Autori: G. Gallavotti, G. Gentile, V. Mastropietro
Titolo: Field theory and KAM tori
Archived: in mp_arc #95-151; (first version)
in chao-dyn@xyz.lanl.gov #9503006; (first version)
in http://chimera.roma1.infn.it (last version)
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 pertubation series and to an
expansion in terms of $l^2$ new parameters forming a
$l\times l$ matrix $\sigma_\varepsilon$ (identified as a
family of renormalization constants). The matrix
$\sigma_\varepsilon$ is an analytic function of the
coupling $\varepsilon$ at small $\varepsilon$: the
breakdown of the tori at large $\varepsilon$ is speculated
to be related to the crossing by $\sigma_\varepsilon$ of a
``critical" surface at a value $\varepsilon=\varepsilon_c$
where the function $\sigma_\varepsilon$ 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
$\varepsilon_c-\varepsilon$ dependence, is proposed.
Fisica, Universita' di Roma La Sapienza,
P.le Moro 2, 00185, Roma, Italia.
e-mail giovanni@ipparco.roma1.infn.it
tel. 6-49914370, fax 6-4957697