Hopf Algebras, Renormalization and Noncommutative Geometry

Abstract

We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of tranverse index theory for foliations.

On Kreimer's Hopf algebra structure of Feynman graphs

“…Our presentation is inspired by an idea of Dirk Kreimer. A detailed discussion of overlapping divergences based on set-theoretical reasoning was given in [4], some remarks can also be found in the appendix of [2].…”

“…The forest formula guiding the renormalization of Feynman graphs with subdivergences is reproduced by a certain interplay of product, coproduct, antipode and counit of that Hopf algebra. Meanwhile Connes and Kreimer elaborated a deep structural link [2] between that Hopf algebra of renormalization and the Hopf algebra emerging in the computation of the local index formula for transverse hypoelliptic operators [3]. This indicates that renormalization provides a mathematical calculus that can be thought of as a refinement of diffeomorphisms.…”

“…As explained by Kreimer in [1,2,4] and in private discussions, overlapping divergences require a special treatment. Overlapping divergences must first be disentangled into a linear combination of terms containing disjoint or nested divergences exclusively.…”

“…We show that one of the antipode axioms recovers the forest formula in its full beauty for any Feynman graph. Following an idea by Dirk Kreimer [2,4] we describe the precise relation between his and our formulations of the Hopf algebra of renormalization. In this way we gain an explicit construction of those primitive elements of Kreimer's Hopf algebra which are different from the graphically primitive elements.…”

“…The box has n B bosonic and n F fermionic external legs. The superficial degree of divergence d[X] of the iPW X is bounded by the power counting theorem (2)…”