Violation of the Robertson-Schrodinger uncertainty principle andnoncommutative quantum mechanics

Catarina Bastos*

Instituto de Plasmas e Fusao Nuclear, Instituto Superior Tecnico, Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal

Orfeu Bertolami†

Departamento de Fısica e Astronomia, Faculdade de Ciencias, da Universidade do Porto, Rua do Campo Alegre,687, 4169-007 Porto, Portugal

Nuno Costa Dias‡,§ and Joao Nuno Prata‡,k

Departamento de Matematica, Universidade Lusofona de Humanidades e Tecnologias, Avenida Campo Grande,376, 1749-024 Lisboa, Portugal

(Received 3 July 2012; published 19 November 2012)

We show that a possible violation of the Robertson-Schrodinger uncertainty principle may signal the

existence of a deformation of the Heisenberg-Weyl algebra. More precisely, we prove that any Gaussian in

phase space (even if it violates the Robertson-Schrodinger uncertainty principle) will always be a quantum

state of an appropriate noncommutative extension of quantum mechanics. Conversely, all canonical

noncommutative extensions of quantum mechanics display states that violate the Robertson-Schrodinger

uncertainty principle.

DOI: 10.1103/PhysRevD.86.105030 PACS numbers: 03.65.Fd

I. INTRODUCTION

Since its inception, quantum mechanics was formulatedin terms of Hilbert spaces and self-adjoint operators actingtherein. In this context Heisenberg’s uncertainty relationsbecome a straightforward consequence of the noncommu-

tativity of the fundamental operators of position Q and

momentum P and the Cauchy-Schwarz inequality. For an-dimensional system, the Heisenberg-Weyl (HW) algebrareads

½Qi; Pj� ¼ i�i;j; i; j ¼ 1; . . . ; n; (1)

and all remaining commutators vanish. Since our resultsare much more general than simple rescalings of Planck’sconstant, we have set ℏ ¼ 1 for the remainder of this workand assumed that position and momentum have the same

units. If we define the phase-space variable Z ¼ ðQ; PÞ, wehave in more compact notation

½Zi; Zj� ¼ iJij; i; j ¼ 1; . . . ; 2n; (2)

where J ¼ ðJijÞ is the standard symplectic matrix

J ¼ �JT ¼ �J�1 ¼ 0 I

�I 0

!: (3)

A simple calculation leads to the inequalities

�Qj� �Pj

� 1

2; j ¼ 1; . . . ; n; (4)

where �Qjand �Pj

denote the mean standard deviations of

Qj and Pj with respect to an arbitrary state. The set of

inequalities (4) are known as the Heisenberg-Weyl-Pauliinequalities. They are not invariant under linear symplectictransformations of the operators nor under metaplectictransformations of the states. This prompted the searchfor an alternative set of inequalities which are strongerthan the Heisenberg inequalities and have the right sym-plectic covariance properties. They are known as theRobertson-Schrodinger uncertainty principle (RSUP) [1]:

� þ i

2J � 0: (5)

Here, � denotes the covariance matrix with respect to

an arbitrary state and has entries �ij ¼ hðZi � hZiiÞ�ðZj � hZjiÞi (i, j ¼ 1; . . . ; 2n), and h�i denotes an expecta-

tion value in a chosen state. In this work we shall focus onthis form of the uncertainty principle, as it implies theHeisenberg uncertainty relations, it is invariant under lin-ear symplectic transformations and, moreover, it consti-tutes the necessary and sufficient condition for a Gaussianto be a quantum mechanical state. In fact, Gaussians areof the utmost importance, not only because they arecompletely determined by their covariance matrix, butalso because experimentally coherent and squeezed states

*catarina.bastos@ist.utl.pt†Also at Instituto de Plasmas e Fusao Nuclear, Instituto

Superior Tecnico, Avenida Rovisco Pais 1, 1049-001 Lisboa,Portugal.orfeu.bertolami@fc.up.pt

‡Also at Grupo de Fısica Matematica, UL, Avenida Prof.Gama Pinto 2, 1649-003 Lisboa, Portugal.

