Lifshitz Quantum Mechanics and Anisotropic Josephson Junction

Abstract: We consider quantum mechanics in spacetime with time anisotropy. Such Lifshitz quantum mechanics is characterized by a kinetic term with fractional derivatives. We show that, contrary to popular belief, local conservation of probability is respected when the probability current is properly identified. As an application we consider a Josephson Junction with an anisotropic insulating layer. We show that anisotropy affects the tunneling rate and can greatly enhance the working of Josephson Junction.