Abstract
It is shown that if the photon propagator is set equal to $\frac{1}{{k}^{2}}$ in the expression for the vacuum polarization, then $Z_{3}^{}{}_{}{}^{\ensuremath{-}1}$ (where ${Z}_{3}$ is the photon wave-function renormalization constant) diverges like a single power of the logarithm of an ultraviolet cutoff in all orders of perturbation theory. The implication of this result upon the possible finiteness of ordinary quantum electrodynamics is discussed.
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