§ncdias@meo.ptkjoao.prata@mail.telepac.pt

PHYSICAL REVIEW D 86, 105030 (2012)

1550-7998=2012=86(10)=105030(6) 105030-1 � 2012 American Physical Society

play an important role in quantum optics [2], quantumcomputation of continuous variables [3] and investigationsof the quantum-classical transition [4].

From a slightly different perspective, since the RSUPaccounts for correlations, it is more suitable to addressseveral interesting problems. The recent work [5] presentsone such example. It is shown that the consideration ofstates with strong position-momentum correlations maylead to greater transparency of the Coulomb barrier duringthe interaction of charged particles. This is very importantin the astrophysics of stars and in controlled nuclear fusion,where the action of the Coulomb barrier leads to a very lowtunneling probability for low-energy particles. It can beshown that for a nonstationary harmonic oscillator withpotential VðtÞ ¼ 1

2m!2ðtÞQ2, a decrease in particle’s

frequency !ðtÞ leads to an increase in the correlationcoefficient:

rðtÞ ¼ hQ PþP Qi2�Q�P

and a change of the uncertainty relation

�Q�P � ð2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� r2

p�1:

When a strongly correlated state with jrj ! 1 is formed,both �Q, �P and their product increase indefinitely. It was

then proved that this leads to a greater barrier transparencyat the same energy [5].

Various authors have tested theoretically and experimen-tally the validity of the inequalities (5) and the consequen-ces of their violation [6]. For instance, Popper’s experimentis usually regarded as a violation of the uncertainty prin-ciple [7]. The type of arguments used by Popper [8] aresimilar in spirit to those of the EPR experiment [9] so thatnonlocality and the uncertainty principle seem to be inex-tricably linked. More precisely, the degree of nonlocalityof any theory is determined by two factors, namely thestrength of the uncertainty principle and the strength of aproperty called ‘‘steering,’’ which determines which statescan be prepared at one location given a measurement atanother [10].

In this work, we show that the breakdown of the uncer-tainty principle may hint that the subatomic world isdescribed not by standard quantum mechanics but by anoncommutative extension [11,12] of it. This extension isobtained by replacing HW algebra (2) with a deformedalgebra. This leads to an extra noncommutativity betweenthe configuration and momentum variables. The mostcommonly used deformed algebra reads

½�i; �j� ¼ i�ij; i; j ¼ 1; . . . ; 2n; (6)

where the matrix � ¼ ð�ijÞ is given by:

� ¼ � I

�I �

!: (7)

Here � ¼ ð�ijÞ and � ¼ ð�ijÞ are real constant skew-

symmetric n� n matrices measuring the strengths of theposition-position and momentum-momentum noncommu-tativities, respectively.One should notice that other (noncanonical) deforma-

tions of the algebra (2) are also possible. These typicallyaccount for nonlinearities and may appear in various sys-tems. For instance, the quantum q-oscillator [13] can beinterpreted as a nonlinear oscillator whose frequency ofvibration depends on the energy of the vibrations througha cosh function [14]. Generalizations thereof, where thefrequency of vibration varies with other functions of theamplitude, are commonly known as f-oscillators [15].Another instance where this kind of algebraic structure iscommon is in the reduced phase-space formulation ofsystems with second class Dirac constraints [16].From a somehow more fundamental point of view,

