  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.
