Notes on path integral reduction in scalar electrodynamics

TL;DR

This paper explores path integral reduction in scalar electrodynamics using the Coulomb gauge, emphasizing the geometric influence of material fields and the necessity of regularization for accurate reduction.

Contribution

It extends a finite-dimensional reduction method to scalar electrodynamics, analyzing the geometry of the reduced space and the role of regularization in the Jacobian's singular behavior.

Findings

Reduced dynamics described by stochastic differential equations

Geometry of the reduced space depends on material fields

Regularization is essential for proper path integral reduction

Abstract

Based on a method developed earlier for a finite-dimensional mechanical system, the problem of path integral reduction for scalar electrodynamics is considered. Using the Coulomb gauge, the stochastic differential equations for the reduced dynamics on the orbit space are obtained. It is shown that the geometry of the reduced space is completely determined by the behaviour of the material fields. Since the main role in the singular behavior of the reduction Jacobian is played by the mean curvature of the orbit, the final solution of the path inyegral reduction problem in the field system under consideration is possible only by carrying out an adequate regularization of the term that determines the volume of the orbit, and on which the additional correction to the interaction potential completely depends.