# On the Hopf algebra structure of perturbative quantum field theories

@article{Kreimer1997OnTH, title={On the Hopf algebra structure of perturbative quantum field theories}, author={Dirk Kreimer}, journal={Advances in Theoretical and Mathematical Physics}, year={1997}, volume={2}, pages={303-334} }

We show that the process of renormalization encapsules a Hopf algebra structure in a natural manner. This sheds light on the recently proposed connection between knots and renormalization theory.

## 416 Citations

Introduction to Hopf algebras in renormalization and noncommutative geometry

- Mathematics
- 1999

We review the appearance of Hopf algebras in the renormalization of quantum field theories and in the study of diffeomorphisms of the frame bundle important for index computations in noncommutative…

Hopf Algebras, Renormalization and Noncommutative Geometry

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- 1998

Abstract:We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of tranverse index theory for foliations.

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We briefly review the Hopf algebra structure arising in the re normalization of quantum field theories. We construct the Hopf algebra explicitly for a simple toy model and show how renormalization is…

Counterterms in the context of the universal Hopf algebra of renormalization

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- 2014

The manuscript discovers a new interpretation of counterterms of renormalizable Quantum Field Theories in terms of formal expansions of decorated rooted trees.

Hopf algebra approach to Feynman diagram calculations

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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…

Hopf algebra approach to Feynman diagram calculations

- Physics
- 2008

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…

Physics and Number Theory

- Mathematics, Physics
- 2006

In this review we discuss the relevance of the Hochschild cohomology of renormalization Hopf algebras for local quantum field theories and their equations of motion.

From Local Perturbation Theory to Hopf and Lie Algebras of Feynman Graphs

- Mathematics
- 2001

We review the algebraic structures imposed on the renormalization procedure in terms of Hopf and Lie algebras of Feynman graphs, and exhibit the connection to diffeomorphisms of physical observables.

Factorization in Quantum Field Theory: An Exercise in Hopf Algebras and Local Singularities

- Mathematics, Physics
- 2007

I discuss the role of Hochschild cohomology in Quantum Field Theory with particular emphasis on Dyson--Schwinger equations.

Renormalization of gauge fields using Hopf algebras

- Mathematics
- 2008

We describe the Hopf algebra structure of Feynman graphs for non-Abelian gauge theories and prove compatibility of the so-called Slavnov-Taylor identities with the coproduct. When these identities…

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