Connection between the renormalization groups of Stückelberg-Petermann and Wilson

Abstract

The Stueckelberg-Petermann renormalization group is the group of finite renormalizations of the S-matrix in the framework of causal perturbation theory. The renormalization group in the sense of Wilson relies usually on a functional integral formalism, it describes the dependence of the theory on a UV-cutoff $\Lambda$; a widespread procedure is to construct the theory by solving Polchinski's flow equation for the effective potential. To clarify the connection between these different approaches we proceed as follows: in the framework of causal perturbation theory we introduce an UV-cutoff $\Lambda$, define an effective potential $V_{\Lambda }$, prove a pertinent flow equation and compare with the corresponding terms in the functional integral formalism. The flow of $V_{\Lambda }$ is a version of Wilson's renormalization group. The restriction of these operators to local interactions can be approximated by a subfamily of the Stueckelberg-Petermann renormalization group.

20 pages; a few formulations improved. To appear in a volume on "Algebraic, geometric and probabilistic aspects of renormalization" in Confluentes Mathematici

References in corpus (3)

• On the Renormalization Group in Curved Spacetime
• Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity
• Removal of violations of the Master Ward Identity in perturbative QFT

Cited by in corpus (1)

• Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization