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Are there interesting (aspects of) problems in modern physics that can be expressed solely in terms of integer numbers? Bonus points for quantum mechanics.

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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 eigenvalues can be put in one-to-one correspondence with the set of non-negative integers (aka the set of eigenvalues is countable). In fact, there is a large class of Hamiltonians in quantum mechanics with discrete spectra, and for each such Hamiltonian the same comments about the spectrum and integers applies.

If you generally interpret your question as a question about "discreteness" in physics; then one can give all sorts of really important examples. Take, for example, lattice field theory in which one can attempt to simulate field theories on a computer.

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  • $\begingroup$ I was hoping for something more advanced or a topic of current research, perhaps elaborate a bit more on lattice field theory? $\endgroup$ – delete000 May 7 '13 at 10:35
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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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  • $\begingroup$ I'd say that the fractional quantum-Hall effect has more interesting "integer" physics than the integer one. Plus, the IQHE is very well understood as a single-particle phenomenon. In the FQHE one has anyons with braiding & fusion properties, topological ground-state degeneracies described by patterns of ones and zeros and so on and so forth. Perhaps more "integer" physics examples there? $\endgroup$ – delete000 May 7 '13 at 10:40

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