I am trying to get my head around the geometric formulation of Quantum Mechanics as a projective Hilbert space (see Ashtekar, <http://arxiv.org/abs/gr-qc/9706069>). So one identifies all the rays $\mathbb{C} \cdot \phi$ with the vector $\phi$ itself.


As the space of square-integrable functions $L^2(\Sigma, \mu)$ is the standard example of a Hilbert space I was wondering whatever there is clear characterization of its projective space. E.g. is there a good "visualization", an explicit notion of the Kähler-structure, etc?


A second question concerning this topic: Let $M$ be a Kähler manifold. Which further properties/conditions on $M$ are required so that $M$ can be realized as a projective Hilbert space? Is the construction than unique?


Thanks,
Tobi


P.s.: Which are the standard references regarding this topic?



Projective Hilbert space: L^2
Look at section 5 of this paper by Helmick and Helminck:
<http://eprints.eemcs.utwente.nl/3487/01/1667.pdf>


