An Introduction to Spectral Regularization for Quantum Field Theory

Abstract

A spectral calculus for the computation of the spectrum of Lorentz invariant Borel complex measures on Minkowski space is introduced. It is shown how problematical objects in quantum field theory (QFT), such as Feynman integrals associated with loops in Feynman graphs, can be given well defined existence as Lorentz invariant tempered Borel complex measures. Their spectral representation can be used to compute an equivalent density which can be used in QFT calculations. As an application the contraction of the vacuum polarization tensor is considered. The spectral vacuum polarization function is shown to have close agreement (up to finite renormalization) with the vacuum polarization function obtained using dimensional regularization/renormalization. The spectral running coupling for QED is computed and shown to manifest no Landau poles.