Reconstruction of higher stage first class constraints into the secondary ones

TL;DR

This paper develops a method to transform higher-stage first class constraints into secondary ones by constructing an extended Lagrangian, clarifying the structure of the Hamiltonian formulation and revealing gauge symmetries.

Contribution

It introduces an improved Hamiltonian and extended Lagrangian that simplify the analysis of higher-stage constraints and establish their relation to gauge symmetries.

Findings

The improved Hamiltonian has the same structure as the extended Hamiltonian but is derived from an extended Lagrangian.

All constraints of the original Lagrangian generate gauge symmetries in the extended formulation.

The method is demonstrated with a model having fourth-stage constraints.

Abstract

We analyze a singular theory with first class constraints of an arbitrary stage. Relation among the formulations of the constrained system in terms of complete and extended Hamiltonians is clarified. We replace the extended Hamiltonian $H_{ext}$ by an improved one. The improved Hamiltonian has the same structure as $H_{ext}$ (higher stage constraints enter into the Hamiltonian in the manifest form), but, in contrast to $H_{ext}$, it arises as the complete Hamiltonian for some Lagrangian $\tilde L$, called the extended Lagrangian. This implies, in particular, that all the quantities appearing in the improved Hamiltonian have a clear meaning in the Dirac framework. $\tilde L$ is obtained in a closed form in terms of quantities of the initial formulation $L$. The formulations with $L$ and $\tilde L$ turn out to be equivalent. As an application of the formalism, we found local symmetries of $\tilde L$ in a closed form. All the constraints of $L$ turn out to be gauge symmetry generators for $\tilde L$. The procedure is illustrated with an example of a model with fourth-stage constraints.