Is the first moment of mass (center of mass times mass) quantized?

Question

Angular momentum is conserved according to Noether's theorem, because the laws of physics are symmetric under rotations. The analogous quantity corresponding to symmetry under boosts is the first moment of mass. Angular momentum is quantized to be an integer (or maybe half integer) multiple of $\hbar$. Is the first moment of mass similarly quantized to be a multiple of $\hbar/c$? If so, what does this look like, and if not, why is there such an asymmetry between space and time?

Answer 1

Position and center-of-mass coordinates are not quantized/discrete in conventional QM (excluding lattice formulations), cf. e.g. this Phys.SE post. In contrast, for the reason why angular momentum is quantized, see e.g. my Phys.SE answer here.