The Gauge Unfixing Formalism and the Solutions of the Dirac Bracket Commutators

TL;DR

This paper introduces a systematic method using the Gauge Unfixing formalism to solve Dirac bracket commutators, converting second class systems into first class ones without enlarging phase space, verified through three physical examples.

Contribution

It presents a novel approach to derive Dirac bracket commutators via gauge unfixing, applicable to various constrained systems without phase space extension.

Findings

Successfully applied to free particle on a sphere

Extended to noncommutative free particle

Demonstrated with doubly special relativity particle

Abstract

We propose a systematic procedure that solves the Dirac bracket commutators. The method is based on the Gauge Unfixing formalism, a procedure that converts second class systems into first class ones without the enlargement of the original phase space variables. We verify that the gauge invariant variables satisfy the Dirac bracket when we strongly impose the discarded second class constraint. Thus, we can derive physical operators that satisfy the Dirac commutators. In order to illustrate our procedure, three second class constrained systems are considered. Firstly, the free particle on the two dimensional sphere is treated. The second case considered is the noncommutative free particle and the third is the doubly special relativity particle.