Twisted statistics in �-Minkowski spacetime

T.R. Govindarajan*

Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India

Kumar S. Gupta+

Theory Division, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700064, India

E. Harikumar‡

School of Physics, University of Hyderabad, Hyderabad 500046, India

S. Meljanacx and D. Meljanack

Rudjer Boskovic Institute, Bijenicka c.54, HR-10002 Zagreb, Croatia(Received 20 February 2008; published 12 May 2008)

We consider the issue of statistics for identical particles or fields in �-deformed spaces, where the

system admits a symmetry group G. We obtain the twisted flip operator compatible with the action of the

symmetry group, which is relevant for describing particle statistics in the presence of the noncommu-

tativity. It is shown that for a special class of realizations, the twisted flip operator is independent of the

ordering prescription.

DOI: 10.1103/PhysRevD.77.105010 PACS numbers: 11.10.Nx, 11.30.Cp

I. INTRODUCTION

Noncommutative geometry is a plausible candidate fordescribing physics at the Planck scale, a simple model ofwhich is given by the Moyal plane [1]. The models ofnoncommutative spacetime that follow from combininggeneral relativity and the uncertainty principle can bemuch more general [2]. An example of this more generalclass is provided by the �-deformed space [3–5], which isbased on a Lie algebra type noncommutativity. Apart fromits algebraic aspects [6–11], various features of field theo-ries and symmetries on �-deformed spaces have recentlybeen studied [12–16]. Such a space has also been discussedin the context of doubly special relativity [17–19].

The issue of particle statistics plays a central role in thequantum description of a many-body system or field the-ory. This issue has naturally arisen in the context of non-commutative quantum mechanics and field theory [20–24].In the noncommutative case, the issue of statistics isclosely related to the symmetry of the noncommutativespacetime on which the dynamics is being studied. If asymmetry acts on a noncommutative spacetime, its copro-duct usually has to be twisted in order to make the sym-metry action compatible with the algebraic structure. In thecommutative case, particle statistics is superselected, i.e. itis preserved under the action of the symmetry. In thepresence of noncommutativity, it is thus natural to demandthat the statistics is invariant under the action of the twisted

symmetry. This condition leads to a new twisted flip op-erator, which is compatible with the twisted coproduct ofthe symmetry group [22,23]. The operators projecting tothe symmetric and antisymmetric sectors of the Hilbertspace are then constructed from the twisted flip operator.While most of the discussion of statistics in the noncom-mutative setup has been done in the context of the Moyalplane, some related ideas for �-deformed spaces have alsoappeared recently [25–27].In this paper we set up a general formalism to describe

statistics in �-deformed spaces. Our formalism presentedhere is applicable to a system with an arbitrary symmetrygroup, which may include Poincare, Lorentz, Euclideanand so on. In particular, we present a general formula forthe twisted flip operator that is applicable for any symme-try group, which is constructed in terms of the commuta-tive flip operator, the undeformed coproduct of thesymmetry group and the twist element for the�-deformed space. In addition, we show that for a specialclass of realization considered here, the twisted flip opera-tor is independent of the choice of ordering. This propertyis valid only for a special class of realizations consideredhere and is not true for any arbitrary realization. This paperis organized as follows. In Sec. II we briefly recall somebasic facts about � space and discuss a special class ofrealizations. In Sec. III we discuss the star product andtwist operator and obtain their expression for the specificclass of realizations discussed here. In Sec. IV we intro-duce the idea of twisted statistics for the � space and obtaina general formula for the twisted flip operator. We alsoobtain the corresponding expression for the special class ofrealizations considered here and discuss the ordering inde-pendence for this case. Section V concludes the paper withsome discussions.

*trg@imsc.res.in+kumars.gupta@saha.ac.in‡harisp@uohyd.ernet.inxmeljanac@irb.hrkdmeljan@irb.hr

PHYSICAL REVIEW D 77, 105010 (2008)

1550-7998=2008=77(10)=105010(6) 105010-1 � 2008 The American Physical Society

II. � SPACE AND ITS REALIZATIONS

In this section we briefly review the formalism that leadsto a derivation of the Drinfeld twist for the �-Minkowskispace. Following [11] we shall first discuss the �-deformedEuclidean space and then analytically continue to thecorresponding Minkowski space.

