Path integrals on a manifold that is a product of the total space of the principal fiber bundle and the vector space

TL;DR

This paper develops a reduction method for Wiener path integrals on a manifold formed by a principal fiber bundle's total space and a vector space, using nonlinear filtering to relate solutions on original and reduced manifolds.

Contribution

It introduces a novel reduction procedure for path integrals on complex manifolds with symmetry, incorporating a Jacobian for zero-momentum reduction.

Findings

Derived integral relation between original and reduced path integrals.

Obtained the reduction Jacobian as an additional potential term.

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

Abstract

Using the path integral measure factorization method based on the nonlinear filtering equation from the stochastic process theory, we consider the reduction procedure in Wiener path integrals for a mechanical system with symmetry that describes the motion of two interacting scalar particles on a special smooth compact Riemannian manifold - the product of total space of the principal fiber bundle and the vector space. The original manifold, the configuration space of this system, is endowed with an isometric free proper action of a compact semisimple unimodular Lie group. The proposed reduction procedure leads to the integral relation between path integrals that represent fundamental solutions of the inverse Kolmogorov equations on the initial and reduced manifolds. For the case of reduction onto the zero-momentum level, the reduction Jacobian is obtained, which is an additional potential term to the Hamiltonian.