Quantum mechanics is not a discrete theory in the sense you seem to be suggesting. There is this idea that is is, which comes from the discrete energy levels it predicts for a simple harmonic oscillator, or the hydrogen atom, but quantities like the momentum, kinetic energy, position etc of a quantum particle are completely allowed to vary continuously.

In some sense (because of the wavefunction) quantum mechanics is less discrete than classical mechanics, a qubit system which would be classicaly described by a single bit is described by a state that sits on a continuous manifold (the Bloch sphere).

I didn't mean to paint it as a discrete theory (and in fact I added the shoutout to the Loop Quantum Gravity stuff at the end in part because I was concerned about this interpretation).

However, across quantum mechanics, there is at least unity on the 'quantum' part. And that's really the point of contention here in terms of integer values, right?

> However, across quantum mechanics, there is at least unity on the 'quantum' part. And that's really the point of contention here in terms of integer values, right?

I don't know what this means, but there is this idea that quantum mechanics leads to integer valued stuff because the first models studied were things like the hydrogen atom and the simple harmonic oscillator which both have a natural integer valued observable (the energy). This is not generally the case, however and in quantum mechanics there are many observables which are not integer valued, for example the kinetic energy of a free particle can be any non-negative real.

To add, perhaps in a very general hand wavy way "quantisation" is something like replacing a commutative algebra A, with a non commutative algebra A_h. I do not necessarily think there is something discrete in this process. In physics functions get sent to operators. States get sent to rays.

It turns out that a lot of interesting physics these operators have physically interesting discrete spectrum.