The answers to [this question](https://mathoverflow.net/questions/6200/what-is-to-quantize-something) do a good job of exploring, at a heuristic level, what "quantization" should be. From my perspective, quantization involves replacing a (commutative) Poisson algebra by some related noncommutative associative algebra. Poisson algebras arise naturally especially as algebras of functions in geometry and physics. Noncommutative algebras arise naturally as algebras of operators on linear spaces.


I've often heard it said that "quantization is not a functor". I'm wondering what a precise statement of this is.


For example, I could imagine statements of the following form.


1. There is no functor from the category of Poisson manifolds (and Poisson maps?) to the (opposite of the) category of associative algebras satisfying some nice property.
2. There is no functor from the category of symplectic manifolds (and Poisson maps?) to the (opposite of the) category of associative algebras satisfying some nice property.
3. Recall that for any smooth manifold, its cotangent bundle is naturally symplectic. There is no functor from the category of smooth manifolds to the category of associative algebras that quantizes the cotangent bundle.
4. Recall that the dual to the universal enveloping algebra of a Lie bialgebra is naturally Poisson Hopf. There is no functor from the category of Lie bialgebras to the category of Hopf algebras satisfying some nice property.


Actually, 4. is false. Indeed, Etingof and Khazdan constructed a functor from bialgebras to Hopf algebras satisfying a host of properties, and Enriquez classified all the ones with nice properties. Note that Kontsevich does give a quantization of any Poisson manifold, but perhaps his isn't functorial?



What does "quantization is not a functor" really mean?
Here is one precise statement of how quantization is not a functor:


5) There is no functor from the classical category $\mathcal C$ of Poisson manifolds and Poisson maps to the quantum category $\mathcal Q$ of Hilbert spaces and unitary operators that is consistent with the cotangent bundle/$\frac12$-density relation (explained below).


The result is due to Van Hove, in "Sur le probleme des relations entre les transformations unitaires de la mecanique quantique et les transformations canoniques de la mecaniques classique." This is an old paper and I can't find a link for it, but the reference I found it in is Weinstein's "Lectures on Symplectic Manifolds."


By "cotangent bundle/$\frac12$-density relation" I mean the following: if $\mathcal M$ is the category of smooth manifolds and diffeomorphisms, we have a cotangent functor $\mathcal M \to \mathcal C$. This assigns to each manifold its cotangent bundle with the canonical symplectic structure, and to each diffeomorphism the induced symplectomorphism of cotangent bundles. 


We also have a natural functor $\mathcal M \to \mathcal Q$. For any smooth manifold $X$ consider the bundle of complex $\frac12$-densities on $X$. (What is the bundle of complex $s$-densities? Well, the fiber over a point $x \in X$ is the set of functions $\delta\_x: \bigwedge^{top} T\_xX \to \mathbb{C}$ such that $\delta(cv) = |c|^{s}\delta(v)$.) If $\delta^1$ and $\delta^2$ are smooth compactly-supported $\frac12$-densities, their pointwise product $\delta^1 \bar{\delta^2}$ is a compactly supported 1-density which we can integrate to get a complex number. This turns the space of all such sections into a pre-Hilbert space, the completion of which is what our functor assigns to the manifold $X$. As we would hope for, the canonical nature of the construction lets us assign unitary operators between Hilbert spaces to diffeomorphisms between smooth manifolds, hence is functorial. 


(Note: If we choose a volume form on $X$, the above procedure produces something isomorphic with the space of $L^2$ functions on $X$ with respect to this form, but to get something functorial we want a canonical construction.)


From this pair of functors $\mathcal M \to \mathcal C$ and $\mathcal M \to \mathcal Q$ we get a product functor $\mathcal M \to \mathcal{C} \times \mathcal{Q}$. The image of this functor is a subcategory of $\mathcal C \times \mathcal Q$ which we will call the "cotangent bundle/$\frac12$-density relation." (The word relation is meant in the same sense that an ordinary relation between two sets is a subset of their product).


Now we can clarify just what is meant by our original statement: there is no functor $\mathcal C \to \mathcal Q$ whose graph contains the cotangent bundle/$\frac12$-density relation. The reasons why this is a desirable condition come from physics and are beyond me, but roughly speaking I think the point is that there exists a good idea of what a quantization functor is supposed to do to cotangent bundles.


