Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1171.81412
Ebrahimi-Fard, Kurusch; Manchon, Dominique; Patras, Frédéric
A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion. (English)
[J] J. Noncommut. Geom. 3, No. 2, 181-222 (2009). ISSN 1661-6952; ISSN 1661-6960/e

Summary: The Bogoliubov recursion is a particular procedure appearing in the process of renormalization in perturbative quantum field theory. It provides convergent expressions for otherwise divergent integrals. We develop here a theory of functional identities for noncommutative Rota-Baxter algebras which is shown to encode, among others, this process in the context of Connes-Kreimer's Hopf algebra of renormalization. Our results generalize the seminal Cartier-Rota theory of classical Spitzer-type identities for commutative Rota-Baxter algebras. In the classical, commutative, case these identities can be understood as deriving from the theory of symmetric functions. Here we show that an analogous property holds for noncommutative Rota-Baxter algebras. That is, we show that functional identities in the noncommutative setting can be derived from the theory of noncommutative symmetric functions. Lie idempotents, and particularly the Dynkin idempotent, play a crucial role in the process. Their action on the pro-unipotent groups such as those of perturbative renormalization is described in detail along the way.

MSC 2000:: *81T15 Perturbative methods of renormalization
05E05 Symmetric functions
16W30 Hopf algebras (assoc. rings and algebras)
17D25 Lie-admissible algebras
81T18 Feynman diagrams
81T17 Renormalization group methods
22E70 Appl. of Lie groups to physics

Keywords: Rota-Baxter relation; Spitzer identity; Bohnenblust-Spitzer identity; Magnus expansion; Dyson-Chen series; Hopf algebra of renormalization

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]