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First quantization refers primarily to the quantum mechanics of a single particle in some classical potential. Many complex problems can be approximately reduced to problems of this form. The usual way to solve such a problem is to find the eigenvectors and eigenvalues of the Hamiltonian function (the Schroedinger equation), which represents the motion of the single particle in the potential. While the time dependence of the Schroedinger equation is linear, the eigenfunctions and eigenvalues are not linear functions of the potential. It is sometimes helpful to think of first quantization as the quantization of angular momentum in integer multiples of the fundamental unit h-bar. Second quantization occurs in quantum field theory. It is the quantization of particle number. For electrons, protons or any other charged particles, quantization of particle number is essentially the same as quantization of the electric charge. But it is also true that other particles, such as photons (the modes of the electromagnetic field), which have no electric charge, are quantized. Quantization of particle numbers is not merely a result of charge conservation. Dirac argued that quantization of the electrical charge is required if magnetic charges exist. Such magnetic charges occur in some popular theories of the early universe. However, despite major efforts to find them, experimental evidence for magnetic charges remains elusive. Other people have suggested that the reason why particle numbers are quantized is that particles are "solitons", localized solutions of some nonlinear field theory. I am not sure what you mean by "nth quantization". Some people believe that in order to find a fully self-consistent theory of quantum gravity we should quantize the fluctuations in the geometry of space and time. As far as I know, this has not been done successfully yet. I do not really understand what you mean when you say that "first quantization was a mistake". It is an approximation which is often useful. In a single particle problem we do not need to worry about the quantization of particle numbers.

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