Locality and renormalization: Universal properties and integrals on trees

@article{Clavier2020LocalityAR,
  title={Locality and renormalization: Universal properties and integrals on trees},
  author={Pierre Clavier and Li Guo and Sylvie Paycha and Bin Zhang},
  journal={Journal of Mathematical Physics},
  year={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 of the universal properties in the locality framework of properly decorated rooted forests. These universal properties are then applied to derive the multivariate regularisation of integrals indexed by rooted forests. We study their renormalisation, along the lines of Kreimer's toy model for… 
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References

SHOWING 1-10 OF 26 REFERENCES
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
Chen’s iterated integral represents the operator product expansion
The recently discovered formalism underlying renormalization theory, the Hopf algebra of rooted trees, allows to generalize Chen’s lemma. In its generalized form it describes the change of a scale in
An algebraic formulation of the locality principle in renormalisation
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 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
Operated semigroups, Motzkin paths and rooted trees
Abstract Combinatorial objects such as rooted trees that carry a recursive structure have found important applications recently in both mathematics and physics. We put such structures in an algebraic
Renormalization and Mellin Transforms
We study renormalization in a kinetic scheme (realized by subtraction at fixed external parameters as implemented in the BPHZ and MOM schemes) using the Hopf algebraic framework, first summarizing
Hopf-algebraic renormalization of Kreimer's toy model
This masters thesis reviews the algebraic formulation of renormalization using Hopf algebras as pioneered by Dirk Kreimer and applies it to a toy model of quantum field theory given through iterated
Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras
The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In
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
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