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.
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.
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.
