The Perturbative Approach to Path Integrals: A Succinct Mathematical Treatment

TL;DR

This paper provides a rigorous mathematical framework for understanding path integrals in quantum field theory, focusing on perturbative expansions, invariance properties, and the implications for both perturbative and nonperturbative definitions.

Contribution

It introduces a precise mathematical treatment of path integrals using Wick's theorem, invariance under symmetries, and explores their asymptotic and formal properties.

Findings

Wick expansion invariance under coordinate changes and symmetries

Analysis of the asymptotic nature of perturbative expansions

Clarification of path integral manipulations like gauge fixing and equations of motion

Abstract

We study finite-dimensional integrals in a way that elucidates the mathematical meaning behind the formal manipulations of path integrals occurring in quantum field theory. This involves a proper understanding of how Wick's theorem allows one to evaluate integrals perturbatively, i.e., as a series expansion in a formal parameter irrespective of convergence properties. We establish invariance properties of such a Wick expansion under coordinate changes and the action of a Lie group of symmetries, and we use this to study essential features of path integral manipulations, including coordinate changes, Ward identities, Schwinger-Dyson equations, Faddeev-Popov gauge-fixing, and eliminating fields by their equation of motion. We also discuss the asymptotic nature of the Wick expansion and the implications this has for defining path integrals perturbatively and nonperturbatively.