider a d-dimensional dynamical system,such that its classical formulation lives in the flat phase space T�M �R2d. A set of global canonicalcoordinates:

� = �R,�� = �Ri,�i�, i = 1, . . . ,d ,

� = R, = 1, . . . ,d and � = �−d, = d + 1, . . . ,2d , �5�

can then be defined in T�M in terms of which the symplectic structure reads w=dRi∧d�i. In thesequel the Latin letters run from 1 to d �e.g., i , j ,k , . . . =1 , . . . ,d�, whereas the Greek letters standfor phase-space indices �e.g., , ,� , . . . =1 , . . . ,2d�, unless otherwise stated. Moreover, summa-

tion over repeated indices is assumed. Upon quantization, the set �� , =1, . . . ,2d� satisfies the

commutation relations of the standard Heisenberg algebra and �Ri , i=1, . . . ,d� constitutes a com-

plete set of commuting observables. Let us denote by �R the general eigenstate of R associatedwith the array of eigenvalues Ri , i=1, . . . ,d and spanning the Hilbert space H=L2�Rd ,dR� ofcomplex valued functions � :Rd→C ���R�= R ���, which are square integrable with respect tothe standard Lesbegue measure dR. The scalar product in H is given by

��, �H =� dR ��R� �R� , �6�

where the overbar denotes complex conjugation.We now introduce the Gel’fand triple of vector spaces:34–37

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S�Rd� � H � S��Rd� , �7�

where S�Rd� is the space of all complex valued functions t�R� that are infinitely smooth and, as�R�→�, they and all their partial derivatives decay to zero faster than any power of 1 / �R�. S�Rd�is the space of Schwartz test functions or of rapid descent test functions,38,39 and S��Rd� is its dual,i.e., the space of tempered distributions. In analogy with �7� let us also introduce the triple

S�R2d� � F = L2�R2d,dR d�� � S��R2d� , �8�

where F is the set of square integrable phase-space functions with scalar product:

�F,G�F =1

�2���d� dR� d� F�R,��G�R,�� . �9�

Finally, let S� be the set of linear operators admitting a representation of the form24

A:S�Rd� → S��Rd�, ��R� → �A���R� =� dR� AK�R,R����R�� , �10�

where AK�R ,R��= R�A�R��S��R2d� is a distributional kernel. The elements of S� are namedgeneralized operators.

The Weyl–Wigner transform is the one-to-one invertible map:22,24,40

W�:S� → S��R2d� ,

A → A�R,�� = W��A� = �d� dy e−i�·yAK R +�

2y,R −

�

2y� = �d� dy e−i�·y

R�R +�

2y�A�R −

�

2y�

R

�11�

where the Fourier transform is taken in the usual generalized way and the second form of theWeyl–Wigner map in terms of Dirac’s bra and ket notation is more standard.1 Note that we have

introduced the subscript R to specify which eigenstates we are referring to. This will be relevantin the sequel.

There are two important restrictions of W�:

�1� The first one is to the vector space F of Hilbert–Schmidt operators on H, which admit arepresentation of the form �10� with AK�R ,R���F, regarded as an algebra with respect to the

standard operator product, which is an inner operation in F. In this space we may also

introduce the inner product �A , B�F� tr�A†B� and the Weyl–Wigner map W� : F→F becomesa one-to-one invertible unitary transformation.

�2� The second one is to the enveloping algebra A�H� of the Heisenberg–Weyl Lie algebra

which contains all polynomials of the fundamental operators R , � and I modulo the idealgenerated by the Heisenberg commutation relations. In this case the Weyl–Wigner transform

W� :A�H�→A�R2d� becomes a one-to-one invertible map from A�H� to the algebra A�R2d�of polynomial functions on R2d. In particular, W��I�=1, W��R�=R, and W����=�.

The previous restrictions can be promoted to isomorphisms if F and A�R2d� are endowed with asuitable product. This is defined by

1The Fourier transform TF of a generalized function B�S��Rn� �for n�1� is another generalized function TF�B��S��Rn� which is defined by TF�B� , t= B ,TF�t� for all t�S�Rn� �Refs. 38 and 39� and where A , t denotes the actionof a distribution A�S��Rn� on the test function t�S�Rn�.

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W��A���W��B� � W��AB� �12�

for A , B� S� such that AB� S�. The �-product admits the kernel representation

A�R,����B�R,�� =1

����2d� dR�� dR�� d��� d�� A�R�,���B�R�,���

�e−2i/����·�R−R��+�·�R�−R��+��·�R�−R�� �13�

and is an inner operation in F as well as in A�R2d�. The previous formula is also valid if we wantto compute A��B with A�F and B�A�R2d�, in which case A��B�S��R2d�. On the other hand,if A�A�R2d� and B�A�R2d��F, the �-product can also be written in the well-known form41,42

A�����B��� = A���e�i�/2����J���B��� , �14�

where ��A���= �� /��

�A��� and J are the components of the symplectic matrix

J = 0d�d Id�d

− Id�d 0d�d� . �15�

Here Id�d denotes the �d�d� identity matrix.The map Eq. �11� yields a phase-space representation of quantum mechanics which is by no

means unique. For instance, the restriction W� :A�H�→A�R2d� is one of infinitely many isomor-

phisms relating the algebras A�H� and A�R2d�. This map is associated with the symmetric order-ing prescription, the Weyl rule, for operators and admits a large class of generalizations which areassociated with different ordering prescriptions.43 Moreover, even within the context of the Weylrule, we may consider canonical operator transformations of the form

T:S� → S�, A��� → T�A���� = A����� = A������� , �16�

where ��= �R� ,��� is a new set of fundamental Heisenberg variables and the action of T istantamount to a change in representation. For the new set of fundamental variables, we may writea new Weyl–Wigner map:

W��:S� → S��R2d� ,

A → A�R�,��� = W���A� = �d� dye−i��·yR��R� +

�

2y�A�R� −

�

2y�

R�, �17�

where now �R�� �� /2�yR� are eigenstates of R�. This map is not equivalent to the original one,W�, given by Eq. �11�. In fact, if we construct the coordinate transformation �=�����=W��������� then, in general, A�����A�������,20 and there is no trivial relation between A and A.

