We studied the
continuity equation in 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 in commutative phase-space cannot
satisfy the condition of current conservation, but with the steady state, in order to solve this problem, we give a new definition of the current density including the contribution due
to the non-local potential. We showed that the calculated current based on the
new definition of current density
maintains the current. As well for the case when the non-commutativity in phase-space considered, we found that the conservation
of the current density completely violated; and the non-commutativity is not
suitable for describing the current density in presence of non-local and local
potentials. Nevertheless, under some conditions, we modified the current
density to solve this problem. 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.
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