Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 086, 31 pages      arXiv:1005.4429      https://doi.org/10.3842/SIGMA.2010.086
Contribution to the Special Issue “Noncommutative Spaces and Fields”

κ-Minkowski Spacetimes and DSR Algebras: Fresh Look and Old Problems

Andrzej Borowiec and Anna Pachol
Institute for Theoretical Physics, University of Wroclaw, pl. Maxa Borna 9, 50-204 Wroclaw, Poland

Received March 30, 2010, in final form October 10, 2010; Published online October 20, 2010

Abstract
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 κ-Minkowski spacetime 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 κ-Minkowski spacetime described as a quantum covariant algebra determining a deformation quantization of the corresponding linear Poisson structure. Formal and conceptual issues concerning quantum κ-Poincaré and κ-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.

Key words: quantum deformations; quantum groups; Hopf module algebras; covariant quantum spaces; crossed product algebra; twist quantization; quantum Weyl algebra; κ-Minkowski spacetime; deformed phase space; quantum gravity scale; deformed dispersion relations; time delay.

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