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Hamiltonian systems, symplectic flows, classical integrable systems

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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: we add a prequantum line …