Renormalization: a quasi-shuffle approach

In recent years, the usual BPHZ algorithm for renormalization in perturbative quantum field theory has been interpreted, after dimensional regularization, as a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs, with values in a Rota-Baxter algebra of amplitudes. We associate...

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Bibliographic Details
Published in:arXiv.org 2018-07
Main Authors: Menous, Frédéric, Patras, Frédéric
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Amplitudes
Feynman diagrams
Field theory
Fields (mathematics)
Graphs
Group theory
Quantum field theory
Quantum theory
Regularization
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Summary:In recent years, the usual BPHZ algorithm for renormalization in perturbative quantum field theory has been interpreted, after dimensional regularization, as a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs, with values in a Rota-Baxter algebra of amplitudes. We associate in this paper to any such algebra a universal semi-group (different in nature from the Connes-Marcolli "cosmical Galois group"). Its action on the physical amplitudes associated to Feynman graphs produces the expected operations: Bogoliubov's preparation map, extraction of divergences, renormalization. In this process a key role is played by commutative and noncommutative quasi-shuffle bialgebras whose universal properties are instrumental in encoding the renormalization process.
ISSN:2331-8422