Renormalization in Quantum Field Theory and the Riemann-Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem

1999/12/13 by Alain Connes, Dirk Kreimer · 1 voice · 487 citations
Mathematics · Physics and Astronomy · #Abelian group #Advanced Topics in Algebra #Feynman diagram #Homotopy and Cohomology in Algebraic Topology #Hopf algebra #Lie algebra #Noncommutative and Quantum Gravity Theories #Quantum field theory #Quantum group #Representation theory of Hopf algebras #Topological quantum field theory #Universal enveloping algebra #hep-th #math-ph #math.MP #math.QA

paper · pdf · doi:10.1007/s002200050779

published in Communications in Mathematical Physics 210(1), 249-273 (Springer Science+Business Media) · 35p, 18 eps-figures, to appear in Commun.Math.Phys

arxiv created 1999/12/13 · arxiv published 1999/12/13 · openalex publication_date 2000/03/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

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) ∈ G z ∈ 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.

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Discussions

• I am diving into the potential minefield that is learning regularization and renormalization, and I am currently lost on dimensional regularization. I understand the intuitive idea using dimension as [stackexchange, 3 points]