noncommutative quantum mechanics (NCQM) is usuallyregarded as a nonrelativistic one-particle sector of the verydiscussed noncommutative quantum field theories [17],which emerge in the context of string theory and quantumgravity [18]. The question of space-time noncommutativityhas a long-standing story. It was put forward by Snyder[19], Heisenberg, Pauli [20] and Yang [21] as a means toregularize quantum field theories. However, the develop-ment of renormalization techniques and certain undesir-able features of noncommutative theories such as thebreakdown of Lorentz invariance [22,23] have hinderedfurther research in this direction. More recently, severalimportant developments in various approaches to the quan-tization of gravity have revived the interest in the conceptof noncommutative space-time. See for instance Ref. [24]in the context of three-dimensional gravity. In the realm ofstring theory, the discovery that the low-energy effectivetheory of a D-brane in the background of a Neveu-SchwarzB field lives on a space with spatial noncommutativity hastriggered an enormous amount of research in this field[17,18,25]. From another perspective, a simple heuristicargument, based on Heisenberg’s uncertainty principle, theequivalence principle and the Schwarzschild metric, showsthat the Planck length seems to be a lower bound on theprecision of a measurement of position [26]. This reenfor-ces the point of view that a new (noncommutative) geome-try of space-time will emerge at a fundamental level[17,27–29]. The simplest way to implement these ideasin quantum field theories is by adding to the phase-spacenoncommutativity of quantum mechanics a new space-time noncommutativity.The implications of considering NC deformations of the

HW algebra have also been investigated in the context ofquantum cosmology. It was shown that this new structureleads to the thermodynamic stability of black holes and apossible regularization of the black hole singularities [30].In this paper, we study some fundamental properties of

states in NCQM. Various features of noncommutative

BASTOS et al. PHYSICAL REVIEW D 86, 105030 (2012)

105030-2

quantum states for the algebra (6) and (7) have alreadybeen derived [11]. Here, we are concerned with the possi-bility of using Gaussian states to identify the correct alge-braic structure for quantum mechanics. More precisely, weshow that (i) NCQM displays states which are not states instandard quantum mechanics (as they violate the RSUP);(ii) conversely, that any Gaussian state (even if it is not astate in standard quantum mechanics) is nevertheless astate of some NC quantum theory; and (iii) this NC theoryis not unique and one can always find a NCQM for whichthe Gaussian is a quantum pure state. We stress that allthese results are true independently of the value of Planck’sconstant, and we set ℏ ¼ 1 for the entire paper.

Our proof takes place in the context of the Weyl-Wignerformulation or deformation quantization framework,where position and momentum variables appear on equalfooting. As shown in Ref. [11], it is only in this formulationthat one can tell whether states correspond to a quantiza-tion of the HW algebra or some of its deformation.

II. WIGNER DISTRIBUTIONS AND GAUSSIANSTATES ON ARBITRARY SYMPLECTIC SPACES

A symplectic form on a real vector space V is abilinear map !: V � V ! R, which is skew-symmetric(!ðv; v0Þ ¼ �!ðv0; vÞ for all v, v0 2 V) and nondegener-ate (!ðv; v0Þ ¼ 0 for all v0 2 V implies v ¼ 0). The ar-chetypal symplectic vector space is V ¼ R2n ¼ Rn

x � Rnp

endowed with the standard symplectic form:

�ðz; z0Þ ¼ z � Jz0 ¼ p � x0 � x � p0; (8)

where J is the standard symplectic matrix and z ¼ ðx; pÞ,z0 ¼ ðx0; p0Þ 2 R2n.

Now, let

!ðz; z0Þ ¼ z ���1z0 (9)

be another arbitrary symplectic form on R2n. Here � issome real, antisymmetric, nonsingular, 2n� 2n matrix[not necessarily (7)]. Awell-known theorem in symplecticgeometry [31] states that all symplectic vector spaces ofequal dimension are symplectically equivalent. In otherwords, there exists a real nonsingular 2n� 2n matrix Ssuch that !ðSz;Sz0Þ ¼ �ðz; z0Þ for all z, z0 2 R2n, ormatrixwise

� ¼ SJST: (10)

Obviously, the matrix S is not unique. Indeed, let P denotea symplectic matrix ½P 2 Spð2n;�Þ�, that is PJPT ¼ J.Then the matrix SP also satisfies (10). We shall call the setof all matrices which satisfy (10) the set of Darbouxmatrices associated with ! and denote it by Dð2n;!Þ.

