Conversion of second-class constraints and resolving the zero curvature conditions in the geometric quantization theory

TL;DR

This paper develops a method to convert second-class constraints into a linear form within geometric quantization, providing new equations and a path integral representation applicable to phase superspaces with Boson and Fermion coordinates.

Contribution

It introduces a novel linearization approach for zero curvature conditions in geometric quantization, extending the framework to superspaces with both Boson and Fermion variables.

Findings

Derived linear equations from zero curvature conditions.

Proved solutions to linear equations satisfy original non-linear conditions.

Provided a path integral representation for the quantization process.

Abstract

In the approach to the geometric quantization, based on the conversion of second-class constraints, we resolve the respective non-linear zero curvature conditions for the extended symplectic potential. From the zero curvature conditions, we deduce new, linear, equations for the extended symplectic potential. Then we show that being the linear equations satisfied, their solution does certainly satisfy the non-linear zero curvature condition, as well. Finally, we give the functional resolution to the new linear equations, and then deduce the respective path integral representation. We do our consideration as to the general case of a phase superspace where both Boson and Fermion coordinates are present on equal footing.