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

Abstract

This paper gives a complete selfcontained proof of our result announced in [6] 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 H which is commutative as an algebra. It is the dual Hopf algebra of the envelopping 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 corresponding Lie group G is the group of characters of H. We shall then show that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loopwhere 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 γ + of the Birkhoff decomposition of γ.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.

The residues of quantum field theory - numbers we should know

“…Even if one were able to resum the perturbative coefficients of a minimal subtraction scheme, the solution so obtained will solve the DSE only with rather meaningless boundary conditions which reflect the presence not only of instanton singularities but, worse, renormalon singularities in the initial asymptotic series. While it is fascinating to import quantum field theory methods into number theory, which suggest to resum perturbation theory amplitudes of a MS scheme making use of the Birkhoff decomposition of [13] combined with progress thanks to Ramis and others in resumation of asymptotic series, as beautifully suggested recently [14], the problem is unfortunately much harder still for a renormalizable quantum field theory. We indeed have almost no handle outside perturbation theory on such schemes, while on the other hand the NP solution of DSE with physical side-constraints like F (α, 1) = 1 is amazingly straightforward and resums perturbation theory naturally once one has recognized the role of the Hochschild closed 1-cocycles [4,11,5].…”

“…For any local QFT this defines a pre-Lie algebra of graph insertions [4]. For a renormalizable theory, the corresponding Lie algebra will be non-trivial for only a finite number of types of 1PI graphs (self-energies, vertex-corrections) corresponding to the superficially divergent graphs, while the superficially convergent ones provide a semi-direct product with a trivial abelian factor [13]. The combinatorial graded pre-Lie algebra so obtained provides not only a Lie-algebra L, but a commutative graded Hopf algebra H as the dual of its universal enveloping algebra U(L), which is not cocommutative if L was non-abelian.…”

Non-commutative Renormalization

“…Classical perturbation theory, like the inductive solution of any deterministic equation, is indexed by trees, whether QFT perturbation theory is indexed by more complicated "Feynman graphs", which contain the famous "loops" of anti-particles responsible for the ultraviolet divergences 5 . But the classical trees hidden inside QFT were revealed in many steps, starting with Zimmermann (which called them forests...) [18] through Gallavotti and many others, until Kreimer and Connes viewed them as generators of Hopf algebras [19,20,21]. Roughly speaking the trees were hidden because they are not just subgraphs of the Feynman graphs.…”

“…The results were given in the form of tables for the discrete divergent sums and combinatoric weights of all 20 We recall that in the ordinary commutative φ 4 4 field theory there is no one loop wave-function renormalization, hence the Landau ghost can be seen directly on the four point function renormalization at one loop. Here we simply reproduce the list of contributing Feynman graphs.…”

Relative locations of subwords in free operated semigroups and Motzkin words

“…Similarly, there are three relative locations of two subwords in a given word, a property that is essential in rewriting systems (critical pairs) and Gröbner bases [2]. Analogous classification of relative locations of combinatorial objects, such as Feynman graphs, plays an important role in combinatorics and physics, for example in the renormalization of quantum field theory [3,9,14,15].…”