Gauge Identities and the Dirac Conjecture

TL;DR

This paper explores the relationship between gauge symmetries in dynamical systems and the Dirac conjecture, demonstrating their connection through a first order Lagrangian formulation and analyzing first class constraints.

Contribution

It clarifies how Dirac's conjecture applies to certain first class constraints and connects Hamiltonian and Lagrangian approaches using examples.

Findings

Dirac's conjecture applies to first class constraints generated iteratively.

Gauge identities reflect gauge symmetries in the Lagrangian approach.

The connection between Hamiltonian and Lagrangian symmetries is clarified.

Abstract

The gauge symmetries of a general dynamical system can be systematically obtained following either a Hamiltonean or a Lagrangean approach. In the former case, these symmetries are generated, according to Dirac's conjecture, by the first class constraints. In the latter approach such local symmetries are reflected in the existence of so called gauge identities. The connection between the two becomes apparent, if one works with a first order Lagrangean formulation. Our analysis applies to purely first class systems. We show that Dirac's conjecture applies to first class constraints which are generated in a particular iterative way, regardless of the possible existence of bifurcations or multiple zeroes of these constraints. We illustrate these statements in terms of several examples.