Noncommutative Relativistic Spacetimes and Worldlines from 2 + 1 Quantum (Anti-)de Sitter Groups

Herranz, Francisco J.  (Departamento de Física, Universidad de Burgos, 09001 Burgos, Spain) ; Ballesteros, Ángel  (Departamento de Física, Universidad de Burgos, 09001 Burgos, Spain) ; Bruno, N. Rossano  (Departamento de Física, Universidad de Burgos, 09001 Burgos, Spain)

28 November 2017

Abstract: The $\kappa$ -deformation of the (2 + 1)D anti-de Sitter, Poincaré, and de Sitter groups is presented through a unified approach in which the curvature of the spacetime (or the cosmological constant) is considered as an explicit parameter. The Drinfel’d-double and the Poisson–Lie structure underlying the $\kappa$ -deformation are explicitly given, and the three quantum kinematical groups are obtained as quantizations of such Poisson–Lie algebras. As a consequence, the noncommutative (2 + 1)D spacetimes that generalize the $\kappa$ -Minkowski space to the (anti-)de Sitter ones are obtained. Moreover, noncommutative 4D spaces of (time-like) geodesics can be defined, and they can be interpreted as a novel possibility to introduce noncommutative worldlines. Furthermore, quantum (anti-)de Sitter algebras are presented both in the known basis related to 2 + 1 quantum gravity and in a new one which generalizes the bicrossproduct one. In this framework, the quantum deformation parameter is related to the Planck length, and the existence of a kind of “duality” between the cosmological constant and the Planck scale is also envisaged.

Published in: Advances in High Energy Physics 2017 (2017) 7876942