Deformed relativistic and nonrelativistic symmetries on canonical noncommutative spaces

Rabin Banerjee*S.N. Bose National Centre for Basic Sciences, JD Block, Sector 3, Salt Lake, Kolkata 700098, India

Kuldeep Kumar†

Department of Physics, Panjab University, Chandigarh 160014, India(Received 28 August 2006; published 9 February 2007)

We study the general deformed conformal-Poincare (Galilean) symmetries consistent with relativistic(nonrelativistic) canonical noncommutative spaces. In either case we obtain deformed generators,containing arbitrary free parameters, which close to yield new algebraic structures. We show that aparticular choice of these parameters reproduces the undeformed algebra. The modified coproduct rulesand the associated Hopf algebra are also obtained. Finally, we show that for the choice of parametersleading to the undeformed algebra, the deformations are represented by twist functions.

DOI: 10.1103/PhysRevD.75.045008 PACS numbers: 11.30.Cp, 02.40.Gh, 11.10.Nx

I. INTRODUCTION

In a series of papers Wess [1] and collaborators [2–4]have discussed the deformation of various symmetries onnoncommutative spaces. The modified coproduct rule ob-tained for the Poincare generators is found to agree with analternative quantum-group-theoretic derivation [5,6] basedon the application of twist functions [7]. The extension ofthese ideas to field theory and possible implications forNoether symmetry are discussed in [8–10]. An attempt toextend such notions to supersymmetry has been done in[11,12]. Recently, the deformed Poincare generators forLie-algebraic � (rather than a constant �) [13] and Snyder[14] noncommutativity [15] have also been analyzed.

In this paper we develop an algebraic method for ana-lyzing the deformed relativistic and nonrelativistic sym-metries in noncommutative spaces with a constantnoncommutativity parameter. By requiring the twin con-ditions of consistency with the noncommutative space andclosure of the Lie algebra, we obtain deformed generatorswith arbitrary free parameters. For conformal-Poincaresymmetries we show that a specific choice of these pa-rameters yields the undeformed algebra, although the gen-erators are still deformed. For the nonrelativistic(Schrodinger [16]) case two possibilities are discussedfor introducing the free parameters. In one of these thereis no choice of the parameters that yields the undeformedalgebra.

A differential-operator realization of the deformed gen-erators is given in the coordinate and momentum represen-tations. The various expressions naturally contain the freeparameters. For the particular choice of these parametersthat yields the undeformed algebra, the deformations in thegenerators drop out completely in the momentumrepresentation.

The modified comultiplication rules (in the coordinaterepresentation) and the associated Hopf algebra are calcu-

lated. For the choice of parameters that leads to the un-deformed algebra these rules agree with those obtained byan application of the Abelian twist function on the primi-tive comultiplication rule. (For the conformal-Poincarecase this computation of modified coproduct rules usingthe twist function already exists in the literature [5–7], buta similar analysis for the nonrelativistic symmetries is newand presented here.) For other choices of the free parame-ters the deformations cannot be represented by twist func-tions. The possibility that there can be such deformationsalso arises in the context of �-deformed symmetries [17].

II. DEFORMED CONFORMAL-POINCAREALGEBRA

Here we analyze the deformations in the full conformal-Poincare generators compatible with a canonical (constant)noncommutative spacetime. We find that there exists a one-parameter class of deformed special conformal generatorsthat yields a closed algebra whose structure is completelynew. A particular value of the parameter leads to theundeformed algebra.

We begin by presenting an algebraic approach wherebycompatibility is achieved with noncommutative spacetimeby the various Poincare generators. This spacetime ischaracterized by the algebra

�x�;x��� i���; �p�;p���0; �x�;p��� i���: (1)

It follows that, for any spacetime transformation,

��x�; x�� � �x�; �x�� � 0 (2)for constant �. Translations, �x� � a�, with constant a�,are obviously compatible with (2). The generator of thetransformation consistent with �x� � ia��P�; x�� isP� � p�. For a Lorentz transformation, �x� � !��x�,!�� � �!��, the requirement (2) implies !�

���� �

!���

�� � 0, which is not satisfied except for two dimen-sions. Therefore, in general, the usual Lorentz transforma-tion is not consistent with (2). A deformation of theLorentz transformation is therefore mandatory. We con-

*Electronic address: rabin@bose.res.in†Electronic address: kuldeepk@pu.ac.in

PHYSICAL REVIEW D 75, 045008 (2007)