Notice that in Eq. �17�,and also in Eq. �11�, we did not specify whether A= A��� or A= A�����. This

is because F�R� ,y��R�R�+ �� /2�y�A�R�− �� /2�yR� is invariant under a change in representation,i.e.,

R��R� +�

2y�A����R� −

�

2y�

R�=

R��R� +�

2y�A������R� −

�

2y�

R�. �18�

We conclude that the following diagram does not close:

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A��� − − − − − − − − − − − − → A����� = A�������

T

W�↓ ↓W��

A��� A����

and that, in general, the two maps W� and W�� yield two distinct phase-space representations.Also arising from the Weyl rule, there is another set of quantum phase-space representations

of the Heisenberg algebra. To construct them explicitly let us consider the more general case

where the transformation T �Eq. �16�� is not required to be canonical and thus the new set of

variables �� may no longer satisfy the Heisenberg commutation relations. In this case we shalldesignate the new variables by z= �q , p�. In general, there is no well-defined map Wz. We demand

that the transformation T be such that �→z=z���=W��z� is a phase-space diffeomorphism and

define the classical counterpart of T by letting it act as a coordinate transformation:

T:S��R2d� → S��R2d�, A��� → T�A���� = A��z� = A���z�� , �19�

where ��z� is the inverse function of z���. �Notice that we have used the notation A in Eq. �17� andwe use the notation A� here. We shall reserve the notation with the prime for the case where thetwo objects are related by a coordinate transformation: A��z�=A���z��.� We may now consider thediagram

A��� − − − − − − − − − − − − → A��z� = A���z��

T

W�↓ ↓Wz�

A��� − − − − − − − − − − − − → A��z� = A���z��

The generalized Weyl–Wigner map Wz� is the covariant generalization of W�. We have

Wz�=T �W� � T−1 or more precisely20

Wz�:S� → S��R2d� ,

A → A��z� = Wz��A� = �d� dx� dy e−i��z�·y��x − R�z��

R�x +�

2y�A�x −

�

2y�

R

, �20�

where again �x� �� /2�yR are eigenstates of R. The map Wz� is still a one-to-one invertible map

between the spaces S��H� and S��R2d� and even if z is a second set of canonical coordinates, in

general, we have Wz��Wz. The map Wz

� can be applied both to a general observable A�A�H� as

well as to the density matrix ��t�= ���t���t��� F. In the first case it yields the z�-Weyl symbol

A��z�=Wz��A� of the original quantum operator and, in the second case, it yields the covariant

generalization of the celebrated Wigner function fW��z , t�= �1 / �2���d�Wz����t��.44 Moreover, the

two restrictions Wz� : F→F and Wz

� :A�H�→A�R2d� can be turned into isomorphisms of Liealgebras once a suitable �-product is introduced in F and A�R2d�.

In analogy with Eq. �12� the covariant star product �z and Moyal bracket �,�z are definedby16,19,20

Wz��A��zWz

��B� � Wz��AB� ,

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�Wz��A�,Wz

��B��z � Wz� 1

i��A,B�� =

1

i��Wz

��A��zWz��B� − Wz

��B��zWz��A�� �21�

for all A , B� S� such that AB , BA� S�. From its definition it follows that the �z-product satisfies

A��z��zB��z� = A���z����B���z�� �22�

and that for A ,B�A�R2d�, it can be written in the explicit form

A��z��zB��z� = A��z�e�i�/2��� z��� zB��z� . �23�

�Notice that the covariant version of the �-product for the configuration variables has been con-sidered in the context of field theory in order to minimally couple a scalar field with gravity.45�This formula truncates at some finite order and is also valid for the case where A �or B� �A�R2d�and B �or A� �F. In Eq. �23� the covariant derivatives are given by

�zA��z� = �z

A��z� = ��/�z�A��z� ,

�z�z

A��z� = �z�z

A��z� − ���z�A��z� , �24�

where the Christoffel symbols read �we assume that the phase space has a flat structure�

�� =�z�

���

�2��

�z � z

, �25�

and the transformed symplectic form � is given by

� =�z

���

�z

���

J�� =�z

�Rk

�z

��k−

�z

��k

�z

�Rk. �26�

We will see in Sec. III that when the transformation �→��z� is linear, the kernel expression of the�-product Eq. �13� also admits a simple covariant generalization.

We complete the presentation of the main structures of the covariant formulation of phase-

space quantum mechanics by introducing the trace formula for a general trace-class operator A

� F:20

Tr�A� =1

�2���d� dz� ��

�z�Wz

��A� =1

�2���d� dz�det �

A��z� , �27�

and for the product of two operators A , B� F, we get

Tr�AB� =1

�2���d� dz�det �

A��z��zB��z� =1

�2���d� dz�det �

A��z�B��z� , �28�

where we have used the fact that �cf. Eq. �22��

� dz�det �

A��z��zB��z� =� d� A�����B��� =� d� A���B��� =� dz�det �

A��z�B��z� . �29�

It follows that

�A,B�F = tr�A†B� =1

�2���d� dz�det �

A��z�B��z� = �A�,B��F �30�

and so Wz� : F→F is a unitary transformation.

