  We develop the Diakonov-Petrov approach to the representation of the
non-Abelian Wilson loop in an Abelian form. First of all, we extend this
approach to the Abelian representation of the parallel transporter
corresponding to the open curve rather than to the closed contour. Next, we
extend this representation to the case of the non-compact groups. The obtained
expressions allow to derive the Abelian form of the Schwinger-Fock
representation for the quark propagator in an external color gauge field.
