Schwinger Terms in Fermion Electrodynamics

Abstract

We investigate the $c$-number Schwinger term and a possible operator Schwinger term in fermion electrodynamics, to the lowest nontrivial orders in perturbation theory. The results of our direct method of calculation agree with results previously obtained from the Bjorken-Johnson-Low high-energy limit. In particular, we find no evidence for the existence of an operator Schwinger term. The structure of our results helps us to understand the calculations by the split-point method of Schwinger, according to which there seems to be an operator Schwinger term. In the examples considered, we discover an analogy between our direct perturbation-theory calculations, which are performed in momentum space, and the split-point calculations, which are performed in coordinate space. Interchanging the equal-time limit with the phase-space integration is analogous to interchanging the equal-time limit with the split-point limit. Pursuing the analogy, we conclude that the split-point definition of the current probably is consistent with perturbation theory, and that the apparent discrepancies are due to the improper interchange of the equal-time limit with the split-point limit in previous calculations by the split-point method.