Building a path-integral calculus: a covariant discretization approach

TL;DR

This paper introduces a covariant discretization method for path integrals that overcomes longstanding issues with non-linear variable changes, enabling a more rigorous path-integral calculus.

Contribution

It presents a novel covariant discretization approach that resolves fundamental mathematical problems in path integral calculus for systems with one degree of freedom.

Findings

Identifies mathematical reasons behind path integral issues.

Develops a covariant discretization scheme.

Provides a method to perform error-free non-linear variable changes.

Abstract

Path integrals are a central tool when it comes to describing quantum or thermal fluctuations of particles or fields. Their success dates back to Feynman who showed how to use them within the framework of quantum mechanics. Since then, path integrals have pervaded all areas of physics where fluctuation effects, quantum and/or thermal, are of paramount importance. Their appeal is based on the fact that one converts a problem formulated in terms of operators into one of sampling classical paths with a given weight. Path integrals are the mirror image of our conventional Riemann integrals, with functions replacing the real numbers one usually sums over. However, unlike conventional integrals, path integration suffers a serious drawback: in general, one cannot make non-linear changes of variables without committing an error of some sort. Thus, no path-integral based calculus is possible. Here we identify which are the deep mathematical reasons causing this important caveat, and we come up with cures for systems described by one degree of freedom. Our main result is a construction of path integration free of this longstanding problem, through a direct time-discretization procedure.