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

This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special insta nce of a general mathematical procedure of extraction of finite values base d 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 alge bra H which is commutative as an algebra. It is the dual Hopf algebra of th e enveloping algebra of a Lie algebra 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 corr esponding Lie group G is the group of characters of H. We shall then show t hat, using dimensional regularization, the bare (unrenormalized) theory giv es rise to a loop gamma(z) epsilon G, z epsilon C, where C is a small circle of complex dimensions around the integer dimensio n 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 de composition 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.