On Kreimer's Hopf algebra structure of Feynman graphs

Abstract

We reinvestigate Kreimer's Hopf algebra structure of perturbative quantum field theories with a special emphasis on overlapping divergences. Kreimer first disentangles overlapping divergences into a linear combination of disjoint and nested ones and then tackles that linear combination by the Hopf algebra operations. We present a formulation where the Hopf algebra operations are directly defined on any type of divergence. We explain the precise relation to Kreimer's Hopf algebra and obtain thereby a characterization of their primitive elements.

Hopf Algebras, Renormalization and Noncommutative Geometry

“…One might suggest to enlargen the Hopf algebra H R of rooted trees to another Hopf algebra, H O say, so that H O directly contains elements which correspond to graphs with overlapping divergences [8].…”

“…Concretely, in the first section we show in Theorem (8) that the algebraic rules of the Hopf algebra H T are the expression of the group law of composition of diffeomorphisms of R in terms of the coordinates δ n given by the Taylor expansion of − log(ψ ′ (x)) at x = 0. In particular this shows that the antipode in H T is, modulo a change of variables, the same as the operation of inversion of a formal power series for the composition law.…”

Graphons and renormalization of large Feynman diagrams

“…For overlapping sub-divergencies, there are some challenges but the corresponding tree representation could be a linear combination of decorated rooted trees. We refer the reader to [1,12,23,24] for further details in this issue.…”

Combinatorics of (perturbative) quantum field theory

“…overlapping divergences in terms of decorated rooted trees determines the appropriate set of primitive elements of the Hopf algebra, which can for example be systematically achieved by making use of Dyson-Schwinger equations [2,15], see also [21]. Then, the set of vertices {1, 2, 3} belongs to a vertex subgraph, as does the set {2, 3, 4}.…”