# Quantizations in quantum mechanics

Perhaps this is not an appropriate question for MO, but having just discovered this site I wanted to ask it.

Is there a rigorous definition as to what "a quantization" of a Hamiltonian dynamical system actually is? I have never been satisfied with the answers my physicist friends have told me. The kind of answer I have in mind is something "a right inverse to the map which sends a $\Psi$DO to its principal symbol".

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See also Weyl quantization (en.wikipedia.org/wiki/Weyl_quantization) and geometric quantization (en.wikipedia.org/wiki/Geometric_quantization). –  Steve Huntsman Feb 18 '11 at 20:59
For Geometric quantization the first reference is Souriau's book "Structure des Systèmes Dynamiques" for french readers, translated in english "Structure of Dynamical Systems" amazon.com/Structure-Dynamical-Systems-Progress-Mathematics/dp/… –  Patrick I-Z Feb 18 '11 at 22:46
Except for in certain contexts, the answer is "no" to whether there is a universal rigorous definition: quantization is an art, not a functor (mathoverflow.net/questions/8606). Rather, there are lots of special cases --- say, from Poisson manifolds to Hilbert spaces --- where you can make precise statements. There is more to be said, but for your question so far I think it's well-covered by op. cit. and questions/6200/. So I'm voting to close as duplicate for now. If it is closed, please revise and flag for moderator attention if you better distinguish this question from earlier ones. –  Theo Johnson-Freyd Feb 18 '11 at 22:52