S(Λ µ ν) = Λ µ ν, (1)

Transcription

1 A NOTE ON GEOMETRY OF -MINKOWSKI SPACE Stefan Giller, Cezary Gonera, Piotr Kosiński, Pawe l Maślanka Department of Theoretical Physics University of Lódź ul. Pomorska 149/153, Lódź, Poland arxiv:q-alg/ v1 2 Feb 1996 Abstract. The infinitesimal action of -Poincaré group on -Minkowski space is computed both for generators of -Poincaré algebra and those of Woronowicz generalized Lie algebra. The notion of invariant operators is introduced and generalized Klein-Gordon equation is written out. I. Introduction In this short note we consider some simple properties of differential operators on -MinkowskispaceM a noncommutativedeformationofminkowskispace-time which depends on dimensionful parameter [1]. We calculate the infinitesimal action of -Poincaré group P [1] on M both for the generators of -Poincaré algebra P [2] this is done using the duality P P described in [3] and for the elements of Woronowicz generalized Lie algebra [4] of -Poincaré group [5]. The result supports the relation between both algebras found in [5]. We introduce also the notion of invariant differential operators on M and write out the generalized Klein-Gordon equation. Let us conclude this section by introducing the notions of -Poincaré group P and algebra P. P is defined by the following relations [1] [x µ,x ν ] = i δµ 0 xν δ ν 0x µ, [Λ µ ν,λ α β] = 0, [Λ µ ν,x ρ ] = i Λµ 0 δ µ 0 Λρ ν +Λ 0 ν δ 0 ν gµρ, Λ µ ν = Λ µ α Λ α ν, x µ = Λ µ α x α +x µ I, SΛ µ ν = Λ µ ν, Sx µ = Λ µ ν x ν, ελ µ ν = δ ν, µ εx µ = 0. 1 * Supported by Lódź University grant N o Typeset by AMS-TEX

2 2 S. GILLER, C. GONERA, P. KOSIŃSKI, P. MAŚLANKA The dual structure, the -Poincaré algebra P, is, in turn, defined as follows [6] [P µ,p ν ] = 0, [M i,m j ] = iε ijk M k, [M i,n j ] = iε ijk N k, [N i,n j ] = iε ijk M k, [M i,p 0 ] = 0, [M i,p j ] = iε ijk P k, [N i,p 0 ] = ip i, [N i,p j ] = iδ ij 2 1 e 2P0/ P 2 i P ip j, M i = M i I +I M i, 2 N i = N i e P0/ +I N i 1 ε ijkm j P k, P 0 = P 0 I +I P 0, P i = P i e P0/ +I P i, SM i = M i, SN i = N i + 3i 2 P i, SP µ = P µ, εp µ,m i,n i = 0. Structures 1, 2 are dual to each other, the duality being fully described in [3]. The analysis given below was suggested to two of the authors P. Kosiński and P. Maślanka by J. Lukierski. II. -Minkowski space The -Minkowski space M [1] is a universal -algebra with unity generated by four selfadjoint elements x µ subject to the following conditions [x µ,x ν ] = i δµ 0 xν δ ν 0x µ. 3a Equipped with the standard coproduct x µ = x µ I +I x µ, 3b antipode Sx µ = x µ and counit εx µ = 0 it becomes a quantum group. OnM onecanconstructabicovariantfive-dimensionalcalculuswhichisdefined

3 by the following relations [5] τ µ dx µ, DIFFERENTIAL CALCULI 3 τ d x 2 + 3i x0 2x µ dx µ, [τ µ,x ν ] = i g0µ τ ν i gµν τ gµν τ, [τ,x µ ] = 4 2τµ, τ µ τ ν = τ ν τ µ, τ τ µ = τ µ τ, τ µ = τ µ, τ = τ, dτ µ = 0, dτ = 2τ µ τ µ. 4 The -Minkowski space carries a left-covariant action of -Poincaré group P [1], ρ L : M P M, given by ρ L x µ = Λ µ ν x ν +a µ I. 5 The calculus defined by 4 is covariant under the action of P which reads ρ L τ µ = Λ µ ν τ ν, ρ L τ = I τ. 6 III. Derivatives, infinitesimal actions and invariant operators The product of generators x µ will be called normally ordered if all x 0 factors stand leftmost. This definition can be used to ascribe a unique element : fx : of M to any polynomial function of four variables f. Formally, it can be extended to any analytic function f. Let us now one define the left partial derivatives: for any f M we write df = µ fτ µ + fτ. 7 It is a matter of some boring calculations using the commutation rules 3a to find the following formula or d : f : =: sin : 4 0 : f :=: i ei f : τ 0 + : e i 0 f 1 cos 0 sin 0 i : f :=: e i 0 f x i : 2 : f :=: ei 0 f : τ i ei f : 1 cos ei 0 f : x i : τi 8 9

