A deformation quantization theory for noncommutative quantum mechanicsNuno Costa Dias, Maurice de Gosson, Franz Luef, and João Nuno Prata Citation: Journal of Mathematical Physics 51, 072101 (2010); doi: 10.1063/1.3436581 View online: http://dx.doi.org/10.1063/1.3436581 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/51/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effective action for noncommutative Bianchi I model AIP Conf. Proc. 1540, 113 (2013); 10.1063/1.4810779 Explicit formulas for noncommutative deformations of C P N and C H N J. Math. Phys. 53, 073502 (2012); 10.1063/1.4731236 Toward an axiomatic formulation of noncommutative quantum field theory J. Math. Phys. 52, 032303 (2011); 10.1063/1.3567411 Free-fall in a uniform gravitational field in noncommutative quantum mechanics J. Math. Phys. 51, 102106 (2010); 10.1063/1.3466812 Deformation quantization in the teaching of quantum mechanics Am. J. Phys. 70, 537 (2002); 10.1119/1.1450573

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A deformation quantization theory for noncommutativequantum mechanics

Nuno Costa Dias,1,a Maurice de Gosson,2,b Franz Luef,2,3,c andJoão Nuno Prata1,d

1Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologias,Av. Campo Grande, 376, 1749-024 Lisboa, Portugal and Grupo de Física Matemática,Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal2NuHAG Fakultät für Mathematik, Universität Wien, Wien 1090, Austria3Department of Mathematics, UC Berkeley, 847 Evans Hall, Berkeley,California 94720-3840, USA

Received 9 April 2010; accepted 4 May 2010; published online 1 July 2010

We show that the deformation quantization of noncommutative quantum mechanicspreviously considered by Dias and Prata “Weyl–Wigner formulation of noncom-mutative quantum mechanics,” J. Math. Phys. 49, 072101 2008 and Bastos, Dias,and Prata “Wigner measures in non-commutative quantum mechanics,” e-printarXiv:math-ph/0907.4438v1; Commun. Math. Phys. to appear can be expressedas a Weyl calculus on a double phase space. We study the properties of the star-product thus defined and prove a spectral theorem for the star-genvalue equationusing an extension of the methods recently initiated by de Gosson and Luef “Anew approach to the -genvalue equation,” Lett. Math. Phys. 85, 173–183 2008.© 2010 American Institute of Physics. doi:10.1063/1.3436581

I. INTRODUCTION

The incorporation of space-time noncommutativity in quantum field theory was consideredoriginally by Snyder,46,47 Heisenberg,32 Pauli,40 and Yang50 as a means to regularize the ubiquitousdivergences in quantum field theories. However, the development of renormalization techniquesand certain undesirable features of noncommutative field theories—such as the breakdown ofLorentz invariance—has hindered further research in this direction. It was only more recently thatnew interest in the topic has emerged due to important developments in various approaches to thequantization of gravity. The most prominent result was the discovery that the low energy effectivetheory of a D-brane in the background of a Neveu–Schwarz–Neveu–Schwarz B field lives on aspace with spatial noncommutativity.14,20,43 This result reinforces the widely accepted idea that thePlanck length constitutes a lower bound on the precision of a measurement of position.42

To implement these ideas at the level of quantum field theory, one usually assumes4,15,17,38,49

that the space-time coordinates do not commute either in a canonical way,

x,x = i, 1

or in a Lie-algebraic way,

aElectronic mail: ncdias@meo.pt.bElectronic mail: maurice.de.gosson@univie.ac.at.cAuthor to whom correspondence should be addressed. Electronic mail: franz.luef@univie.ac.at.dElectronic mail: joao.prata@mail.telepac.pt.

