Non-zero momentum level reduction in path integrals for dynamical systems with symmetry given on a product manifold consisting of the total space of the principal fiber bundle and a vector space

TL;DR

This paper develops a method for non-zero momentum level reduction in Wiener path integrals for systems with symmetry on product manifolds, linking solutions of Kolmogorov equations on total and base spaces.

Contribution

It introduces a new integral relation connecting path integrals on the total space of a principal fiber bundle to those on the base manifold, extending reduction techniques to non-zero momentum levels.

Findings

Derived integral relation between path integrals on total and base spaces.

Applied the method to a system of two interacting scalar particles.

Extended the reduction framework to non-zero momentum levels.

Abstract

The case of non-zero momentum level reduction in Wiener path integrals for a mechanical system with symmetry describing the motion of two scalar particles with interaction on a Riemannian product manifold with the given action a compact semisimple Lie group is considered. The original product manifold consists of the vector space and a smooth compact finite-dimensional Riemannian manifold, which, due to the action of the group, can be regarded as the total space of the principal fiber bundle. The integral relation between the path integrals representing the fundamental solutions of the backward Kolmogorov equation defined on the total space of the principal fiber bundle (the original Riemannian product manifold) and the corresponding backward Kolmogorov equation gion the space of the sections of the associated covector bundle is obtained.