Constraints and Anomalies in Finite Quantum Electrodynamics

Abstract

It is argued that the leading singularity of the electron-photon vertex is inversion-invariant in finite quantum electrodynamics. Crewther's method is then applied to this vertex to obtain a number of consistency relations among the $c$-number coefficients that appear in various short-distance expansions. As a result, if there is a Gell-Mann-Low eigenvalue which makes ${Z}_{3}$ finite, then either (1) the leading singularity of the electron-photon Green's function vanishes in the limit obtained from the short-distance behavior of the two fermion fields, or (2) the scale-invariant short-distance expansion for the $q$-number part of $T{\overline{\ensuremath{\psi}}(0){\ensuremath{\gamma}}^{\ensuremath{\mu}}\ensuremath{\psi}(x)}$ has infinite $c$-number coefficients. If the first possibility occurs, then either ${Z}_{2}=0$ or the amputated vertex vanishes in the corresponding limit. The structure of the electron-photon vertex also permits one to characterize the possible $q$-number anomaly for the equal-time commutator $[\ensuremath{\psi}(x),{j}^{i}(0)]\ensuremath{\delta}({x}^{0})$ without reference to perturbation theory.