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51, 072101-10022-2488/2010/517/072101/12/$30.00 © 2010 American Institute of Physics

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x,x = iCx, 2

where and C are real constants for , ,=0,1 , . . . ,d. In general, one chooses 0i=0 or

C0i=0 i.e., time is an ordinary commutative parameter to avoid problems with the lack of

unitarity.Equations 1 and 2 can be regarded as a particular case of the more general deformation

x ,x= ifx, where fx are real functions of x such that antisymmetry and the Jacobiidentity are respected.15 This occurs when the laboratory frame has a space-time dependent mo-tion. As suggested in Ref. 34, it is likely that f have a fixed value in the cosmic microwavebackground radiation frame, which may be considered as approximately fixed in the celestialsphere. For this reason, we shall adopt the simpler canonical situation 1 in this work. Moreover,there is an additional bonus. This choice will simplify drastically the theory in technical termssince—as we shall see—one is able to construct symplectomorphisms to the usual canonicalHeisenberg algebra.

Adopting this point of view, many authors addressed the problem of testing the existence ofspace-time noncommutativity in nature by resorting to a nonrelativistic, one particle quantummechanical approximation to the full quantum field theory. This generalization of quantum me-chanics obtained by considering canonical extensions of the Heisenberg algebra is usually referredto as noncommutative quantum mechanics NCQM.26 Many physical applications of NCQMhave been considered, in particular, the quantum Hall effect,16,23,48 quantum mechanics on thenoncommutative sphere,39 gravitational quantum well,9 noncommutative supersymmetric quantummechanics,31 noncommutative Chern–Simmons theory,22,48 the Landau problem,33 and the hydro-gen atom.13

We shall thus consider time to be a commutative parameter and the remaining spatial coordi-nates to be canonically noncommutative. We shall also assume noncommuting momenta. Thereare several reasons for this. First of all, it seems more symmetrical. However, perhaps moreimportantly, there are various instances where momentum noncommutativity leads to importanteffects. In Refs. 1 and 9 it was shown that momentum noncommutativity yields larger correctionsthan the configurational counterpart, and it is therefore more susceptible to experimental verifica-tion. Moreover, recent results in the context of noncommutative quantum cosmology4,2,3 haverevealed that momentum noncommutativity may be a decisive ingredient for the solution of thesingularity problem of black holes.

Here and in a series of papers, we will develop various mathematical aspects of NCQM,notably the deformation quantization, spectral results,28 the geometry of the uncertaintyprinciple,11,35,36 and the connection with modulation spaces.24,29

In Refs. 5 and 6, two of us discussed various aspects of NCQM related to Flato–Sternheimerdeformation quantization.7,8 In this paper, we propose an operator theoretical approach, based onprevious work by de Gosson and Luef28 the remaining two of us. In that article, it was shownthat the Moyal–Groenewold product ab of two functions on R2n can be interpreted in terms of aWeyl calculus on R2n. In fact,

a b = Ab , 3

where A is the phase space operator with Weyl symbol a defined on R2n R2n by

az, = az − 12J 4

J is the standard symplectic matrix.In this paper, we show that this redefinition of the star-product can be modified so that it leads

to a natural notion of deformation quantization for the NCQM associated with an antisymmetricmatrix of the type

072101-2 Dias et al. J. Math. Phys. 51, 072101 2010

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= −1 I

− I −1N , 5

where and N measure the noncommutativity in the position and momentum variables, respec-tively. We define a new star-product by replacing formula 3 by

ab = Ab , 6

where A is the operator with Weyl symbol

az, = az − 12 . 7

Of course, 7 reduces to 3 when =N=0.In this article, we are going to rigorously justify the definition above and study the properties

of this new star-product . The difficulty associated with the fact that the symplectic formassociated with depends on will be resolved we will show that ab is well-defined as astar-product, thanks to supplementary conditions on and N, which are physically meaningful.In fact, coincides with the star-product defined in terms of the generalized Weyl–Wignermap19 in Eq. 21 of Ref. 5, and where it was shown that it is related to the standard star-product.In the same paper, it was concluded in Eq. 53 that the generalized star-product between twopolynomials can be represented as a kind of “Bopp shift,” which also turns out to be identical to.

Notation 1: The generic point of phase space R2n is denoted as z= x , p. We denote bySp2n ,R the standard symplectic group, defined as the group of linear automorphisms of R2n

equipped with the symplectic form z ,z=Jz ·z, J= 0 I−I 0

. We use the standard notations SRmand SRm for the Schwartz space of test functions on Rm and its dual.

