Abstract

We present some clues to the study of the renormalization group, at graduate level, as well as some bibliographical pointers to classical resources. Just the kind of things one had liked to hear when starting to study the subject. This was going to be a notice on “recent advances on renormalization group theory”, from which advanced students can get bibliography to please their teachers. But I happened to visit Jacques Gabay’s printing house, and I decided to take a wider view. At least, wider than usual treatises on Quantum Field Theory. Gabay’s mission is to keep in print old mathematics texts from the late XIXth and early XXth, and his work helps to keep the perspective. Fact is, all generations of physicists since Euler times have been used to live with divergences. Students are supposed to become exposed to the subject gradually, but this gradation varies strongly across schools and faculties, and the balance keeps more in the room of Cauchy than in Borel quarters. There is even a darker side, about if everything which is legal in Mathematics should be legal in Physics, but this question keeps usually in the philosophical level. Still, one should point that classical mechanics, the science of newtonian limits, is mathematically legal but physically ruled out! The usual scenario for divergences is: we have a differential equation. We look for a solution from power series expansion. Most times, the solution is known to exist, say from Picard’s fixed point method, say from other convergence theorem. But we are forced to pick up a power series expansion on the “wrong ” parameter, so that the convergence radius is zero. Or (change x −> 1/z if necesary) we could to know only a asymptotic series around infinity. Poincare treatises are the first ones showing all of this. Note, still, that we are here in a purely classical matter. We can do perturbation theory with a small coupling constant in a perturbation potential, and then expand the solution as a power series of the coupling constant. The series diverges, and then we have only an asymptotic series. Borel transform, or other resummation techniques, can be invoked to get a better expansion. Sometimes even a convergent series is known, for instance for the three-body problem