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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.


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.

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    $\begingroup$ I believe you are confusing two theories with the most popular description pictures utilized for them. QM has a perfectly fine description in phase space , as well, and, conversely, classical mechanics in Hilbert space, as well; and contrasting the structure of the two descriptions does not fully contrast the two theories. $\endgroup$ – Cosmas Zachos Feb 12 at 16:48
  • $\begingroup$ By the way, the nitpicking definition in Dirac's book is $q_i\to |q_i\rangle= \delta (\hat x -q_i) \rangle$, where $\rangle$ is Dirac's "standard ket", the translationally invariant vacuum, i.e. the $p\to 0$ limit of $|p\rangle$. $\endgroup$ – Cosmas Zachos Feb 12 at 17:34
  • $\begingroup$ State space is what you just might be after; also see. $\endgroup$ – Cosmas Zachos Feb 12 at 22:38
  • $\begingroup$ If you are fearless of mathematical notation, your best bet is Foundations of Quantum Theory: From Classical Concepts to Operator Algebras by Klaas Landsman, Springer 2017. $\endgroup$ – Cosmas Zachos Feb 13 at 15:32
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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.

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