Renormalization in Quantum Field Theory and the Riemann-Hilbert Problem I. The211 Hopf Algebra Structure of Graphs and the Main Theorem

摘要

This paper gives a complete self-contained proof of the authors result announced211u001ein showing that renormalization in quantum field theory is a special instance of 211u001ea general mathematical procedure of extraction of finite values based on the 211u001eRiemann-Hilbert problem. the authors shall first show that for any quantum field 211u001etheory, the combinatorics of Feynman graphs gives rise to a Hopf algebra H which 211u001eis a cummutative as an algebra. It is the dual Hopf algebra of the enveloping 211u001ealgebra of Lie algera G whose basis is labelled by the one particle irreducible 211u001eFeynman graphs. The Lie bracket of two such graphs is computed from insertions of 211u001eone graphs in the other and vice versa. The corresponding Lie group G is the 211u001egroup of characters of H.