On the coordinate (in)dependence of the formal path integral

TL;DR

This paper investigates how the formal path integral in quantum field theory depends on coordinate choices, showing it is invariant under coordinate transformations when a fiberwise volume form is fixed, especially in the absence of divergences.

Contribution

It provides a proof that the formal path integral's dependence is limited to the choice of fiberwise volume form, extending understanding beyond vector bundle fields to curved fiber bundles.

Findings

Path integral invariance under coordinate changes with fixed volume form

Clarification of the role of fiberwise volume forms in path integrals

Proof validity in divergence-free (ultraviolet finite) cases

Abstract

When path integrals are discussed in quantum field theory, it is almost always assumed that the fields take values in a vector bundle. When the fields are instead valued in a possibly-curved fiber bundle, the independence of the formal path integral on the coordinates becomes much less obvious. In this short note, aimed primarily at mathematicians, we first briefly recall the notions of Lagrangian classical and quantum field theory and the standard coordinate-full definition of the "formal" or "Feynman-diagrammatic" path integral construction. We then outline a proof of the following claim: the formal path integral does not depend on the choice of coordinates, but only on a choice of fiberwise volume form. Our outline is an honest proof when the formal path integral is defined without ultraviolet divergences.