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

TL;DR

This paper develops a unified quantum group framework for (2+1)D anti-de Sitter, Poincaré, and de Sitter spacetimes, introducing non-commutative geometries and worldlines with implications for quantum gravity.

Contribution

It provides a unified approach to $mbda$-deformations of (2+1)D spacetime groups, explicitly constructs the underlying Poisson-Lie structures, and introduces non-commutative worldlines.

Findings

Derived non-commutative (2+1)D spacetimes generalizing $mbda$-Minkowski space.

Defined non-commutative spaces of geodesics as worldlines.

Established a duality between the cosmological constant and Planck scale.

Abstract

The $\kappa$-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-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 non-commutative (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 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.