Gauge Symmetry from Integral Viewpoint

TL;DR

This paper examines the gauge symmetry in classical field theories using an integral formalism, revealing that non-Abelian gauge symmetry is generally broken non-perturbatively, while Abelian gauge symmetry remains intact.

Contribution

It provides a detailed analysis of gauge invariance in integral formalism, highlighting the non-perturbative breaking of non-Abelian gauge symmetry and confirming the invariance of Abelian theories.

Findings

Non-Abelian gauge symmetry is broken non-perturbatively.

Abelian gauge symmetry remains strictly invariant.

Interference patterns in Aharonov-Bohm effect demonstrate Abelian invariance.

Abstract

The purpose of this paper is to investigate the gauge symmetry of classical field theories in integral formalism. A gauge invariant theory is defined in terms of the invariance of the physical observables under the coordinate transformations in principal bundle space. Through the detailed study on the property of non-Abelian parallel transportor under gauge transformations, we show that it is not generally a two-point spinor, i.e. an operator to be affected only by the gauge group elements at the two end points of the parallel transport path, except for the pure gauge situation, and therefore the local gauge symmetry for non-Abelian models is found to be broken in non-perturbative domain. However, an Abelian gauge theory is proved to be strictly invariant under local gauge transformation, as it is illustrated by the invariance of the interference pattern of electrons in Aharonov-Bohm effect. The related issues of the phenomenon are discussed.