Noncommutative spacetime symmetries from covariant quantum mechanics

TL;DR

This paper characterizes a broad class of single-particle noncommutative spacetime models, showing they can be derived from standard Minkowski space and Poincaré transformations, and finds noncommutativity affects coordinate transformations but not dispersion relations.

Contribution

It provides a complete classification of physically sensible noncommutative spacetime models in covariant quantum mechanics and demonstrates their relation to standard Minkowski space via variable changes.

Findings

Noncommutative models can be derived from Minkowski space through variable transformations.

Spacetime noncommutativity alters transformation properties, not dispersion relations.

The class of models includes all physically sensible single-particle noncommutative spacetimes.

Abstract

In the last decades, noncommutative spacetimes and their deformed relativistic symmetries have usually been studied in the context of field theory, replacing the ordinary Minkowski background with an algebra of noncommutative coordinates. However, spacetime noncommutativity can also be introduced into single-particle covariant quantum mechanics, replacing the commuting operators representing the particle's spacetime coordinates with noncommuting ones. In this paper we provide a full characterization of a wide class of physically sensible single-particle noncommutative spacetime models and the associated deformed relativistic symmetries. In particular, we prove that they can all be obtained from the standard Minkowski model and the usual Poincar\'e transformations via a suitable change of variables. Contrary to previous studies, we find that spacetime noncommutativity does not affect the dispersion relation of a relativistic quantum particle, but only the transformation properties of its spacetime coordinates under translations and Lorentz transformations.