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An algebraic formulation of the locality principle in renormalisation

  • 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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Metadaten
Author details:Pierre J. ClavierGND, Li Guo, Sylvie PaychaORCiDGND, Bin Zhang
DOI:https://doi.org/10.1007/s40879-018-0255-8
ISSN:2199-675X
ISSN:2199-6768
Title of parent work (English):European Journal of Mathematics
Publisher:Springer
Place of publishing:Cham
Publication type:Article
Language:English
Date of first publication:2019/06/05
Publication year:2019
Release date:2021/02/03
Tag:Algebraic Birkhoff factorisation; Hopf algebra; Lattice cones; Locality; Multivariate meromorphic functions; Partial algebra; Renormalisation; Rota-Baxter algebra
Volume:5
Issue:2
Number of pages:39
First page:356
Last Page:394
Funding institution:Natural Science Foundation of ChinaNational Natural Science Foundation of China [11521061, 11771190]; German Research Foundation (DFG)German Research Foundation (DFG) [PA 1686/6-1]
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik
DDC classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Peer review:Referiert
Publishing method:Open Access / Green Open-Access
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