The classical basis for κ-deformed Poincaré (super)algebra and the second κ-deformed supersymmetric Casimir.

Transcription

1 The classical basis for -deformed Poincaré super)algebra and the second -deformed supersymmetric Casimir. arxiv:hep-th/941114v1 13 Dec 1994 P. Kosiński, J. Lukierski, P. Maślanka, and J. Sobczyk Abstract We present here the general solution describing generators of -deformed Poincaré algebra as the functions of classical Poincaré algebra generators as well as the inverse formulae. Further we present analogous relations for the generators of N = 1 D = 4 -deformed Poincaré superalgebra expressed by the classical Poincaré superalgebra generators. In such a way we obtain the - deformed Poincaré super)algebras with all the quantum deformation present only in the coalgebra sector. Using the classical basis of -deformed Poincaré superalgebra we obtain as a new result the -deformation of supersymmetric covariant spin square Casimir. Institute of Physics, University of Lódź, ul. Pomorska 149/153, Lódź, Poland. Institute for Theoretical Physics, University of Wroc law, pl. Maxa Borna 9, Wroc law, Poland. Dept. of Functional Analysis, Institute of Mathematics, University of Lódź, ul. S. Banacha, Lódź, Poland. Partially supported by KBN grant P Partially supported by KBN grant P

2 1 Introduction The quantum -deformation with Hopf algebra structure, and mass-like deformation parameter is known for D = 4 Poincaré algebra [1 5] as well as for D = 4 Poincaré superalgebra [6 8]. We would like to point out that recently both the -deformed Poincaré algebra [5] as well as the -deformed Poincaré superalgebra [8] were presented in bicrossproduct basis [9, 10]. In such a basis the algebra is simplified, with free super)lorentz generators and -deformed relations appearing only in the cross-sector of the super)lorentz and super)translation generators. The aim of this paper is to simplify further the algebraic sector. In particular 1. we introduce the basis of -deformed Poincaré super)algebra with classical super)poincaré generators,. we calculate the up to now unknown second Casimir of -deformed D = 4 Poincaré superalgebra. In Sect. we shall consider the most general formula, expressing the bicrossproduct basis of -deformed Poincaré algebra [5] in terms of classical Poincaré generators. The only part of the algebra which undergoes the nonlinear change are the fourmomentum generators, because the Lorentz generators are already classical in the bicrossproduct basis. We consider also the inverse formulae, expressing the classical Poincaré generators in terms of -deformed ones, and select two special cases of general formulas. In Sect. 3 we introduce the classical basis for -deformed Poincaré superalgebra, in particular for the one given in [8]. It appears that for such a purpose one should perform in bicrossproduct basis the nonlinear change of the fourmomenta as well as of two chiral supercharges. The classical basis of -Poincaré superalgebra is used in Sect. 4 for obtaining the -deformed supersymmetric spin Casimir supersymmetrized square of Pauli-Lubanski fourvector). With two known Casimirs of -deformed Poincaré superalgebra it is possible to develop further the theory of its irreducible representations, generalizing e.g. the results obtained in [11]. Classical -Poincaré basis Let us write the -Poincaré algebra as the following cross-product [5] in its algebraic sector U P 4 ) = UO3, 1)) < UT 4 ),.1) where UT 4 ) is the algebra of classical functions of the Abelian fourmomentum generators P µ = P i, P 0 ) and UO3, 1)) describes the enveloping classical Lorentz algebra. The cross relations between the O3, 1) generators M µν = M i, N i ) and T 4 generators P µ = P i, P 0 ) are the following [5]: [M i, P j ] = iǫ ijk P k, [M i, P 0 ] = 0,.a) 1

3 [N i, P j ] = iδ ij [ 1 e P 0 ) + 1 P ] i P ip j,.b) [N i, P 0 ] = ip i..c) Let us derive the generators of the crossproduct basis of -Poincaré algebra satisfying.a)-.c) as functions of classical Poincaré generators M 0) i, N 0) i, i, 0 ). Because in the crossproduct basis.1) the Lorentz algebra is classical we put where M i = M 0) i, N i = N 0) i, P 0 = f 0, M0 ) P i = g 0, M0 0) )P i.3) M 0 = ) ).4) is the mass Casimir of the classical Poincaré algebra. One obtains from the substitution of the relations.3) in the relations.a)-.c) that f f ) The general solutions of equation.5) are the following 0 g = 1 g, f = g..5) g = where from the relations 0 + C, f = ln 0 + C A,.6) one gets Besides one obtains the relation which implies that lim P i = i, lim P 0 = 0,.7) A lim = lim C = 1..8) g 0 = 1 e P 0 ) + 1 P i,.9) C A = M 0..10) We see therefore that one obtains the family of solutions depending on one arbitrary function A; M 0), satisfying the condition.8). The -deformed Casimir in Majid-Ruegg basis takes the form sinh P ) 0 e P 0 P i = M..11)

