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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?

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  • $\begingroup$ I don't understand the question: For a Kahler manifold to be a projective hilbert it has to be a projective space $\mathbb{C} P^n.$ For pretty much all you can say on the subject, see en.wikipedia.org/wiki/Projective_Hilbert_space $\endgroup$
    – Igor Rivin
    Commented Dec 23, 2011 at 20:24
  • $\begingroup$ Yes for the finite-dimensional case this is easy. But is there something as $\mathbb{C}P^\infty$? $\endgroup$
    – Tobias Diez
    Commented Dec 23, 2011 at 22:53

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Look at section 5 of this paper by Helmick and Helminck: http://eprints.eemcs.utwente.nl/3487/01/1667.pdf

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Feynman, in his lecture notes, argues convincingly that you will understand the guts of Quantum Mechanics [QM] as best you can by looking at the two-slit experiment whose Hilbert space is two-dimensional. Or you could look at the Stern-Gerlach experiment, where it is three-dimensional. I recommend you stick to these cases, drop $L^2$, and see carefully what all the concepts boil down to there in terms of the Kahler geometry of projective space. Feynman does this, in physics language, in his 1957 paper with Vernon titled ‘Geometrical Representation of the Schrodinger Equation for Solving Maser Problems’: they solve the needed Schrodinger equation by drawing circles on the two-sphere = $CP^1$.

The ‘Berry Phase’ is a post-Feynman idea that is essentially the curvature of the canonical connection for the canonical line bundle over $CP^n$. For a dictionary from standard QM to Kahler geometry and connections you could look at ‘Heisenberg and isoholonomic inequalites’.

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