  This paper presents a spectral calculus for computing the spectrum of a
causal Lorentz invariant Borel complex measure on Minkowski space, thereby
enabling one to compute the density for such a measure with respect to Lebesque
measure. It is proved that the convolution of arbitrary causal Lorentz
invariant Borel measures exists and the product of such measures exists in a
wide class of cases. Techniques for their computation are presented. Divergent
integrals in quantum field theory (QFT) are shown to have a well defined
existence as Lorentz covariant measures. The case of vacuum polarization 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 the exact Uehling potential function is
derived. The spectral running coupling constant is computed and is shown to
converge for all energies while the integral defining the running coupling
constant obtained using dimensional regularization / renormalization is shown
to diverge for all non-zero energies.
