Relation between quantum $\kappa$-Poincar\'{e} framework and Doubly Special Relativity

Abstract

We describe firstly the basic features of quantum $\kappa$-Poincar\'{e} symmetries with their Hopf algebra structure. The quantum $\kappa$-Poincar\'{e} framework in any basis relates rigidly the quantum $\kappa$-Poincar\'{e} algebra with quantum $\kappa$-Poincar\'{e} group, noncommutative space-time and $\kappa$-deformed phase space. Further we present the approach of Doubly Special Relativity (DSR) theories, which introduce(in the version DSR1) kinematically the frame - independent fundamental mass parameter as described by maximal three-momentum $|\overrightarrow{p}|=\kappa c$. We argue why the DSR theories in one-particle sector can be treated as the part of quantum $\kappa$-Poincar\'{e} framework. The DSR formulation has been extended to multiparticle states either in a way leading to nonlinear description of classical relativistic symmetries, or providing the identification of DSR approach with full quantum $\kappa$-Poincar\'{e} framework.