Poincare-Snyder Relativity with Quantization

Based on a linear realization formulation of a quantum relativity – the proposed relativity for quantum space-time', we introduce the Poincar\'e-Snyder relativity and Snyder relativity as relativities in between the latter and the well known Galilean and Einstein cases. We discuss how the Poincar\'e-Snyder relativity may provide a stronger framework for the description of the usual (Einstein) relativistic quantum mechanics and beyond. In particular, we discuss a geometric quantization picture through the U(1) central extension of the relativity group, which had been establish to work well for the Galilean case but not for the Einstein case. We discuss similarities and differences between our Poincar\'e-Snyder picture with a still not fully understood $\sigma$ variable as the evolution’ parameter and some use of an invariant time or the proper time parameter in some earlier formulations with very similar mathematical structure. The study is a first step towards the investigation of physics of the $\sigma$ variable at the Poincar'e-Snyder setting, plausible leading to experimental signatures to be studied.