II. DESCRIPTION OF THE PROBLEM

Let us begin by explaining what we mean by noncommutativity in the present context. Thestudy of noncommutative field theories and their connections with quantum gravity see Refs. 10,17, 21, 43, and 49 and references therein leads to the consideration of commutation relations ofthe type

z, z = i, 1 , 2n , 8

where = 1,2n is the 2n 2n antisymmetric matrix defined by 5, where = 1,n and N= 1,n are antisymmetric matrices measuring the noncommutativityin the position and momentum variables. We have set here z= x if 1n and z= p−n if n+12n, where

x = x +1

2i

x+

1

2ip

, 9

p = p −1

2ix

+1

2i

p. 10

It turns out that, as proven in Ref. 6, is invertible if

2 for 1 n and 1 n . 11

We will assume from now on that these conditions are satisfied; that this requirement is physicallymeaningful is well-known it is fulfilled, for instance, in the case of the noncommutative quantumwell see, for instance, Refs. 9 and 12. Since we will be concerned with a deformation quanti-zation with parameter →0, we will furthermore assume that and N depend smoothly on insuch a way that

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= o2 and N = o2 12

recall that f=om means that lim→0f /m=0. We thus have

lim→0

= J = 0 I

− I 0

the standard symplectic matrix. It turns out that the condition 12 is compatible with thenumerical results in Refs. 9 and 12, where it is shown that the estimates 4 10−40 m2 and1.76 10−61 kg2 m2 s−2 hold. Moreover, the analysis of NCQM in the context of dissipativeopen systems reveals that a transition →0 occurs prior to →0.18

These facts, and the theory developed in Ref. 28, suggest that we represent z= z1 , . . . , z2n bythe vector operator

z = z + 12 iz 13

which acts on functions defined on the phase space R2n. Notice that the condition 12 show thatin the limit →0, we have the asymptotic formulas

x = x + 12 ip

+ o2, p = p − 12 ix

+ o2 . 14

The “quantization rules” 13 lead us to the consideration of pseudodifferential operators formallydefined by 7.

The underlying symplectic structure we are going to use is defined as follows. We will denoteby s a linear automorphism of R2n such that =s; equivalently, sJsT=. Thus, s is a symplec-tomorphism s : R2n ,→ R2n ,. Note that the mapping s is sometimes called the “Seiberg–Witten map” in the physical literature; its existence is, of course, mathematically a trivialitybecause it is just a linear version of Darboux’s theorem see Ref. 27, Sec. 1.1.2. Writing s in ablock-matrix form A B

C D, the condition sJsT= is equivalent to

ABT − BAT = −1, CDT − DCT = −1N, ADT − BCT = I .

Of course, the automorphism s is not uniquely defined: if s=s, then s−1sSp2n ,R. Alsonote that in the limit →0, the matrices −1 and −1N vanish and s becomes, as expected,symplectic in the usual sense, that is, sSp2n ,R.

III. DEFINITION OF THE STAR-PRODUCT Ω

Let be the symplectic form on R2n defined by z ,z=z ·−1z; it coincides with thestandard symplectic form when =J.

We will need the two following unitary transformations.

• The -symplectic transform F defined, for aSR2n, by

Faz = 1

2n

det −1/2R2n

e−i/z,zazdz. 15

• It extends into an involutive automorphism of SR2n also denoted by F and whoserestriction to L2R2n is unitary.

• The unitary operator Tz0 defined, for SR2n, by the formula

Tz0z = e−i/z,z0z − 12z0 . 16

Notice that when =J, we have Tz0= Tz0, where Tz0 is defined by formula 8 in Ref.28.

Let us express the operator A=az+ 12 iz in terms of Fa and Tz0.

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Proposition 2: Let A be the operator on R2n with Weyl symbol

az, = az − 12 . 17

We have

A = 1

2n

det −1/2R2n

FazTzdz . 18

Proof: Let us denote by B the right-hand side of 18. We have, setting u=z− 12z0,

Bz = 1

2n

det −1/2R2n

Faz0e−i/z,z0z −1

2z0dz0

= 2

n

det −1/2R2n

Fa2z − ue2i/z,uudu .

