The first version is the usual one we're taught. But there's this other version too : A quantum non-relativistic field theory.
Take a non-relativistic classical field, like the non-relativistic limit of the Klein Gordon field. And then quantise this field.
You end up with a theory that, unlike the usual NR quantum mechanics, allows for a probabilistic particle number. We have a Fock space in this theory.
If our universe were Galilean, would it follow the usual Non Relativistic QM, or this Quantum Non Relativistic Field theory?
OR MAYBE Both these theories are identical in practice. The second theory does have a Fock space. But I think most of the vectors in that Fock space are useless in practice.
Because to actually create or annihilate a particle, you'd need infinite energy in this theory by $E=mc^2$, $c=\infty$. So this theory does have a Fock space, but the interactions don't actually allow for particle creation and annihilation. So we're left with the usual Hilbert space of a fixed number of particles, identical to the ordinary NR Quantum Mechanics. Is this reasoning correct?