I am reading the nice discussion on this MO thread on the idea behind Quantization mathematically. This answer is quite nice, and for my question, I take some quotes from it:
"quantum mathematics" is when you try to take geometric facts, written algebraically, and interpret them in a noncommutative algebra. .
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Then quantization is the process of reversing the above quotient. In particular, lots of spaces that we care about come with canonical Poisson structures. For example, for any manifold, the algebra of functions on its cotangent bundle has a Possion bracket. "Quantizing a manifold" normally means finding a noncommutative algebra so that some quotient (like the one above) gives the original algebra of functions on the cotangent bundle. The standard way to do this is to use Hilbert spaces and bounded operators, as I think another answerer described.
Mathematically speaking, I think I can understand at least some of this, but I am totally confused how it relates to physical fact. By such a procedure as said above, how do classical variables become describe by probability wave functions?