Consider the Lie algebra type relations

½x�; x�� ¼ iC���x� ¼ iða�x� � a�x�Þ; (1)

defining a noncommutative (NC) space with coordinatesx1; x2; . . . ; xn, and where a1; a2; . . . ; an are real constantsparametrizing the deformation of the Euclidean space. Wechoose a1 ¼ a2 ¼ . . . ¼ an�1 ¼ 0 and an ¼ a ¼ 1

� as a

very small length scale. Here we use Latin indices for thesubspace ð1; 2; . . . ; n� 1Þ and Greek indices for the wholespace ð1; 2; . . . ; n� 1Þ. Then the algebra of the NC coor-dinates becomes

½xi; xj� ¼ 0; ½xn; xi� ¼ iaxi;

i; j ¼ 1; 2; . . . ; n� 1:(2)

The NC coordinates x� admit realizations in terms of

ordinary commutative coordinates x1; x2; . . . ; xn and theirderivatives @1; @2; . . . ; @n, where @� ¼ @

@x�, which generate

the Heisenberg algebra

½@�; x�� ¼ ���: (3)

The realizations we want to find can be written in the form

x � ¼ x����ð@xÞ: (4)

They will satisfy the condition

½@�; x�� ¼ ���ð@xÞ: (5)

Here we shall concentrate on the special case of the algebra(5), given by the relations

½@i; xj� ¼ �ij’ðAÞ; ½@i; xn� ¼ ia@i�ðAÞ; (6)

½@n; xi� ¼ 0; ½@n; xn� ¼ 1; (7)

where ’ and � are functions of A ¼ ia@n. An explicit formof a class of realizations satisfying these conditions isgiven by

x i ¼ xi’ðAÞ; xn ¼ xn ðAÞ þ iaxk@k�ðAÞ; (8)

where the summation over repeated indices is understood.The functions ’ and satisfy the conditions ’ð0Þ ¼ 1, ð0Þ ¼ 1 and they are assumed to be positive. The function

� satisfies ’0’ ¼ �� 1, where’0 ¼ d’

dA . It is also assumed

that the quantity �ð0Þ ¼ ’0ð0Þ þ 1 is finite. Further con-ditions on the functions ’ and are obtained by consid-ering the Lorentz generators in this spacetime. Imposingthe conditions that the Lorentz generators be linear in xwith an infinite series in @, leads to two families of realiza-

tions given by ¼ 1 and ¼ 1þ 2A. Here we consideronly the case ¼ 1, for which the realizations are parame-trized by an arbitrary positive function ’ðAÞ, ’ð0Þ ¼ 1.The coproduct 4’ corresponding to partial derivatives

appearing in (8) is given by

4’ ð@nÞ ¼ @n � 1þ 1 � @n ¼ @xn þ @yn;

4’ ð@iÞ ¼ @xi’ðAx þ AyÞ’ðAxÞ þ @yi

’ðAx þ AyÞ’ðAyÞ eAx ;

(9)

where Ax ¼ ia@xn and similarly for Ay. The last equation

can also be written in a more compact way as

4’ ð@iÞ ¼ ’ðA � 1þ 1 � AÞ�@i’ðAÞ � 1þ eA � @i

’ðAÞ�:

(10)

There is a well-defined correspondence between therealizations defined above and the ordering prescriptionin the �-space [11]. To describe it, first note that we takethe vacuum state to be represented by 1, i.e. j0i � 1, so thatit is annihilated by all derivatives @�. We now introduce

’-ordering : eik�x’ :’ defined by the relation

: eik�x’ :’j0i ¼ eik�x: (11)

As special cases, the expressions for the left, right, andsymmetric ordering prescriptions are, respectively, givenby

: eik�x� :L � eiknxneikixi ¼ eiknxnþikixi’Sð�aknÞe�akn ;

: eik�x� :R � eikixieiknxn ¼ eiknxnþikixi’Sð�aknÞ;

: eik�x� :S ¼ eik�x� ;

