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

TL;DR

This work provides a complete self-contained proof that renormalization in perturbative quantum field theory is an instance of the Riemann–Hilbert problem, realized through a Hopf‑algebraic encoding of Feynman graph combinatorics. It constructs a commutative Hopf algebra $\mathcal{H}$ generated by disjoint unions of 1PI graphs, dual to the enveloping algebra of a Lie algebra $\mathfrak{g}$ with group of characters $G$, and shows that a regularized loop $\gamma(z)\in G$ near $z=D$ admits a Birkhoff decomposition $\gamma(z)=\gamma_-(z)^{-1}\gamma_+(z)$ whose holomorphic part at $z=D$ yields the renormalized values. The group structure is established as a semidirect product $G=G_0\rtimes G_c$, with $G_0$ abelian and $G_c$ connected to diffeomorphism groups; the Milnor–Moore theorem identifies $\mathcal{H}$ with the dual of the enveloping algebra, and connections to the Hopf algebra of rooted trees are noted, setting the stage for the sequel to address the renormalization group and anomalous dimensions.

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.