Realizations of observables in Hamiltonian systems with first class constraints

TL;DR

This paper demonstrates the isomorphism between two algebraic structures of observables in constrained Hamiltonian systems and introduces a new realization of these algebras using the original Poisson bracket.

Contribution

It establishes the isomorphism between the quotient Poisson and Dirac bracket algebras and provides a novel realization via the original Poisson bracket.

Findings

The two observable algebras are isomorphic.

A new realization of the observable algebra through the original Poisson bracket is presented.

Explicit calculations of generators, brackets, and products are provided.

Abstract

In a Hamiltonian system with first class constraints observables can be defined as elements of a quotient Poisson bracket algebra. In the gauge fixing method observables form a quotient Dirac bracket algebra. We show that these two algebras are isomorphic. A new realization of the observable algebras through the original Poisson bracket is found. Generators, brackets and pointwise products of the algebras under consideration are calculated.