Non-covariance of the generalized holonomies: Examples

TL;DR

This paper demonstrates that generalized holonomies in a proposed quantum Yang-Mills and gravity framework are not gauge covariant, challenging their role as observables and highlighting potential issues in using generalized loops in physics.

Contribution

It reveals a key technical subtlety showing generalized holonomies are not gauge covariant, questioning their suitability as observables in the generalized loop representation.

Findings

Generalized holonomies are not covariant under small gauge transformations.

Traces of generalized holonomies are not gauge observables.

This poses a serious complication for the generalized loop approach in physics.

Abstract

A key aspect of a recent proposal for a {\em generalized loop representation} of quantum Yang-Mills theory and gravity is considered. Such a representation of the quantum theory has been expected to arise via consideration of a particular algebra of observables -- given by the traces of the holonomies of {\em generalized loops}. We notice, however, a technical subtlety, which prevents us from reaching the conclusion that the generalized holonomies are covariant with respect to small gauge transformations. Further analysis is given which shows that they are {\em not} covariant with respect to small gauge transformations; their traces are {\em not} observables of the gauge theory. This result indicates what may be a serious complication to the use of generalized loops in physics.