Quantum Electrodynamics at Small Distances

The renormalized propagation functions $D_{\mathrm{FC}}$ and $S_{\mathrm{FC}}$ for photons and electrons, respectively, are investigated for momenta much greater than the mass of the electron. It is found that in this region the individual terms of the perturbation series to all orders in the coupling constant take on very simple asymptotic forms. An attempt to sum the entire series is only partially successful. It is found that the series satisfy certain functional equations by virtue of the renormalizability of the theory. If photon self-energy parts are omitted from the series, so that $D_{\mathrm{FC}}=D_F$, then $S_{\mathrm{FC}}$ has the asymptotic form $A{\left[\frac{p^2}{m^2}\right]}^n{\left[i\gamma ·p\right]}^{-1\mathrm{}}$, where $A=A\left({e_1}^2\right)$ and $n=n\left({e_1}^2\right)$. When all diagrams are included, less specific results are found. One conclusion is that the shape of the charge distribution surrounding a test charge in the vacuum does not, at small distances, depend on the coupling constant except through a scale factor. The behavior of the propagation functions for large momenta is related to the magnitude of the renormalization constants in the theory. Thus it is shown that the unrenormalized coupling constant $\frac{{e_0}^2}{4\pi \hslash c}$, which appears in perturbation theory as a power series in the renormalized coupling constant $\frac{{e_1}^2}{4\pi \hslash c}$ with divergent coefficients, may behave in either of two ways:

(a) It may really be infinite as perturbation theory indicates;

(b) It may be a finite number independent of $\frac{{e_1}^2}{4\pi \hslash c}$.