The E(2) Particle

TL;DR

This paper explores the E(2) subgroup of the Lorentz group, constructing a non-commutative phase space model that preserves Einstein's dispersion relation, providing insights into minimal symmetry frameworks in physics.

Contribution

It introduces a consistent non-commutative phase space structure based on the E(2) subgroup and develops a point particle action that maintains the Einstein dispersion relation.

Findings

Non-commutative phase space constructed for E(2) symmetry

Point particle action consistent with Einstein dispersion relation

Model exploits dual canonical phase space scheme

Abstract

Recently it has been advocated [1] that for describing nature within the minimal symmetry requirement, certain subgroups of Lorentz group may play a fundamental role. One such group is E(2) which induces a Lie algebraic Non-Commutative spacetime [4] where translation invariance is not fully maintained. We have constructed a consistent structure of Non-commutative phase space for this system and furthermore we have studied an appropriate point particle action on it. Interestingly, the Einstein dispersion relation $p^2=m^2$ remains intact. The model is constructed by exploiting a dual canonical phase space following the scheme developed by us earlier [8].