# Does symplectic morphism after geometric quantization induce Hilbert spaces morphism?

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.

• I suggest as a reading : "Functorial geometric quantization and van Hove's Theorem" by Mark Gotay: you can find it online. A related question is: mathoverflow.net/questions/8606/… – Nicola Ciccoli Apr 26 at 17:13

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.