Aristotelian physics, shorn of whatever the historical Aristotle actually believed, is pretty similar to Newtonian physics. Instead of "An object in motion stays in motion unless acted on by an unbalanced force", we have "An object at rest stays at rest unless acted on by an unbalanced momentum." Newton's F=ma, which is a second order differential equation, becomes p=mv, which is a first order differential equation. Otherwise, we have business as usual. My question is, can we quantize this theory? Instead of constructing Hilbert space operators using the representation theory of the full Galilei group, we just use the representation theory of the Galilei group excluding Galilean transformations, i.e. just consisting of spatial translations, spatial rotations, and time translations. What would such a quantum theory look like? I can tell right off the bat there will probably be fewer conserved quantities, but not much more. Any help would be greatly appreciated. Thank You in Advance.