Renormalization in Quantum Field Theory: An Improved Rigorous Method

TL;DR

This paper introduces a simplified, rigorous renormalization method in quantum field theory that unifies and clarifies existing approaches, reducing computational complexity and enhancing conceptual understanding.

Contribution

It proposes a modified BPHZ procedure that simplifies perturbation theory, aligns with Epstein-Glaser regularization, and clarifies the relationship with dimensional regularization.

Findings

Reduction in complexity of multi-loop diagram calculations

Equivalence of the modified BPHZ method with Epstein-Glaser approach

Enhanced understanding of dimensional regularization parameters

Abstract

The perturbative construction of the S-matrix in the causal spacetime approach of Epstein and Glaser may be interpreted as a method of regularization for divergent Feynman diagrams. The results of any method of regularization must be equivalent to those obtained from the Epstein-Glaser (EG) construction, within the freedom left by the latter. In particular, the conceptually well-defined approach of Bogoliubov, Parasuk, Hepp, and Zimmermann (BPHZ), though conceptually different from EG, meets this requirement. Based on this equivalence we propose a modified BPHZ procedure which provides a significant simplification of the techniques of perturbation theory, and which applies equally well to standard quantum field theory and to chiral theories. We illustrate the proposed method by a number of examples of various orders in perturbation theory. At the level of multi-loop diagrams we find that subdiagrams as classified by Zimmermann's forest formula in BPHZ may be restricted to subdiagrams in the sense of Epstein-Glaser, thus entailing an important reduction of actual computations. Furthermore, the relationship of our approach to the method of dimensional regularization is particularly transparent, without having to invoke analytic continuation to unphysical spacetime dimension, and sheds new light on certain parameters within dimensional regularization.