TITLE: What is Quantization ?
QUESTION [78 upvotes]: I would like to know what quantization is, I mean I would like to have some elementary examples, some soft nontechnical definition, some explanation about what do mathematicians quantize?, can we quantize a function?, a set?, a theorem?, a definition?, a theory?

REPLY [7 votes]: A very basic answer: think about the classical Hamiltonian,
$$
a(x,\xi)=\vert \xi\vert^2-\frac{\kappa}{\vert x\vert},\quad \text{$\kappa>0$ parameter}.
$$
The classical motion is described by the integral curves of the Hamiltonian vector field of $a$,
$$
\dot x=\frac{\partial a}{\partial\xi},\quad
\dot \xi=-\frac{\partial a}{\partial x}.
$$
The attempt of describing the motion of an electron around a proton by classical mechanics leads to the study of the previous integral curves and is extremely unstable since the function $a$ is unbounded from below. If classical mechanics were governing atomic motion, matter would not exist, or would be so unstable that could not sustain its observed structure for a long time, with electrons collapsing onto the nucleus.
Now, you change the perspective and you decide, quite arbitrarily that atomic motion will be governed by the spectral theory of the quantization of $a$, i.e. by the selfadjoint operator
$$
-\Delta-\frac{\kappa}{\vert x\vert}=A.
$$
It turns out that the spectrum of that operator is bounded from below by some fixed negative constant, and this a way to explain stability of matter. Moreover the eigenvalues of $A$ are describing with an astonishing accuracy the levels of energy of an electron around a proton
(hydrogen atom).
My point is that, although quantization has many various mathematical interpretations, its success is linked to a striking physical phenomenon: matter is existing with some stability and no explanation of that fact has a classical mechanics interpretation. The atomic mechanics should be revisited, and quantization is quite surprisingly providing a rather satisfactory answer. For physicists, it remains a violence that so refined mathematical objects
(unbounded operators acting on - necessarily - infinite dimensional space) have so many things to say about nature. It's not only Einstein's "God does not play dice", but also Feynman's "Nobody understands Quantum Mechanics" or Wigner's "Unreasonable effectiveness of Mathematics."