Conversely, given any real, nonsingular, 2n� 2nmatrixS, let � be defined by (10). Then ! given by (9) is a

symplectic form on R2n. This simple observation will bethe crux of our main result.

A. Quantization on the standard symplectic space

To quantize a system on the standard symplectic space,one resorts to the HW operators

U �ðzÞ ¼ ei�ðz;ZÞ ¼ eiðp�Q�x�PÞ (11)

with z ¼ ðx; pÞ 2 R2n, and where Z ¼ ðQ; PÞ denotes thequantum mechanical position and momentum operators.These operators constitute a unitary irreducible repre-

sentation of the HW algebra. Indeed, they satisfy therelations

U �ðzÞU�ðz0Þ ¼ ei2�ðz;z0ÞU�ðzþ z0Þ ¼ ei�ðz;z0ÞU�ðz0ÞU�ðzÞ;

(12)

which can be readily obtained from the HW algebra (2)through the Baker-Campbell-Hausdorff formula.A generic linear operator acting on the Hilbert space of

the system [in this case L2ðRnÞ] can then be represented by

A ¼ ð2�Þ�nZ

dz~��ðJ�1zÞU�ðzÞ; (13)

where ~�� denotes the Fourier transform of some suitabletempered distribution ��ðzÞ on the phase space, commonly

known as the Weyl symbol of the operator A.

Since TrðU�ðzÞÞ ¼ ð2�Þn�ðzÞ, we obtain from (12) that

TrðU�ðzÞU�ðz0ÞÞ ¼ ð2�Þn�ðzþ z0Þ, and (13) can be read-

ily inverted. Thus, ~��ðzÞ ¼ TrðAU�ðJ�1zÞÞ, and the Weyl

symbol of A reads:

A � W�A ¼ ��ðzÞ ¼ ð2�Þ�nRdz0 TrðAU�ðz0ÞÞei�ðz0;zÞ

¼Z

dy

�xþ y

2jAjx� y

2

�e�ip�y: (14)

This procedure establishes a one-to-one map W�—calledWeyl correspondence—between linear operators and theassociated symbols.The state of a quantum system is represented by a

positive trace-class operator, the density matrix �. Whenapplied to �, the Weyl correspondence yields (up to anormalization constant) the celebrated Wigner functionon ðR2n; �Þ:

W��ðx; pÞ ¼ ð2�Þ�nZ

dy

�xþ y

2j�jx� y

2

�e�ip�y: (15)

In particular, for a pure state � ¼ jc ihc j for c 2 L2ðRnÞ:

W�c ðx; pÞ ¼ ð2�Þ�nZ

dyc

�xþ y

2

�c

�x� y

2

�e�ip�y:

(16)

In general, it is very difficult to assess whether a functionFðx; pÞ in phase space is the Wigner function of some

VIOLATION OF THE ROBERTSON-SCHRODINGER . . . PHYSICAL REVIEW D 86, 105030 (2012)

105030-3

density matrix (see e.g., Ref. [32]). A notable exception arethe Gaussians

G �;� ðzÞ ¼ 1

ð2�Þn ffiffiffiffiffiffiffiffiffiffidet�

p exp

�� 1

2ðz� �Þ ���1ðz� �Þ

�;

(17)

where z ¼ ðx; pÞ 2 R2n, � 2 R2n and � is the covariancematrix, which is a real, positive-definite 2n� 2n matrix. Itis a well documented fact [2,32] that such a Gaussian is aWigner function on ðR2n;�Þ if and only if it satisfies theRSUP (5).

