Does symplectic morphism after geometric quantization induce Hilbert spaces morphism?

Question

Let us have two symplectic manifolds $(M, \,\omega)$ and $(N, \,\omega')$ and morphism between them: $$ \varphi \ :\ M \to N.$$ Then we geometrically quantize these systems:

1. we add a prequantum line bundle, $ L_M \to M$ and $L_N \to N$ correspondingly;

2. choose some real polarizations $P_M$ and $P_N$ such that $(\varphi_*P_M)(x) \neq P_N(x), \, \text{for }x \in \varphi(M)$;

3. let the quotient maps $\pi_{M}\ :\ M \to M/P_M$ and $\pi_N\ :\ N \to N/P_N$ be fibration.

4. Then the space of compactly supported sections of $L_M/P_M \otimes |T(M/P_M)|^{1/2} $ after completion realise Hilbert space $H_M$. The same procedure for $N$ gives $H_N$.

I don't understand how to build morphism between $H_M$ and $H_N$ from $\varphi$. Is there any natural way to do it?

I'm not sure whether it's a trivial question or not. I don't have deep understanding of geometric quantization, I've only read this review by Lerman. So if some comprehensible text about the question exists, I'll gladly accept a reference.

P.S. I'm reading this paper by Nekrasov which describes morphism between classical Calogero-Moser and Calogero-Sutherland models, and derives from it morphism between their quantum versions. Nekrasov briefly describes how he gets quantum morphism from classical one in the beginning of 4th section. But I don't understand it. So I'm interested whether there is a general way to do it. If it is possible only in this particular case, I'd like to get a more thoroughly explanation of Nekrasov's procedure.

Answer 1

The slogan to keep in mind is that "quantization is not a functor". For details on this slogan, see the (decade old!) Mathoverflow question What does “quantization is not a functor” really mean?. Basically, "is not a functor" means that there is no canonical way to build morphisms of Hilbert spaces from morphisms of symplectic manifolds.

Of course, in many practical cases there may be some "best" morphism of Hilbert spaces for a given "nice" morphism of symplectic manifolds, or some family of "best" morphisms. For instance, perhaps in your practical case you have some control over how your $\varphi$ interacts with the prequantum and polarization data.