BFFT Nonlinear Constraints Abelianization of a Prototypical Second-Class System

TL;DR

This paper applies the BFFT formalism to convert second-class constraints into first-class constraints in a prototypical system, enabling broader applications in mechanical and field theory models with nonlinear constraints.

Contribution

It introduces a generalized approach for Abelianizing second-class constraints with nonlinearities, providing explicit closed-form solutions for complex systems.

Findings

Successfully Abelianized constraints for a particle on a torus

Derived closed-form involutive Hamiltonian for a Lorentz symmetry breaking model

Extended BFFT formalism to nonlinear and arbitrary constraints

Abstract

We apply the BFFT formalism to a prototypical second-class system, aiming to convert its constraints from second- to first-class. The proposed system admits a consistent initial set of second-class constraints and an open potential function providing room for applications to mechanical models as well as field theory such as the non-linear sigma model. The constraints can be arbitralily non-linear, broadly generalizing previously known cases. We obtain a sufficient condition for which a simple closed expression for the Abelian converted constraints and modified involutive Hamiltonian can be achieved. As explicit examples, we discuss a particle on a torus and a spontaneous Lorentz symmetry breaking vector model, obtaining in both situations the full first-class abelianized constraints in closed form and the corresponding involutive Hamiltonian.