Renormalization as a functor on bialgebras

TL;DR

This paper models the algebraic structure of renormalization in quantum field theory using bialgebras and explores its connections with known algebraic frameworks, providing a unifying mathematical perspective.

Contribution

It introduces a functorial bialgebra framework for renormalization, linking it to symmetric and Hopf algebras, and relates it to existing algebraic structures like the Faa di Bruno and Connes-Moscovici algebras.

Findings

The bialgebra S(S(B)^+) models renormalization for scalar fields.

Connections established between renormalization bialgebras and classical algebraic structures.

Standard renormalization results are recovered within this algebraic framework.

Abstract

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T(T(B)^+), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B)^+), the double symmetric algebra of B, into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of B are not renormalised, i.e. when Feynman diagrams containing one single vertex are not renormalised. When B is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S(S(B)^+) and the Faa di Bruno bialgebra of composition of series. The relation with the Connes-Moscovici Hopf algebra of diffeomorphisms is given. Finally, the bialgebra S(S(B)^+) is shown to give the same results as the standard renormalisation procedure for the scalar field.