The Faddeev-Popov trick in the presence of boundaries

TL;DR

This paper establishes criteria for applying the Faddeev-Popov method to gauge theories on manifolds with boundaries, demonstrating gauge independence under these criteria and gauge dependence otherwise, with implications for boundary conditions.

Contribution

It formulates specific criteria for the applicability of the Faddeev-Popov trick in boundary gauge theories and illustrates their importance using Euclidean Maxwell theory.

Findings

Path integral gauge independence when criteria are met

Gauge dependence arises if criteria are violated

Gauge dependent boundary conditions are necessary in some cases

Abstract

We formulate criteria of applicability of the Faddeev-Popov trick to gauge theories on manifolds with boundaries. With the example of Euclidean Maxwell theory we demonstrate that the path integral is indeed gauge independent when these criteria are satisfied, and depends on a gauge choice whenever these criteria are violated. In the latter case gauge dependent boundary conditions are required for a self-consistent formulation of the path intgral.