11 research outputs found
The well-behaved Catalan and Brownian averages and their applications to real resummation
The aim of this expository paper is to introduce the well-behaved uniformizing averages, which are useful in resummation theory. These averages associate three essential, but often antithetic, properties: respecting convolution; preserving realness; reproducing lateral growth. These new objects are serviceable in real resummation and we sketch two typical applications: the unitary iteration of unitary diffeomorphisms and the real normalization of real, local, analytic, vector fields
Renormalization: a quasi-shuffle approach
In recent years, the usual BPHZ algorithm for renormalization in perturbative
quantum field theory has been interpreted, after dimensional regularization, as
a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs,
with values in a Rota-Baxter algebra of amplitudes. We associate in this paper
to any such algebra a universal semi-group (different in nature from the
Connes-Marcolli "cosmical Galois group"). Its action on the physical amplitudes
associated to Feynman graphs produces the expected operations: Bogoliubov's
preparation map, extraction of divergences, renormalization. In this process a
key role is played by commutative and noncommutative quasi-shuffle bialgebras
whose universal properties are instrumental in encoding the renormalization
process
Combinatorial Hopf algebras from renormalization
In this paper we describe the right-sided combinatorial Hopf structure of
three Hopf algebras appearing in the context of renormalization in quantum
field theory: the non-commutative version of the Fa\`a di Bruno Hopf algebra,
the non-commutative version of the charge renormalization Hopf algebra on
planar binary trees for quantum electrodynamics, and the non-commutative
version of the Pinter renormalization Hopf algebra on any bosonic field. We
also describe two general ways to define the associative product in such Hopf
algebras, the first one by recursion, and the second one by grafting and
shuffling some decorated rooted trees.Comment: 16 page
The well-behaved Catalan and Brownian averages and their applications to real resummation
The aim of this expository paper is to introduce the well-behaved uniformizing averages, which are useful in resummation theory. These averages associate three essential, but often antithetic, properties: respecting convolution; preserving realness; reproducing lateral growth. These new objects are serviceable in real resummation and we sketch two typical applications: the unitary iteration of unitary diffeomorphisms and the real normalization of real, local, analytic, vector fields
FROM DYNAMICAL SYSTEMS TO RENORMALIZATION
Abstract. We study in this paper logarithmic derivatives associated to derivations on graded complete Lie algebra, as well as the existence of inverses. These logarithmic derivatives, when invertible, generalize the exp–log correspondence between a Lie algebra and its Lie group. Such correspondences occur naturally in the study of dynamical systems when dealing with the linearization of vector fields and the non–linearizability of a resonant vector fields corresponds to the non–invertibility of a logarithmic derivative and to the existence of normal forms. These concepts, stemming from the theory of dynamical systems, can be rephrased in the abstract setting of Lie algebra and the same difficulties as in perturbative quantum field theory (pQFT) arise here. Surprisingly, one can adopt the same ideas as in pQFT with fruitful results such as new constructions of normal forms with the help of the Birkhoff decomposition. The analogy goes even further (locality of counter terms, choice of a renormalization scheme) and shall lead to more interactions between dynamical systems and quantum field theory. hal-00794157, version 1- 25 Feb 2013 1. Introduction. Since the work of A. Connes and D.K Kreimer (see [3], [4]) in perturbative quantum field theory (pQFT), it has been possible to havea purely algebraicinterpretation of some renormalizationschemes