  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.
