Gallavotti, G., Gentile, G., Mastropietro, V.
Field theory and KAM tori
(52K, TeX)
ABSTRACT. The parametric equations of KAM tori for a $l$ degrees of freedom quasi
integrable system, are shown to be one point Schwinger functions of a
suitable euclidean quantum field theory on the $l$ dimensional
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 $l^2$ new parameters forming a $l\times l$ matrix $\s_\e$
(identified as a family of renormalization constants). The matrix
$\s_\e$ is an analytic function of the coupling $\e$ at small $\e$: the
breakdown of the tori at large $\e$ is speculated to be related to the
crossing by $\s_\e$ of a ``critical" surface at a value $\e=\e_c$ where
the function $\s_\e$ is 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
$\e_c-\e$ dependence, is proposed.}