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When formulated in terms of these structures, phase-space quantum mechanics becomes fullyinvariant under the action of general coordinate transformations. The covariant generalization ofthe Moyal and stargenvalue equations read

� fW�

�t= �H�, fW��z, � A��z��z�a,b� �z� = a�a,b� �z�

�a,b� �z��zA��z� = b�a,b� �z� ,� �31�

where �a,b� �z� is the a-left and b-right �z-genfunctions of A��z�, �a,b� �z�= �2���−dWz���ab��, and H�

is the z�-symbol of the Hamiltonian H. Finally, the probability distributions for a general observ-

able is given by

P�A��z,t� = a� =� dz�det �

�a,a� �z�fW��z,t� . �32�

A final remark is in order: From the trace formula, it follows that the covariant Wignerfunction fW��z , t�= �1 / �2���d�Wz

����t�� does not satisfy the normalization condition �dz fW��z , t�=1. Instead we have

� dz�det �

fW��z,t� = 1. �33�

In order to prevent this situation, we shall introduce the following convention, which is the onethat renders our future results simpler. If det � is a constant, which is the case for the D map, weredefine the Wigner function as

fW��z,t� →1

�det �fW��z,t� , �34�

so that the quasidistribution is normalized. On the other hand, if det � is not a constant, then weshall keep the original definition in order not to be forced to change the main equations that shouldbe satisfied by the Wigner function, such as the Moyal and stargenvalue equations.

III. NONCOMMUTATIVE QUANTUM MECHANICS IN PHASE SPACE

In this section we shall consider a quantum system in d dimensions with both spatial andmomentum noncommutativities. This is expressed in terms of the extended Heisenberg algebrasatisfying the commutation relations

�qi, qj� = i�ij, �qi, pj� = i��ij, �pi, pj� = i�ij, i, j = 1, . . . ,d , �35�

where �ij and �ij are antisymmetric real constant �d�d� matrices. We shall assume that thesematrices are both invertible and that

�ij � �ij +1

�2�ik�kj �36�

is equally an invertible matrix. This will certainly happen if for any matrix elements � and �, theirproduct is considerably smaller than �2:

�� � �2. �37�

We will tacitly assume this to be the case. Notice that this condition is experimentally fulfilled, forinstance, in the case of the noncommutative gravitational quantum well.11 It is well known thatunder a linear transformation, this algebra can be mapped to the usual Heisenberg algebra:

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�Ri,Rj� = 0, �Ri,� j� = i��ij, ��i,� j� = 0, i, j = 1, . . . ,d , �38�

via the D map,15 which can be cast in the form

qi = AijRj + Bij� j, pi = CijRj + Dij� j , �39�

where A ,B ,C ,D are real constant matrices. We shall assume that this transformation is invertible.The transformation is easily shown to obey the following lemma:33

Lemma 3.1: The matrices A ,B ,C ,D are solutions of the equations

ADT − BCT = Id�d, ABT − BAT =1

��, CDT − DCT =

1

�N , �40�

where the superscript T stands for matrix transposition. A ,B ,C ,D ,� ,N are the matrices withentries Aij ,Bij ,Cij ,Dij ,�ij ,�ij, respectively.

The D transformation ��39� and �40�� can be expressed more compactly as follows. If as

before, we write �= �R ,��, z= �q , p�, we have

��, �� = i�J, �z, z� = i��, , = 1, . . . ,2d ,

where �J� and ��� are the elements of the 2d�2d matrices J �cf. �15�� and �:

� =�1

�� Id�d

− Id�d1

�N � .

For a D map,

z = S�,

we conclude that the constant real matrix S= �S� satisfies

SJST = � ,

and so S is manifestly not a symplectic matrix.Thanks to this linear transformation, the noncommutative algebra Eq. �35� admits a represen-

tation in terms of the Hilbert space of ordinary quantum mechanics. Notice that the D map is notunique. Indeed, the Heisenberg algebra �38� is only defined up to a unitary transformation. Con-sequently, there should be an infinity set of solutions to the set of constraints �40�. Indeed, thereare four �d�d� matrices to be determined, i.e., 4d2 real parameters. However, in �40� there ared�3d−1� /2 independent equations, leaving a total of d�5d+1� /2 free parameters.

The aim of this section is to find a phase-space representation of the commutation relations

�35� and, by doing so, provide a phase-space formulation of NCQM. Let A�q , p� , B�q , p�� S� be

two generalized operators. Two important subspaces of S� are the enveloping algebra of the

extended Heisenberg algebra AE�H� and the algebra of Hilbert–Schmidt operators F. It follows

from the D map that AE�H�=A�H�. We are looking for a one-to-one invertible linear map

V : S�→S��R2d� satisfying

�i� V�I�=1,�ii� V�q�=q,�iii� V�p�= p, and

�iv� V�A�q , p�B�q , p��=V�A�q , p���V�B�q , p�� for some suitable product � and provided AB

� S�.

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Notice that these properties immediately impose the extended Heisenberg commutation rela-tions in phase space:

V��z, z�� = V�z� � V�z� − V�z� � V�z� = �V�z�,V�z���, �41�

where we defined the bracket associated with the product �.There is a large set of maps V that satisfy properties �i�–�iv�. This set can be divided into

several subsets, each one being characterized by a particular ordering prescription for operators.The class of maps V which are associated with the Weyl rule can be constructed by resorting to thestandard Weyl–Wigner map:

�42�

and since the D transformation is linear, there are no ordering ambiguities and thus D and D havethe same functional form. One can easily see that Wz

� provides the desired phase-space represen-tation for NCQM. Indeed the map Wz

� satisfies the four required properties and defines a new�-product in phase space through condition �iv�. In Secs. III A and III B we will study thealgebraic structure of the new theory and in Sec. III C we shall characterize its states. Finally, inSec. III D we shall also show that the map Wz

�, and thus the entire structure of the phase-spacerepresentation of NCQM, is independent of the particular choice for the D map.

