First class models from linear and nonlinear second class constraints

Abstract

Two models with linear and nonlinear second class constraints are considered and gauged by embedding in an extended phase space. These models are studied by considering a free non-relativistic particle on the hyperplane and hypersphere in the configuration space. The gauged theory of the first model is obtained by converting the very second class system to the first class one directly. In contrast, the first class system related to the free particle on the hypersphere is derived with the help of the infinite Batalin–Fradkin–Tyutin (BFT) embedding procedure. We propose a practical formula, based on the simplified BFT method, which is practical in gauging linear and some nonlinear second class systems. As a result of gauging these two models, we show that in the conversion of second class constraints to the first class ones, the minimum number of phase space degrees of freedom for both systems is a pair of phase space coordinates. This pair is made up of a coordinate and its conjugate momentum for the first model, but the corresponding Poisson structure of the embedded non-relativistic particle on hypersphere is a nontrivial one. We derive infinite correction terms for the Hamiltonian of the nonlinear constraints and an interacting gauged Hamiltonian is constructed by summing over them. At the end, we find an open algebra for three first class objects of the embedded nonlinear system.