Renormalization, Diagrams, and Gauge Invariance

A compact formula for the renormalized transition probability in quantum electrodynamics is derived and its interpretation in terms of the diagrams is given. This formula is used next to prove the equivalence of the Coulomb gauge and the Feynman gauge. It is shown that, contrary to widespread belief, the problem of the gauge invariance becomes a very delicate one when the radiative corrections are included. The common practice of dropping $k_{\mu }$ and $k_{\nu }$ terms in the photon propagator $D_{\mu \nu }\mathrm{}\left(k\right)\mathrm{}$ is justified, but the justification requires a nontrivial analysis of the gauge transformations of the propagators involved.