4 Substituting the formulae.3) one gets ) C M = A 1..1) Using.10) one obtains the following relations between two Casimirs.10) and.11) ) M0 = A M ) The general inverse formulae look as follows: 0 = A e P 0 ) C = A A e P 0 M ) 1,.14a) i = A e P 0 Pi..14b) We shall consider the following two particular solutions: a) b) We get A = M 0 4, C = + M ) P 0 and the inverse formulae P i = = ln M 0 4 M M 0 4 ) 0 = e P 0 1 M 0 ) 4 i = 1 M 0 4 j, e P ),.16) P i e P 0.17) where in.17) one should use the formulae.13) giving for the choice.15) the result One obtains P 0 P i = M 0 = M 1 + M 4,.18) A =, C = M 0 + ) 1..19) [ = ln 0 + M0 + ) ] M 0 + ) 1 3 i ln,.0)

5 and The formula.13) takes the form 0 = sinh P e P0 P, i = e P 0 Pi. M 0 = M ) 1 + M 4.1)..) It should be mentioned that the formulae.1)-.) have been already obtained in [1], and it has been observed by Ruegg [13] that the formulae.1) applied to bicrossproduct basis provide the classical Poincaré algebra. In the basis given by.14a)-.14b) see also.17) and.1)) all the quantum deformation is contained in the crossproduct. The relations are quite complicated and the coproduct for the energy operator 0 does not lead to the additivity of the energy. 3 Classical -superpoincaré basis Recently it has been shown [8] that the Majid-Ruegg basis with the cross-relations.a)-.c) can be supersymmetrized. The relation.1) is supersymmetrized as follows 1 : U P 4;1 ) = UO3, 1; )) < UT 4; ), 3.1) where UT 4; ) is the graded algebra of functions on the trivial chiral superalgebra: T 4; : {Q α, Q β} = [P µ, Q α ] = [P µ, P ν ] = 0 3.) and UO3, 1; )) describes the enveloping algebra of the graded Lorentz algebra O3, 1; ) = M µν, Q α ), with the Lorentz algebra supplemented by the relations [M µν, Q α ] = 1 σ µν) α β Q β, 3.3a) {Q α, Q β } = b) The cross-relations in 3.1) are given by the following extension of the set of relations.a)-.c) M µν M i = 1 ǫ ijkm jk, N i = M 0i )): [M i, Q α ] = 1 Qσ i) α, 3.4a) [N i, Q α ] = i e P 0 Qσi ) α + 1 ǫ ijkp j Qσ k ) α, 3.4b) 1 We discuss here only one explicite way of describing the -deformed Poincaré superalgebra as a graded bicrossproduct. For other ways see [8]. 4

6 [Q α, P µ ] = 0, 3.4c) {Q α, Q β} = 4δ α β sinh P 0 P iσ i ) α βe P0. 3.4d) It is known from Sect. that the fourmomentum operators P µ satisfying.a)-.c) can be expressed as functions of the fourmomenta P µ 0) belonging to the classical Poincaré algebra. Similarly, one can express the generators Q α, Q α in terms of classical N = 1 Poincaré superalgebra generators Q 0) α, P µ 0) ), satisfying the relations {Q 0) α, Q 0) β } = δ 0) α β P 0 σ i ) α β i, {Q 0) α, Q0) β } = {Q0) α, Q0) β } = 0, [Q 0) α, P µ] = [Q 0) α, P µ] = ) We obtain that Q α = AA + C) Q0) α, 3.6a) Q α = Q 0) β 0) P k [expα P 0) σ k)] β α, 3.6b) where sinh α = cosh α = [ 0 + C)A + C)] C + A [ 0 + C)A + C)] 1, 3.7a). 3.7b) It is easy to check that cosh α sinh α = 1 follows from the relation.10). In order to obtain the formulae expressing the classical supercharges Q 0) α, Q0) α in terms of -deformed generators Q α, Q α, P µ one should substitute in 3.6b) the relations.14a)-.14b) and put the exponential on the other side of the equation 3.6b). 4 The second Casimir for the N = 1 -Poincaré superalgebra In [6] it was observed that the -deformed mass Casimir.11) of the -Poincaré algebra becomes the Casimir of -deformed N = 1 superpoincaré algebra. The 5

