There is an well-known infamous DICTUM:
-Second Quantization is a functor, First Quantization is a mystery-.
Indeed, second quantization is the "Fock functor", which builds the Fock space in a canonical way out of the Hilbert space of a single particle.
But, what about first quantization? There is probably no hope to canonically associate an Hilbert space to the manifold of states of a classical particle (mathematically, there seem to be an inherent element of choice as far as turning functions into operators).
However, there is (I suspect) some functorial description for going the other way around, FROM the quantum scenario INTO the classical one (corresponding to the limit $h\rightarrow 0$). If this is true, there maybe a "fiber" of candidate quantum descriptions, all collapsing into the same classical one.
Any place where this has been worked out clearly?