The current density in quantum electrodynamics in external potentials

TL;DR

This paper reviews various definitions of current density in quantum electrodynamics with external potentials, highlighting issues in popular methods and proposing Dirac's approach as the most satisfactory, with applications to vacuum polarization.

Contribution

It introduces Dirac's method for defining current density in external potentials, addressing deficiencies in previous prescriptions and providing new formulas for vacuum polarization.

Findings

Dirac's method is conceptually superior for current density in external fields.

Identified deficiencies in Schwinger and mode sum prescriptions.

Derived a general formula for vacuum polarization in static external potentials.

Abstract

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.