Dependent coordinates in path integral measure factorization

TL;DR

This paper studies how the path integral measure transforms when reducing dynamical systems with symmetry, using stochastic filtering to relate original and reduced kernels on Riemannian manifolds.

Contribution

It introduces a method to factorize the path integral measure in symmetric systems using nonlinear filtering, linking original and reduced semigroup kernels.

Findings

Derived integral relation between original and reduced kernels.

Applied stochastic filtering to path integral measure transformation.

Provided a framework for diffusion processes on manifolds with symmetry.

Abstract

The transformation of the path integral measure under the reduction procedure in the dynamical systems with a symmetry is considered. The investigation is carried out in the case of the Wiener--type path integrals that are used for description of the diffusion on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple unimodular Lie group. The transformation of the path integral, which factorizes the path integral measure, is based on the application of the optimal nonlinear filtering equation from the stochastic theory. The integral relation between the kernels of the original and reduced semigroup are obtained.