Confluentes Mathematici

The Stückelberg–Petermann renormalization group is the group of finite renormalizations of the S-matrix in the framework of causal perturbation theory. The renormalization group in the sense of Wilson relies usually on a functional integral formalism, it describes the dependence of the theory on a UV-cutoff Λ; a widespread procedure is to construct the theory by solving Polchinski's flow equation for the effective potential.

To clarify the connection between these different approaches we proceed as follows: in the framework of causal perturbation theory we introduce an UV-cutoff Λ, define an effective potential V_Λ, prove a pertinent flow equation and compare with the corresponding terms in the functional integral formalism. The flow of V_Λ is a version of Wilson's renormalization group. The restriction of these operators to local interactions can be approximated by a subfamily of the Stückelberg–Petermann renormalization group.

A formalism for the renormalization procedure is given.

We define two coproducts for cycle-free oriented graphs, thus building up two commutative connected graded Hopf algebras, such that one is a comodule-coalgebra on the other, thus generalizing the result obtained in [2] for Hopf algebras of rooted trees.

Nous développons dans cet article les principaux arguments constructifs utilisés en théorie quantique des champs, en nous cantonnant aux théories bosoniques, pour lesquelles il n'existe pas de présentation générale récente. L'article s'adresse d'abord et avant tout à des mathématiciens ou physiciens mathématiciens connaissant les arguments de base de la théorie perturbative des champs, et souhaitant connaître un cadre général dans lequel ils peuvent être rendus rigoureux. Il fournit également un aperçu d'une série d'articles récents [50, 51] visant à donner une définition constructive des chemins rugueux et du calcul stochastique fractionnaire.

We develop in this article the principal constructive arguments used in quantum field theory, limiting us to bosonic theories, for which there does not exist any recent general presentation. The article is primarily written for mathematicians or mathematical physicists knowing the basic arguments of quantum field theory, and desiring to discover a general framework in which they can be made rigorous. It also provides a glimpse of a recent series of articles [50, 51] whose aim is to give a constructive definition of rough paths and fractionary stochastic calculus.