Lifshitz Quantum Mechanics and Anisotropic Josephson Junction

We consider quantum mechanics in spacetime with anisotropy in time. Such Lifshitz quantum mechanics is characterized by a kinetic term with fractional derivatives. We show that, contrary to a common claim in the literature, local conservation of probability is respected when the probability current is properly identified. As an application we consider a Josephson Junction with an insulating layer exhibiting Lifshitz anisotropy. We show that anisotropy modifies the tunneling rate and can significantly enhance the performance of the Josephson Junction.