Combinatorial Hopf algebraic description of the multiscale renormalization in quantum field theory
Abstract
We define in this paper several Hopf algebras describing the combinatorics of the so-called multi-scale renormalization in quantum field theory. After a brief recall of the main mathematical features of multi-scale renormalization, we define assigned graphs, that are graphs with appropriate decorations for the multi-scale framework. We then define Hopf algebras on these assigned graphs and on the Gallavotti-Nicol\`o trees, particular class of trees encoding the supplementary informations of the assigned graphs. Several morphisms between these combinatorial Hopf algebras and the Connes-Kreimer algebra are given. Finally, scale dependent couplings are analyzed via this combinatorial algebraic setting.
Keywords
Cite
@article{arxiv.1211.4429,
title = {Combinatorial Hopf algebraic description of the multiscale renormalization in quantum field theory},
author = {Thomas Krajewski and Vincent Rivasseau and Adrian Tanasa},
journal= {arXiv preprint arXiv:1211.4429},
year = {2014}
}
Comments
26 pages, 3 figures; the presentation of the results has been reorganized. Several details of various proofs are given and some references have been added