Lie algebraic noncommuting structures from reparametrisation symmetry

TL;DR

This paper demonstrates how reparametrisation symmetry in particle models can produce Lie algebraic noncommuting structures in space-time and space-space sectors, with specific parameter choices affecting noncommutativity.

Contribution

It explicitly shows that certain reparametrisation parameters induce Lie algebraic noncommutative structures in relativistic particles, expanding previous work and connecting to existing models.

Findings

Certain reparametrisation parameters yield Lie algebraic noncommutativity.

Noncommutativity can be tuned to vanish for specific parameter values.

Angular momentum algebra remains consistent within the framework.

Abstract

We extend our earlier work of revealing both space-space and space-time noncommuting structures in various models in particle mechanics exhibiting reparametrisation symmetry. We show explicitly (in contrast to the earlier results in our paper \cite{sg}) that for some special choices of the reparametrisation parameter $\epsilon$, one can obtain space-space noncommuting structures which are Lie-algebraic in form even in the case of the relativistic free particle. The connection of these structures with the existing models in the literature is also briefly discussed. Further, there exists some values of $\epsilon$ for which the noncommutativity in the space-space sector can be made to vanish. As a matter of internal consistency of our approach, we also study the angular momentum algebra in details.