Dynkin operators and renormalization group actions in pQFT

Abstract

Renormalization techniques in perturbative quantum field theory were known, from their inception, to have a strong combinatorial content emphasized, among others, by Zimmermann's celebrated forest formula. The present article reports on recent advances on the subject, featuring the role played by the Dynkin operators (actually their extension to the Hopf algebraic setting) at two crucial levels of renormalization, namely the Bogolioubov recursion and the renormalization group (RG) equations. For that purpose, an iterated integrals toy model is introduced to emphasize how the operators appear naturally in the setting of renormalization group analysis. The toy model, in spite of its simplicity, captures many key features of recent approaches to RG equations in pQFT, including the construction of a universal Galois group for quantum field theories.

References in corpus (8)

• Hopf Algebras, Renormalization and Noncommutative Geometry
• Renormalization in quantum field theory and the Riemann-Hilbert problem II: the $\beta$-function, diffeomorphisms and the renormalization group
• A Lie theoretic approach to renormalization
• New identities in dendriform algebras
• From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory
• Rota-Baxter Algebras in Renormalization of Perturbative Quantum Field Theory
• Quantum fields and motives
• The combinatorics of Bogoliubov's recursion in renormalization