Spectral regularization and a QED running coupling without a Landau pole

Divergent integrals in quantum field theory (QFT) can be given well defined existence as Lorentz covariant complex measures, which may be analyzed by means of a spectral calculus. The case of the photon self energy is considered and the spectral vacuum polarization function is shown to have very close agreement with the vacuum polarization function obtained using dimensional regularization / renormalization in the timelike domain. Using the spectral vacuum polarization function a potential function defined in the timelike domain is derived. The Uehling potential function, from which the Uehling contribution to the Lamb shift may be computed, is derived from an analytic continuation into the spacelike domain of this potential function. The spectral running coupling for QED is computed from this analytically continued potential function. The integral defining the spectral running coupling constant is shown to converge for all non-zero energies while that for the running coupling constant computed using dimensional regularization / renormalization is shown to diverge for all non-zero energies. It is seen that the spectral running coupling does not have a Landau pole and agrees both qualitatively and quantitatively with the results of scattering experiments at all energies.