The KAM theorem and renormalization group

Simone, Emiliano De; Kupiainen, Antti

doi:10.1017/s0143385708080450

Cited by 6 publications

(2 citation statements)

References 13 publications

(9 reference statements)

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“…Ultimately, the cancellation mechanism for the resonances is deeply related to that assuring the formal solubility of the equations of motions, which in turn is due to a symmetry property, as already shown by Poincaré [19]. Subsequently, stressing further the analogy with quantum field theory, Bricmont et al showed that the cancellations can be interpreted as a consequence of suitable Ward identities of the corresponding field theory [4] (see also [7]). In the isochronous case, in terms of Cartesian coordinates the cancellation mechanism works in a completely different way with respect to action-angle coordinates.…”

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confidence: 87%

KAM Theory in Configuration Space and Cancellations in the Lindstedt Series

Corsi

Gentile

Procesi

2010

Commun. Math. Phys.

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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 contributions of the same perturbation order. In turn, this is due to symmetries in the problem. Such symmetries are easily visualised in action-angle coordinates, where the KAM theorem is usually formulated by exploiting the analogy between Lindstedt series and perturbation expansions in quantum field theory and, in particular, the possibility of expressing the solutions in terms of tree graphs, which are the analogue of Feynman diagrams. If the unperturbed system is isochronous, Moser's modifying terms theorem ensures that an analytic quasi-periodic solution with the same Diophantine frequency vector as the unperturbed Hamiltonian exists for the system obtained by adding a suitable constant (counterterm) to the vector field. Also in this case, one can follow the alternative approach of studying the perturbation expansion for both the solution and the counterterm, and again convergence of the two series is obtained as a consequence of deep cancellations between contributions of the same order. In this paper, we revisit Moser's theorem, by studying the perturbation expansion one obtains by working in Cartesian coordinates. We investigate the symmetries giving rise to the cancellations which makes possible the convergence of the series. We find that the cancellation mechanism works in a completely different way in Cartesian coordinates, and the interpretation of the underlying symmetries in terms of tree graphs is much more subtle than in the case of action-angle coordinates. © 2010 Springer-Verlag

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confidence: 87%

KAM Theory in Configuration Space and Cancellations in the Lindstedt Series

Corsi

Gentile

Procesi

2010

Commun. Math. Phys.

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“…Ward identities play a crucial role in quantum field theory. The analogy between KAM theory and quantum field theory has been widely stressed in the literature [11,2,6]; in particular the cancellations which assure the convergence of the perturbation series for maximal KAM tori are deeply related to a Ward identity, as shown in [2], which can be seen as a remarkable identity between classes of graphs. In the case studied in this paper, we have a similar situation, made fiddlier by the fact that we have to deal with nonconvergent series to be resummed, and it is well known that identities which are trivial on a formal level can turn out to be difficult to prove rigorously [19].…”

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confidence: 99%

Response solutions for arbitrary quasi-periodic perturbations with Bryuno frequency vector

Corsi,

Gentile

2010

Preprint

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We study the problem of existence of response solutions for a real-analytic onedimensional system, consisting of a rotator subjected to a small quasi-periodic forcing. We prove that at least one response solution always exists, without any assumption on the forcing besides smallness and analyticity. This strengthens the results available in the literature, where generic non-degeneracy conditions are assumed. The proof is based on a diagrammatic formalism and relies on renormalisation group techniques, which exploit the formal analogy with problems of quantum field theory; a crucial role is played by remarkable identities between classes of diagrams.

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