1550-7998=2007=75(4)=045008(5) 045008-1 © 2007 The American Physical Society

sider the minimal O��� deformation:

�x� � !��x� � n1!�����p� � n2!�

����p�

� n3!�����p�;

where n1, n2, and n3 are coefficients to be determined byconsistency arguments. Now the generator,

J �� � x�p� � �1���p�p

� � 12�2�

��p2 � h��i;

where h��i denotes the preceding terms with � and �interchanged, is consistent with �x����i=2�!���J

��;x�� for n1�n2�1��1, n3 � ��2, aresult which follows on using (1). It is therefore clearthat n1 � n2 � 0 is not possible, which necessitates themodification of the transformation as well as the generator.It turns out that

�J ��; J ��� � i���J �� � ���f�2�1 � 1�p�p�

� �2p2��g� � h����i;

with the notation Z��� � h����i �Z��� � h��i� � h��i.The closure of the normal Lorentz algebra is obtained onlyfor �1 � 1=2 and �2 � 0 [4].

Similarly, the usual scale transformation, �x� � x�, isnot consistent with (2). A minimally deformed form of thetransformation is �x� � x� � n���p�. The consis-tency, �x� � i�D; x��, is achieved only for n � 1 byD � x�p�. Likewise, starting with the minimally de-formed form of the special conformal transformation wefind that the generator,

K�� 2x�x�p� � x2p� � 1���p� � 2���x�p�p�

� 3���x�p�p� (3)

is consistent with �x� � i!��K�; x�� for 2 � 3 � 0.

This completes our demonstration of the compatibilityof the various transformation laws with the basic noncom-mutative algebra. However, achieving consistency with thetransformation and closure of the algebra are two differentthings. It turns out that the minimal O��� deformation,while preserving consistency, does not yield a closed alge-bra. Indeed we find that �K�; D� algebra does not close,necessitating the inclusion of O��2� terms in the deformedtransformation and the deformed generator. An appropri-ately deformed form,

K� � �right-hand side of Eq:�3�� � 4����p�p�pp

� 5����p2pp � 6����p�p2;

involves 6 free parameters. However, the closure of the�K�; D� algebra fixes 5 parameters, 2 � �3 ��44 � 1, 5 � 6 � 0, leaving only one, 1, as free.The final form of the deformed generators,

P�� p�; J ��� x�p�� 12���p�p

��h��i; D� x�p�;

K��2x�x�p�� x2p��1���p�����x�p�p�

����x�p�p��14����p�p�p�; (4)

involves one free parameter. The algebra satisfied by thegenerators is such that the Poincare sector remains unaf-fected changing only the conformal sector:

�D; P�� � iP�; �D; J ��� � 0; �K�; P�� � 2i���D� J ���;

�K�; J ��� � �i���K�� �i� 1�����P

� � �����P��� � h��i; �K�; D� � i�K�� 2�i� 1����P��;

�K�; K�� � �2i�i� 1���

��D� ���J ��� � h��i:

(5)

A one-parameter class of closed algebras is found. Fixing 1 � �i yields the usual (undeformed) Lie algebra. In that casethe deformed special conformal generator also agrees with the result given in [12].

The relations in (1) are easily reproduced by representing

x� � x�; p� � �i@� �i@@x�

:

In this coordinate representation, the generators read

P� � �i@�; J �� � �ix�@� � 12���@�@

� � h��i; D � �ix�@�;

K�� �2ix�x�@� � ix2@� � i1�

��@� � ���x�@�@� � �

��x�@�@� �

i

4���

�@�@�@�:

(6)

One may also choose the momentum representation, p� �p�, x� � i~@� � 1

2���p� i@=@p� �

12���p�, which

gives

P �� p�; J ��� i�p� ~@�� p� ~@��; D� ip� ~@�� iN;

K�� p� ~@2�2p� ~@� ~@��2N~@���1� i����p�;

where N � ��� is the number of spacetime dimensions.For 1 � �i, when the generators satisfy the undeformedalgebra, the deformation in K� drops out and all thegenerators in momentum representation have exactly thesame structure as in the commutative description.

The deformed generators lead to new comultiplicationrules. The coproduct rules for the Poincare sector were

BANERJEE, RABIN AND KUMAR, KULDEEP PHYSICAL REVIEW D 75, 045008 (2007)

045008-2

earlier derived in [1,4,5] and for the conformal sector in[6,12]. The free parameter appearing in K� does notappear explicitly in the coproduct ��K�

�. Computingthe basic Hopf algebra, it turns out that the Hopf algebracan be read off from (5) by just replacing the generators bythe coproducts. For example,

���K��;��D�� � i���K�

� � 2�i� 1������P���:

III. DEFORMED SCHRODINGER ANDCONFORMAL-GALILEAN ALGEBRAS

Now we consider separately the Schrodinger symmetryand the conformal-Galilean symmetry, both of which areextensions of the Galilean symmetry.

The standard Schrodinger algebra is given by extendingthe Galilean algebra, which involves Hamiltonian (H ),

translations (P i), rotations (J ij), and boosts (Gi), with thealgebra of dilatation (D) and expansion or special confor-mal transformation (K). The standard free-particle repre-sentation of this algebra is given by H � p2=2m,P i � pi, J ij � xipj � xjpi, Gi � mxi � tpi, D �pixi � tp2=m, and K � m�x� tp=m�2=2.

Now we introduce noncommutativity in space:�xi; xj� � i�ij, �pi; pj� � 0, �xi; pj� � i�ij. Like the de-formed conformal-Poincare case, we follow a two-stepalgebraic process. First, by requiring the compatibility oftransformations with the noncommutative space, a generaldeformation of the generators is obtained. A definite struc-ture emerges after demanding the closure of the algebra.The linear momentum and the Hamiltonian retain theiroriginal forms because the algebra of pi is identical topi. For other generators we consider the minimal deforma-tion. The final form of the generators,

H �1

2mp2; P i � pi; J ij � xipj � 1

2�ikpkpj � 1

2�2�ijp2 � hiji;

Gi� mxi � tpi � �3m�

ijpj; D � pixi �tm

p2; K �m2

�x�

tm

p�

2�m4�ijxipj;

(7)

leads to a nonstandard closure of the algebra:

�J ij; J k‘� � i��ikJ j‘ � 2�ik�2mH�j‘� � hijk‘i; �Gi; Gj� � im2�1� 2�3��ij;

�Gi; J jk� � i��ikGj�m�12� �3���

ijP k � �ik�jmPm� �m�2�jkP i� � hjki; �J ij; D� � �4im�2�

ijH ;

�D; Gi� � �i�Gi

�m�1� 2�3��ijPj�; �K; P i� � i�Gi � �14� �3�m�ijP

j�;

�J ij;K� � i�m4��ikJ kj � �jkJ ki� � 2�2m�ijD

�; �K; Gi

� � �im�ij��34� �3�Gj� ��2

3 � �3 �14�m�

jkP k�;

�D; K� � �i�2K�

m4�ijJ ij � 1

2�2m2�ij�ijH

�;

(8)

the other brackets remaining unaltered.Some comments are in order. We have obtained the

deformed Schrodinger algebra involving two parameters,�2 and �3. For �! 0, the deformed algebra reduces to theundeformed one. A distinctive feature is that there is nochoice of the free parameters for which the standard (un-deformed) algebra can be reproduced. This is an obviousand important difference from the Poincare treatment.

It is however possible to obtain an alternative deforma-tion which, for a particular choice of parameters, yields theundeformed algebra. We notice that the brackets involvingall generators other than K reduce to the standard ones byfixing �2 � 0 and �3 � 1=2, although the generators aredeformed. So we allow O��2� terms in K. Demanding the

closure of �H ; K� and �D; K� brackets yieldsO��2�-deformed K:

K�m2

�x�

tm

p�

2��6m�

ijxipj�m�1

8��6

2

��ij�jkpipk:

(9)

The brackets involving this K turn out to be

�H ;K���iD; �K;P i�� i�Gi���6��3�m�ijPj�;

�J ij;K�� i��12��6�m��ikJ kj��jkJ ki��2�2m�

ijD�;

�K;Gi���im�ij��1��3��6�G

j

���23��3�

14�m�

jkP k�;

�D;K�� i��2K��12��6�m�ij�2�2m�ijH � J ij��

(10)

which give us another deformed Schrodinger algebra in-volving three parameters, �2, �3, and �6. It is easily seenfrom (10) that the particular choice of parameters, �2 � 0and �3 � �6 � 1=2, reproduces the standard algebra. Thisagrees with [18]. Now onwards we shall restrict to K givenby (9) whenever expansions are considered.

