Convolution regularization in quantum electrodynamics

Abstract

We consider the divergent graphs of quantum electrodynamics and interpret the undefined products of two or more propagators as follows: we first modify the argument of one propagator by adding a small spacelike four-vector of magnitude ɛ; we then compute the integral and average the resulting expression over all angles and, finally, take the limit ɛ=0. In momentum space each diagram turns into the Fourier transform of the usual integrand. The computation of any loop reduces to simple iterated convolution products of the causal delta Δc(ɛ) by itself. The procedure is fully covariant and the Ward identity is satisfied for all ɛ; however it is not gauge invariant, not even in the limit ɛ=0. This invariance may be recovereda posteriori by renormalizing the photon mass to zero. To show how easily the method works, the electron self-energy diagram in lowest order is explicitly evaluated. The result coincides exactly with the conventional way of handling infinite integrals, both for the finite part and for the ultraviolet and infra-red divergences. Feynman’s auxiliarly massM turns out to be inversely proportional to ɛ.