7 second relativistic spin square Casimir of the -Poincaré algebra takes the form [, 3] C = cosh P 0 P 4 )W 0 W 4.1) where W 0 = P M, 4.a) W k = M k sinh P 0 + ǫ ijkp i L j 4.b) We construct the supersymmetric extension of the -deformed Casimir C as follows: i) Using the classical superalgebra basis of N = 1 -superpoincaré algebra see Sect.3) the supersymmetric extension C N=1 of the relativistic spin square Casimir is described by the length of the supersymmetric Pauli-Lubanski vector C N=1 = Wµ N=1 W µn=1 W0 N=1 ) W N=1 W N=1 4.3) where [14] and see [6]) W N=1 0 = M + T 0) 0, 4.4) W N=1 i = M i 0 + ǫ kij i N 0) j + T 0) i 4.5) T µ 0) = Q 0) α σ α βq 0) µ β. 4.6) ii) In the crossproduct basis given in [8] see also Sect. 3) the second Casimir 4.3) can be expressed in terms of the -deformed generators provided that the formulae.14a)-.14b) and 3.6a)-3.6b) are properly employed. One gets T 0) 0 = A cosh P 0 T 0 + A T 0) j = A cosh P 0 e P0 Pj T j, T j + A P0 e Pj T 0 Pk T l, i A ǫ jkle P0 4.7a) 4.7b) where T µ = T i, T 0 ) is given by the formulae 3.6a)-3.6b) with the supercharges Q 0) α, Q0) α replaced with Q α, Q α. One obtains W0 N=1 = A [e P 0 Pj M j + cosh P 0 )T e P0 Pk T k ], 4.8a) 6

8 Wi N=1 = A [M je P 0 + cosh P 0 )T j + 1 e P0 i ǫ ikle P0 Pk T l ]. C A ) + ǫ ikle P0 Pk N l + Pj T 0 The relations 4.8a)-4.8b) should be subsequently substituted in 4.3) 4.8b) iii) In the conventional basis given in [6] the second Casimir 4.3) is obtained if we substitute in 4.8a)-4.8b) the formulae relating standard basis see [6]) with the bicrossproduct basis see [8]). 5 Final Remarks In this note we discussed the -Poincaré super)algebra with the basis described by the generators satisfying classical super-)poincaré algebra. We recall that in such a basis all the deformation is present in the coalgebra sector. We would also like to mention that for any choice of the fourmomenta in classical basis see.17)) the energy ceases to be additive. It is known that the coalgebra sector of the quantum Lie algebra determines the algebraic relations satisfied by the generators of dual quantum Lie group. For the bicrossproduct basis of -Poincaré algebra [5] it was recently shown [15] that the quantum Poincaré group is described by the relations firstly obtained by Zakrzewski [16] via quantization of the Lie-Poisson structure. It would be very interesting to generalize this result to the case of -deformed Poincaré superalgebra for which the super-extension of the quantum Poincaré group of Zakrzewski was given in [7]. Another problem which is interesting to solve is to provide a basis of the quantum Poincaré group dual to the classical basis.17) of the -Poincaré algebra. Because the coproduct for -Poincaré algebra with classical basis is complicated, one should expect complicated algebraic relation for these Poincaré group generators. The question which remains still not well understood in the framework of - deformed symmetry is the physical meaning of -deformed coproducts. One can say only very generally that it describes some kind of basic interaction dictated by quantum deformation of underlying relativistic symmetry. References [1] J. Lukierski, A. Nowicki, H. Ruegg and V. Tolstoy, Phys. Lett. B64, ) [] S. Giller, J. Kunz, P. Kosiński, M. Majewski and P. Maślanka, Phys. Lett. B86, ) 7

9 [3] J. Lukierski, A. Nowicki and H. Ruegg, Phys. Lett. B93, ) [4] J. Lukierski, H. Ruegg and W. Rühl, Phys. Lett. B313, ) [5] S. Majid and H. Ruegg, Phys. Lett. B334, ) [6] J. Lukierski, A. Nowicki and J. Sobczyk, J. Phys. A6, L ) [7] P. Kosiński, J. Lukierski, P. Maślanka and J. Sobczyk, J. Phys. A7, ) [8] P. Kosiński, J. Lukierski, P. Maślanka and J. Sobczyk, SISSA preprint 134/94/FM, November 1994, submitted to Journ. of Phys. A [9] S. Majid, Journ. Alg. 130, ) [10] S. Majid, Foundations of Quantum Groups, Chapt. 6, Cambridge Univ. Press, in press [11] S. Giller, C. Gonera, P. Kosiński, J. Kuntz and M. Majewski, Mod. Phys. Lett. A8, ) [1] H. Ruegg and V.N. Tolstoy, Lett. Math. Phys. 3, ) [13] H. Ruegg, private communication, June 1994 [14] S.J. Gates, M.T. Grisaru, M. Rocek and W. Siegel, Superspace, ed. Benjamin- Cummings, Reading, MA, 1983, p.7 [15] P. Kosiński and P. Maślanka, Lódź Univ. preprint, November 1994; hep-th [16] S. Zakrzewski, Journ. of Phys. A7, ) 8