Hence, the kernel of B is given by

Kz,u = 2

n

det −1/2Fa2z − ue2i/z,u.

It follows that the Weyl symbol b of B is given by

bz, = R2n

e−i/·Kz +1

2,z −

1

2d

= 2

n

det −1/2R2n

e−i/·Fa2e−2i/z,d,

that is, using the obvious relation

· + 2z, = 2z − ,

together with the change of variables z=2,

bz, = 2

n

det −1/2R2n

e−i/2z−,Fa2d

= 1

2n

det −1/2R2n

e−i/z−1/2,zFazdz,

that is, using the fact that FF is the identity,

bz, = az − 12 = az,

which concludes the proof.

The result above motivates the following definition.Definition 3: Let aSR2n and bSR2n. The -star-product of a and b is the element of

SR2n defined by

ab = Ab . 19

Note that it is not yet clear from the definition above that is a bona fide star-product. For

instance, while it is obvious that 1b=b because the operator A with symbol a=1 is theidentity, the formula b1=b is certainly not, and it is even less clear that is associative!

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IV. A NEW STAR-PRODUCT IS BORN

It turns out that we can reduce the study of the newly defined star-product to that of the usualGroenewold–Moyal product . For this we, will need Lemma 4 below.

Lemma 4: Let s be a linear automorphism such that =s and define an automorphismMs :SR2n→SR2n by

Msz = det ssz . 20

[hence, Ms is unitary on L2R2n]. We have

MsA = AMs, 21

where A= AJ corresponds to the operator A acting on L2Rn with Weyl symbol az=asz, andhence

Msab = det sa b , 22

where bz=bsz.Proof: Formula 22 immediately follows from formula 21. To prove formula 21, one first

checks the identities

MsTz0 = Ts−1z0Ms, MsF = FJMs

the verification of which is purely computational and therefore left to the reader; using theseidentities, we have

MsA = 1

2n

det −1/2R2n

Faz0Ts−1z0Msdz0 = 1

2n

det s

det −1/2R2n

FaszTzMsdz = 1

2n

det s1/2det −1/2R2n

MsFazTzMsdz

= 1

2n

det s1/2det −1/2R2n

FJMsazTzMsdz = AMs.

The double equality

1a = a1 = a 23

now immediately follows from formula 22: we have

Ms1a = det s1 a = 1 Msa = Msa ,

hence we recover the equality 1a=a; similarly,

Msa1 = det sa 1 = Msa 1 = Msa ,

hence a1=a.Let us now prove the associativity of the -star-product.Proposition 5: Assume that the star-products abc and abc are defined. We then

have

abc = abc . 24

Proof: It is, of course, sufficient to show that

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Msabc = Msabc . 25

We have, by repeated use of 22 together with the definition of Ms,

Msabc = det sa bc = a Msbc = det sa b c .

A similar calculation yields

Msabc = det sa b c ,

hence the equality 25 in view of the associativity of the Groenewold–Moyal product.

That we have a deformation of a Poisson bracket follows from the following considerations.Let us define an -Poisson bracket · , · by

a,b = − Xa,,Xb, , 26

where the vector fields Xa, and Xb, are given by

Xa, = za, Xb, = zb . 27

In particular, a ,b is the usual Poisson bracket a ,b, and Xa,J and Xa,J are the usual Hamiltonvector fields when =N=0. We have the following asymptotic formula relating both notions ofPoisson brackets:

a,b = a,b + o for → 0. 28

In fact, by definitions 26 and 27,

a,b = − Xa, · −1Xb, = − za · zb ,

that is,

a,b = a,b − −1xa · xb + Npa · pb

from which 28 follows in view of condition 12.Proposition 6: We have

ab − ba = ia,b + O2 . 29

Proof: We have, since Ms is linear,

Msab − ba = det sa b − b a = det sia,b + O2 ,

where, as usual, az=asz and bz=bsz. Now, by the chain rule and the relation JsT

=s−1 ,

Xa = JsTzasz = s−1zasz ,

Xb = JsTzbsz = s−1zbsz ,

and hence, using the identities a ,b=JXa ·Xb and sT−1J−1 s−1=−1,

det sa,b = − det sJs−1 zasz · s−1 zbsz = det szasz · zbsz

= − det szasz · −1zbsz = − MsXa,,Xb, .

