This question discusses discretization in some sense, and this question talks about how quantization and Hilbert Spaces are related (the answer seems to to be not at all), but what I'm curious about is how the uncertainty principle, quantization (as in quantized values for observables), and non-commutation are related- mathematically and physically. I'm not far enough along in physics to see the `obvious' answers to these questions, and I'm not far enough along in math to read through some of the more dense articles talking about Lie Algebras.
However, I feel as if there must be some relation between the generally quantization of values in quantum mechanics, and the non-commutativity of operators; when people discuss moving from the quantum realm to the classical realm they talk about the commutator going to zero. So what is it about the commutation/non-commutation of operators that makes a system quantized mathematically? The answer here attempts to give some link between the two, but I did not find it very cohesive or convincing.
I also understand that there may be a different meaning to the phrase quantization, but what I'm referring to is the appearance of quantized values of angular momentum, energy levels, etc.