We may be more specific and determine whether theGaussian is the Wigner function of a pure state. Indeed, ithas been proven [4] that (17) is the Wigner function ofa pure state if and only if there exists P 2 Spð2n;�Þsuch that

� ¼ 1

2PTP: (18)

B. Quantization on nonstandard symplectic spaces

NCQM results from quantizing the classical theory on anonstandard symplectic space ðR2n;!Þ [11,12]. In thiscase, the HW algebra (2) is replaced by the modifiedalgebra (6). The HW operators (11) become

U !ðÞ ¼ ei!ð;�Þ ¼ ei���1� (19)

and, concomitantly,

U !ðÞU!ð0Þ ¼ ei2!ð;0ÞU!ðþ 0Þ

¼ ei!ð;0ÞU!ð0ÞU!ðÞ: (20)

Since � ¼ SZ for some S 2 Dð2n;!Þ, from (10):

U!ðÞ ¼ U�ðS�1Þ;TrðU!ðÞU!ð0ÞÞ ¼ ð2�Þn

ffiffiffiffiffiffiffiffiffiffiffidet�

p�ðþ 0Þ:

(21)

Substituting (21) into (13), we obtain

A ¼ ð2�ffiffiffiffiffiffiffiffiffiffiffidet�

p�n

Zdz0~a!ð��1z0ÞU!ðz0Þ; (22)

where (10) and ~a!ðuÞ ¼ ~a�ðSTuÞ have been used. Hence,

the Weyl symbol of A on ðR2n;!Þ is given by

A � W!A ¼ a!ðÞ ¼ ðffiffiffiffiffiffiffiffiffiffiffidet�

pÞ�1a�ðS�1Þ

¼ ðð2�Þn det�Þ�1Z

d0 TrðAU!ð0ÞÞei!ð0;Þ:(23)

In particular, if W�� denotes the Wigner function of adensity matrix � on ðR2n;�Þ, then the correspondingWigner function on ðR2n;!Þ is given by [11]

W!�ðÞ ¼ 1ffiffiffiffiffiffiffiffiffiffiffidet�

p W��ðS�1Þ: (24)

C. Main result

From (24) one can derive the counterparts of the RSUPand Littlejohn’s Theorem [4] for Gaussians on ðR2n;!Þ.From (5) and (24), we conclude that the Gaussian (17) is aquantum state on ðR2n;!Þ if and only if

� þ i

2� � 0: (25)

Likewise, from (18) and (24), the Gaussian is a Wignerfunction on ðR2n;!Þ of a pure state if and only if thereexists a matrix C 2 Dð2n;!Þ such that

� ¼ 1

2CCT: (26)

We are now in a condition to prove our main result.Let G�;� be a Gaussian of the form (17). Then there

exists a matrix � associated with a symplectic form (9),such that G�;� is a quantum state of the NCQM based on

the deformed Heisenberg algebra (6).Indeed, since � is positive definite, there exists a real,

nonsingular, 2n� 2n matrix C for which (26) holds.Define a matrix � by � ¼ CJCT . Clearly, the form !defined by (9) is a symplectic form and C 2 Dð2n;!Þ.According to Littlejohn’s Theorem on ðR2n;!Þ [cf. (26)],then G�;� is a Wigner function on ðR2n;!Þ of a pure state.This concludes the proof.Notice that as a byproduct of the proof, for any

Gaussian, one can always find a NCQM for which theGaussian is a pure state. In a certain sense, this procedureamounts to a purification of the state.There is a converse result of the previous one. Namely,

given a NCQM, we can always find quantum states thatviolate the standard RSUP. A general proof of this result isbeyond the scope of the present work. Here, we shallillustrate explicitly this result for NCQM in two dimen-sions with algebra (6) and (7). That is

� ¼ �E I

�I �E

!; E ¼ 0 1

�1 0

!; (27)

where �, �> 0 are real constants such that ¼ �� < 1.A simple Darboux matrix is C 2 Dð2n;!Þ given by

C ¼ I � �2E

�2�E �I

0@

1A; (28)

where �, are real parameters such that 2� ¼1þ ffiffiffiffiffiffiffiffiffiffiffiffi

1� p

. By construction the Gaussian with covariancematrix� ¼ 1

2CCT is a Wigner function on the nonstandard

BASTOS et al. PHYSICAL REVIEW D 86, 105030 (2012)

105030-4

symplectic space ðR4; !Þ. However, a straightforward cal-culation reveals that it violates the standard RSUP (5).

