Renormalization functional relationships in quantum electrodynamics

Gell-Mann–Low type asymptotic functional relationships for Green's functions are deduced for the electron mass renormalization $\delta m=mF(\Lambda/m, \alpha)$ and the photon and electron wave function renormalization constants $Z_3(\Lambda/m,\alpha)$ and $Z_2(\Lambda/m,\alpha)$, respectively. These are regarded as functions of the cutoff parameter $\Lambda$, the renormalized mass $m$, and the renormalized coupling constant $\alpha$. The relationships obtained are used to investigate the following problems: the existence of a solution of the “superconducting type” for the self-energy of the electron; the nature of the divergences in quantum electrodynamics; and the self-consistency of the theory. An approximation scheme is developed in order to determine the above functions; it is based on the functional relationships obtained and perturbation theory.