An algebraic formulation of the locality principle in renormalisation

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.

Locality and renormalization: Universal properties and integrals on trees

“…Thus a locality relation ⊤ on a set X induces a relation on the power set P(X), which we denote by the same symbol ⊤ Let (X, ⊤) be a locality set. Then the relation locality induced on P(X) is symmetric, so that (P(X), ⊤) is a locality set [CGPZ1]. Furthermore, P(X) ⊤ = P(X ⊤ ), as can be checked directly.…”

“…The purpose of this paper is to apply to the study of concrete models an algebraic formulation of the locality principle developed in [CGPZ1] in the context of the (co)algebraic approach to perturbative quantum field theory initiated by Connes and Kreimer [CK, K]. The concrete model we focus on is Kreimer's rooted forests toy model, which serves as an experimental model to study renormalisation and related renormalisation groups.…”

Locality and causality in perturbative algebraic quantum field theory

“…In this article we show how the different meanings of the term "locality" appearing in the quantum field theory (QFT) context can be described by properties of a common mathematical structure that we call group with causality. This is a generalization of the ideas proposed in [CGPZ18a] and applied to renormalization in Euclidean QFT in [CGPZ18b].…”

“…In [CGPZ18a] the authors introduce an abstract notion of locality captured by the definition of the locality set:…”

“…However, in our context, we will need to equip a locality set (G, ⊥ ) with a full group structure (without restricting to ⊥ ), compatible with ⊥ . To make a distinction to [CGPZ18a] we will call this a group with locality.…”