Renormalization Theory for Use in Convergent Expansions of Euclidean Quantum Field Theory

Abstract

Renormalization is necessary for perturbation and cluster expansion methods of Euclidean quantum field theory. Bounds for Feynman graphs show that divergencies for renormalizable theories come from subgraphs with a small number of low frequency external lines [1,2]. Renormalization group suggests a way to avoid such divergencies by introducing running coupling constants. The tree expansion introduced by G. Gallavotti and F. Nicolo [3-6] represents the expansion in powers of running coupling constants instead of renormalized coupling constants. Running coupling constants are related by renormalization group equations and the bare coupling constants have to be chosen such that the renormalized coupling constants are finite if an ultraviolet cutoff is removed.