Space-Time Noncommutativity from Particle Mechanics

TL;DR

This paper demonstrates how reparametrization symmetry in relativistic particle mechanics can induce space-time noncommutativity, which can be interpreted differently without altering the underlying quantum algebra, and extends this to particles in electromagnetic fields.

Contribution

It introduces a gauge-based method to generate space-time noncommutativity in relativistic particles and shows the equivalence of different gauges at the quantum level.

Findings

Space-time noncommutativity can be derived from reparametrization symmetry.

A gauge transformation maps noncommutative and commutative descriptions.

Standard dynamical systems can be made noncommutative via variable changes.

Abstract

We exploit the reparametrization symmetry of a relativistic free particle to impose a gauge condition which upon quantization implies space-time noncommutativity. We show that there is an algebraic map from this gauge back to the standard `commuting' gauge. Therefore the Poisson algebra, and the resulting quantum theory, are identical in the two gauges. The only difference is in the interpretation of space-time coordinates. The procedure is repeated for the case of a coupling with a constant electromagnetic field, where the reparametrization symmetry is preserved. For more arbitrary interactions, we show that standard dynamical system can be rendered noncommutative in space and time by a simple change of variables.