Renormalization without infinities

TL;DR

This paper presents a conceptual scheme for renormalization in quantum field theories that eliminates ultraviolet divergences by reformulating Feynman diagrams with dressed vertices, linking to renormalization group ideas.

Contribution

It introduces a divergence-free renormalization scheme based on dressed irreducible vertices and unambiguous diagram paths, enhancing conceptual understanding of infinities.

Findings

Diagrams and equations contain no ultraviolet divergences.

Original Lagrangian enters as adjustable constants.

Scheme relates to renormalization group equations.

Abstract

Most renormalizable quantum field theories can be rephrased in terms of Feynman diagrams that only contain dressed irreducible 2-, 3-, and 4-point vertices. These irreducible vertices in turn can be solved from equations that also only contain dressed irreducible vertices. The diagrams and equations that one ends up with do not contain any ultraviolet divergences. The original bare Lagrangian of the theory only enters in terms of freely adjustable integration constants. It is explained how the procedure proposed here is related to the renormalization group equations. The procedure requires the identification of unambiguous "paths" in a Feynman diagrams, and it is shown how to define such paths in most of the quantum field theories that are in use today. We do not claim to have a more convenient calculational scheme here, but rather a scheme that allows for a better conceptual understanding of ultraviolet infinities. Dedicated to Paul Frampton's 60th birthday