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An algebraic formulation of the locality principle in renormalisation

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Abstract

We study the mathematical structure underlying the concept of locality which lies at the heart of classical and quantum field theory, and develop a machinery used to preserve locality during the renormalisation procedure. Viewing renormalisation in the framework of Connes and Kreimer as the algebraic Birkhoff factorisation of characters on a Hopf algebra with values in a Rota–Baxter algebra, we build locality variants of these algebraic structures, leading to a locality variant of the algebraic Birkhoff factorisation. This provides an algebraic formulation of the conservation of locality while renormalising. As an application in the context of the Euler–Maclaurin formula on lattice cones, we renormalise the exponential generating function which sums over the lattice points in a lattice cone. As a consequence, for a suitable multivariate regularisation, renormalisation from the algebraic Birkhoff factorisation amounts to composition by a projection onto holomorphic multivariate germs.

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Notes

  1. In the Epstein–Glaser formalism, the method of analytic regularisation à la Speer yields Feynman amplitudes obeying amongst other axioms, a factorisation condition reflecting the locality principle [12, Theorem 6.2].

  2. As a special case of partial algebras [18], the terminology “partial semigroup” is used for a set equipped with a partial associative product defined only for certain pairs of elements in the set. The condition for a locality semigroup is more restrictive than that of a partial semigroup in that the former requires that the pairs for which the partial product is defined stem from a symmetric relation and that the partial product should be compatible with the locality relation in the sense of (8).

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Acknowledgements

The authors thank Christian Brouder for very enlightening comments on a preliminary version of this paper. They also thank the referee for helpful suggestions.

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Authors and Affiliations

  1. Institute of Mathematics, University of Potsdam, 14476, Potsdam, Germany

    Pierre Clavier

  2. Department of Mathematics, Jiangxi Normal University, Nanchang, 330022, Jiangxi, China

    Li Guo

  3. Department of Mathematics and Computer Science, Rutgers University, Newark, NJ, 07102, USA

    Li Guo

  4. Institute of Mathematics, University of Potsdam, 14469, Potsdam, Germany

    Sylvie Paycha

  5. Laboratoire de Mathématiques Blaise Pascal, Université Clermont-Auvergne, 63001, Clermont-Ferrand, France

    Sylvie Paycha

  6. School of Mathematics, Yangtze Center of Mathematics, Sichuan University, Chengdu, 610064, China

    Bin Zhang

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  1. Pierre Clavier
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  2. Li Guo
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  3. Sylvie Paycha
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  4. Bin Zhang
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Correspondence to Sylvie Paycha.

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Sylvie Paycha, on leave from Université Clermont-Auvergne.

The authors acknowledge supports from the Natural Science Foundation of China (Grant Nos. 11521061 and 11771190) and the German Research Foundation (DFG Grant PA 1686/6-1). They are grateful to the hospitalities of Sichuan University and University of Potsdam where parts of the work were completed.

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Clavier, P., Guo, L., Paycha, S. et al. An algebraic formulation of the locality principle in renormalisation. European Journal of Mathematics 5, 356–394 (2019). https://doi.org/10.1007/s40879-018-0255-8

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  • DOI: https://doi.org/10.1007/s40879-018-0255-8

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