Notice also that because the generalized Weyl–Wigner map is covariant under phase-spacediffeomorphisms, we may without effort write formulas, e.g., the star product or the NCWF if wechoose, for instance, another set of unitarily equivalent variables:

q� = UqU−1, p� = UpU−1, UU† = 1.

Obviously q� , p� also obey the extended Heisenberg algebra �1�.

A. The � product

Using the generalized Weyl–Wigner map �42�, we now construct a �-product for the NCQMin phase space. In A�R2d� we can use Eq. �23� to write the �-product in terms of the variables z.In this context in order to keep the notation simple, we shall write A�z� and � instead of A��z� and�z, respectively. Since the D map is a linear transformation, the Christoffel symbols in �25� vanishand the covariant derivatives �24� reduce to the standard derivatives. As for the symplectic form,we get from Eqs. �26�, �39�, and �40�

�qiqj= AikBjk − BikAjk =

1

��ij . �43�

Likewise:

�qipj= �ij, �pipj

=1

��ij . �44�

Altogether:

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� =�1

�� Id�d

− Id�d1

�N � . �45�

We have thus proved the following theorem:Theorem 3.2: For A ,B�A�R2d�, the �-product in the noncommutative variables z= �q , p�

reads

A�z� � B�z� = A�z�e�i�/2���z���zB�z� = A�z�������B�z� , �46�

where

A�z���B�z� � A�z�e�i�/2���zJ��zB�z� , �47�

A�z���B�z� � A�z�e�i/2���� /�qi��ij��� /�qj�B�z� , �48�

A�z���B�z� � A�z�e�i/2���� /��pi�ij��� /�pj�B�z� . �49�

We can cast all these products in a compact notation:

A�����B��� = A���e�i/2����R�RS���SB��� . �50�

In the case of � and ��, the variable � stands for �=z= �q , p� in the 2d-dimensional phase space�R ,S=1, . . . ,2d�. The symplectic matrices in these cases read

� = �� if�� = � , � = �J if �� = ��. �51�

In the case of �� and ��. the variable � stands for �=q or �= p �R ,S=1, . . . ,d�, respectively, andthe remaining variables in Eqs. �48� and �49�, p or q, are just regarded as fixed parameters in thecorresponding d-dimensional configuration or momentum spaces. The symplectic matrices thusread

� = � if �� = ��, � = N if �� = ��. �52�

Then we can derive two additional representations for these �-products.Lemma 3.3: A �-product of the form �50� acting on the space of polynomials on phase space

���=� or ��=���, configuration space ���=���, or momentum space ���=��� can be representedas a Bopp shift:

A�����B��� = A � +i

2�����B��� = A���B � −

i

2����� . �53�

Proof: Assuming that A���, B��� are polynomials, then they admit a generalized Fouriertransform. We may thus write

A��� =� d� f���ei�·�, B��� =� d� g���ei�·�. �54�

Here �=z �for ��=� or ��=���, �=q �for ��=���, or �= p �for ��=���. The coefficientsf��� ,g��� are singular. From �50� we get

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A�����B��� =� d�� d� f���g���ei�·�e�i/2����R�RS���Sei�·� =� d�� d� f���g���ei�R��R+�i/2��RS���S

�ei�·�

= A � +i

2�����B��� . �55�

The second identity in �53� is proved in a similar way. �

Lemma 3.4: Let the matrix � be invertible. Then the �-product of the form �50� admits akernel representation:

A�����B��� =1

�n�det ��� d��� d�� A����B����e�2i�� − ���T�−1���−���, �56�

for A ,B�A�Rn�. Moreover, this kernel representation is equally valid for A ,B�A�R2d��F if�=��. Here n stands for 2d or d depending on whether the �-product is �, ��, or ��, ��,respectively.

Proof: Let us prove the equivalence of �50� and �56� for polynomials A ,B. Again, A��� andB��� admit the Fourier transforms �54� with inverse forms given by

f��� =1

�2��n� d� A���e−i�·�,g��� =1

�2��n� d� B���e−i�·�. �57�

From �53� we get

A�����B��� =� d�� d� f���g���ei�R��R+�i/2��RS���S�ei�·� =� d�� d� f���g���e�i�·�+i�·�−�i/2��T���.

�58�

Upon substitution of Eq. �57� into Eq. �58�, we get

A�����B��� =1

�2��2n� d�� d�� d��� d�� A����B����exp − i� · �� − i� · �� + i� · � + i� · �

−i

2�T��� . �59�

Let us first perform the integration over �:

� d� e�−i�·��+i�·�−�i/2��T��� = �2��n� � − �� −1

2�T�� =

�4��n

�det ����� − 2�−1��� − ��� . �60�

From Eqs. �59� and �60� we recover �56� upon integration over �.Finally, we prove the last statement in the lemma. Let C��� and D��� be the Weyl symbols of

some operators C, D in A�H�� F. The kernel representation of the ��-product Eq. �13� can bewritten in the form

C�����D��� =1

����2d� d��� d�� C����D����e�−�2i/���� − ���TJ���−���. �61�

If we perform the D transformation:

� → z = T�, C��� → C��z� = C���z��, D��� → D��z� = D���z�� , �62�

with T=�z /��. The symplectic matrix transforms as in �26�:

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� = TJTT, �63�

and det T= �det ��1/2. Under the D map, the kernel representation �61� transforms according to�cf. �22��

�C�����D�����=��z� = C��z� � D��z�

=1

����2d� dz�� dz��det T�−2C��z��D��z��e�−�2i/���z − z��T�T−1�TJT−1�z�−z��.

