Schwinger's Propagator Is Only A Green's Function

Abstract

Schwinger used an analytic continuation of the effective action to correctly compute the particle production rate per unit volume for QED in a uniform electric field. However, if one simply evaluates the one loop expectation value of the current operator using his propagator, the result is zero! We analyze this curious fact from the context of a canonical formalism of operators and states. The explanation turns out to be that Schwinger's propagator is not actually the expectation value of the time-ordered product of field operators in the presence of a time-independent state, although it is of course a Green's function. We compute the true propagator in the presence of a state which is empty at $x_+ = 0$ where $x_+ \equiv (x^0+x^3)/\sqrt{2}$ is the lightcone evolution parameter. Our result can be generalized to electric fields which depend arbitrarily on $x_+$.