III. CONCLUSIONS

Let us close our discussion with some clarifyingremarks. The symplectic form ! such that G�;� is a quan-

tum state on ðR2n;!Þ is not unique. Indeed, a phase-spacefunction may be a Wigner function upon quantizationon several distinct symplectic spaces (see Ref. [11] forexamples). Moreover, although our result proves that aGaussian will always be associated with some pure stateon an appropriate symplectic space, it may neverthelessbe a Wigner function on another symplectic space, thisinstance associated with a mixed state.

Finally, one could wonder whether any reasonable func-tion on phase space will always be a Wigner function onsome appropriate symplectic space ðR2n;!Þ. By a reason-able function, we mean a function that satisfies someobvious a priori requirements of Wigner functions: being

real, normalized, uniformly continuous, bounded, squareintegrable, etc. The answer is negative. Indeed, if a func-tion FðzÞ is a Wigner function on some ðR2n;!Þ, then therehas to exist a matrix S such that j detSjFðSzÞ is a Wignerfunction on ðR2n;�Þ. But one may easily construct func-tions that are reasonable but are not Wigner functions onðR2n;�Þ, even admitting rescalings of Planck’s constant.The derivation of such examples is difficult and beyondthe scope of the present work. It requires the concept ofNarcowich-Wigner spectrum [32]. We shall get back to thisissue in a future work.

ACKNOWLEDGMENTS

The work of C. B. is supported by Fundacao paraa Ciencia e a Tecnologia (FCT) under the GrantNo. SFRH/BPD/62861/2009. The work of O. B. is partiallysupported by the FCT Project No. PTDC/FIS/111362/2009. N. C. D. and J. N. P. have been supported by theFCT Grant No. PTDC/MAT/099880/2008.

[1] H. P. Robertson, Phys. Rev. 34, 163 (1929); E. Schrodingerand Sitzungsber, Sitzungsber. Preuss. Akad. Wiss. Phys.Math. Kl. 24, 296 (1930).

[2] R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys.Rev. A 36, 3868 (1987).

[3] G. Giedke, Ph.D. thesis, Innsbruck, 2001.[4] R. G. Littlejohn, Phys. Rep. 138, 193 (1986).[5] V. I. Vysotskii, M.V. Vysotskyy, and S.V. Adamenko,

JETP 114, 243 (2012); J. Surf. Invest. X-ray, Synch.and Neutron Tech. 6, 369 (2012); V. I. Vysotskii andS. V. Adamenko, Tech. Phys. 55, 613 (2010).

[6] L. Rozema, A. Darabi, D. Mahler, A. Hayat, Y. Soudagar,and A. Steinberg, Phys. Rev. Lett. 109, 100404 (2012).

[7] Y.-H. Kim and Y. Shih, Found. Phys. 29, 1849 (1999).[8] K. R. Popper, Naturwissenschaften 22, 807 (1934);

Quantum Theory and the Schism in Physics(Hutchinson, London, 1983).

[9] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777(1935).

[10] J. Oppenheim and S. Wehner, Science 330, 1072 (2010).[11] C. Bastos, O. Bertolami, N. C. Dias, and J. N. Prata, J.

Math. Phys. (N.Y.) 49, 072101 (2008); C. Bastos, N. C.Dias, and J. N. Prata, Commun. Math. Phys. 299, 709(2010); N. C. Dias, M. de Gosson, F. Luef, and J. N.Prata, J. Math. Phys. (N.Y.) 51, 072101 (2010); J. Math.Pures Appl. 96, 423 (2011).