�64�

From �63� we have �−1=−�T−1�TJT−1, and thus:

C��z� � D��z� =1

����2d�det ��� dz�� dz� C��z��D��z��e��2i/���z − z��T�−1�z�−z��. �65�

If we redefine A�z��C��z�, B�z��D��z�, we recover �56�. �

Remark 3.5: Notice that for the expression �56� to make sense, it is crucial that the matrix �is invertible. According to our assumptions, Eq. �37�, and the preceding discussion, this is cer-tainly the case for the matrices J, �, and N. The matrix � also has an inverse:

�−1 =�1

�N�−1 − ��−1�T

�−1 1

����−1�T� , �66�

where �−1 is the inverse of the matrix �36�.Theorem 3.6: For a �-product of the form �56�, the following identity holds:

� d� A�����B��� =� d� A���B��� . �67�

Proof: From the kernel representation, we have

� d� A�����B��� =1

�n�det ��� d�� d��� d�� A����B����e�2i�� − ���T�−1���−���

=1

�n�det ��� d��� d�� �n�det ������ − ���A����B����e−2i��T�−1��

=� d� A���B��� , �68�

where the antisymmetry of � has been used in the last step. Notice that this formula is inagreement with �28�. �

B. The Moyal bracket

The extended Moyal bracket is defined according to Eq. �21�.Definition 3.7: For A ,B�S��R2d� such that A�B and B�A are also in S��R2d�, the noncom-

mutative Moyal bracket is defined by

�A�z�,B�z��� �1

i��A�z� � B�z� − B�z� � A�z�� . �69�

In particular, this definition is valid for A ,B�A�R2d��F. �

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For A�A�R2d� and B�A�R2d��F, one easily realizes that the Moyal bracket can be writtenin the form

�A�z�,B�z��� �1

i��A�z� � B�z� − B�z� � A�z�� =

1

i��A�z�e�i�/2���z

���zB�z� − B�z�e�i�/2���z���zA�z��

=1

i�A�z��e�i�/2���z

���z − e−�i�/2���z���z�B�z� =

2

�A�z�sin �

2��z

���z�B�z� , �70�

where in the penultimate step, we have used the antisymmetry of the matrix �.

C. The Wigner function

Following the strategy of Sec. III A, we can use the generalized Weyl–Wigner map to definethe NCWF. Since the D map is a linear transformation, the Jacobian is a constant and we mayabsorb it in the definition of the NCWF to get a normalized distribution �cf. the remark at the endof Sec. II�:

Definition 3.8: Let the system be in a pure or mixed state represented by a density matrix �

� F. The NCWF in the noncommutative variables z= �q , p� is defined by

fNC�z� �1

�det ��1/2�2���dWz���� , �71�

where det � is the determinant of �45�. �

It follows from the definition of fNC�z� that if �= �R ,�� is the set of Heisenberg variablesobtained via the D map �39� and we compute the ordinary Wigner function fW�����1 / �2���d�W���� associated with �, then fNC�z�= ��det ���−1/2fW���z��, where ��z� is the inversetransformation of the D map. Moreover, since the D map is a linear transformation, the Jacobiandet �−1/2 is a constant, and it follows from Eq. �33� that the NCWF is normalized:

� dz fNC�z� = 1. �72�

Other properties of the NCWF follow directly from application of the generalized Weyl–Wignermap. For instance, if we apply this map to the von Neumann equation

i�� �

�t= �H, �� , �73�

we immediately get the following theorem:Theorem 3.9: The dynamics of the NCWF is dictated by the noncommutative von Neumann–

Moyal equation

� fNC

�t�z,t� = �H�z�, fNC�z,t���, �74�

where

H�z� � Wz��H��z� . �75�

Likewise, suppose that ��a , ��b are normalized eigenstates of a certain operator A�z��A�H�with eigenvalues a ,b:

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A��a = a��a, A��b = b��b . �76�

If we apply the generalized Weyl–Wigner map to the nondiagonal density matrix element

��a�b�� F, we obtain the nondiagonal NCWF

fabNC�z� = �det ��−1/2fab

W ���z�� , �77�

where

fabW ��� �

1

����d� dy e−2ip·y/�R + y��a�b�R − y . �78�

We then get the following:Lemma 3.10: The nondiagonal NCWF �77� is a solution of the �-genvalue equations

A�z� � fabNC�z� = afab

NC�z�, fabNC�z� � A�z� = bfab

NC�z� , �79�

where A�z��Wz��A�.

Proof: This result can be easily proved if we apply the generalized Weyl–Wigner map to theequations

A��a�b� = a��a�b�, ��a�b�A = b��a�b� �80�

and multiply them by �2���−d�det ��−1/2. �

Lemma 3.11: Let ���a ,a� I� be a complete orthonormal basis of the Hilbert space. Let�fab

NC�z� ,a ,b� I� be the nondiagonal NCWF as in Eqs. �77� and �78�. Then the following identitieshold for all a ,b ,c ,d� I:

fabNC�z� � fcd

NC�z� =�bc

�2���d�det ��1/2 fadNC�z� . �81�

Proof: The nondiagonal density matrix elements ���a�b� , �a ,b� I�� obey the orthogonalityconditions

��a�b���c�d� = �bc��a�d� . �82�

If we multiply these equations by �2���−2d�det ��−1 and apply the generalized Weyl–Wignermap, we recover �81�. �

Lemma 3.12: If a system is in a pure state, then the corresponding NCWF satisfies

fNC�z� � fNC�z� =1

�2���d�det ��1/2 fNC�z� . �83�

Proof: If the system is in a pure state, then the corresponding density matrix satisfies �2= �.Again, we multiply this equation by �2���−2d�det ��−1 and apply the generalized Weyl–Wignermap. �

Lemma 3.13: A phase-space function fNC�z� is a NCWF if and only if there exists a complexvalued phase-space function g�z� such that

�i� �dz�g�z��2=1 and�ii� fNC�z�= g�z��g�z�.