We have thus proven that

Msab − ba = − iMsXa,,Xb, + O2 .

From this and 28, Eq. 29 follows.

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More generally, using the approach above, it is easy to show that

fg = k0

Bk,f ,gk,

where the Bk,f ,g are bidifferential operators. In particular, B0,f ,g=1 and B1,f ,g= i2

f ,g, but for k2, they differ from those of the usual Moyal product. We leave these techni-calities aside in this article.

V. THE INTERTWINING PROPERTY

In Ref. 28, two of us defined a family of partial isometries W :L2Rn→L2R2n indexed by

SRn, and intertwining the operator A= AJ and the usual Weyl operator A,

AW = WA and W A = AW

. 30

These intertwiners are defined by

W = 2n/2W, , 31

where W , is the cross-Wigner distribution,

W,z = 1

2n

Rne−i/p·yx +

1

2yx −

1

2ydy 32

and W denotes the adjoint of W.

The following result is an extension of Proposition 2 in Ref. 28.Theorem 7: Let s be a linear automorphism of R2n such that s=.

i The mappings Ws, :SRn→SR2n defined by the formula,

Ws, = Ms−1W 33

are partial isometries L2Rn→L2R2n and we have

AWs, = Ws,A and Ws, A = AWs,

, 34

where A is the operator with Weyl symbol a=asz and Ws, denotes the adjoint of Ws,.

ii The replacement of s by s such that =s is equivalent to the replacement of A by

S−1AS

and of Ws, by Ws,S−1S

−1, where S is any of the two operators in the metaplectic

group Mp2n ,R whose projection on Sp2n ,R is s=s−1s.

Proof:

i We have, using the first formula 30 and definition 33,

AWs, = Ms−1AMsMs

−1W ,

that is,

AWs, = Ms−1AW = Ms

−1WA = Ws,A,

the equality Ws, A= AWs,

is proven in a similar way. That Ws, is a partial isometry isobvious since W is a partial isometry and Ms is unitary.

ii We have s=s−1sSp2n ,R, hence asz=assz=asz. Let A be the operator withWeyl symbol az=asz. In view of the symplectic covariance property of Weyl opera-

tors, we have A=S−1AS

. Similarly,

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Ws,z = Ms−1MsMs

−1Wz = Ms−1Ws

−1z = Ws,s−1z ,

hence Ws,=Ws,SS

in view of the symplectic covariance of the cross-Wigner trans-form 32; the result follows.

An important property of the mappings Ws, is that they can be used to construct orthonormalbases in L2R2n starting from an orthonormal basis in L2Rn.

Proposition 8: Let j jF be an arbitrary orthonormal basis of L2Rn; the functions j,k

=Ws,jk with j ,kF F form an orthonormal basis of L2R2n, and we have j,kH j Hk,

with H j =Ws,jL2Rn.

Proof: In Ref. 7, the property was proven for the mappings Wj=WI,j

; the lemma followssince Ws,=Ms

−1W and Ms is unitary.

VI. THE Ω-GENVALUE EQUATION: SPECTRAL RESULTS

Let us consider the star-genvalue equation for the star-product ,

a = , 35

here a can be viewed as some Hamiltonian function whose properties are going to be describedand is a “phase-space function.” Following definition 19, the study of this problem is equiva-lent to that of the eigenvalue equation

A = 36

for the pseudodifferential operator A. Using the intertwining relations 34, it is easy to relate the

eigenvalues of A to those of A following the lines in Ref. 28; for instance, one sees, adapting

mutatis mutandis the proof of Theorem 4 in the reference, that the operators A and A have thesame eigenvalues see Theorem 9 below. Note that it follows from Theorem 7, part ii, that the

eigenvalues of A do not depend on the choice of s such that s=.

