Continuity Equation in Presence of a Non-Local Potential in Non-Commutative Phase-Space

We studied the continuity equation in the presence of a local potential, and a non-local potential arising from electron-electron interaction in both commutative and non-commutative phase-space. Furthermore we examined the influence of the phase-space non-commutativity on both the locality and the non-locality, where the definition of current density cannot satisfy the condition of current conservation, but with the steady state, in order to solve this problem, we give a new definition of current density including the contribution due to the non-local potential. We show that the calculated current based on the new definition of current density maintains the current. But for the case when the non-commutativity considered, we find that the conservation of the current density completely violated. Subsequently, as an application we studied the Frahn-Lemmer non-local potential, taking into account that the employed methods concerning the phase-space non-commutativity are both of Bopp-shift linear transformation through the Heisenberg-like commutation relations, and the Moyal-Weyl product.