Physical motivation of quantization I am a mathematics student recently looking into (geometric and deformation) quantization. I'd like to know more about their physical motivations. Here by "quantization" I mean any process that transforms a classical system (modeled on a symplectic or Poisson manifold) into a quantum system. Here is my question:
What role does geometric and deformation quantization play in physics? Are there physics theories that crucially depend on a well-developed theory of geometric/deformation quantization? I'd be especially interested in the latter case. I've heard things like "quantum field theory is the quantization of classical field theory", but that seems to be a different usage of the word "quantization".
More generally, for the role of quantization in physics: from a naive aspect, it is quantum physics that's more fundamental. The similarities between the Heisenberg picture in QM and the Hamiltonian formalism in CM seems to me more like a consequence of some statistical behavior. It also seems to me that the inverse process would be more natural: try to construct a classical system out of a quantum one. Is there a good reason that we start with classical systems (except that it is better understood)?
 A: It is said (I forget by whom) that the  process of of quantization -- of turning a classical phase space into a Hilbert space -- is not a functor. There is no set recipe, but there is instead  a variety of recipes.
Canonical quantization,  for example, consists of making the classical system look like a bunch of quantized harmonic oscilators. This works for for any system undergoing small (hence linear)  oscillations and so is good for many systems, and in  particular weakly interacting  fields.
A cartoon description  of geometric quantization is that it tries to make eveything look like the quantization of spin. In particular it nicely generates the representation spaces of compact groups from the Kirillov-Constant symplectic structure --- and of course this representation theory is everywhere in particle physics.
I think it fair to say that nearly all physicists working on actual physical systems are familiar with the first of these processes but  very few are familiar with the second. Therefore the  motivation for geometric quantization  comes mostly from the mathematics side.
Some of the ideas of Moyal algebras and star products  are used in the quantum Hall effect as a low energy/high-magnetic-field approximation, but again this is a niche area.
