MPEJ Volume 1, No.5, 13pp
Received: March 24, 1995, Revised: September 11, 1995, Accepted: September 29, 1995
G. Gallavotti, G. Gentile, V. Mastropietro
Field theory and KAM tori
ABSTRACT: The parametric equations of KAM tori for an 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.
The KAM theorem is equivalent to an ultraviolet stability theorem.
A renormalization group treatment of the field theory leads to a resummation
of the formal perturbation series and to an expansion in terms of
$l^2$ new parameters forming a $l\times l$ matrix $\sigma_\epsilon$
(identified as a family of renormalization constants).
The matrix $\sigma_\epsilon$ is an analytic function of the coupling $\epsilon$
at small $\epsilon$: the breakdown of the tori at large $\epsilon$
is speculated to be related to the crossing by $\sigma_\epsilon$ of a
"critical" surface at a value $\epsilon=\epsilon_c$ where the function
$\sigma_epsilon$ 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 $\epsilon_c-\epsilon$ dependence,
is proposed.