A Lie theoretic approach to renormalization

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.

typos corrected, references updated, accepted for publ. in Commun. Math. Phys

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Cited by in corpus (5)

• Recursion and growth estimates in renormalizable quantum field theory
• Feynman integrals and motives
• Feynman Diagrams and Lax Pair Equations
• A Zassenhaus-type algorithm solves the Bogoliubov recursion
• Dynkin operators and renormalization group actions in pQFT