Path integrals and stochastic calculus
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.
Cite
@article{arxiv.2211.09470,
title = {Path integrals and stochastic calculus},
author = {Thibaut Arnoulx de Pirey and Leticia F. Cugliandolo and Vivien Lecomte and Frédéric van Wijland},
journal= {arXiv preprint arXiv:2211.09470},
year = {2023}
}
Comments
96 pages, 4 figures. New title, expanded introduction and additional references. Version accepted in Advandes in Physics