Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem

TL;DR

This paper demonstrates that renormalization in quantum field theory can be understood as a special case of the Riemann-Hilbert problem, using Hopf algebra structures of Feynman graphs and Birkhoff decomposition.

Contribution

It establishes a mathematical framework linking renormalization to the Riemann-Hilbert problem via Hopf algebras and Lie groups, providing a self-contained proof of this connection.

Findings

Feynman graph combinatorics form a Hopf algebra structure.

Renormalization corresponds to the Birkhoff decomposition of a loop in a Lie group.

The group of characters is a semi-direct product involving diffeomorphism groups.

Abstract

This paper gives a complete selfcontained proof of our result announced in hep-th/9909126 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra $\Hc$ which is commutative as an algebra. It is the dual Hopf algebra of the envelopping algebra of a Lie algebra $\ud G$ whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group $G$ is the group of characters of $\Hc$. We shall then show that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop $$ \g (z) \in G \qquad z \in C $$ where $C$ is a small circle of complex dimensions around the integer dimension $D$ of space-time. Our main result is that the renormalized theory is just the evaluation at $z = D$ of the holomorphic part $\g_+$ of the Birkhoff decomposition of $\g$. We begin to analyse the group $G$ and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. The analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title.