Proof: A linear, trace-class operator � is a density matrix if and only if there exists an operatora such that �1� tr�a†a�=1 and �2� �= a†a.17 If we apply the generalized Weyl–Wigner map to thesecond condition, we get

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fNC�z� = a�z��2���d/2�det ��1/4� � a�z�

�2���d/2�det ��1/4� , �84�

where obviously a�z�=W�z�a�. If we define g�z���2���−d/2�det ��−1/4a�z�, we recover condition

�ii� in the lemma. Moreover, from Eqs. �27� and �28�, we have

1 = tr�a†a� =1

�2���d�det ��1/2� dz a�z� � a�z� =� dz a�z��2���d/2�det ��1/4�

� a�z��2���d/2�det ��1/4� =� dz�g�z��2, �85�

and we recover condition �i� in the lemma. Conversely, suppose that fNC�z� is such that �i� and �ii�hold. Then, let us define

a � �2���d/2�det ��1/4�Wz��−1�a�z��, � � �2���d�det ��1/2�Wz

��−1�fNC�z�� , �86�

where �Wz��−1 is the inverse generalized Weyl–Wigner map. Then from �i� and �ii�, we conclude

that tr�a†a�=1 and �= a†a. This means that fNC is the generalized Weyl–Wigner transform of aquantum density matrix, i.e., a NCWF. �

Theorem 3.14: The expectation value is an observable A�z��A�H� in a state �� F, such that

A�� F is evaluated according to

A =� dz A�z�fNC�z� , �87�

where A�z�=W�z�A� and fNC�z� is the NCWF associated with �.

Proof: From Eqs. �27� and �28� we have

A = Tr�A�� =1

�2���d� dz�det �

A�z� � W�z��� =� dz A�z� � fNC�z� =� dz A�z�fNC�z� .

�88�

�

D. Independence of Wz� from the particular choice of the Darboux map

It may strike the reader that the NCWF as defined by Eq. �71� seems to be explicitly depen-dent on the D map. Clearly this is physically unacceptable. Also, from a mathematical point ofview, this seems somewhat paradoxical. Indeed, suppose a NCWF fNC�z� is a solution of thenoncommutative �-genvalue Eq. �79� or of the noncommutative von Neumann–Moyal Eq. �74�.Since neither the �-product �46� nor the Moyal bracket �70� depends on the D map, this iscontradictory with the fact that the solution fNC�z� has the form �71� which apparently depends onthe D transformation. However, this is only apparently so. In this section we will prove that theextended Weyl–Wigner map is independent of the particular choice of the D transform. Hence theentire phase-space formulation of NCQM is invariant under a modification of the D map.

Suppose that � and �� are two sets of Heisenberg variables �38�, related by a unitary trans-formation:

� = U��U†, UU† = 1. �89�

We may define two Weyl–Wigner maps W�, W�� �cf. �11� and �17�� and consequently two gener-

alized Weyl–Wigner maps Wz�, Wz

��. Since the two D maps are linear diffeormorphisms, then theunitary transformation �89� is itself a linear diffeomorphism:

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� = �����, ��z� = �����z�� . �90�

The two generalized Weyl–Wigner maps act on a generic element A of the algebra, which may bean operator or a density matrix, as

Wz��A� = A���z�� � A1�z�, Wz

���A� = A�����z�� � A2�z� , �91�

where

A��� = W��A�, A����� = W���A� . �92�

The obvious question is then whether A1�z�=A2�z�. From the unitary transformation �89� and �90�,we may write

A��� = A�U��U†� = UA����U†, �93�

and thus,

A����� � W���A� = W���UA����U†� = U��A������U , �94�

where U=U����=W���U� and the product �� is the Moyal product �12� with respect to the vari-ables ��. Since the unitary transformation �89� is linear �cf. �39� and �90��, it acts as a localcoordinate transformation in phase space,20 i.e.,

A����� = A�U������U� = A������� . �95�

Hence,

A2�z� = A�����z�� = A������z��� = A���z�� = A1�z� , �96�

where we used �90�, �91�, and �95�. This proves that the result is the same irrespective of the

particular D map. Hence Wz��=Wz

�. If we now go back to our definition of the NCWF, we come tothe conclusion that in order for the NCWF fNC�z� to be independent of the D map, the associatedWigner function fW��� must be based on a density matrix R+y���R−y or a wave function ��R�which depends explicitly on the D map. This is explicitly illustrated in Sec. IV for the harmonicoscillator �cf. Eqs. �115�–�121��.

