Elementary Systems of (1 + 1) Kinematical Groups: Contraction and Quantization

Abstract

We present the (algebra) group contraction chain SU(1, 1) P(1, 1) G(1, 1), where P(1, 1) and G(1, 1) are the Poincare and the Galilei groups, respectively, in (1 + 1) dimensions. We have paid attention to the contraction of the pseudo-extended Poincare group to the central extended Galilei group. Objects like group laws, coadjoint orbits and representations of the contracted groups have been obtained in terms of their noncontracted counterparts. As an application we study the Moyal quantization of classical systems, having those groups as symmetry groups, by means of the contraction of the so called Stratonovich-Weyl kernels which provide such quantization.