Theorem 9: The operators A and A have the same eigenvalues.

i Let be an eigenvector of A: A=. Then =Ws, is an eigenvector of A corre-

sponding to the same eigenvalue: A=.

ii Conversely, if is an eigenvector of A, then =Ws, is an eigenvector of A corre-

sponding to the same eigenvalue.

Proof:

i That every eigenvalue of A also is an eigenvalue of A is clear: if A= for some0, then

AWs, = Ws,A = Ws,

and =Ws,0; this proves at the same time that Ws, is an eigenvector of A

because Ws, is injective.

ii Assume conversely that A= for L2R2n, 0, and R. For every , wehave

AWs, = Ws,

A = Ws, ,

hence is an eigenvalue of A and an eigenvector if =Ws, 0. We have Ws,

=Ws,Ws, = Ps,, where Ps, is the orthogonal projection on the range Hs, of Ws,.

Assume that =0; then Ps,=0 for every SRn, and hence =0 in view of Propo-sition 8.

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Let us give an application of the above result. Assume that the symbol a belongs to the Shubinclass H

m1,m0R2n; recall44 that aHm1,m0R2n m0 ,m1R and 01 if aCR2n and if

there exist constants C0 ,C10 and, for every Nn, 0, a constant C0, such that for zsufficiently large

C0zm0 az C1zm1, zaz Cazz−. 37

The following result Ref. 44, Chap. 4 is important in our context.Theorem 10: Let aH

m1,m0R2n be real and m00. Then the formally self-adjoint opera-

tor A with Weyl symbol a has the following properties:

i A is essentially self-adjoint in L2Rn and has discrete spectrum.ii There exists an orthonormal basis of eigenfunctions j SRn j=1,2 , . . . with eigenval-

ues j R such that limj→ j=.

Corollary 11: Let aHm1,m0R2n be real and m00.

i The star-genvalue equation a= has a sequence of real eigenvalues j such that

limj→ j=, and these eigenvalues are those of the operator A with Weyl symbolaz=asz.

ii The star-eigenvectors of a are in one-to-one correspondence with the eigenvectors j

SRn of A by the formula k,j =Ws,k j.

Proof: It is an immediate consequence of Theorems 9 and 10.

VII. CONCLUDING REMARKS

The results using the generalized Weyl–Wigner map19 seem to be quite general since they alsoapply to the case of nonlinear transformations of R2n for a review, see also Sec. II of Ref. 5. Inparticular, a more general star-product than the one of noncommutative quantum mechanics wasobtained in Ref. 5 see Eq. 23 in this reference. A future project could be to extend the approachof the present paper to this case. Another important topic we have not addressed in this article isthe characterization of the optimal symbol classes and function spaces associated with the star-product . As two of us have shown elsewhere,30 Feichtinger’s modulations spaces see Refs. 24and 25 for a review and the closely related Sjöstrand classes45 are excellent candidates in the case

of Landau-type operators which are a variant of the operators A corresponding to the case =J. It seems very plausible that these function spaces are likely to play an equally important rolein the theory of the star-product . Another future project concerns a discussion of the star-product and its connection to Rieffel’s work in deformation quantization as outlined in Ref. 41,and the methods introduced in Ref. 37 in a different context.

ACKNOWLEDGMENTS

M.d.G. was financed by the Austrian Research Agency FWF Projektnummer P20442-N13.N.C.D. and J.N.P. were supported by Grant Nos. PDTC/MAT/69635/2006 and PTDC/MAT/099880/2008 of the Portuguese Science Foundation FCT. F.L. was financed by the Marie CurieOutgoing Fellowship under Grant No. PIOF 220464.

1 Bastos, C. and Bertolami, O., “Berry phase in the gravitational quantum well and the Seiberg-Witten map,” Phys. Lett.A 372, 5556 2008.

2 Bastos, C., Bertolami, O., Dias, N. C., and Prata, J. N., “Black holes and phase-space noncommutativity,” Phys. Rev. D80, 124038 2009.

3 Bastos, C., Bertolami, O., Dias, N. C., and Prata, J. N., “Phase-space noncommutative quantum cosmology,” Phys. Rev.D 78, 023516 2008.

