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Integrable systems, symmetries, and quantization

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Abstract

These notes are an expanded version of a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularities and expose recent research in connection with semi-toric systems. The quantum and semiclassical counterparts are also presented, in the viewpoint of the inverse question: from the quantum mechanical spectrum, can one recover the classical system?

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Notes

  1. So much so that Weinstein once wrote ‘Everything is a Lagrangian submanifold!’ (cf. [86]).

  2. The only element that acts as the identity at all points of M is the identity of the group.

  3. A priori there is no guarantee that \(F_1(M_1)\) is an open subset of \({\mathbb {R}}^n\). Throughout these notes, if \(A \subset {\mathbb {R}}^n\) is any subset, a map \(H: A \rightarrow {\mathbb {R}}^m\) is said to be smooth if for every \(x \in A\), there exist an open neighborhood W and a smooth map \(H_W : W \rightarrow {\mathbb {R}}^m\) extending \(H|_{A \cap W}\). A diffeomorphism is therefore a smooth map whose inverse is also smooth in the above sense.

  4. This means that all the standard axioms for actions are verified whenever the compositions are possible.

  5. It is possible to introduce a local model for a compact orbit of any Williamson type (cf. [54]), but this is beyond the scope of these notes.

  6. The discussion holds mutatis mutandis in the more general context of \(\left( G,X\right) \)-structures in the sense of [71].

  7. Throughout, we fix an identification \(\mathrm {Lie}\left( \mathbb {T}^2 \right) \cong {\mathbb {R}}^2\).

  8. This corresponds to the fact that the volume invariants of [62, 63] are necessarily positive.

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  1. Instítuto de Matemática, Departamento de Matemática Aplicada, Rua Professor Marcos Waldemar de Freitas Reis, s/n, Campus do Gragoatá, Niterói, CEP 24210-201, Brazil

    Daniele Sepe

  2. IRMAR (UMR 6625), Université de Rennes 1, Campus de Beaulieu, 35042, Rennes Cedex, France

    San Vũ Ngọc

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Sepe, D., Vũ Ngọc, S. Integrable systems, symmetries, and quantization. Lett Math Phys 108, 499–571 (2018). https://doi.org/10.1007/s11005-017-1018-z

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