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We studied the continuity equation in presence of a local potential, and anon-local potential arising from electron-electron interaction in bothcommutative and non-commutative phase-space. Furthermore, we examined theinfluence of the phase-space non-commutativity on both the locality and thenon-locality, where the definition of current density in commutativephase-space cannot satisfy the condition of current conservation, but with thesteady state, in order to solve this problem, we give a new definition of thecurrent density including the contribution due to the non-local potential. Weshowed that the calculated current based on the new definition of currentdensity maintains the current. As well for the case when the non-commutativityin phase-space considered, we found that the conservation of the currentdensity completely violated; and the non-commutativity is not suitable fordescribing the current density in presence of non-local and local potentials.Nevertheless, under some conditions, we modified the current density to solvethis problem. Subsequently, as an application we studied the Frahn-Lemmernon-local potential, taking into account that the employed methods concerningthe phase-space non-commutativity are both of Bopp-shift linear transformationthrough the Heisenberg-like commutation relations, and the Moyal-Weyl product.

Continuity Equation in Presence of a Non-local potential in Non-Commutative Phase-Space