IV. COMPARISON WITH OTHER PROPOSALS

All the results presented in Sec. III for the deformation quantization of noncommutativesystems are valid for an arbitrary number of dimensions and general noncommutativity, i.e., bothfor position and momentum. The aim of this section is to compare our general results to knownproposals that can be found in the literature. In Refs. 30–32 the authors consider the two-dimensional plane with only spatial noncommutativity. In this case, the extended Heisenbergalgebra simplifies considerably:

�qi, qj� = i��ij, �qi, pj� = i��ij, �pi, pj� = 0, i, j = 1,2, �97�

where �12=−�21=1, �11=�22=0 and � is the only noncommutativity real parameter. A possible Dmap for this algebra is given by

Ri = qi +�

�2���ijpj,�i = pi. �98�

The Jacobian is simply given by

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det � = 1. �99�

The authors in Refs. 30 and 32 proposed the following expression for the NCWF:

fNC�q,p� =1

����2� dy e−2ip·y/���q + y�����q − y� , �100�

where ��q��L2�R2 ,dq� and the �� product is given by Eq. �48� with d=2 and �ij =��ij:

A�q���B�q� = A�q�e�i�/2���� /�qi��ij��� /�qj�B�q� . �101�

The following lemma explains how Eq. �100� relates to our general formula Eq. �71�.Lemma 4.1: If we choose the linear transformation �98� as our D map, then we can cast our

formula �71� in the form �100�.Proof: Let us compute the �� product using the kernel representation �56�. We denote by E the

matrix with entries �ij. Notice that the ��-product looks exactly like the usual ��-product, whereq1 ,q2 ,� ,E play the role of q , p ,� ,J, respectively. The kernel form �56� is then directly applicable.It thus follows that

��q + y�����q − y� =1

����2� dq�� dq� ��q� + y���q� − y�e�−�2i/���q − q��TE�q�−q��

=1

����2� du� dv��u� ��v�e�−�2i/���q − u + y�TE�v+y−q��. �102�

Substituting this expression in Eq. �100� and integrating over y, we obtain

fNC�q,p� =1

����2� du� dv � v − 2q −�

�Ep + u���u���v�e�−�2i/���q − u�TE�v−q��

=1

����2� du ��u�� 2q +�

�Ep − u�e−�2i/��p·�u−q�, �103�

where we have used the antisymmetry of the matrix �ij. Finally we perform the substitution u=q+ �� /2��Ep+y to obtain

fNC�q,p� =1

����2� dy � q +�

2�Ep + y�� q +

�

2�Ep − y�e−2ip·y/�. �104�

Notice that from Eq. �98�, we can rewrite this as

fNC�q,p� =1

����2� dy ��R�q,p� + y���R�q,p� − y�e−2i��q,p�·y/� = fW�R�q,p�,��q,p�� ,

�105�

in agreement with �71�. We recall that the Jacobian of the transformation is equal to 1. �

In Ref. 31 the authors considered again the plane with spatial noncommutativity �97�. Since,e.g., q1 and p2 commute, they derived the deformation quantization of such systems by resortingto the ket basis ��q1 , p2�. They thus obtained the following expression for the NCWF:

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fNC�q,p� �1

�2���2� dx1� dy2 q1 +x1

2+

�y2

2�,p2 −

y2

2� q1 −

x1

2−

�y2

2�,p2

+y2

2�e�i/��x1p1+�i/��y2q2. �106�

Here �q1 , p2� is the q1 , p2 representation of some Hilbert space state � , i.e., �q1 , p2��q1 , p2 � . Again it is possible to relate their expression to our general formalism:

Lemma 4.2: If we define ��q1 ,q2��L2�R2 ,dq1dq2� by

��q1,q2� =1

�2��� dp2 q1 −

�

2�p2,p2�e�i/��q2p2, �107�

�q1,p2� =1

�2��� dq2 � q1 +

�

2�p2,q2�e−�i/��q2p2, �108�

then formula �106� in Ref. 33 is equivalent to �71� and �100�.Proof: Let us substitute the expression �108� into Eq. �106� and perform the change in variable

x1=−2y1−�y2 /2�. The result is

fNC�q,p� =2

�2���3� dy1� dy2� dq2�� dq2� � q1 +�

2�p2 − y1,q2��� q1 +

�

2�p2

+ y1,q2��e��i/���q2�p2−�1/2�q2�y2−q2�p2−�1/2�y2q2�−2p1y1−��/2��p1y2+y2q2��. �109�

The integration over y2 yields 4����−q2�−q2−�p1 /�+2q2�. Finally, if we integrate over q2� andperform the substitution q2�=q2− �� /2��p1−y2, we obtain

fNC�q,p� =1

����2� dy1� dy2 � q1 +�

2�p2 − y1,q2 −

�

2�p1 − y2�� q1 +

�

2�p2 + y1,q2 −

�

2�p1

+ y2�e�−�2i/���p1y1+p2y2��, �110�

which has the form �104�. �

In Ref. 33 the following strategy for solving eigenvalue problems in noncommutative phasespace is adopted:

�i� Start from a quantum Hamiltonian H�z� written in terms of the noncommutative variablesz= �q , p�.

�ii� Perform a D transformation z→ z��� and define a new Hamiltonian in terms of the Heisen-

berg set �= �R ,��: H����� H�z����;�iii� Apply the ordinary Weyl–Wigner map W� to the eigenvalue equation

H������E = E��E �111�

to obtain

H������fEW��� = EfE

W��� , �112�

where obviously, H�����W��H�� and

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fEW��� � �2���−dW����E�E�� =

1

����d� dy e−2i�·y/��E�R − y��E�R + y� . �113�

Examining diagram 2, we realize that the authors of Ref. 33 did not complete the square:

B�z� − − − − − − − − − − − − − → B����

D−1

↓W�

B����

as the inverse D transformation in phase space was not performed. Hence, their solutions areclearly explicitly dependent on the particular D map chosen. This is indeed what happens as canbe seen from the specific example of the harmonic oscillator:

