Operadic construction of the renormalization group

TL;DR

This paper introduces an operadic framework to construct groups related to symmetric operads, including the renormalization group in quantum field theory, providing a new algebraic perspective on renormalization processes.

Contribution

It develops a functorial method to associate groups to symmetric operads and interprets the renormalization group operadically, linking algebraic structures to quantum field theory.

Findings

Constructs a group from any symmetric operad functorially.

Associates a symmetric operad to decorated graphs, modeling the renormalization group.

Provides an operadic interpretation of the Connes-Kreimer Hopf algebra.

Abstract

First, we give a functorial construction of a group associated to a symmetric operad. Applied to the endomorphism operad it gives the group of formal diffeomorphisms. Second, we associate a symmetric operad to any family of decorated graphs stable by contraction. In the case of Quantum Field Theory models it gives the renormalization group. As an example we get an operadic interpretation of the group of "diffeographisms" attached to the Connes-Kreimer Hopf algebra.