4 4 S. GILLER, C. GONERA, P. KOSIŃSKI, P. MAŚLANKA Let us now define the infinitesimal action of P on M. Let X be any element of the Hopf algebra dual to P the -Poincaré algebra P cf. [3] for the proof of duality. The corresponding infinitesimal action is defined as follows: for any f M, X : M M Using the standard duality rules [3], we conclude that etc. One can show that, in general, Xf = X id ρ L f. 10 P µ x α = iδ α µ, P µ : x α x β := iδ β µx α +iδ α µx β 11 P µ : f :=: i f x µ : 12 Also, usingthe fact that ρ L is aleft actionofp on M togetherwith the duality P P, we conclude that F P µ : f :=: F i f x µ f : 13 Formulae have the following interpretation. In [5] the fifteen-dimensionalbicovariantcalculusonp hasbeen constructedusingthe methods developed by Woronowicz [4]. The resulting generalized Lie algebra is also fifteen-dimensional, the additional generators being the generalized mass square operator and the components of generalized Pauli-Lubanski fourvector. All generators of this Lie algebra can be expresses in terms of the generators P µ, M αβ of P [5]. In particular, the translation generators χ µ as well as the mass squared operator χ are expressible in terms of P µ only. The relevant expressions are given by formulae 20 of [5]. Comparing them with 9, 13 above, we conclude that χ µ µ, χ. 14 These relations, obtained here by explicit computations, follow also from 7 if one takes into account that M is a quantum subgroup of P. It is also not difficult to obtain the action of Lorentz generators. Combining 1 and 3a with the duality P P described in detail in [5], we conclude first that the action of M i and N i coincides with the proposal of Majid and Ruegg [6]; the actual computation is then easy and gives M i : fx µ :=: iε ijl x j fxµ x l : N i : fx µ :=: ix 0 x i +xi 1 e 2i xk x k x i fx µ : x

5 DIFFERENTIAL CALCULI 5 Let us now pass to the notion of invariant operator; Ĉ is an invariant operator on M if ρ L Ĉ = id Ĉ ρ L. 16 We shall show that if C is a central element of P, then Ĉf = C id ρ L f 17 is an invariant operator. To prove this let us take any Y P, then or, in other words, for any a P, YC = CY 18 Ya 1 Ca 2 = Ca 1 Ya 2 19 where a = a 1 a 2. Let us fix a and write 19 as Ya 1 Ca 2 = Ya 2 Ca As 20 holds for any Y P we conclude that for any a P Ca 2 a 1 = Ca 1 a Now let The identity ρ L x = a 1 x 1, ρ L x 1 = a 12 x 2, 22 a 1 = a 1 1 a2 1. id ρ L ρ L = id ρ L 23 implies a 1 a 12 x 2 = a 1 1 a2 1 x Applying to both sides id C id and C id id, we get Ca 12 a 1 x 2 = Ca 2 1 a1 1 x 1, Ca 1 a 12 x 2 = Ca 1 1 a2 1 x 1. It follows from 21 applied to a 1 that the right-hand sides of 25 are equal. So, Ca 12 a 1 x 2 = Ca 1 a 12 x 2 26 i.e. id Ĉ ρ Lx = ρ L Ĉx. 27 Using the above result we can easily construct the deformed Klein-Gordon equation. Namely, we take as a central element the counterpart of mass squared Casimir operator χ [5]. Due to 14 the generalized Klein-Gordon equation reads + m2 f = 0; 28 8 the coefficient 1 8 is dictated by the correspondence with standard Klein-Gordon equation in the limit. Let us note that 28 can be written, due to 9, in the form ] [ i +m2 1+ m2 4 2 f = 0; 29 here 0, i are the operators given by 9. It seems therefore that the Woronowicz operators χ µ are better candidates for translation generators than P µ s. Note that the operators χ µ already appeared in [7], [8]. 25

6 6 S. GILLER, C. GONERA, P. KOSIŃSKI, P. MAŚLANKA References [1] S. Zakrzewski, J. Phys. A , [2] J. Lukierski, A. Nowicki, H. Ruegg, Phys. Lett. B , 344. [3] P. Kosiński, P. Maślanka, The duality between -Poincaré algebra and -Poincaré group, hep-th. [4] S.L. Woronowicz, Comm. Math. Phys , 125. [5] P. Kosiński, P.Maślanka, J. Sobczyk, in: Proc. of IV Coll. on Quantum Groups and Integrable Sysytems, Prague 1995, to be published. [6] S. Majid, H. Ruegg, Phys. Lett. B , 348. [7] H. Ruegg, V. Tolstoy, Lett. Math. Phys , 85. [8] J. Lukierski, H. Ruegg, V. Tolstoy, in: Quantum Groups. Formalism and Applications. Proc. of XXX Karpacz Winter School of Theoretical Physics, ed. J. Lukierski, Z. Popowicz, J. Sobczyk, PWN 1995.