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

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.

• 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. – Cosmas Zachos Feb 12 at 16:48
• 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 $|p\rangle$. – Cosmas Zachos Feb 12 at 17:34
• State space is what you just might be after; also see. – Cosmas Zachos Feb 12 at 22:38
• 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. – Cosmas Zachos Feb 13 at 15:32