Axial-Vector Currents in Finite Theories of Quantum Electrodynamics

We investigate the consequences of assuming that there is an observable axial-vector current $J_{5\mu }$ in theories of finite quantum electrodynamics (QED). Positivity constraints for the matrix elements of $J_{5\mu }$ can be neatly formulated for the massive theory via the Schwarz inequality. In order that these requirements be satisfied in the presence of a nonvanishing anomalous constant $S$, it is essential that $d$, the dimension of $J_{5\mu }$, be greater than 3. In that case, we can explain why the condition $J_{\mu }|0\mathrm{}〉=0\mathrm{}$ satisfied by the electromagnetic current $J_{\mu }$ in the zero-mass theory does not contradict the Adler-Bardeen theorem ($S\ne \mathrm{O}$). Further, we explain why the photon-photon scattering subgraphs which cause the anomaly in $d$ are not asymptotically negligible in the Johnson-Baker-Willey (JBW) and Adler versions of finite QED: The argument for $d=3\mathrm{}$ fails because it involves the use of an illegal skeleton expansion. The value of $d$ depends on the number of fundamental fermion species in the theory. Vector and axial-vector currents which generate a non-Abelian internal symmetry of the fermions cannot be introduced in the JBW and Adler theories, because the relevant positivity conditions cannot be satisfied. We examine evidence that the infinite perturbative sums are sufficiently irregular to permit this situation. In an appendix, we solve the Callan-Symanzik equations in the asymptotic region when the zero ${\alpha }_e$ of the Callan-Symanzik function $\beta \left(\alpha \right)$ is not simple. In general, violations of asymptotic scale invariance occur if the physical coupling constant is not equal to ${\alpha }_e$. In the JBW model, this phenomenon can be readily interpreted diagrammatically for matrix elements of $J_{5\mu }$ and the electron propagator.