An algebraic formulation of the locality principle in renormalisation

Pierre Clavier, Li Guo, Sylvie Paycha, Bin Zhang

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)356-394
Number of pages39
JournalEuropean Journal of Mathematics
Volume5
Issue number2
DOIs
StatePublished - Jun 15 2019

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • Algebraic Birkhoff factorisation
  • Hopf algebra
  • Lattice cones
  • Locality
  • Multivariate meromorphic functions
  • Partial algebra
  • Renormalisation
  • Rota–Baxter algebra

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