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When I read physics explanations of "quantization", I am confused, because they talk about particles, momentum, and other specific things. It seems to me that quantum formalism is much more general than this (e.g. in quantum computing there are no "particles").

What is the most general statement of "quantization", i.e. "turning a classical model into a 'quantum version' of the same model"? i.e. what does quantization mean formally?

user56834
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  • What you quantize aren't particles, but systems. I never quite understood it myself, though. – Arthur Aug 22 '19 at 10:53

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Todorov's delightful article in the refs of your link emphasizes the arbitrary and non-unique version of such projects.

The proper definition is "turning a classical model into some quantum model whose classical limit is the classical model you started from". The procedure is heuristic and ill-defined, but, of course, in an overwhelming majority of cases addressed, the problem is rather straightforward to solve.

A classical model obeys Hamilton's differential equations to have its time-evolution specified, and this time evolution is summarized in classical trajectories.

A quantum model cannot have trajectories, but, instead describes how "states", vectors in a Hilbert space, evolve in time, through unitary evolution, i.e. acted upon by linear operators properly specified. Ultimately, suitable scalar products of such states, yield probabilities or expectation values of "observables" of use in physics, computing, etc... Physical systems involve a crucial "deformation parameter" $\hbar$, to connect to classical systems, but it is not crucial in purely formal discussions of QM.

Through the Weyl-Wigner invertible map, one may translate Hilbert space entities into phase-space ones (which is the ambit of Hamiltonian classical mechanics) so you may contrast a quantum model to the classical model resulting by taking a deformation parameter $\hbar\to 0$. As indicated, the quantum system contains much more information, in some sense, than its classical limit, which is then reachable from several different quantum models.

Cosmas Zachos
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  • "turning a classical model into some quantum model whose classical limit is the classical model you started from". Is there not a general way to do it? (The fact that people treat it as a heuristic, doesnt mean there might not also be a general exact way). The reason why I thought there might be is because I think of the quantum version of a classical state space as something very general, namely a complex vector space whose basis elements are elements of the classical state space (with normalization etc). This suggests to me there should be a very general method – user56834 Aug 23 '19 at 05:55
  • No, quantization is not a functor. The respective Lie algebras differ, as Todorov reiterates. – Cosmas Zachos Aug 23 '19 at 12:14