# What happened to phase space?

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?

• en.wikipedia.org/wiki/Geometric_quantization this might be helpful Jan 16, 2021 at 6:41
• You may stay in phase space and do your quantum mechanics just fine. Equivalently, you may map Phase space to Hilbert space invertibly, and get the very same answers: expectation values. Jan 16, 2021 at 11:14
• Arguably linked. Jan 16, 2021 at 22:20

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 $${\cal H}=L^2(Q)$$.