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I beg your pardon in advance if this question is naive.

In Quantum Mechanics, discrete values of measurements occur only in relation to bound states. This is because of the general solution for the time-independent Shroedinger's equation.

Similarly, in order to quantise space-time, do we need to make space-time compact? How would a compact space-time look like? Would we simply need close it by including $\infty$? Is that physical (infinity doesn't sound very physical to me)?

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up vote 6 down vote accepted

Discrete eigenvalues for measurable observables in QM occur more broadly than just for bound states. For example angular momentum, whether orbital or spin, is always restricted to discrete eigenvalues for all systems, free and unbound as well as bound systems. It is the case, however, that discrete spectra is 'associated' with compact geometries. Thus the angular momentum operators are the generators of rotations and the group of rotations is a compact group.

At this stage of our understanding, one can not make a final statement about possible discrete quantization of space-time since we have no established/corroborated theory that entails such quantization. However, recent research in the Loop Quantum Gravity research program looking for a quantum theory of gravity, has led to the result that if the LQG approach holds, then the boundary surface areas and the internal volumes of compact space-like regions of space-time must have discrete spectra.

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