VARIATIONS OF THE PARALLEL PROPAGATOR AND HOLONOMY OPERATOR AND THE GAUSS LAW CONSTRAINT

A simple derivation is presented of the equations for the variation of the parallel propagator and the holonomy operators of Yang-Mills (YM) connections caused by variations of both the connection and the path. The derivation does not make any direct use of functional derivatives and is based on the solution of the varied parallel transport equatio n. In particular, the different forms that these equations take for a two parameter family of curves in E3 are discussed. As an example of t his formalism, it is shown how any congruence defines a solution of th e Hamilton-Jacobi version of the Gauss law constraint of YM theories, or equivalently, of the Dirac quantum-Gauss law constraint.