Resummation of Feynman Diagrams and the Inversion of Matrices

In many field theoretical models one has to resum two- and four-legged subdiagrams in order to determine their behaviour. In this article we present a novel formalism which does this in a nice way. It is based on the central limit theorem of probability and an inversion formula for matrices which is obtained by repeated application of the Feshbach projection method. We discuss applications to the Anderson model, to the many-electron system and to the phi^4-model. In particular, for the many-electron system with attractive delta-interaction, we find that the existence of a BCS gap and a macroscopic value of the Hubbard-Stratonovich field for zero momentum enforce each other.