First class models from linear and nonlinear second class constraints

TL;DR

This paper develops a practical method for converting second class constraints into first class constraints in models like a free particle on a hyperplane or hyper sphere, revealing the structure of the embedded systems.

Contribution

It introduces a simplified BFT embedding procedure for gauging linear and some nonlinear second class systems, providing explicit formulas and analyzing the resulting algebraic structures.

Findings

Gauging reduces the phase space to a minimal pair of conjugate variables.

Embedded nonlinear systems have a non-trivial Poisson structure.

Infinite correction terms are derived for the nonlinear Hamiltonian.

Abstract

Two models with linear and nonlinear second class constraints are considered and gauged by embedding in an extended phase space. These models are the free non-relativistic particle on a hyperplane and hyper sphere in configuration space. For the first model we construct its gauged corresponding by the condition of converting second class system to first class one, directly. In contrast the first class system related to the free particle on hyper sphere is derived by the BFT embedding procedure, where its steps are infinite. We give a practical formula for gauging linear and some of the nonlinear second class systems, based on the simplified BFT method. As a result of the gauging two models, we show that in the conversion of second class to the first class constraints the minimum number of phase space degrees of freedom for both systems is a pair of phase space coordinate. This pair for first system is a coordinate and its momentum conjugate, but Poisson structure of embedded non-relativistic particle on hyper sphere is a non-trivial one. We derive infinite correction terms for the Hamiltonian of the nonlinear constraints. The summation is done and constructs an interacting gauged Hamiltonian. We find an open algebra for three first class objects of the embedded nonlinear system.