# The current density in quantum electrodynamics in external potentials

January 23, 2015

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.

Keywords:

locally covariant quantum field theory, external potentials