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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)?

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    $\begingroup$ “Are there serious applications of quantization in physics?” Honestly, I’m not understanding this question at all. $\endgroup$
    – Gilbert
    Dec 29, 2021 at 3:50
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    $\begingroup$ Let me admit that my physics knowledge involves only basic QM, and for me it seems that quantization plays no critical role in it. There are some analogies from classical mechanics, but I imagine that the theory can be developed only based on experimental facts. So I'm wondering if there is a certain point in physics where quantizations become crucially important. $\endgroup$
    – lw h
    Dec 29, 2021 at 3:56
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    $\begingroup$ You seem to need to read about the history and development of quantum theory. That's way beyond the scope of an answer here. The only physical motivation is that the quantum theory models are the ones that fit the experimental data and the classical ones don't. $\endgroup$ Dec 29, 2021 at 4:51
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    $\begingroup$ Here is my understanding of the history: (1). there is classical mechanics (2).experiment shows that classical mechanics is not correct at microscopic level (3).physicists develop quantum theory based on classical heuristics (adding hats on classical observables to "quantize" them). To rephrase my question: is "quantization" developed just to make step (3) rigorous, or do physicists quantize classical systems by e.g. geometric quantization in their daily lives? $\endgroup$
    – lw h
    Dec 29, 2021 at 4:56
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    $\begingroup$ Physics is not mathematics. I think you'd probably be better served by looking at how Richard Feynman compared physics and mathematics](cantorsparadise.com/…). Rigor is kind of an optional extra in physics. We use it when it's useful, but mostly we don't care. We care about matching the result to experimental data and many theories that are rigorous fail because they don't match reality. Mathematics is not physics. Put another way : you can't get there from here. $\endgroup$ Dec 29, 2021 at 5:38

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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.

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