First class functions in constrained second class systems

TL;DR

This paper investigates the algebraic structure of first class functions in systems with second class constraints, revealing their properties and relationships within the Dirac bracket formalism.

Contribution

It explicitly constructs generators and brackets of these algebras, demonstrating their isomorphism to phase variable algebras in the Dirac formalism.

Findings

First class functions form algebras under Dirac brackets and pointwise multiplication.

Subspace of functions vanishing on the constraint surface are ideals of these algebras.

Quotient algebras are isomorphic to phase variable algebras in the Dirac bracket formalism.

Abstract

Generators of the algebra of first class functions in a system with second class constraints are found. It is shown that first class functions form algebras with respect to the Dirac bracket and pointwise multiplication.The subspace of functions vanishing on constraint surface are ideals of these algebras. The corresponding quotient algebras are isomorphic to the algebras of phase variables in the Dirac bracket formalism. Explicite expressions for generators and brackets of the algebras under consideration are obtained.