Constraints and Anomalies in Finite Quantum Electrodynamics

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 Z₃ 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 Tψ(0)γ^μψ(x) has infinite c-number coefficients. If the first possibility occurs, then either Z₂=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 [ψ(x),jⁱ(0)]δ(x⁰) without reference to perturbation theory.