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 $Z_3$ 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 $T\left\{\overline{\psi }\mathrm{}\right(0\left)\mathrm{}{\gamma }^{\mu }\psi \mathrm{}\right(x\left)\right\}$ has infinite $c$-number coefficients. If the first possibility occurs, then either $Z_2=0\mathrm{}$ 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 $\left[\psi \mathrm{}\right(x),j^i\mathrm{}(0\left)\right]\delta \left(x^0\right)$ without reference to perturbation theory.