Dynkin operators, renormalization and the geometric $\beta$ function

TL;DR

This paper explores the relationship between renormalization in quantum field theory and a generalized Dynkin operator, introducing a geometric $eta$ function that captures the theory's scale dependence through Lie group dynamics.

Contribution

It establishes a novel connection between renormalization, logarithmic derivations, and Dynkin operators, providing a geometric perspective on the $eta$ function in QFT.

Findings

The geometric $eta$ function is defined by a complete vector field on a Lie group.

A generalized Dynkin operator is introduced via logarithmic derivations.

The work links renormalization group flow to Lie group geometry.

Abstract

In this paper, I show a close connection between renormalization and a generalization of the Dynkin operator in terms of logarithmic derivations. The geometric $\beta$ function, which describes the dependence of a Quantum Field Theory on an energy scale defines is defined by a complete vector field on a Lie group $G$ defined by a QFT. It also defines a generalized Dynkin operator.