(12)

where

’SðAÞ ¼ A

eA � 1; A ¼ ia@n � �akn: (13)

Corresponding to the realizations of the form (8), there alsoexists a family of ordering prescriptions interpolating be-tween left and right, parametrized by the function ’ ¼e��A. This ordering prescription looks like

: eik�x� :� ¼ ei�knxneikixieið1��Þknxn ; (14)

For � ¼ 0, we obtain a right ordering, while for � ¼ 1, aleft ordering emerges. For � ¼ 1

2 , the same factor appears

on the left- and right-hand side symmetrically [28]. In thissense it can be named as symmetric ordering, but it shouldbe kept in mind that it is completely different from thetotally symmetric Weyl ordering.More generally, corresponding to the ’ ordering given

by (8), the ordering prescription can be written as

: eik�x� :’ ¼ eiknxnþikixiðð’Sð�aknÞÞ=ð’ð�aknÞÞÞ; (15)

which contains all the special cases described above. This

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gives the direct correspondence between the various real-izations and the corresponding ordering prescriptions.

III. STAR PRODUCTAND TWIST IN � SPACE

For any’ ordering, a star product can be defined in the �space. This star product can be expressed in terms of thecorresponding twist element F ’ as

f ?’ g ¼ m0ðF ’f � gÞ ¼ m’ðf � gÞ; m’ ¼ m0F ’

(16)

where m0 is the usual pointwise multiplication map in thecommutative algebra of smooth functions A0, namely,m0ðf � gÞ ¼ fg. and f, g 2 A0.

The star product of two functions f and g in the ’realization is given by

ðf ?’ gÞðxÞ ¼ limu!xt!xm0ðex�ð4’�40Þ@�fðuÞ � gðtÞÞ; (17)

where 4’ is given in (10) and

40 ð@Þ ¼ @ � 1þ 1 � @ (18)

is the untwisted coproduct. The corresponding twist ele-ment is given by

F ’ ¼ ex�ð�’��0Þ@�; (19)

where 4’ satisfies the relation

4’ ¼ F�1’ 40 F ’: (20)

Using (9) and (17), the star product in the �-space can bewritten as

ðf ?’ gÞðxÞ ¼ limu!xt!x

exj@uj ðð’ðAuþAtÞÞ=ð’ðAuÞÞ�1Þþxj@tjðð’ðAuþAtÞÞ=ð’ðAtÞÞeAu�1ÞfðuÞgðtÞ; (21)

where we use the notation Ax ¼ ia @@xn

, etc. The corresponding twist operator (19) can be written as

F ’ ¼ eNx lnð’ðAxþAyÞÞ=ð’ðAxÞÞþNyðAxþlnð’ðAxþAyÞÞ=’ðAyÞÞÞ � eðN�1Þ lnð’ðA�1þ1�AÞÞ=ð’ðA�1ÞÞþð1�NÞðA�1þlnð’ðA�1þ1�AÞÞ=ð’ð1�AÞÞÞ; (22)

where Nx ¼ xi@=@xi and similarly for Ny. In writing theabove expression of the twist operator we have used theidentity [11] [see eqs (A.16),(A.17)]

limu!xe

xi@ui ½�ðAu;AyÞ�1� ¼ exi@

xi ln�ðAx;AyÞ; (23)

valid for arbitrary function �ðAu; AyÞ.We point out that the star product and the twist element

depend explicitly on the choice of the ordering. For ex-ample, the twist operator corresponding to the left orderingis given by

F L ¼ e�NxAy ¼ e�N�A; (24)

while that corresponding to the right ordering is

F R ¼ eAxNy ¼ eA�N: (25)

For �ðAÞ ¼ e�cA where c 2 R, we obtain a simple inter-polation between right ordering (c ¼ 0) and left ordering(c ¼ 1), with the twist operator given by

F c ¼ e�cN�Aþð1�cÞA�N: (26)

Using 4N ¼ N � 1þ 1 � N, 4A ¼ A � 1þ 1 � A, and½N;A� ¼ 0, it is easy to verify that the above class of twistoperators F c satisfies the cocycle condition

ðF c � 1Þð4 � 1ÞF c ¼ ð1 �F cÞð1 �4ÞF c; (27)

for all c 2 R.The above results can be continued to the Minkowski

space by redefining the momenta as

P0 ¼ i@0; Pi ¼ �i@i; i ¼ 1; 2; . . . ; N � 1; (28)

which satisfy

½P�; x�� ¼ �i��; �; � ¼ 0; 1; . . . ; N � 1; (29)

�� ¼ diagð�1;þ1;þ1; . . . ;þ1Þ. The algebra of the var-iables x�, @� remains unaltered,

½@�; x�� ¼ ���; �; � ¼ 0; 1; . . . ; N � 1: (30)

Accordingly, the commutation relation describing�-deformed Minkowski space takes the form

½xi; xj� ¼ 0; ½x0; xi� ¼ iaxi;

i; j ¼ 1; 2; . . . ; N � 1;(31)

where xn ¼ ix0 and the old deformation parameter an isrelated to the new one a0 by an ¼ ia0. For simplicity, weremove the index 0 of it, but in all subsequent considerationone should keep in mind that it is the time component ofthe N-vector in Minkowski space and that it is real. Thedefinition of the star product also remains unchanged,provided in (21) we replace Ax by Ax ¼ ia@x0.In terms of the �-Minkowski space variables, as defined

above, we can write the coproduct and the twist operator inthe momentum space as well. The coproduct K’ðp; qÞ inmomentum space is given by

½K’ðp; qÞ�� ¼ �i4’ ð@�Þ;K’ðp; qÞx ¼ �ðk0 þ q0Þx0 þ ’ðak0 þ aq0Þ

��kixi’ðak0Þ þ

eak0

’ðaq0Þqixi�: (32)

Let F denote the corresponding twist operator in the

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momentum space. Its defining relation is

F fðxÞgðyÞ � F ðx; px; y; pyÞfðxÞgðyÞ¼ F

Zd4keikx ~fðkÞ

Zd4qeiqy~gðqÞ

¼Zd4kd4qðF eikxeiqyÞ~fðkÞ~gðqÞ

¼Zd4kd4qeikxeiqyF

�i@

@k; k; i

@

@q; q

�

� ~fðkÞ~gðqÞ:From this we obtain

F ~fðkÞ � ~gðqÞ ¼ F�i@

@k; k; i

@

@q; q

�~fðkÞ~gðqÞ: (33)

IV. TWISTED STATISTICS IN � SPACE

Consider a system of particles or fields described by anaction principle. Very often the action describing such asystem remains invariant when acted upon by a suitablegroup, which is called the symmetry group of the system.For example, the symmetry group could be the Poincaregroup, Lorentz group, Euclidean group and so on. In whatfollows we assume that the system under consideration isinvariant under a symmetry group denoted by G.

In the commutative case, the identical particles or fieldsusually satisfy either bosonic or fermionic statistics. On atwo particle Hilbert space, the bosonic or fermionic statis-tics is implemented by restricting to the symmetrized andantisymmetrized sectors, respectively. Normally, the twosubsectors of the Hilbert space corresponding to bosonic orfermionic statistics are individually preserved under theaction of the symmetry group G. This is to say that thestatistics is superselected.

Let � be an element of G which acts with some repre-sentation D. In the commutative case with a ¼ 0, theaction of G on the two particle Hilbert space is describedby the coproduct �0 where

�0: � ! � ��;

f � g! ðD �DÞ�0ð�Þf � g: (34)

The coproduct �0ð�Þ is compatible with the usual point-wise multiplication map m0, i.e. it satisfies the condition

m0ððD �DÞ�0ð�Þf � gÞ ¼ Dð�Þm0ðf � gÞ: (35)

Statistics of identical particles is usually defined in termsof a flip operator 0. On an element f � g 2 A0 �A0, ithas the action

0ðf � gÞ ¼ g � f: (36)

Symmetrization or antisymmetrization on a two particleHilbert space is carried out by the operators 1

2 ð1� 0Þ. Inthe commutative case, the flip operator 0 commutes withthe coproduct �0ð�Þ of the symmetry group G, i.e.

