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

*(English)*Zbl 1032.81026Summary: This paper gives a complete selfcontained proof of our result announced in [J. High Energy Phys. 1999, No. 9, Paper No. 24 (1999; Zbl 0957.81011)] 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 first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra \({\mathcal H}\) which is commutative as an algebra. It is the dual Hopf algebra of the enveloping algebra of a Lie algebra \(\underline G\) 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 \({\mathcal H}\). We shall then show that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop \(\gamma(z)\in G\), \(z\in C\), 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 \(\gamma_+\) of the Birkhoff decomposition of \(\gamma\). 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.

##### MSC:

81T18 | Feynman diagrams |

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

46N50 | Applications of functional analysis in quantum physics |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

22E70 | Applications of Lie groups to the sciences; explicit representations |