Functional integration and gauge ambiguities in generalized abelian gauge theories

TL;DR

This paper investigates the covariant quantization of generalized abelian gauge theories on compact manifolds, revealing topological obstructions to gauge fixing and providing explicit expressions for physical observables and Wilson operators.

Contribution

It introduces a novel framework for quantizing generalized abelian gauge theories, accounting for topological bundle structures and gauge ambiguities, with explicit formulas for observables.

Findings

Topological obstructions prevent global gauge fixing.

A regularized functional integral measure is proposed.

Explicit expressions for Green's functions and Wilson operators are derived.

Abstract

We consider the covariant quantization of generalized abelian gauge theories on a closed and compact n-dimensional manifold whose space of gauge invariant fields is the abelian group of Cheeger-Simons differential characters. The space of gauge fields is shown to be a non-trivial bundle over the orbits of the subgroup of smooth Cheeger-Simons differential characters. Furthermore each orbit itself has the structure of a bundle over a multi-dimensional torus. As a consequence there is a topological obstruction to the existence of a global gauge fixing condition. A functional integral measure is proposed on the space of gauge fields which takes this problem into account and provides a regularization of the gauge degrees of freedom. For the generalized p-form Maxwell theory closed expressions for all physical observables are obtained. The Greens functions are shown to be affected by the non-trivial bundle structure. Finally the vacuum expectation values of circle-valued homomorphisms, including the Wilson operator for singular p-cycles of the manifold, are computed and selection rules are derived.