Quantum Electrodynamics at Small Distances

In an earlier paper, it was shown that if the vacuum polarization in quantum electrodynamics is calculated neglecting corrections to internal photon lines, then the charge renormalization diverges as a single power of the logarithm of a cutoff in every order of perturbation theory. It was further shown that if the coefficient of the logarithm $f\left({\alpha }_0\right)$ has a zero, and a further positivity condition of $f$ is satisfied, then the complete unrenormalized-photon Green's function is finite. These results we now relate to the Gell-Mann-Low investigation of renormalized quantum electrodynamics at small distances. In particular, the vanishing of the Gell-Mann-Low function $\psi$ is shown to follow from the vanishing of $f$. This demonstrates why it is sufficient, from the point of view of the renormalized theory, to study quantum electrodynamics without photon self-energy corrections in order to examine the predictions of the theory at small distances.