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In relativistic quantum field theory with local interactions, charge islocally conserved. This implies local conservation of probability for the Diracand Klein-Gordon wavefunctions, as special cases; and then in turn fornon-relativistic quantum field theory and for the Schroedinger andGinzburg-Landau equations, regarded as low energy limits. Quantum mechanics,however, is wider than quantum field theory, as an effective model of reality.For instance, fractional quantum mechanics and Schroedinger equations withnon-local terms have been successfully employed in several applications. Thenon-locality of these formalisms is strictly related to the problem of time inquantum mechanics. We compute explicitly for continuum wave packets the termsof the fractional Schroedinger equation and of the non-local Schroedingerequation by Lenzi et al. which break the local current conservation, anddiscuss their physical significance. The results are especially relevant forthe electromagnetic coupling of these wavefunctions. A connection with thenon-local Gorkov equation for superconductors and their proximity effect isalso outlined.

Time in quantum mechanics and the local non-conservation of the probability current