A Lie Theoretic Approach to Renormalization

Abstract

Abstract. Motivated by recent work of Connes and Marcolli, based on the Connes-Kreimer approach to renormalization, we augment the latter by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras, and on properties of Hopf algebras encapsulated in the notion of associated descent algebras. Besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures, the new techniques give rise to an algebraic approach to the Galois theory of renormalization. In particular, they do not depend on the geometry underlying the case of dimensional regularization and the Riemann-Hilbert correspondence. This is illustrated with a discussion of the BPHZ renormalization scheme.

A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion

“…As an application we present a closed formula for the Bogoliubov recursion in the context of ConnesKreimer's Hopf algebra approach to perturbative renormalization. This last finding is complementary to the main result of the recent article [20], where two of the present authors together with J. M. Gracia-Bondía proved that the mathematical properties of locality and the so-called beta-function in pQFT could be derived from the properties of the Dynkin operator. Our new findings reinforce the idea that the Dynkin operator and its algebraic properties have to be considered as one of the building blocks of the modern mathematical theory of renormalization.…”

“…This is a result dual to Theorem 4.1 in [20], which established the same formula for characters and infinitesimal characters of graded connected commutative Hopf algebras when the Lie series was the Dynkin series. The assertion on pro-unipotency and pro-nilpotency follows, for instance, from the classical equivalence between group schemes and commutative Hopf algebras.…”

“…is a right inverse (and in fact also a left inverse) to l. The particular formula for the Dynkin idempotent follows from [25] or from Lemma 2.1 in [20].…”

“…To start with, let us first recall from [43], [25], [41] and [20] some definitions and properties relating the classical Dynkin operator to the fine structure theory of Hopf algebras. Some details are needed, since the results gathered in the literature are not necessarily stated in a form convenient for our purposes.…”

“…Here we will restrict our attention to the cocommutative case, we refer to [20] for applications of the Dynkin operator formalism to the commutative but noncocommutative Hopf algebras appearing in the Connes-Kreimer Hopf algebraic theory of renormalization in pQFT, see [10], [11], [12], [22]. …”