½�0ð�Þ; 0� ¼ 0: (37)

Physically this means that the process of symmetrization orantisymmetrization is frame independent. The statisticsthus remains invariant under the action of the symmetrygroup G.In the noncommutative case, when a ¼ 1

� � 0, the mul-

tiplication map in the algebra is defined by the star product,denoted by m’ in the case of a given ’ realization. In this

case, the coproduct �0 of the symmetry group G is nolonger compatible with m’. This implies that G cannot be

an automorphism of the algebra (2) with the coproduct �0.We can define a twisted coproduct for the symmetry groupG, denoted by �’ where

�’ð�Þ ¼ F�1’ �0ð�ÞF ’; (38)

andF ’ is the twist element. This new twisted coproduct is

compatible with the multiplication map m’, i.e. it satisfies

the equation

m’ððD �DÞ�’ð�Þf � gÞ ¼ Dð�Þm’ðf � gÞ: (39)

Henceforth we work with the twisted coproduct �’. The

symmetry group G with the twisted coproduct (38) is anautomorphism of the algebra (2) and is often called thetwisted symmetry group.In the presence of � deformation, it is also desirable that

the statistics be left unchanged under the action of thetwisted symmetry group. This would ensure that even inthe presence of the � deformation, the symmetrizationprocedure would be left invariant under the action of thetwisted symmetry group and that the twisted statisticsremains superselected. However, it turns out that the flipoperator 0 does not commute with the twisted coproduct,i.e.

½�’ð�Þ; 0� � 0: (40)

This means that the symmetrization or antisymmetrizationcarried out by 0 would not be preserved by the action ofthe twisted symmetry group. We must therefore seek a newflip operator, called the twisted flip operator ’ which

would satisfy the condition

½�’ð�Þ; ’� ¼ 0: (41)

Such a twisted flip operator can be defined and is given by

’ ¼ F�1’ 0F ’; (42)

where F ’ is the twist element in (19). The new flip

operator ’ commutes with the twisted coproduct of the

symmetry group G, i.e. it satisfies the condition (41).Having defined the twisted flip operator, the symmetriza-tion and antisymmetrization can now be carried out withthe projection operators 1

2 ð1� ’Þ, respectively. This

would ensure that the twisted statistics of a two particlestate in the � space would remain unchanged under theaction of the twisted symmetry group G.

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The expression of the twisted flip operator ’ in (42) is

valid for an arbitrary’ realization of the �-deformed spaceand for a general symmetry group G for the dynamics. Wenow restrict our attention to the special cases of the ’realizations given in (8). For this particular special class ofrealizations, the twist element is given in (22). Using (22)and (42), we obtain an explicit expression for the twistedflip operator ’ for the class (8) of realizations as

’ ¼ eiðxiPi�A�A�xiPiÞ0: (43)

Note that as A is directly proportional to the deformationparameter a, the twisted flip operator ’ goes over

smoothly to the untwisted flip operator 0 as the deforma-tion parameter a! 0. It may be noted that although thetwist elementF ’ depends explicitly on the choice of the�

realization, it is a remarkable fact that the twisted flipoperator ’ as obtained in (43) is independent of the class

of realizations (8) and the corresponding orderings. Thisconclusion follows naturally from the properties of thetwist element (22) and the twisted flip operator (43) inthe � space. By construction, any symmetrization or anti-symmetrization of two particle states carried out using theflip operator ’ would be preserved under the action of the

Poincare group with a twisted coproduct. Thus, the twistedflip operator obtained above is a good candidate to describeparticle statistics in the � space. We however point out thatthe ’ independence of the twisted flip operator is validonly for the class of realization (8) considered in this paperand will not hold in the case of an arbitrary realization. Itmay also be noted that the twisted flip operator given in(42) or (43) can be identified with the R matrix and thecorresponding r-matrix in our notation is given by r ¼ 1

a �ðN ^ AÞ. This r-matrix satisfies classical Yang-Baxterequation, since ½N;A� ¼ 0. Further properties of theR-matrix are presently under investigation.

