Feynman path integrals on compact Lie groups with bi-invariant Riemannian metrics

TL;DR

This paper extends the Feynman path integral framework to compact Lie groups with bi-invariant metrics, demonstrating its connection to the Schrödinger equation via the Laplace-Beltrami operator.

Contribution

It introduces a method to define Feynman path integrals on compact Lie groups using Cartan development, oscillatory integrals, and Chernoff approximation, linking to quantum mechanics.

Findings

Feynman map solves the Schrödinger equation on compact Lie groups.

The Laplace-Beltrami operator equals the second order Casimir operator.

The approach applies to a dense subspace of the relevant Hilbert space.

Abstract

In this work we consider a suitable generalization of the Feynman path integral on a specific class of Riemannian manifolds consisting of compact Lie groups with bi-invariant Riemannian metrics. The main tools we use are the Cartan development map, the notion of oscillatory integral, and the Chernoff approximation theorem. We prove that, for a class of functions of a dense subspace of the relevant Hilbert space, the Feynman map produces the solution of the Schr\"odinger equation, where the Laplace-Beltrami operator coincides with the second order Casimir operator of the group.