Non-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti-)de Sitter groups

Abstract

The κ-deformation of the (2+1)D anti-de Sitter, Poincar´e 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–doubleand the Poisson–Lie structure underlying the κ-deformation are explicitly given, andthe three quantum kinematical groups are obtained as quantizations of such Poisson–Lie algebras. As a consequence, the non-commutative (2+1)D spacetimes that generalize the κ-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 non-commutative worldlines. Furthermore, quantum (anti-)de Sitter algebras are presented both in the known basis related with 2+1 quantum gravity and in a new one which generalizes the bicrossproduct one. In this framework, the quantum deformation parameter is related with the Planck length, and the existence of a kind of “duality” between the cosmological constant and the Planck scale is also envisaged.