Interacting Relativistic Particle: Time-Space Noncommutativity And Symmetries

TL;DR

This paper explores how symmetry considerations in an interacting relativistic particle model reveal time-space noncommutativity in spacetime structure, linking it to gauge symmetries and deformations of the Poincaré algebra.

Contribution

It demonstrates the emergence of time-space noncommutativity from symmetry analysis and connects it to a unique gauge symmetry and Poincaré algebra deformation.

Findings

Time-space noncommutativity arises from symmetry considerations.

Noncommutative and commutative properties are linked to a single gauge symmetry.

Deformation of the Poincaré algebra is discussed in detail.

Abstract

We discuss the symmetry properties of the reparametrization invariant model of an interacting relativistic particle where the electromagnetic field is taken as the constant background field. The direct coupling between the relativistic particle and the electromagnetic {\it gauge} field is a special case of the above with a specific set of subtleties involved in it. For the above model, we demonstrate the existence of a time-space noncommutativity (NC) in the spacetime structure from the symmetry considerations alone. We further show that the NC and commutativity properties of this model are different aspects of a unique continuous {\it gauge} symmetry that is derived from the non-standard gauge-type symmetry transformations by requiring their consistency with (i) the equations of motion, and (ii) the expressions for the canonical momenta, derived from the Lagrangians. We provide a detailed discussion on the noncommutative deformation of the Poincar{\'e} algebra.