What exactly is the relationship between the algebraic formulation of Quantum Mechanics and the geometric formulation of Classical Mechanics?

Question

Okay so if we consider a particular physical system, the classical description of the system starts by first introducing a symplectic manifold, which is the cotangent bundle of a configuration manifold. There is the canonical Poisson Bracket which forms an algebra of functions on the manifold (i.e. the observables) and the time evolution of an observable $f$ is just $\{H,f\}$ where $H$ is the Hamiltonian.

If we want to describe the same physical system from a Quantum Mechanical perspective, we introduce a Hilbert space whose equivalence class of normalised elements correspond to the physical states. The algebra of observables on the Hilbert space form a non-commutative algebra with the time evolution described by Schrodinger's Equation. So I understand the motivation for introducing both formalisms but how do I relate these constructions?

How is the Hilbert Space related to the phase space manifold? The process of quantization replaces the poisson brackets with commutators but I don't understand what this implies for the underlying structures. Most people say that it's a hopeless task to go from Classical Mechanics to Quantum Mechanics because you've lost a lot of information but it's not at all clear to me whether this is actually the case. If anything it feels like Quantum Mechanics arises from discarding away some information from classical description, thereby quantizing it. And even if this is not the case and you do in fact lose information, the process of quantization proves that there is a systematic way of recovering this information because Quantization works independent of the specific system under consideration.

Now I know that every manifold can be understood through its algebra of functions but the algebra of functions on a manifold is always defined pointwise so its naturally commutative. In Quantum Mechanics we start with non-commutative algebra of functions so I'm assuming you can't think of it as an algebra of functions over some geometric manifold. If they're not related in any way, then how can anyone convincingly say that Quantum Mechanics reduces to Classical Mechanics on the macroscopic scale.

Answer 1

There is a quite satisfactory picture of quantization and classical limit, at least for quantum mechanical systems (for quantum fields, especially fermionic, it is less clear).

First of all, it is not possible to quantize every real-valued classical symbol on the phase-space to a self-adjoint element affiliated to a given C*-algebra (as it would be suitable to prove that the quantum theory provides no additional information than the classical theory). Therefore, it is not possible to interpret all the quantum observables and dynamical maps straightforwardly as the quantization of classical phase-space counterparts.

On the other hand, the classical limit of quantum states and observables is pretty well understood for a huge class of interesting quantum (bosonic) systems, and it is possible to recover essentially all reasonable classical states and symbols as the limit of quantum states and quantum observables respectively (it is even possible to give a suitable topology on which the limit is a true mathematical limit). It is also possible to link the unitary quantum dynamics to possibly non-linear classical Hamiltonian flows in the phase space.

There is a huge mathematical literature on the subject, check e.g. classical textbooks in pseudodifferential calculus and semiclassical analysis (and references thereof contained).