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.
Author details: | Pierre J. ClavierGND, Li Guo, Sylvie PaychaORCiDGND, Bin Zhang |
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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 |