From Bargmann's superselection rule to quantum Newtonian spacetime

Bargmann's superselection rule, which forbids the existence of superpositions of states with different mass and, therefore, implies the impossibility of describing unstable particles in non-relativistic quantum mechanics, arises as a consequence of demanding Galilean covariance of Schr\"odinger's equation. However, the usual Galilean transformations inadequately describe the symmetries of non-relativistic quantum mechanics since they fail to take into account relativistic time contraction effects which can produce non-relativistic phases in the wavefunction. In this paper we describe the incompatibility between Bargmann's rule and Lorentz transformations in the low-velocities limit, we analyze its classical origin and we show that the Extended Galilei group characterizes better the symmetries of the theory. Furthermore, we claim that a proper description of non-relativistic quantum mechanics requires a modification of the notion of spacetime in the corresponding limit, which is noticeable only for quantum particles.