The Hopf algebra approach to Feynman diagram calculations

2005/10/31 by Kurusch Ebrahimi‐Fard, Kurusch Ebrahimi-Fard, Dirk Kreimer
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Diagram #Feynman diagram #Feynman integral #Hopf algebra #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Physics #Pure mathematics #Statistics #hep-ph #hep-th #math.QA

paper · pdf · doi:10.1088/0305-4470/38/50/r01

published as J.Phys.A38:R385-R406,2005 · 30 pages, review, final version

openalex publication_date 2005/11/30 · arxiv created 2005/12/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

The Hopf algebra structure underlying Feynman diagrams which governs the process of renormalization in perturbative quantum field theory is reviewed. Recent progress is briefly summarized with an emphasis on further directions of research.

Citations

• Matrix Representation of Renormalization in Perturbative Quantum Field Theory
• On the Structure of Hopf Algebras
• Über die Multiplikation der Kausalfunktionen in der Quantentheorie der Felder
• Combinatorics of renormalization as matrix calculus
• The Electromagnetic Shift of Energy Levels
• Spitzer's identity and the algebraic Birkhoff decomposition in pQFT
• Renormalization in Quantum Field Theory and the Riemann-Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem
• TheSMatrix in Quantum Electrodynamics
• The Radiation Theories of Tomonaga, Schwinger, and Feynman
• Coalgebras of words and phrases
• Quantum Groups

Cited by

• Anatomy of a gauge theory