Gauge Covariant Fermion Propagator in the Presence of Arbitrary External Gauge Field and Its Schwinger--Dyson Equation

Gauge covariance for Green's functions of a gauge theory through a fermion propagator in the presence of arbitrary external gauge field is proven and a formalism of gauge and Lorentz covariant Schwinger--Dyson equation for the fermion propagator with external gauge field is built up within ladder approximation.