The current density in quantum electrodynamics in external potentials

We review different definitions of the current density for quantized fermions in the presence of an external electromagnetic field. Several deficiencies in the popular prescription due to Schwinger and the mode sum formula for static external potentials are pointed out. We argue that Dirac's method, which is the analog of the Hadamard point-splitting employed in quantum field theory in curved space-times, is conceptually the most satisfactory. As a concrete example, we discuss vacuum polarization and the stress-energy tensor for massless fermions in 1+1 dimension. Also a general formula for the vacuum polarization in static external potentials in 3+1 dimensions is derived.