Noncommutative space-time models

TL;DR

This paper develops models of noncommutative space-time using quantum Euclidean spaces and Cayley-Klein contractions, proposing quantum analogs of classical space-time geometries with fundamental length and time scales.

Contribution

It introduces a framework for noncommutative space-time models via quantum Cayley-Klein spaces and constructs quantum analogs of (1+3) space-time geometries.

Findings

Quantum (anti) de Sitter, Newton, Galilei kinematics with fundamental scales are proposed.

Noncommutative spaces are formulated as spheres in quantum Cayley-Klein spaces.

Part of the models serve as noncommutative analogs of classical space-time models.

Abstract

The FRT quantum Euclidean spaces $O_q^N$ are formulated in terms of Cartesian generators. The quantum analogs of N-dimensional Cayley-Klein spaces are obtained by contractions and analytical continuations. Noncommutative constant curvature spaces are introduced as a spheres in the quantum Cayley-Klein spaces. For N=5 part of them are interpreted as the noncommutative analogs of (1+3) space-time models. As a result the quantum (anti) de Sitter, Newton, Galilei kinematics with the fundamental length and the fundamental time are suggested.