Inadequacy of the Usual Galilean Transformation in Quantum Mechanics

We show that the superselection rule in the Galilean transformation, forbidding the superposition of states of different mass, in inconsistent with the nonrelativistic limit of the Lorentz transformation. We also point out that the extra Galilean phase is merely the residue of the “twin-paradox” effect, which does not vanish nonrelativistically. In general, there are phase effects due to proper time differences, and effects due to mass-energy equivalence, that do not vanish nonrelativistically but that are not handled adequately by the Galilean transformation.