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Could anyone comment on possible references concerning infinite dimensionsal symplectic manifolds?. I am mainly concerned with hilbert spaces, so i am not interested in the convenient analysis approach to the topic given, for instance, by Michor and collaborators,

More spectifically, I am interested in an infinite-dimensional analogue of the Marsden-Weinstein reduction and applications. I am specially concerned with its possible uses in quantum mechanics.

Thank you in advance.

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    $\begingroup$ Marsden and Weinstein's original paper already contains infinite-dimensional examples (pp. 125-126, and lemma p. 123). $\endgroup$ – Francois Ziegler Aug 4 '17 at 9:02
  • $\begingroup$ Haller, Stefan; Vizman, Cornelia Non-linear Grassmannians as coadjoint orbits. Math. Ann. 329 (2004), no. 4, 771–785. arxiv version $\endgroup$ – Jarek Kędra Aug 4 '17 at 9:05
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Aspects of symplectic topology hold in the infinite dimensional setting. A notable difference is that there are different notions for what a symplectic form should be: there is the notion of a strong symplectic form where $\omega$ is required to induce an isomorphism $\mathbb H\rightarrow \mathbb H^*$, or a weak one, where this map is only injective. In finite dimensions this cannot occur as an injective map from a finite dimensional space to a space with the same dimension is necessarily bijective.

For strong symplectic forms some things are known. The following paper "A non-squeezing theorem for convex symplectic images of the Hilbert ball" of Abbondandolo and Majer contains a non-squeezing result in infinite dimensions and starts with a nice discussion of its history.

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A standard way of going from a symplectic manifold to quantum mechanics is Fedosov quantization. Now if you want to do this in infinite dimension, that means you are interested in quantum field theory. A good place to start is the thesis by Giovanni Collini (a former student of Stefan Hollands) which develops such a Fedosov quantization in the QFT context.

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    $\begingroup$ One should note that the infinite-dimensional spaces in Collini's thesis definitely go beyond Hilbert spaces, requiring Fréchet or locally convex topologies. $\endgroup$ – Igor Khavkine Aug 4 '17 at 17:40
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    $\begingroup$ @IgorKhavkine: You are right, but I think you can't really do anything serious in infinite dimensional analysis/geometry with just Hilbert or Banach spaces. $\endgroup$ – Abdelmalek Abdesselam Aug 6 '17 at 18:03

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