Gauge Invariance and Current Definition in Quantum Electrodynamics

Abstract

In order to study the problems of gauge invariance, Lorentz covariance, and the operator properties of the "Schwinger terms" in the current commutators, spin-½ quantum electrodynamics is written as the limit of a nonlocal theory. The conditions on such a theory are discussed and the nonlocal equations in the case of an external vector potential derived. The gauge-invariant, Lorentz-covariant limit of these equations is then discussed, and it is found that (in the case of spin ½) the "Schwinger terms" are purely $c$-number. The quantized vector potential is considered by means of a Feynman path integral and its gauge structure determined. It is found that an automatically gauge-covariant theory results and that the $c$-number character of the Schwinger terms apparently persists.