Infrared catastrophe averted by Hertz potential

A computational scheme for quantum electrodynamics is developed from first principles in which infrared divergences are not encountered. The gist of the method is that the photon propagator is properly $-i{g}_{\mu \nu }{(k^2+i\epsilon )}^{-1\mathrm{}}{\left[\right(\frac{1}{2}\left)\frac{k^{\sigma }\partial }{\partial k^{\sigma }}\right)]}_w\ln (-{\lambda }^{-2\mathrm{}}{k}^2-i\epsilon )$. Here $\lambda$ is a parameter which plays the role of the photon mass, but which cancels out of cross sections for any finite value ($\lambda \to 0$ is never taken). The subscript $w$ means weak derivative, so a partial integration on $k$, with neglect of boundary terms, is always understood in Feynman integrals. This makes integrals converge which otherwise would be logarithmically divergent. For sufficiently convergent integrals the partial integration may be undone and the conventional value results because $\left[\right(\frac{1}{2}\left)\frac{k^{\sigma }\partial }{\partial k^{\sigma }}\right)\left]\ln \right(-{\lambda }^{-2\mathrm{}}{k}^2-i\epsilon )=1\mathrm{}$.. It is shown that the above propagator is not merely an artificial device, but results if the vector potential $A^{\mu }$ is derived from a Hertz potential ${\Pi }^{\lambda \mu }=-{\Pi }^{\mu \lambda }$, $A^{\mu }={\partial }_{\mu }{\Pi }^{\lambda \mu }$, which is a local field. The real-photon infrared divergences do not occur when the integral over real photons is obtained from the discontinuity in the $k^0$ plane of the above propagator. A contribution to the real-photon integral at strictly zero frequency, resembling a third polarization state of the photon, is isolated and evaluated explicitly. The $S$ matrix and physical subspaces are defined.