An Efficient Method for the Solution of Schwinger--Dyson equations for propagators

Efficient computation methods are devised for the perturbative solution of Schwinger--Dyson equations for propagators. We show how a simple computation allows to obtain the dominant contribution in the sum of many parts of previous computations. This allows for an easy study of the asymptotic behavior of the perturbative series. In the cases of the four-dimensional supersymmetric Wess--Zumino model and the $\phi_6^3$ complex scalar field, the singularities of the Borel transform for both positive and negative values of the parameter are obtained and compared.Comment: 9 pages, no figures. Match of the published version, with the corrections in proo

Lectures on the three--dimensional non--commutative spheres

These are expanded notes for a short course given at the Universidad Nacional de La Plata. They aim at giving a self-contained account of the results of Alain Connes and Michel Dubois--Violette.Comment: 17 page

Approximate Differential Equations for Renormalization Group Functions in Models Free of Vertex Divergencies

I introduce an approximation scheme that allows to deduce differential equations for the renormalization group $\beta$-function from a Schwinger--Dyson equation for the propagator. This approximation is proven to give the dominant asymptotic behavior of the perturbative solution. In the supersymmetric Wess--Zumino model and a $\phi^3_6$ scalar model which do not have divergent vertex functions, this simple Schwinger--Dyson equation for the propagator captures the main quantum corrections.Comment: Clarification of the presentation of results. Equations and results unchanged. Match the published version. 12 page

Alien Calculus and non perturbative effects in Quantum Field Theory

In many domains of physics, methods are needed to deal with non-perturbative aspects. I want here to argue that a good approach is to work on the Borel transforms of the quantities of interest, the singularities of which give non-perturbative contributions. These singularities in many cases can be largely determined by using the alien calculus developed by Jean \'Ecalle. My main example will be the two point function of a massless theory given as a solution of a renormalization group equation.Comment: 4 pages, double-colum

The Standard Model of Leptons as a Purely Vectorial Theory

We propose a way to reconcile the Standard Model of leptons with a purely vectorial theory. The observed neutrino is predicted to be massless. The unobservability of its partner and the $V-A$ structure of the weak currents are given the same origin.Comment: 10 pages. Latex, 8 postscript figures included. We have corrected 2 (cancelling) sign misprints, and made explicit that we also recover the usual couplings of the U(1) gauge field B. The conclusions are unchanged. PAR-LPTHE 93/1

Integrable mappings and polynomial growth

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds of transformations on $q \times q$ matrices: the inversion of the $q \times q$ matrix and an (involutive) permutation of the entries of the matrix. We concentrate on the case where these permutations are elementary transpositions of two entries. In this case the birational transformations fall into six different classes. For each class we analyze the factorization properties of the iteration of these transformations. These factorization properties enable to define some canonical homogeneous polynomials associated with these factorization properties. Some mappings yield a polynomial growth of the complexity of the iterations. For three classes the successive iterates, for $q=4$, actually lie on elliptic curves. This analysis also provides examples of integrable mappings in arbitrary dimension, even infinite. Moreover, for two classes, the homogeneous polynomials are shown to satisfy non trivial non-linear recurrences. The relations between factorizations of the iterations, the existence of recurrences on one or several variables, as well as the integrability of the mappings are analyzed.Comment: 45 page

Conventionalisation? Organic farming bites back!

This is a summary of the discussion during the workshop 2.6 on conventionalisation of organic farming and how farmers or farmers' associations avoid conventionalisation. It also includes the abstracts of the papers that were presented during the workshop

The quantum Neumann model: asymptotic analysis

We use semi--classical and perturbation methods to establish the quantum theory of the Neumann model, and explain the features observed in previous numerical computations.Comment: 14 pages, 3 figure

The quantum Neumann model: refined semiclassical results

We extend the semiclassical study of the Neumann model down to the deep quantum regime. A detailed study of connection formulae at the turning points allows to get good matching with the exact results for the whole range of parameters.Comment: 10 pages, 5 figures Minor edit