KAM Theorem and Quantum Field Theory
نویسندگان
چکیده
We give a new proof of the KAM theorem for analytic Hamiltonians. The proof is inspired by a quantum field theory formulation of the problem and is based on a renormalization group argument treating the small denominators inductively scale by scale. The crucial cancellations of resonances are shown to follow from the Ward identities expressing the translation invariance of the corresponding field theory.
منابع مشابه
KAM theory in configuration space and cancellations in the Lindstedt series
The KAM theorem for analytic quasi-integrable anisochronous Hamiltonian systems yields that the perturbation expansion (Lindstedt series) for any quasi-periodic solution with Diophantine frequency vector converges. If one studies the Lindstedt series by following a perturbation theory approach, one finds that convergence is ultimately related to the presence of cancellations between contributio...
متن کاملJ un 1 99 5 Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics A review with some applications
This paper consists in a unified exposition of methods and techniques of the renormalizationgroup approach to quantum field theory applied to classical mechanics, and in a review ofresults: (1) a proof of the KAM theorem, by studing the perturbative expansion (Lindstedtseries) for the formal solution of the equations of motion; (2) a proof of a conjecture byGallavotti about ...
متن کاملField Theory and Kam Tori
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 eld theory on the l dimensional torus. KAM theorem is equivalent to a ultraviolet stability theorem. A renormalization group treatment of the eld theory leads to a resummation of the formal pertubation series and to an expansion i...
متن کاملQuantum analogue of the Kolmogorov–Arnold–Moser transition in field-induced barrier penetration in a quartic potential
Quantum signatures of the Kolmogorov–Arnold– Moser (KAM) transition from the regular to chaotic classical dynamics of a double-well oscillator in the presence of an external monochromatic field of different amplitudes are analysed in terms of the corresponding Bohmian trajectories. It is observed that the classical chaos generally enhances the quantum fluctuations, while the quantum nonclassica...
متن کاملA KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations
We establish an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field, which could be applied to a large class of Hamiltonian PDEs containing the derivative ∂x in the perturbation. Especially, in this range of application lie a class of derivative nonlinear Schrödinger equations with Dirichlet boundary conditions and perturbed Benjamin-Ono equation with peri...
متن کامل