An algebraic formulation of the locality principle in renormalisation

TL;DR

This paper develops an algebraic framework to preserve the principle of locality during renormalisation in quantum field theory, using Hopf algebras and Birkhoff factorisation, with applications to lattice point enumeration.

Contribution

It introduces a locality variant of algebraic Birkhoff factorisation, providing a new algebraic formulation to maintain locality in renormalisation procedures.

Findings

Established a locality-preserving algebraic Birkhoff factorisation.

Applied the framework to renormalise exponential generating functions on cones.

Demonstrated the approach's effectiveness in a multivariate regularisation context.

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 cones, we renormalise the exponential generating function which sums over the lattice points in convex cones. For a suitable multivariate regularisation, renormalisation from the algebraic Birkhoff factorisation amounts to composition by a projection onto holomorphic multivariate functions.