Locality in Abelian gauge theories over globally hyperbolic spacetimes

TL;DR

This paper examines the violation of the locality axiom in Abelian gauge theories like Maxwell $k$-forms and $U(1)$ Yang-Mills over globally hyperbolic spacetimes, highlighting the challenges in recovering locality at classical and quantum levels.

Contribution

It provides a detailed analysis of the locality axiom in Abelian gauge theories, demonstrating the inherent global features that prevent its coherent recovery.

Findings

Maxwell $k$-forms cannot coherently recover locality.

In $U(1)$ Yang-Mills, locality can be recovered but misses Aharonov-Bohm effects.

A no-go theorem shows locality cannot be fully recovered in the Wilson loop approach.

Abstract

The thesis investigates the locality axiom of general local covariance for Abelian gauge theories. Two models, Maxwell $k$-forms (higher analogues of the electromagnetic vector potential) and the $U(1)$ Yang-Mills model are analyzed over globally hyperbolic spacetimes. Our attention is mainly focused on the locality axiom of general local covariance, which states that a causal embedding between spacetimes should induce an inclusion at the level of observables. Both at the classical and at the quantum level, it turns out that the models we consider violate locality depending on certain global features of the background spacetime. For Maxwell $k$-forms, we prove that there is no coherent way to recover the locality axiom. For the $U(1)$ Yang-Mills model we adopt two different approaches: in the first one locality can be recovered coherently, but the class of observables we consider fails in detecting those field configurations which correspond to the Aharonov-Bohm effect; conversely, in our second approach observables are defined in the spirit of Wilson loops (hence capturing also Aharonov-Bohm configurations), but a no-go theorem shows that locality cannot be recovered in a coherent way.