# Infinite dimensional symplectic geometry

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.

• Marsden and Weinstein's original paper already contains infinite-dimensional examples (pp. 125-126, and lemma p. 123). – Francois Ziegler Aug 4 '17 at 9:02
• Haller, Stefan; Vizman, Cornelia Non-linear Grassmannians as coadjoint orbits. Math. Ann. 329 (2004), no. 4, 771–785. arxiv version – Jarek Kędra Aug 4 '17 at 9:05

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.