In the coordinate representation, D � �ixi@i ��t=m�r2 � iN0 and

DEFORMED RELATIVISTIC AND NONRELATIVISTIC . . . PHYSICAL REVIEW D 75, 045008 (2007)

045008-3

K �m2

x2 �t2

2mr2 � itxi@i � i

tN0

2� i�6m�

ijxi@j

�m�1

8��6

2

��ij�jk@i@k;

where N0 � �ii is the number of space dimensions. Themomentum representation of D is D � ipi ~@i � tp2=m.The representation for K involves a deformation which,expectedly, drops out for �6 � 1=2 that corresponds to thestandard algebra.

The comultiplication rules, using the coordinate repre-sentation, turn out to be

��H � � H 1� 1 H �1

mP i P i; (11)

��Pi� � P 1� 1 P i; (12)

��J ij�� 12�J

ij1�1 J ij��im�P j Pm� Pm P j��

��2�ijPm Pm�hiji; (13)

��Gi� � 1

2�Gi 1� 1 Gi

� t�P i 1� 1 P i�

�m�ijf��3 � 1�P j 1� �31 P jg�; (14)

��D� � D 1� 1 D�2tmP i P i

� iN0

21 1� �ijP i P j; (15)

��K��K1�1K� itN0

211�

1

2m�t2P i P i

� Gi Gi

� t�P iGi� Gi

P i��

�1

2�ij�tP i P j���3�1�P iGj

��3Gj P i�

�m2�ij�ik

��2

3��3�1

2

�P j P k: (16)

Among the free parameters �2, �3, and �6 appearing in thedefinition of the deformed generators, only the first twooccur in the expressions for the deformed coproducts. Theparameter �6, which is present in K, however, does notoccur in ��K�. Expectedly, it turns out that the Hopfalgebra can be directly read off from the algebra

(Eqs. (8), etc.) by just replacing the generators by thecoproducts.

There is an alternative method, based on quantum-group-theoretic arguments, of computing the coproducts[5,6]. This is obtained for the particular case when thedeformed generators satisfy the undeformed algebra. Inour analysis it corresponds to the choice �2 � 0, �3 ��6 � 1=2. The essential ingredient is the application ofthe Abelian twist function, F � exp��i=2��ijP i P j�, asa similarity transformation on the primitive coproduct ruleto abstract the deformed rule. After some calculations itcan be shown that the deformed coproduct rule (14), forexample, for the specific values of the free parametersalready stated, is obtained by identifying

��Gi� � �F��Gi�F�1�Gi!Gi;P j!P j :

The coproducts for other generators can similarly be ob-tained from the same twist element.

Strictly speaking, the algebra obtained by enlarging theGalilean algebra by including dilatations and expansions isnot a conformal algebra since it does not inherit some basiccharacteristics like vanishing of the mass, equality of thenumber of translations and the special conformal trans-formations, etc. However since it is a symmetry of theSchrodinger equation, this enlargement of the Galileanalgebra is appropriately referred to as the Schrodingeralgebra. It is possible to discuss the conformal extensionof the Galilean algebra by means of a nonrelativistic con-traction of the relativistic conformal-Poincare algebra.Recently this was discussed for the particular case of threedimensions [19]. This algebra is different from theSchrodinger algebra discussed earlier. We scale the gen-erators and the noncommutativity parameter as

D � �D; K�� �K0; Ki

� � �c �K; c2 �Ki�;

P� � �P 0; P i� � � �H =c; �P i�;

J �� � �J 0i; J ij� � �c �Gi; �J ij�;

��� � ��0i; �ij� � �0; c2 ��ij�;

where c is the velocity of light. We use this scaling in (5)and take the limit c! 1. Finally we redefine to choose thesame symbols for the nonrelativistic case ( �D! D, etc.).Then we get the deformed algebra

�D;H � � iH ; �D; P i� � iP i; �D; J ij� � 0; �D; Gi� � 0; �K;H � � 2i00D;

�K; P i� � 2iGi; �K; D� � iK; �K; K� � 0; �K; Gi� � �i00�Ki� �i� 1��ijP j�;

�K; Ki� � 2i�i� 1��ijGj; �K; J ij� � 0; �Ki;H � � �2iGi; �Ki; P j� � 0; �Ki; Gj

� � 0;

�Ki; J jk� � �i�ijKk� �i� 1���

ijP k � ij�k‘P ‘�� � hjki; �Ki; D� � i�Ki� 2�i� 1��

ijP j�;

(17)

which also contains a free parameter. Restricting to three dimensions and the specific choice 1 � �i reproduces theresults obtained recently in [19].

