Noncommutativity In A Simple Toy Model

TL;DR

This paper explores how symmetry transformations influence the commutative and noncommutative properties of a toy model of a free particle, revealing that noncommutativity arises from underlying symmetries and differs between commutative and noncommutative geometries.

Contribution

It demonstrates the origin of noncommutativity from symmetry transformations and establishes a cohomological equivalence between NC and commutative cases in a simple toy model.

Findings

Noncommutativity is an artifact of symmetry transformations.

Mass parameter becomes noncommutative in NC geometry.

Quantum group connections exist only in the commutative case.

Abstract

We discuss various symmetry properties of the reparametrization invariant toy model of a free non-relativistic particle and show that its commutativity and noncommutativity (NC) properties are the artifact of the underlying symmetry transformations. For the case of the symmetry transformations corresponding to the noncommutative geometry, the mass parameter of the toy model turns out to be noncommutative in nature. By exploiting the Becchi-Rouet-Stora-Tyutin (BRST) symmetry transformations, we demonstrate the existence of the NC and show its cohomological equivalence with its commutative counterpart. A connection between the usual gauge symmetry transformations corresponding to the commutative geometry and the quantum groups, defined on the phase space, is also established for the present model at the level of Poisson bracket structure. We show that, for the NC geometry, such a kind of quantum group connection does not exist.