What happened to phase space?

Question

I need a simple explanation, as I am still a BSc student. I learned about Hamiltonian formalism and also I know much about quantum mechanics.

As far as I have heard/read, we can quantize the classical system by changing the Poisson brackets to commutator brackets and observable by equivalent operators. And yes it seems true as the Poisson bracket algebra for angular momentum is very identical to commutator algebra in QM.

But these operators are defined on Hilbert spaces, while the Hamiltonian formalism of classical physics uses $2n$ dimensional phase space. Where did the phase space go? Or do the phase space becomes a Hilbert space? But I have also heard about phase space formalism of quantum mechanics. If I am right then the quantum mechanics that was introduced at the very beginning is not exactly the one obtained out of quantization. Am I right?

Answer 1

Quantization is a huge topic, so we will only sketch the basic idea.

Let the classical phase space be a symplectic manifold $(M,\omega)$. As OP already knows, Poisson brackets should, roughly speaking, be replaced by commutators, cf. e.g. this Phys.SE post. Also we know from the HUP that we can only measure commuting observables simultaneously. A basic idea is to consider a Lagrangian submanifold $Q\subseteq M$, and pick the Hilbert space as ${\cal H}=L^2(Q)$.

For more information, see e.g. geometric quantization and deformation quantization.