BANERJEE, RABIN AND KUMAR, KULDEEP PHYSICAL REVIEW D 75, 045008 (2007)

045008-4

IV. CONCLUSIONS

We have analyzed the deformed conformal-Poincare,Schrodinger, and conformal-Galilean symmetries compat-ible with the canonical (constant) noncommutative space-time and found new algebraic structures.

For the conformal-Poincare case we found a one-parameter class of deformed special conformal generatorsthat yielded a closed algebra whose structure was com-pletely new. Fixing the arbitrary parameter reproduced theusual (undeformed) Lie algebra.

Next we considered the Schrodinger symmetry. Here weobtained the deformed Schrodinger algebra involving twoparameters. The closure of this algebra yielded new struc-tures. The generators involved O��� deformations. For �!0, the deformed algebra reduced to the undeformed one.However a distinctive feature was that there was no choiceof the free parameters for which the standard (undeformed)algebra could be reproduced. Exploring other possibilities,then we obtained an alternative deformation which, for aparticular choice of parameters, indeed reproduced the

undeformed algebra. In this case the modified specialconformal generator involved O��2� terms. The deformedSchrodinger algebra now involved three parameters, aparticular choice of which reproduced the standardalgebra.

Finally we discussed the conformal extension of theGalilean algebra by means of a nonrelativistic contractionof the relativistic conformal-Poincare algebra. This algebrais different from the Schrodinger algebra, both in thecommutative and noncommutative descriptions. Thepresent analysis can be extended to other (nonconstant)types of noncommutativity. Some results in this directionhave already been provided for the Snyder space [15].

ACKNOWLEDGMENTS

K. K. thanks the Council of Scientific and IndustrialResearch, Government of India, for financial support dur-ing the period when the major part of this work wascompleted at S. N. Bose National Centre for BasicSciences, Kolkata.

[1] J. Wess, hep-th/0408080.[2] M. Dimitrijevic and J. Wess, hep-th/0411224.[3] P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P.

Schupp, and J. Wess, Classical Quantum Gravity 22, 3511(2005).

[4] F. Koch and E. Tsouchnika, Nucl. Phys. B717, 387 (2005).[5] M. Chaichian, P. P. Kulish, K. Nishijima, and A. Tureanu,

Phys. Lett. B 604, 98 (2004); M. Chaichian, P. Presnajder,and A. Tureanu, Phys. Rev. Lett. 94, 151602 (2005).

[6] P. Matlock, Phys. Rev. D 71, 126007 (2005).[7] R. Oeckl, Nucl. Phys. B581, 559 (2000).[8] R. Banerjee, B. Chakraborty, and K. Kumar, Phys. Rev. D

70, 125004 (2004).[9] C. Gonera, P. Kosinski, P. Maslanka, and S. Giller, Phys.

Lett. B 622, 192 (2005).[10] X. Calmet, Phys. Rev. D 71, 085012 (2005).[11] Y. Kobayashi and S. Sasaki, Int. J. Mod. Phys. A 20, 7175

(2005); B. M. Zupnik, Phys. Lett. B 627, 208 (2005); M.Ihl and C. Saemann, J. High Energy Phys. 01 (2006) 065.

[12] R. Banerjee, C. Lee, and S. Siwach, Eur. Phys. J. C 48, 305(2006).

[13] J. Lukierski and M. Woronowicz, Phys. Lett. B 633, 116(2006).

[14] H. S. Snyder, Phys. Rev. 71, 38 (1947).[15] R. Banerjee, S. Kulkarni, and S. Samanta, J. High Energy

Phys. 05 (2006) 077.[16] U. Niederer, Helv. Phys. Acta 45, 802 (1972); C. R.

Hagen, Phys. Rev. D 5, 377 (1972); G. Burdet and M.Perrin, Lett. Nuovo Cimento Soc. Ital. Fis. 4, 651 (1972).

[17] J. Lukierski, hep-th/0604083.[18] R. Banerjee, Eur. Phys. J. C 47, 541 (2006).[19] J. Lukierski, P. C. Stichel, and W. J. Zakrzewski, Phys.

Lett. A 357, 1 (2006).

DEFORMED RELATIVISTIC AND NONRELATIVISTIC . . . PHYSICAL REVIEW D 75, 045008 (2007)

045008-5