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In Jose & Saletan's "Classical Dynamics", they show the global structure of Hamiltonian mechanics: you then have a $Q$ manifold (configuration space), and the phase space structure is precisely the cotangent bundle $T^{*}Q$. Even more, if we take the phase space functions $f(T^{*}Q)$ together with the Poisson brackets $\{\cdot ,\cdot \}$, we build the lie algebra for all classical mechanics.

In Berndt's "Introduction to Symplectic Geometry" in Section 5.5 on page 129, the phase space is said to be replaced by the projective space $\mathbb{P}(\mathcal{H})$, where $\mathcal{H}$ is the hilbert space. This sounds strange since $\mathcal{H}$ isn't a manifold.

My question is: what is quantum phase space geometrical structure?

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  • $\begingroup$ When $\mathcal{H}$ is finite-dimensional, the space is just $\mathbb{CP}^{d-1}$ where $d$ is the dimension of $\mathcal{H}$. For example, when $d=2$ this gives the famous Bloch sphere. $\endgroup$
    – Meng Cheng
    Commented Jun 28, 2022 at 19:00

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Berndt is talking about geometric quantization. In the simplest model the quantum Hilbert space ${\cal H}=L^2(Q)$ is the space of square integrable wavefunctions on the classical configuration manifold $Q$.

(More generally, if one instead starts from the notion of classical phase space, i.e. a symplectic manifold $(M,\omega)$, one needs to introduce a polarization, and the quantum Hilbert space ${\cal H}\subset L^2(M)$.)

The projective Hilbert space $\mathbb{P}({\cal H})$ is the corresponding ray space.

References:

  1. N.M.J. Woodhouse, Geometric Quantization, 1992.
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Contra your assertion, the Hilbert space is a smooth manifold. In QM, due to the Born rule, the quantum phase space should be a projective Hilbert space. This is not usually done in standard run of the mill textbooks because the ordinary Hilbert space is easier to work with than the projective version. The work around there is then to focus on unit length vectors, the Hilbert sphere.

Classical mechanics, that is Newtonian mechanics, is traditionally done in a Euclidean space. However, there is a generally covariant version of classical mechanics and this is what geometric classical mechanics is more or less about. It's doing Newtonian mechanics on any smooth manifold - your configuration manifold $Q$. It takes two forms, the Lagrangian and the Hamiltonian. The first is based on the tangent bundle, $TQ$ and the second on the cotangent bundle, $T^*Q$. They are related in good cases by the fibre derivative, aka the Legendre transformation.

In QM, we have an algebra of observables and the space of states. And here, in this classical world, we have their classical analogues. The classical algebra of observables and the classical space of states.

There are quite a few geometric models of quantisation and the quantum state space. One to take note of is deformation quantisation which focuses on the deforming the classical algebra of observables into a quantum algebra of observables - this is the Heisenberg picture.

Now the classical algebra of observables on $T^*Q$ turns it into a Poisson manifold. Whilst Kontsevich '97 gave a proof that any Poisson manifold of finite dimension has a deformation quantisation this does not generalise to field theory. However, Fedosov '94 identified a deformation quantisation of any finite dimensional symplectic manifold - now called Fedosov Quantisation - and a variant of this does generalise to the infinite dimensions symplectic manifolds of field theory where according to Collini '16 it is equivalent to causal perturbation quantisation which is the UV divergence free form of standard perturbative QFT.

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