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Field theory and KAM tori. (English) Zbl 0849.58038

Summary: The parametric equations of KAM tori for an \(\ell\) degrees of freedom quasi integrable system, are shown to be one point Schwinger functions of a suitable Euclidean quantum field theory on the 1 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 an \(l \times l\) 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.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
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