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Renormalization: A Quasi-shuffle Approach

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Book cover Computation and Combinatorics in Dynamics, Stochastics and Control (Abelsymposium 2016)

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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 semigroup (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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Acknowledgements

We acknowledge support from the CARMA grant ANR-12-BS01-0017, “Combinatoire Algébrique, Résurgence, Moules et Applications”and the CNRS GDR “Renormalisation”. We thank warmly K. Ebrahimi-Fard, from whom we learned some years ago already the meaningfulness of Rota–Baxter algebras and their links with quasi–shuffle algebras.

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Authors and Affiliations

  1. Laboratoire de Mathmatiques d’Orsay, University of Paris-Sud, CNRS, Universit Paris-Saclay, Orsay, France

    Frédéric Menous

  2. Laboratoire J.A. Dieudonné, Université de la Côte d’Azur, CNRS, UMR 7531, Nice Cedex 2, France

    Frédéric Patras

Authors
  1. Frédéric Menous
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  2. Frédéric Patras
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Correspondence to Frédéric Menous .

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Editors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Elena Celledoni

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Giulia Di Nunno

  3. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  4. Department of Mathematics, University of Bergen, Bergen, Norway

    Hans Zanna Munthe-Kaas

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Menous, F., Patras, F. (2018). Renormalization: A Quasi-shuffle Approach. In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-01593-0_21

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