Path integral representation of the quantum evolution in dynamical systems with a symmetry for the non-zero momentum level reduction

TL;DR

This paper derives a path integral representation for quantum evolution of a scalar particle on a Riemannian manifold with symmetry, specifically for reduction onto the non-zero momentum level, linking total space and reduced system solutions.

Contribution

It provides a novel path integral formulation for quantum systems with symmetry, explicitly relating total space and reduced space solutions in the non-zero momentum case.

Findings

Derived the path integral representation of the Green's function for reduced quantum motion.

Established the integral relation between path integrals on total space and reduced space.

Extended the path integral approach to systems with symmetry and non-zero momentum reduction.

Abstract

For the case of reduction onto the non-zero momentum level, in the problem of the path integral quantization of a scalar particle motion on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimle Lie group, the path integral representation of the matrix Green's function, which describes the quantum evolution of the reduced motion, has been obtained. The integral relation between the path integrals representing the fundamental solutions of the parabolic differential equation defined on the total space of the principal fiber bundle and the linear parabolic system of the differential equations on the space of the sections of the associated covector bundle has been derived.