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arXiv · 1703.07304

Renormalization: a quasi-shuffle approach

Frédéric Menous·Frédéric Patras
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Abstract

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.

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Frédéric Menous, Frédéric Patras. 2017-03-21. Renormalization: a quasi-shuffle approach. https://arxiv.org/abs/1703.07304

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