\(\kappa\)-Minkowski spacetimes and DSR algebras: fresh look and old problems. (English) Zbl 1219.81159
Summary: Some classes of Deformed Special Relativity (DSR) theories are reconsidered within the Hopf algebraic formulation. For this purpose we shall explore a minimal framework of deformed Weyl-Heisenberg algebras provided by a smash product construction of DSR algebra. It is proved that this DSR algebra, which uniquely unifies \(\kappa \)-Minkowski space-time coordinates with Poincaré generators, can be obtained by nonlinear change of generators from undeformed one. Its various realizations in terms of the standard (undeformed) Weyl-Heisenberg algebra opens the way for quantum mechanical interpretation of DSR theories in terms of relativistic (Stückelberg version) quantum mechanics. On this basis we review some recent results concerning twist realization of \(\kappa \)-Minkowski space-time described as a quantum covariant algebra determining a deformation quantization of the corresponding linear Poisson structure. Formal and conceptual issues concerning quantum \(\kappa \)-Poincaré and \(\kappa \)-Minkowski algebras as well as DSR theories are discussed. Particularly, the so-called “\(q\)-analog” version of DSR algebra is introduced. Is deformed special relativity quantization of doubly special relativity remains an open question. Finally, possible physical applications of DSR algebra to description of some aspects of Planck scale physics are shortly recalled.
MSC:
81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
81R60 | Noncommutative geometry in quantum theory |
81T75 | Noncommutative geometry methods in quantum field theory |
83C65 | Methods of noncommutative geometry in general relativity |
16T05 | Hopf algebras and their applications |
17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
46L65 | Quantizations, deformations for selfadjoint operator algebras |
53D55 | Deformation quantization, star products |