Multiscale block averaging for QED in d=3

Abstract

We continue the study of the ultraviolet problem for QED in d=3. The model is defined on a fine toroidal lattice and we seek control as the lattice spacing goes to zero. The problem is analyzed using Balaban's formulation of the renormalization group. This involves a sequence of transformations consisting of a split into large and small field regions, then block averaging, and then scaling. The the effective actions generated by this method depend strongly on certain multi-scale propagators and minimizers. The study of these objects both for fermions and for gauge fields is content of this paper. Earlier work on the subject is reviewed. In addition for fermions a polymer expansion is obtained for the determinants of the fermion propagators. For the gauge field a detailed local regularity result is obtained for the minimizers.