Noncommutative Spaces and Poincar\'e Symmetry

TL;DR

This paper develops a unified framework for noncommutative spacetimes using deformed Heisenberg algebras, analyzing Lorentz symmetry actions and their implications for momentum composition.

Contribution

It introduces a comprehensive approach to describe noncommutative geometries with linear coordinate commutation relations and explores Lorentz transformations in this context.

Findings

Lorentz transformations act nontrivially on tensor products of momenta.

Different Lorentz group elements act on the left and right of momentum compositions.

Two examples illustrate the momentum-dependent Lorentz actions.

Abstract

We present a framework which unifies a large class of non-commutative spacetimes that can be described in terms of a deformed Heisenberg algebra. The commutation relations between spacetime coordinates are up to linear order in the coordinates, with structure constants depending on the momenta plus terms depending only on the momenta. The possible implementations of the action of Lorentz transformations on these deformed phase spaces are considered, together with the consistency requirements they introduce. It is found that Lorentz transformations in general act nontrivially on tensor products of momenta. In particular the Lorentz group element which acts on the left and on the right of a composition of two momenta is different, and depends on the momenta involved in the process. We conclude with two representative examples, which illustrate the mentioned effect.