4 Bastos, C., Bertolami, O., Dias, N. C., and Prata, J. N., “The singularity problem and phase-space noncanonical non-commutativity,” e-print arXiv:hep-th/0912.4027v1.

5 Bastos, C., Bertolami, O., Dias, N. C., and Prata, J. N., “Weyl–Wigner formulation of noncommutative quantum me-chanics,” J. Math. Phys. 49, 072101 2008.

072101-10 Dias et al. J. Math. Phys. 51, 072101 2010

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155.33.16.124 On: Mon, 24 Nov 2014 12:24:22

6 Bastos, C., Dias, N. C., and Prata, J. N., “Wigner measures in non-commutative quantum mechanics,” e-printarXiv:0907.4438v1 math-ph; Commun. Math. Phys. to appear.

7 Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D., “Deformation theory and quantization. I.Deformation of symplectic structures,” Ann. Phys. 111, 61 1978.

8 Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D., “Deformation theory and quantization. II.Physical applications,” Ann. Phys. 111, 111 1978.

9 Bertolami, O., Rosa, J. G., de Aragão, C. M. L., Castorina, P., and Zappalà, D., “Noncommutative gravitational quantumwell,” Phys. Rev. D 72, 025010 2005.

10 Binz, E., Honegger, R., and Rieckers, A., “Field-theoretic Weyl quantization as a strict and continuous deformationquantization,” Ann. Henri Poincare 5, 327 2004.

11 Bolonek, K. and Kosińsky, P., “On uncertainty relations in noncommutative quantum mechanics,” Phys. Lett. B 547, 512002.

12 Carroll, S. M., Harvey, J. A., Kostelecký, V. A., Lane, C. D., and Okamoto, T., “Noncommutative field theory andLorentz violation,” Phys. Rev. Lett. 87, 141601 2001.

13 Chaichian, M., Sheikh-Jabbari, M. M., and Tureanu, A., “Hydrogen atom spectrum and the Lamb shift in noncommu-tative QED,” Phys. Rev. Lett. 86, 2716 2001.

14 Connes, A., Douglas, M. R., and Schwarz, A., “Noncommutative geometry and matrix theory: Compactification on tori,”J. High Energy Phys. 02 1998 3.

15 Daszkiewicz, M. and Walczyk, C. J., “Newton equation for canonical, Lie-algebraic and quadratic deformation ofclassical space,” Phys. Rev. D 77, 105008 2008.

16 Dayi, O. F. and Jellal, A., “Hall effect in noncommutative coordinates,” J. Math. Phys. 43, 4592 2002.17 Delduc, F., Duret, Q., Gieres, F., and Lefrançois, M., “Magnetic fields in noncommutative quantum mechanics,” J. Phys.:

Conf. Ser. 103, 012020 2008.18 Dias, N. C. and Prata, J. N., “Exact master equation for a noncommutative Brownian particle,” Ann. Phys. 324, 73

2009.19 Dias, N. C. and Prata, J. N., “Generalized Weyl-Wigner map and Vey quantum mechanics,” J. Math. Phys. 42, 5565

2001.20 Douglas, M. R. and Hull, C. “D-branes and the noncommutative torus,” J. High Energy Phys. 02 1998 8.21 Douglas, M. R. and Nekrasov, N. A., “Noncommutative field theory,” Rev. Mod. Phys. 73, 977 2001.22 Dunne, G. and Jackiw, R., “‘Peierls substitution’ and Chern-Simmons quantum mechanics,” Nucl. Phys. Proc. Suppl C

33, 114 1993.23 Duval, C. and Horváthy, P. A., “Exotic Galilean symmetry in the non-commutative plane, and the Hall effect,” J. Phys.

A 34, 10097 2001.24 Feichtinger, H. G., “Modulation spaces: Looking back and ahead,” Sampl. Theory Signal Image Process. 5, 109 2006.25 Feichtinger, H. G. and Luef, F., “Wiener amalgam spaces for the fundamental identity of Gabor analysis,” Collect. Math.

