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.