Field theory and KAM tori

TL;DR

This paper links KAM tori in quasi-integrable systems to Euclidean quantum field theory, using renormalization group techniques to analyze stability and potential breakdown of invariant tori at large coupling.

Contribution

It introduces a novel quantum field theoretic framework for studying KAM tori and applies renormalization group methods to analyze their stability and singularities.

Findings

KAM tori correspond to Schwinger functions of a quantum field theory.

Renormalization group analysis resums perturbation series and identifies parameters controlling stability.

Potential universality of singularities in the breakdown of tori at large coupling.

Abstract

The parametric equations of KAM tori for a quasi integrable system, are shown to be one point Schwinger functions of a suitable euclidean quantum field theory on the torus. KAM theorem is equivalent to a ultraviolet stability theorem. A renormalization group treatment of the field theory leads to a resummation of the formal pertubation series and to an expansion in terms of new parameters (identified as a family of renormalization constants). The new parameters are analytic in the coupling (at small coupling): the breakdown of the tori at large coupling is speculated to be related to the crossing of a "critical" surface at a value where the renormalization constants are still finite. A mechanism for the possible universality of the singularities of parametric equations for the invariant tori, in their parameter dependence as well as in the coupling dependence, is proposed.}