In the paper by Daszkiewicz, Lukierski andWoronowicz[27], the quantized scalar field on �-spacetime was con-sidered. Calculations performed in [27], especially the

coproduct, correspond to our realization �ðAÞ ¼ e�A=2.In classical limit their results are equivalent to our statisticsflip operator equation (43). Arzano and Marciano [25]construct their twist operator to be consistent with kappaPoincare algebra, whereas in our construction we have anarbitrary symmetry algebra G. In the limit kappa going toinfinity both lead to commutative theory. It may also benoted that in our approach the Lorentz algebra is unde-formed and corresponding coalgebra is � deformed. ThePoincare group together with the twisted coproduct pro-vides an automorphism of the algebra where the multi-plication map is given by the star product in (17) or (21).The effect of this will be reflected in correlation functions.The quantization of scalar field on �-Minkowski spacealong within our framework and its relation to [25,27] ispresently under investigation.

We end this section with some remarks about the CPTsymmetry and spin-statistics theorem for the �-deformedspace. For the Moyal plane with space-time noncommuta-tivity these issues have been discussed in [29]. Let usconsider a system described by the �-deformedMinkowski space (31). The parity operator P, which islinear, can be defined as

P: x0 ! x0; xi ! �xi; ia! ia: (44)

This is an automorphism of the algebra (31). Hence, unlikein the Moyal case [29], parity is a symmetry of the�-deformed Minkowski space.The time reversal operator T, which is antilinear, is

defined by

T: x0 ! �x0; xi ! xi; ia! �ia: (45)

Thus T and is also an automorphism of the algebra (31) andthe same holds for PT. In case the charge conjugation isdefined and is an automorphism, then CPT can be realizedas an automorphism on the �-deformed Minkowski space.The star product for the �-deformed Minkowski space,

just as in the Moyal case, is a nonlocal quantity. The lack oflocality leads to a failure of causality [30], and the spin-statistics theorem also fails to be valid. In particular, thePauli exclusion principle is not expected to be valid fortheories based on the algebra (31), and bounds on thedeformation parameter a can be obtained from the boundson the validity of the Pauli exclusion principle [22,23].

V. CONCLUSION

In this paper we have discussed the issue of statistics fora system of identical particles or a field theory in � de-formed space, where the symmetry group of the systemunder consideration is described by a general group G.Because of the noncommutative structure of the � de-formed space, the coproduct of the symmetry group Ghas to be twisted. The flip operator describing the statisticsin the commutative case no longer commutes with thetwisted coproduct of the symmetry group. As a result, thesymmetrization or antisymmetrization carried out by theprojection operators obtained from the commutative flipoperator are no longer preserved under the action of thesymmetry group. This leads us to seek a new twisted flipoperator that would commute with the twisted coproduct ofG, and we have presented a general expression of thetwisted flip operator for an arbitrary realization of the �deformed space and for a general symmetry group G. Thetwisted statistics obtained from the twisted flip operatorremains superselected.A � deformed space admits many different realizations.

We have restricted our attention to a particular class in thispaper. It has been shown that within this class, the twistedflip operator is independent of the choice of the realizationand the corresponding ordering. The projection operatorscorresponding to the symmetric and antisymmetric sectors

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of the Hilbert space also share this property for this specialclass. This property is not valid for any arbitrary realizationof the � deformed space, not belonging to the class de-scribed here.

The analysis here provides a general framework forconsidering statistics in � deformed spaces which wouldbe relevant for the analysis of field theories in such spaces.The implications of the twisted statistics for quantum fieldtheories on � deformed spaces including the issues con-cerning the R matrix and Yang-Baxter equations are cur-rently under investigation.

ACKNOWLEDGMENTS

The authors would like to thank A. Samsarov and Z.Skoda for discussions. This work was supported by theMinistry of Science and Technology of the Republic ofCroatia under Contract No. 098-0000000-2865. This workwas done within the framework of the Indo-Croatian JointProgramme of Cooperation in Science and Technologysponsored by the Department of Science and Technology,India (DST/INT/CROATIA/P-4/05), and the Ministry ofScience, Education and Sports, Republic of Croatia.

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