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

From Bargmann's superselection rule to quantum Newtonian spacetime