Path integrals and stochastic calculus

TL;DR

This paper reviews and develops methods for defining and manipulating path integrals in physics and stochastic calculus, including geometric and discretized approaches, to improve their mathematical rigor and usability.

Contribution

It introduces new discretized approaches for path integrals and compares them with existing geometric methods, enhancing the mathematical foundation of path integral formulations.

Findings

Discretized methods achieve proper calculus rules for path integrals.

Geometric approaches based on Riemannian geometry are effective in arbitrary dimensions.

Examples demonstrate the application to diffusion on a sphere.

Abstract

Path integrals are a ubiquitous tool in theoretical physics. However, their use is sometimes hindered by the lack of control on various manipulations -- such as performing a change of the integration path -- one would like to carry out in the light-hearted fashion that physicists enjoy. Similar issues arise in the field of stochastic calculus, which we review to prepare the ground for a proper construction of path integrals. At the level of path integration, and in arbitrary space dimension, we not only report on existing Riemannian geometry-based approaches that render path integrals amenable to the standard rules of calculus, but also bring forth new routes, based on a fully time-discretized approach, that achieve the same goal. We illustrate these various definitions of path integration on simple examples such as the diffusion of a particle on a sphere.