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… 

Renormalisation and locality: branched zeta values

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

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

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

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 Galois groups of meromorphic germs in several variables

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

A topological splitting of the space of meromorphic germs in several variables and continuous evaluators

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

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

. 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

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

From Orthocomplementations to Locality

After some background on lattices, the locality framework introduced in earlier work by the authors is extended to cover posets and lattices. We then extend the correspondence between Euclidean

Several locality semigroups, path semigroups and partial semigroups

Locality semigroups were proposed recently as one of the basic locality algebraic structures, which are studied in mathematics and physics. Path semigroups and partial semigroups were also developed

References

SHOWING 1-10 OF 29 REFERENCES

Perturbative Algebraic Quantum Field Theory and the Renormalization Groups

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

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

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

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

Hopf algebras, from basics to applications to renormalization

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

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

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

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

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

Algebraic Birkhoff Factorization and the Euler-Maclaurin Formula on Cones

We equip the space of lattice cones with a coproduct which makes it a connected cograded colagebra. The exponential sum and exponential integral on lattice cones can be viewed as linear maps on this