On Hilbert's sixth problem

TL;DR

This paper explores the mathematical foundations of Feynman path integrals in quantum physics and differential geometry, highlighting their diverse applications and the need for rigorous analysis.

Contribution

It provides an overview of historical developments and mathematical approaches to Feynman path integrals in nonrelativistic quantum mechanics and scalar quantum fields.

Findings

Different applications of path integrals in flat space-time and topological quantum field theory

Mathematical approaches to path integrals in quantum mechanics and scalar fields

Emphasis on the need for rigorous mathematical treatment

Abstract

Feynman path integrals are now a standard tool in quantum physics and their use in differential geometry leads to new mathematical insights. A logical treatment of quantum phenomena seems to require a sustained mathematical analysis of path integrals. The subject is complicated by the fact that their application in flat space-time is quite different from how path integrals are used in, say, topological quantum field theory, where there is no natural notion of time translation. An historical background is given in this paper and a few mathematical approaches to Feynman path integrals in the context of nonrelativistic quantum mechanics and scalar quantum fields with polynomial self-interactions are outlined.