H�q, p� =p2

2m+

1

2m�2q2, �114�

on the plane and with spatial and momentum noncommutativity,

�qi, qj� = i��ij, �qi, pj� = i��ij, �pi, pj� = i��ij, i, j = 1,2. �115�

The following D map was used:

qi = �Ri −�

2���ij� j, pi = ��i +

�

2���ijRj , �116�

where the parameters � ,� are subject to the constraint

��

4�2 = ���1 − ��� . �117�

The corresponding Jacobian reads

��q,p���R,��

= �det ��1/2 = 1 −��

�2 . �118�

The D transformation is thus invertible provided ����2, in which case we have

Ri = � 1 −��

�2 �−1/2 qi +�

2����ijpj� ,

�i = � 1 −��

�2 �−1/2 pi −�

2����ijqj� . �119�

In our notation, the resulting stargenfunctions �112� and �113� are given by

fn1,n2

W �R,�� = e−��/�R2+�/��2�Ln1 2

��+�Ln2

2

��−� , �120�

where Ln are the Laguerre polynomials, n1 ,n2 are non-negative integers, and

2 ��2m�2

2+

�2

8m�2�2 ,

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2 ��2

2m+

m�2�2

8�2�2 ,

�� =

R2 +

�2 � 2R�� . �121�

One can immediately realize that, as mentioned above, the stargenfunctions depend explicitly onthe parameters � ,� of the D map. It is not difficult to see that if one uses the inverse D transfor-mation �119� and the constraint �117� and multiplies by the normalization �det ��−1/2= �1−�� /�2�−1, the resulting NCWF fn1,n2

NC �q , p� is independent of � ,�.

V. CONCLUSIONS

In this paper we have developed the Weyl–Wigner formalism in the context of a phase-spacenoncommutative extension of quantum mechanics. Our strategy relies on a covariant formulationof deformation quantization, which is based on an extension of the Weyl–Wigner map. Using thisformalism, we constructed the extended �-product, the extended Moyal bracket, as well as thenoncommutative Wigner function. Moreover, we derived several properties of the extended Weyl–Wigner map and explicitly proved that albeit apparently dependent on the particular choice of theD transformation, it is in fact invariant under different choices of such maps. This in agreementwith the fact that neither the extended �-product nor the extended Moyal bracket depends on theD map. The generality of our results allow us to obtain the NCWF for a large class of two-dimensional spaces and compare our results to other models discussed in the literature.

The main advantage of our approach is that it incorporates and generalizes all the previousproposals in a unified framework. In fact, all the models discussed in the literature are particularcases of our formalism, for instance, d=2, spatial noncommutativity. However, our proposal forthe Wigner function �71� is valid for any arbitrary dimension and for both spatial as well asmomentum noncommutativities. The results of Refs. 30–32 are nevertheless useful. We arguedthat our formula �71� is apparently dependent on the choice of the D map but not the physicalpredictions. Expression �106� of Ref. 31 has the advantage of being explicitly independent of theD map. This is because the authors there resorted to the �q1 , p2 representation. Moreover, usingthis representation, their method can be easily extended to the case where noncommutativity ofmomenta is included. However, we can anticipate some difficulties with this approach if weconsider higher-dimensional �d�3� systems. For instance, in three dimensions, we cannot findthree observables among �qi , pj , i , j=1,2 ,3� which are mutually commutative, as was the case ofq1 , p2 or q2 , p1 in two dimensions. Probably, one will be forced to construct linear combinations ofthese.

By contrast, as we have shown in Lemma 4.1, expression �100� of Ref. 30 corresponds to aspecific choice �98� of the D map. In a forthcoming paper46 we will show that this expression alsohas its advantages. Indeed, in dimension d with spatial noncommutativity:

�qi, qj� = i�ij, �qi, pj� = i��ij, �pi, pj� = 0, i, j = 1, . . . ,d , �122�

we may easily generalize the D map �98�:

Ri = qi +1

2��ijpj, �i = pi, i, j = 1, . . . ,d . �123�

Following the proof of Lemma 4.1, we can show that our expression �71� can be cast in the form

fNC�q,p� =1

����d� dy e−2ip·y/���q + y�����q − y� , �124�

where the ��-product is given by �48�, and so, the formula of Ref. 30 is easily generalized tod-dimensions with spatial noncommutativity �122�.

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This formula has an immediate consequence. If we integrate over the momenta, we obtain

� dp fNC�q,p� = ��q�����q� .

Because of the ��-product inserted between the two wave functions, this marginal distribution maytake on negative values. This means that it cannot be interpreted as a joint probability distributionfor q1 ,q2 , . . . ,qd. This is in contrast with ordinary Wigner functions where the same integrationwould yield the spatial probability distribution ���q��2�0. This shows explicitly that the sets ofWigner and noncommutative Wigner functions do not coincide. This is a very important point.Remember that, in the Heisenberg–Schrödinger operator formulation, the Hilbert space for NCQMis the same as that of standard quantum mechanics. In contrast with this situation, the sets ofWigner and noncommutative Wigner functions do not coincide. This means that the Weyl–Wignerformulation is more suited to determine whether a state �a phase-space function� represents astandard quantum mechanical system or a noncommutative quantum mechanical system. It willalso allow us to device criteria for assessing whether a transition from noncommutative to ordinaryquantum mechanics has taken place �in analogy with the classical limit of standard quantummechanics�. This research program was initiated in Ref. 47, where some of us derived the non-commutative version of the Hu–Paz–Zhang equation for a Brownian particle linearly coupled to abath of oscillators at thermal equilibrium on the plane with spatial noncommutativity. We willreturn to these issues in a forthcoming paper.46

ACKNOWLEDGMENTS

The work of C.B. is supported by Fundação para a Ciência e a Tecnologia �FCT� underFellowship No. SFRH/BD/24058/2005. The work of N.C.D. and J.N.P. was partially supported byGrant Nos. POCTI/MAT/45306/2002 and POCTI/0208/2003 of the FCT.

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