Let $E$ be a set, and let $\mathcal{F}$ be a set of maps $E \rightarrow \mathbb{R}$ that "carries all the information on $E$", that is, the map $x \in E \mapsto (f(x))_{f \in \mathcal{F}}$ is injective.
Let us call "classical things" the elements of $E$. Let us call "a representation of quantum things" a Hilbert space $\mathcal{H}$, and a map $\widehat{ \cdot } : \mathcal{F} \rightarrow Obs(\mathcal{H})$, where we denote by $Obs(\mathcal{H})$ the set of self-adjoint operators on $\mathcal{H}$, such that the map $\hat{ \cdot }$ satisfies some conditions.
Example 1: Let $E$ be $\mathbb{R}^3$, and let $\mathcal{F}$ be the set of maps $w \mapsto \langle v,w\rangle$ for $w \in \mathbb{R}^3$. Then $\mathcal{F}$ carries all the information on $\mathbb{R}^3$ since a vector is uniquely defined by the data consisting of the scalar products with all vectors.
Then every vector $\phi$ in $\mathcal{H}$ can be seen as a state of a "quantum vector", to be thought as having coordinate $a$ in the $v$-direction if, when measuring the $\widehat{( w\mapsto \langle v,w\rangle)}$ observable, one gets $a$.
As a particular case, spin $\frac{1}{2}$ can be considered as a kind of a "quantum vector with all coordinates that whenever measured, give a result in $\{\pm \frac{1}{2}\}$".
Example 2: the phase space of a classical $1$-dimensional particle is $\mathbb{R}\times \mathbb{R}$, and one can consider the set of maps carrying all the information to be $\mathcal{F} := \{q,p\} = \{(x,y) \mapsto x, (x,y) \mapsto y\}$.
To this classical representation corresponds the quantum representation where $\hat{q} := Q$ is the position operator, and $\hat{p} := P$ is the momentum operator.
My question is the following: I think this definition probably already exists, maybe in categorical terms, maybe in the particular case where $E$ is a manifold with enough structure. I would like to know the name of this construction; that is, the name of the general abstract framework where the two examples fit.
Best Answer
One mathematical concept OP might be looking for is quantization, which applies to any process turning a classical mechanical system into a quantum one. The emphasis on mechanical is important, as the symplectic structure of phase space is often a necessary data for the quantization process.
For specific methods of quantization for finite dimensional phase spaces (ie regular quantum mechanics and not field theory), which are well understood mathematically, you can look up deformation quantization and geometric quantization.
However, those quantization processes start with a phase space (ie a symplectic manifold), so cannot be applied to example 1. In particular, I don't think finite dimensional Hilbert spaces can be obtained this way.