@MISC{Rivasseau99constructiverenormalization, author = {Vincent Rivasseau}, title = {Constructive Renormalization Theory}, year = {1999} }

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Abstract

These notes are the second part of a common course on Renormalization Theory given with Professor P. da Veiga 1. I emphasize here the rigorous non-perturbative or constructive aspects of the theory. The usual formalism for the renormalization group in field theory or statistical mechanics is reviewed, together with its limits. The constructive formalism is introduced step by step. Taylor forest formulas allow to perform easily the cluster and Mayer expansions which are needed for a single step of the renormalization group in the case of Bosonic theories. The iteration of this single step leads to further difficulties whose solution is briefly sketched. The second part of the course is devoted to Fermionic models. These models are easier to treat on the constructive level, so they are very well suited to beginners in constructive theory. It is shown how the Taylor forest formulas allow to reorganize perturbation theory nicely in order to construct the Gross-Neveu2 model without any need for cluster or Mayer expansions. Finally applications of this technique to condensed matter and renormalization group around Fermi surface are briefly reviewed. 1 The Renormalization Group: an overview 1.1 Functional Integration and its problems In this section we restrict ourselves to the bosonic φ4 field theory in d Euclidean space time dimensions. The model, introduced in P. Da Veiga’s lectures, is defined by the (formal) measure dµC(φ)e −S(φ) ∫, S(φ) = λ φ 4 (x)d d x (1.1) where dµC(φ) represents the Gaussian measure for the free field. Gaussian measures are characterized by their covariance, or propagator, which for a massive theory is, in Fourier space: C(p) = (p 2 + m 2) −1, (1.2) and S is the (bare) action. In dimension d = 2,3 the model is superrenormalizable, and its rigorous construction was the first major achievement of constructive theory [GJ]. In