# An algebraic formulation of the locality principle in renormalisation

@article{Clavier2017AnAF, title={An algebraic formulation of the locality principle in renormalisation}, author={Pierre Clavier and Li Guo and Sylvie Paycha and Bin Zhang}, journal={European Journal of Mathematics}, year={2017}, volume={5}, pages={356-394} }

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…

## 14 Citations

### Renormalisation and locality: branched zeta values

- Mathematics
- 2018

Multivariate renormalisation techniques are implemented in order to build, study and then renormalise at the poles, branched zeta functions associated with trees. For this purpose, we first prove…

### Renormalisation via locality morphisms

- MathematicsRevista Colombiana de Matemáticas
- 2019

This is a survey on renormalisation in algebraic locality setup highlighting the role that locality morphisms can play for renormalisation purposes. After describing the general framework to build…

### Renormalization of Feynman amplitudes on manifolds by spectral zeta regularization and blow-ups

- MathematicsJournal of the European Mathematical Society
- 2020

Our goal in this paper is to present a generalization of the spectral zeta regularization for general Feynman amplitudes. Our method uses complex powers of elliptic operators but involves several…

### Tensor products and the Milnor-Moore theorem in the locality setup

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- 2022

The present exploratory paper deals with tensor products in the locality framework developed in previous work, a natural setting for an algebraic formulation of the locality principle in quantum ﬁeld…

### Locality and renormalization: Universal properties and integrals on trees

- MathematicsJournal of Mathematical Physics
- 2020

The purpose of this paper is to build an algebraic framework suited to regularise branched structures emanating from rooted forests and which encodes the locality principle. This is achieved by means…

### Locality Galois groups of meromorphic germs in several variables

- Mathematics
- 2023

A bstract . Meromorphic germs in several variables with linear poles naturally arise in mathematics in various disguises. We investigate their rich structures under the prism of locality, including…

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- 2022

We prove a topological decomposition of the space of meromorphic germs at zero in several variables with prescribed linear poles as a sum of spaces of holomorphic and polar germs. Evaluating the…

### A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles

- MathematicsPacific Journal of Mathematics
- 2020

We use convex polyhedral cones to study a large class of multivariate meromorphic germs, namely those with linear poles, which naturally arise in various contexts in mathematics and physics. We…

### Quivers and path semigroups characterized by locality conditions

- Mathematics
- 2022

. The notion of locality semigroups was recently introduced with motivation from locality in convex geometry and quantum ﬁeld theory. We show that there is a natural correspondence between locality…

### Locality and causality in perturbative AQFT

- Mathematics
- 2019

In this paper we introduce a notion of a group with causality, which is a natural generalization of a locality group, introduced by P. Clavier, L. Guo, S. Paycha, and B. Zhang. We also propose a…

## References

SHOWING 1-10 OF 29 REFERENCES

### Perturbative Algebraic Quantum Field Theory and the Renormalization Groups

- Physics
- 2009

A new formalism for the perturbative construction of algebraic quantum field theory is developed. The formalism allows the treatment of low dimensional theories and of non-polynomial interactions. We…

### Renormalisation and locality: branched zeta values

- Mathematics
- 2018

Multivariate renormalisation techniques are implemented in order to build, study and then renormalise at the poles, branched zeta functions associated with trees. For this purpose, we first prove…

### Renormalization and the Euler-Maclaurin formula on cones

- Mathematics
- 2013

The generalized algebraic approach of Connes and Kreimer to perturbative quantum field theory is applied to the study of exponential sums on lattice points in convex rational polyhedral cones. For…

### Renormalization in Quantum Field Theory and the Riemann–Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem

- Mathematics
- 2000

Abstract:This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special instance of a general mathematical procedure…

### Renormalization of Feynman amplitudes on manifolds by spectral zeta regularization and blow-ups

- MathematicsJournal of the European Mathematical Society
- 2020

Our goal in this paper is to present a generalization of the spectral zeta regularization for general Feynman amplitudes. Our method uses complex powers of elliptic operators but involves several…

### Hopf algebras, from basics to applications to renormalization

- Mathematics
- 2001

These notes are an extended version of a series of lectures given at Bogota from 2nd to 6th december 2002. They aim to present a self-contained introduction to the Hopf-algebraic techniques which…

### Noncommutative version of Borcherds' approach to quantum field theory

- Mathematics
- 2015

Richard Borcherds proposed an elegant geometric version of renormalized perturbative quantum field theory in curved spacetimes, where Lagrangians are sections of a Hopf algebra bundle over a smooth…

### The Role of locality in perturbation theory

- Mathematics
- 1973

It is shown how an inductive construction of the renormalized perturbation series of quantum field theory automatically yields, at each order, finite terms satisfying the requirements of locality.…

### Exponential Renormalization II: Bogoliubov's R-operation and momentum subtraction schemes

- Physics
- 2011

This article aims at advancing the recently introduced exponential method for renormalisation in perturbative quantum field theory. It is shown that this new procedure provides a meaningful recursive…

### On the Locality Ideal In the Algebra of Test Functions for Quantum Fields

- Mathematics
- 1984

Some basic properties of the locality ideal in Borchers's tensor algebra are established. It is shown that the ideal is a prime ideal and that the corresponding quotient algebra has a faithful…