Why?
[1] https://physics.stackexchange.com/questions/16889/non-comput...
Don't they have, in the sense that to keep probing ever more precisely, you need ever more energy, and at some point too much mass~energy in a too small volume is going to form a black hole and you cannot go further ?
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Argh, I should have known that this discussion would start to involve operators at some point, especially ones with an infinite number of dimensions... XD
Though this seems related mathematically to how the uncertainty principle can be interpreted through Fourier transforms : localized <=> spread out ; quantized <=> continuous ; finite (+ conditions on potential) <=> infinite (+ other conditions on potential) ?
This may or may not be true - we can't probe those length scales yet, and maybe ever, so we don't really know - but it has no bearing on ordinary QM, which is nonrelativistic.
> Though this seems related mathematically to how the uncertainty principle can be interpreted through Fourier transforms
Not especially. They're both results in functional analysis, but that's about it.
> And would you get quantification of mass~energy from that of the impulse (through v having dimensions of space and time), or do you need to use a different approach and assume black holes ? (Or both, and pick the biggest value ?)
Neither, though the impulse thing is at least in the right neighborhood. (Black holes have absolutely nothing to do with this). You get quantized energy whenever the Hamiltonian of your system has a pure point spectrum. Typically this happens for finite dimensional systems, and infinite dimensional systems with potentials that grow rapidly with increasing distance. Generic infinite dimensional systems will usually have continuous spectra.