What happens to the configuration manifold when one quantizes the Hamiltonian?

Question

A system in classical mechanics can be described by a configuration manifold $Q$ and a Lagrangian \begin{equation} L:TQ\rightarrow \mathbb{R} \end{equation} where $TQ$ is the tangent bundle or a Hamiltonian \begin{equation} L:{ T }^{ * }Q\rightarrow \mathbb{R} \end{equation} where ${ T }^{ * }Q$ is the cotangent bundle.

If one now quantizes the Hamiltonian by "replacing the variables with operators"
\begin{equation} H=T+V\quad \rightarrow \quad \hat { H } =\hat { T } +\hat { V } \end{equation} what exactly happens with the configuration manifold $Q$? Which dimension has $Q$ after the quantization and what is a point $a \in Q$ when we choose local coordinates.

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My guess is that a point $({ q }_{ 1 },...,{ q }_{ n },{ p }_{ 1 },...,{ p }_{ n })$ transforms in the following way: \begin{equation} { q }_{ i }\quad \rightarrow \quad \left| { x }_{ i } \right> =\delta({ x }_{ i }-x) \quad \quad \quad \quad { p }_{ i }\quad \rightarrow \quad { -i\hbar\partial }_{ { x }_{ i } } \end{equation}

But then the position/momentum-coordinates become distributions respectively differential operators and I don't think that such a mathematical set has the structure of a manifold.

Answer 1

There is no "configuration manifold" in quantum mechanics. "Quantization" does not map points in the classical configuration manifold to anything in the quantum theory. Quantization produces a quantum theory from a classical theory, it is not a map that turns classical states into quantum states.

"Quantization" is, in general, not a well-defined process, see e.g. this answer of mine. There are several heuristic quantization procedures of varying sophistication, such as canonical quantization and geometric quantization, but none of these preserve the classical setup of a configuration manifold - quantum states are rays in a Hibert space or points in a projective Hilbert space or quasi-probability distributions on (classical Hamiltonian) phase space, but there is never a map from classical to quantum states. In naive canonical quantization the quantum space of states is the space of wavefunctions on the classical configuration space, but that doesn't mean that individual points in the configuration space would be mapped to specific wave functions - that's simply not what quantization is supposed to do.