The Quantum Deformed Dirac Equation from the κ–Poincaré Algebra

In this letter we derive a deformed Dirac equation invariant under the κ–Poincaré quantum algebra. A peculiar feature is that the square of the κ–Dirac operator is related to the second Casimir (the κ–deformed squared Pauli–Lubanski vector). The “spinorial” realization of the κ–Poincaré is obtained by a contraction of the coproduct of the real form of SOq(3, 2) using the 4–dimensional representation which results to be, up some scalar factors, the same of the undeformed algebra in terms of the usual γ–matrices. Preprint DFF 177/12/92 Firenze, December 1992. On leave of absence from Institute of Physics, Pedagogical University, Plac S lowiański 6, 65029 Zielona Góra, Poland. In recent papers the structure of quantum deformations of the Poincaré algebra has been found and studied in detail [1, 2, 3, 4, 5], one of the approach used is based on the generalization of the contraction procedure of Lie algebras to Hopf structure introduced in [6, 7]; in the contest of quantum algebras the main new point is to rescale, beside the generators, the quantum deformation parameter. The contraction we consider in this paper is performed from the standard real form of SOq(3, 2) using q real and defining κ −1 = R log q as the rescaled quantum parameter (R is the de–Sitter curvature and goes to infinity in the contraction limit, κ then has the dimension of the inverse of a length)[5]. This is precisely the choice proposed in [7] to obtain the quantum Euclidean algebra from the contraction of SOq(4), the contracted algebra in both the cases is a real form under an involution having the standard properties at the level of algebra and coalgebra. In [4] it has been found a non linear transformation of the boost sector that simplifies the algebra of the κ–Poincaré and leads to a more symmetric form for the coalgebra. Let us summarize this final structure, the algebra reads: [Pi, Pj] = 0 , [Pi, P0] = 0 , [Mi, Pj] = iǫijk Pk , [Mi, P0] = 0 , [Li, P0] = iPi , [Li, Pj] = iδij κ sinh(P0/κ) , (1) [Mi,Mj] = iǫijk Mk , [Mi, Lj ] = iǫijk Lk , [Li, Lj] = −iǫijk(Mk cosh(P0/κ)− 1/(4κ ) Pk PlMl) . Where Pμ ≡ {P0, Pi} are the deformed energy and momenta, the Mi are the spatial rotation generators (they close an undeformed Hopf subalgebra), the Li are the deformed boost generators. The coalgebra and the antipode result: ∆Mi = Mi ⊗ I + I ⊗Mi , ∆P0 = P0 ⊗ I + I ⊗ P0 , ∆Pi = Pi ⊗ exp( P0 2κ ) + exp(− P0 2κ )⊗ Pi , ∆Li = Li ⊗ exp( P0 2κ ) + exp(− P0 2κ )⊗ Li + (2) 1 2κ ǫijk ( Pj ⊗Mk exp( P0 2κ ) + exp(− P0 2κ )Mj ⊗ Pk )