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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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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