Reduction of path integrals for interacting systems: The case of using dependent coordinates in the description of reduced motion on the orbit space

TL;DR

This paper develops a reduction method for Wiener path integrals in systems with symmetry, using dependent coordinates and stochastic filtering to analyze the measure's behavior and Jacobian in the context of interacting particles on a manifold.

Contribution

It introduces a novel reduction procedure for path integrals in systems with symmetry, employing dependent coordinates and stochastic filtering techniques.

Findings

The measure in the path integral is shown to be non-invariant under reduction.

The Jacobian of the reduction is related to the projection of the mean curvature vector.

Dependent coordinates facilitate the description of reduced motion on orbit space.

Abstract

We consider a reduction procedure in Wiener-type path integral for a finite-dimensional mechanical system with a symmetry representing the motion of two interacting scalar particles on a manifold that is the product of the total space of the principal bundle and a vector space. By analogy with what is done in gauge theories, the local description of the reduced motion on orbit space is carried out using dependent coordinates. The factorization of the measure in the path integral, which is necessary for the reduction, is based on the application of the stochastic differential equation of the optimal nonlinear filtering from the theory of stochastic processes. The non-invariance of the measure in the path integral under the reduction is shown. The Jacobian of the reduction is generated by the projection of the mean curvature vector field of the orbit onto the submanifold, which is used to determine the adapted coordinates in the principal fiber bundle associated with the problem under study.