Rota-Baxter Algebras in Renormalization of Perturbative Quantum Field Theory

Abstract

Recently, the theory of renormalization in perturbative quantum field theory underwent some exciting new developments. Kreimer discovered an organization of Feynman graphs into combinatorial Hopf algebras. The process of renormalization is captured by a factorization theorem for regularized Hopf algebra characters. In this context the notion of Rota-Baxter algebras enters the scene. We review several aspects of Rota-Baxter algebras as they appear in other sectors also relevant to perturbative renormalization, for instance multiple-zeta-values and matrix differential equations.

improved version, based on a talk given at the workshop Renormalization and Universality in Mathematical Physics held at the Fields Institute for Research in Mathematical Sciences, October 18-22, 2005

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Cited by in corpus (22)

• Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras
• Rota-Baxter algebras and new combinatorial identities
• Entropy algebras and Birkhoff factorization
• Algebraic Birkhoff decomposition and its applications
• The Hopf algebra of (q)multiple polylogarithms with non-positive arguments
• Generalized shuffles related to Nijenhuis and TD-algebras
• BiHom-pre-alternative algebras and BiHom-alternative quadri-algebras
• Rota-Baxter operators on BiHom-associative algebras and related structures
• Representations of Rota-Baxter algebras and regular singular decompositions
• Renormalization of gauge fields using Hopf algebras
• Representations and Modules of Rota-Baxter Algebras
• Renormalisation of q-regularised multiple zeta values
• Double shuffle relations and renormalization of multiple zeta values
• Structure theorems of mixable shuffle algebras and free commutative Rota-Baxter algebras
• Renormalization, isogenies and rational symmetries of differential equations
• An Asymptotic Expansion for the Number of 2-Connected Chord Diagrams
• Motivic renormalization and singularities
• Universal Hopf Algebra of Renormalization and Hopf Algebras of Rooted Trees
• On Differential Rota-Baxter Algebras
• Dynkin operators and renormalization group actions in pQFT
• Riemann-Hilbert Problem and Quantum Field Theory: Integrable Renormalization, Dyson-Schwinger Equations
• Rota-Baxter Hom-Lie-admissible algebras