Renormalization in quantum field theory: an improved rigorous method

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, Parasiuk, 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 confirm that subdiagrams as classified by Zimmermann's forest formula in BPHZ can be restricted to subdiagrams in the sense of Epstein–Glaser, thus entailing an important reduction of actual computations. The relationship of our approach to the method of dimensional regularization (and renormalization) is particularly transparent, without having to invoke analytic continuation to unphysical spacetime dimension. It sheds new light on the role of some parameters that appear within the dimensional regularization, and thus establishes a direct link of this traditional method to the BPHZ scheme.