[12] V. P. Nair and A. P. Polychronakros, Phys. Lett. B 505, 267(2001); J. Gamboa, M. Loewe, and J. C. Rojas, Phys. Rev.D 64, 067901 (2001); M. Demetrian and D. Kochan, ActaPhys. Slovaca 52, 1 (2002); P. A. Horwathy, Ann. Phys.(Berlin) 299, 128 (2002); O. Bertolami, J. G. Rosa,C.M. L. de Aragao, P. Castorina, and D. Zappala, Phys.Rev. D 72, 025010 (2005).

[13] L. C. Biedenharn, J. Phys. A 22, L873 (1989); A. J.Macfarlane, J. Phys. A 22, 4581 (1989).

[14] V. I. Man’ko, G. Marmo, S. Solimeno, and F. Zaccaria, Int. J.Mod. Phys. A 08, 3577 (1993); Phys. Lett. A 176, 173 (1993).

[15] V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F.Zaccaria, Phys. Scr. 55, 528 (1997).

[16] M. Henneaux and C. Teitelboim, Quantization of GaugeFields (Princeton University Press, Princeton, NJ, 1992).

[17] M. R. Douglas and N.A. Nekrasov, Rev. Mod. Phys. 73,977 (2001).

[18] A. Connes, M. R. Douglas, and A. Schwarz, J. HighEnergy Phys. 02 (1998) 003; N. Seiberg and E. Witten,J. High Energy Phys. 09 (1999) 032.

[19] H. S. Snyder, Phys. Rev. 71, 38 (1947); 72, 68 (1947).[20] Letter of Heisenberg to Peierls (1930), Letter of Pauli to

Oppenheimer (1946), Wolfgang Pauli, Scientific corre-spondence, edited by K. von Meyenn (Springer-Verlag,Berlin, 1993).

[21] C. N. Yang, Phys. Rev. 72, 874 (1947).[22] S.M. Carroll, J. A. Harvey, V.A. Kostelecky, C. D. Lane,

and T. Okamoto, Phys. Rev. Lett. 87, 141601 (2001).[23] O. Bertolami and L. Guisado, J. High Energy Phys. 12

(2003) 013.[24] L. Freidel and E. R. Livine, in 4-th International

Symposium Quantum Theory and Symmetries, (HeronPress, Sofia, 2006).

[25] M. R. Douglas and C. Hull, J. High Energy Phys. 02(1998) 008.

[26] M. Rosenbaum and J. D. Vergara, Gen. Relativ. Gravit. 38,607 (2006).

[27] A. Connes, Noncommutative Geometry (Academic Press,London, 1994).

[28] P. Martinetti, Mod. Phys. Lett. A 20, 1315 (2005).

VIOLATION OF THE ROBERTSON-SCHRODINGER . . . PHYSICAL REVIEW D 86, 105030 (2012)

105030-5

[29] R. J. Szabo, Phys. Rep. 378, 207 (2003).[30] C. Bastos, O. Bertolami, N. C. Dias, and J. N. Prata, Phys.

Rev. D 78, 023516 (2008); 80, 124038 (2009); 82, 041502(2010); 84, 024005 (2011); P. Nicolini, A. Smailagic, andE. Spallucci, Phys. Lett. B 632, 547 (2006); O. Obregonand I. Quiros, Phys. Rev. D 84, 044005 (2011); B.Malekolkalami and M. Farhoudi, Classical QuantumGravity 27, 245009 (2010).

[31] A. Cannas da Silva, Lectures on Symplectic Geometry(Springer, New York, 2001); M. de Gosson, SymplecticGeometry and Quantum Mechanics (Birkhauser, NewYork, 2006).

[32] N. C. Dias and J. N. Prata, Ann. Phys. (Amsterdam) 313,110 (2004); Rep. Math. Phys. 63, 43 (2009); F. J.Narcowich, J. Math. Phys. (N.Y.) 29, 2036 (1988); Phys.Rev. A 34, 1 (1986).

BASTOS et al. PHYSICAL REVIEW D 86, 105030 (2012)

105030-6