57, 233 2006.26 Gamboa, J., Loewe, M., and Rojas, J. C., “Noncommutative quantum mechanics,” Phys. Rev. D 64, 067901 2001.27 de Gosson, M., Symplectic Geometry and Quantum Mechanics, Operator Theory: Advances and Applications Vol. 166

Birkhäuser, Basel, 2006.28 De Gosson, M. and Luef, F., “A new approach to the -genvalue equation,” Lett. Math. Phys. 85, 173 2008.29 de Gosson, M. and Luef, F., “On the usefulness of modulation spaces in deformation quantization,” J. Phys. A: Math.

Theor. 42, 315205 2009.30 de Gosson, M. and Luef, F., “Spectral and regularity properties of a pseudo-differential calculus related to Landau

quantization,” J. Pseudo-Differential Operators and Applications 1, 3 2010.31 Harikumar, E., Kumar, V. S., and Khare, A., “Supersymmetric quantum mechanics on noncommutative plane,” Phys.

Lett. B 589, 155 2004.32 Heisenberg, W., in Letter of Heisenberg to Peierls (1930), Wolfgang Pauli, Scientific Correspondence, edited by von

Meyenn, K. Springer-Verlag, Berlin, 1993.33 Horváthy, P. A., “The noncommutative Landau problem,” Ann. Phys. 299, 128 2002.34 Kamoshita, J.-I., “Probing noncommutative space-time in the laboratory frame,” Eur. Phys. J. C 52, 451 2007.35 Kosińsky, P. and Bolonek, K., “Minimalization of uncertainty relations in noncommutative quantum mechanics,” Acta

Phys. Pol. B 34, 2575 2003.36 de Gosson, M. and Luef, F., “Symplectic capacities and the geometry of uncertainty: The irruption of symplectic

topology in classical and quantum mechanics,” Phys. Rep. 484, 131 2009.37 Luef, F. and Manin, Yu. I., “Quantum theta functions and Gabor frames for modulation spaces,” Lett. Math. Phys. 88,

131 2009.38 Martinetti, P., “What kind of noncommutative geometry for quantum gravity?,” Mod. Phys. Lett. A 20, 1315 2005.39 Nair, V. P. and Polychronakos, A. P., “Quantum mechanics on the noncommutative plane and sphere,” Phys. Lett. B 505,

267 2001.40 Pauli, W., in Letter of Pauli to Oppenheimer (1946), Wolfgang Pauli, Scientific Correspondence, edited by von Meyenn,

K. Springer-Verlag, Berlin, 1993.41 Rieffel, M. A., “Deformation quantization for actions of Rd,” Mem. Am. Math. Soc. 106, 506 1993.42 Rosenbaum, M. and Vergara, J. D., “The star-value equation and Wigner distributions in noncommutative Heisenberg

algebra,” Gen. Relativ. Gravit. 38, 607 2006.43 Seiberg, N. and Witten, E. “String theory and noncommutative geometry,” J. High Energy Phys. 09 1999 32.44 Shubin, M. A., Pseudodifferential Operators and Spectral Theory, 2nd ed. Springer-Verlag, Berlin, 2001 original work

published 1987 original Russian edition in Nauka, Moskva, 1978.

072101-11 Deformation quantization and noncommutative ... J. Math. Phys. 51, 072101 2010

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:

155.33.16.124 On: Mon, 24 Nov 2014 12:24:22

45 Sjöstrand, J., “An algebra of pseudodifferential operators,” Math. Res. Lett. 1, 185 1994.46 Snyder, H. S., “Quantized space-time,” Phys. Rev. 71, 38 1947.47 Snyder, H. S., “The electromagnetic field in quantized space-time,” Phys. Rev. 72, 68 1947.48 Susskind, L., “The quantum Hall fluid and noncommutative Chern-Simmons theory,” e-print arXiv:hep-th/0101029.49 Szabo, R., “Quantum field theory on noncommutative spaces,” Phys. Rep. 378, 207 2003.50 Yang, C. N., “On quantized space-time,” Phys. Rev. 72, 874 1947.

072101-12 Dias et al. J. Math. Phys. 51, 072101 2010

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:

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