Gauge invariant actions for the noncommutative phase-space relativistic particle

TL;DR

This paper investigates gauge invariance in the noncommutative relativistic particle by converting second-class constraints into first-class ones, deriving gauge-invariant actions via Noether and Batalin-Fradkin-Fradkina-Tyutin formalisms.

Contribution

It introduces two methods to obtain gauge-invariant actions for the noncommutative relativistic particle, enhancing understanding of gauge symmetry in such systems.

Findings

Derived a gauge-invariant action using the Noether procedure.

Converted second-class constraints to first-class constraints.

Obtained a second gauge-invariant Lagrangian via Batalin-Fradkin-Fradkina-Tyutin formalism.

Abstract

Our main interest here is to analyze the gauge invariance issue concerning the noncommutative relativistic particle. Since the analysis of the constraint set from Dirac's point of view classifies it as a second-class system, it is not a gauge theory. Hence, the objective here is to obtain gauge invariant actions linked to the original one. However, we have two starting points, meaning that firstly we will begin directly from the original action and, using the Noether procedure, we have obtained a specific dual (gauge invariant) action. Following another path, we will act toward the constraints so that we have carried out the conversion of second to first-class constraints through the Batalin-Fradkin-Fradkina-Tyutin formalism, obtaining the second gauge invariant Lagrangian.