Contextuality of the probability current in quantum mechanics

We revisit an argument proposed by Hardy \cite{Hardy-1} concerning local realistic theories, but in terms of the motion of the probability fluid and its current within standard quantum mechanic. We emphasize surprising properties of the flux lines of this current in configuration space, in particular (quasi) discontinuous variations when the context (the experimental setup) is changed. This occurs in both Galilean and Einsteinian relativity. In the latter case, discontinuous variations also appear without changing the setup, when the Lorentzian reference frame is changed. A relativistically consistent definition of the motion of a quantum probability fluid is therefore impossible.