On the quantum Lorentz group

Abstract

: The quantum analogues of Pauli matrices are introduced and investigated. From these matrices and an appropriate trace over spinorial indices we construct a quantum Minkowsky metric. In this framwork we show explicitly the correspondence between the SL(2, C) and Lorentz quantum groups. The R matrices of the quantum Lorentz group are constructed in terms of the R matrices of SL(2, C) group. These R matrices satisfy adequate properties as Yang-Baxter equations, Hecke relations and quantum symmetrization of the metric. It is also shown that the Minkowsky metric leads to an invariant and central norm.

The quantum spheres and their embedding into quantum Minkowski space‐time

“…From the hermiticity of the Minkowskian metric and the orthogonality conditions we can see that the four-vector length G N M X N X M = −τ 2 is real and invariant. It was also shown in [12] that τ 2 is central, it commutes with the Minkowski space-time coordinates and the quantum Lorentz group generators. Λ M N and X N are subject to the commutation rules controlled by the R N M P Q matrix as:…”

“…Before embedding the different quantum spheres into quantum Minkowski space-time, let us recall briefly some properties of the noncommutative special relativity presented in [10]. First it was shown in [12] that the generators Λ M N (N, M = 0, 1, 2, 3) of quantum Lorentz group may be written in terms of those of quantum SL(2, C) group as…”

“…and the relations (13) for L = ∞ where the orthonormal states |L, n and |n , L, n ′ |L, n = δ n ′ ,n , n ′ , n = 0, 1, ..., L, and n ′ |n = δ n ′ ,n , n ′ , n = 0, 1, ..., ∞, denote the states satisfying (11,12) and ( 13) respectively. The unique state |t, 0, 0 corresponding to L = 0 describes a particle at rest at the origin of the spacial coordinate system, for t = 0 then τ 2 = 0 it representes the origin of the four coordinate system of the quantum Minkowski space-time, X N |0, 0, 0 = 0|0, 0, 0 .…”

“…These constraints correspond to slices of the past time-like region for finite L or a slice of the past light-cone region for L = ∞. They fit with the quantum spheres S 2 qc with c = c(n) ≤ 0 which are described by states satisfying (11,12,14) for finite L ≥ 1 and (13) and (15) for L = ∞.…”

“…In this case we obtain from (18) a length of the velocity [10]. Note that from (18), (11,12) and (13) we have…”