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The semiclassical/WKB quantization rule is discussed in numerous textbooks. The discrete quantization condition follows from requiring single-valueness of the wavefunction. Note that quantization formula gets modified by the metaplectic correction/Maslov index because of turning points. For an elementary treatment, see e.g. Refs 1 and 2 below. For a ...

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From what we understand today, p-branes are honest degrees of freedom, on equal footing with strings. Shop you have a good question. But I don't think anyone had so far managed to consistently quantize a p-brane. Loosely, a brane has much more degrees of frerdom than a string and it's difficult to get them under control. So quantizing it is a technical ...

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Yes, although it depends on how one interprets the terms "aspects" and "solely". Take, for example, the Hamiltonian for the one-dimensional quantum harmonic oscillator. The spectrum of the Hamiltonian (set of possible energies for the oscillator) is $$E_n = (n+\tfrac{1}{2})\hbar\omega, \qquad n = 0,1,2,\dots$$ The spectrum is discrete, and the ...

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I would say that a nice example of this is the Quantum Hall Effect. Here the conductivity $\sigma$ come in integer multiples of $e^2/h$. In this case, the integers here have a topological origin.

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