Notes on Feynman path integral-like methods of quantization on Riemannian manifolds

TL;DR

This paper introduces a new approach for Feynman path integrals on compact Riemannian manifolds using shortest path action integrals, proving convergence and including curvature effects for specific manifolds.

Contribution

It presents an alternative method for path integrals on Riemannian manifolds and proves convergence with curvature considerations for rank 1 locally symmetric cases.

Findings

Proves strong convergence of time slicing products for low energy functions.

Includes Dewitt curvature in the limit expression.

Applies to rank 1 locally symmetric Riemannian manifolds.

Abstract

We propose an alternative method for Feynman path integrals on compact Riemannian manifolds. Our method employs action integrals along the shortest paths. In the case of rank 1 locally symmetric Riemannian manifolds, we prove the strong convergence of time slicing products of oscillatory integrals for low energy functions. Moreover, the strong limit includes Dewitt curvature $R/6$, where $R$ denotes the scalar curvature of a Riemannian manifold.