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

## TL;DR

This paper investigates the continuity equation with local and non-local potentials in both commutative and non-commutative phase-space, proposing a new current density definition to ensure current conservation, and analyzing the effects of phase-space non-commutativity.

## Contribution

It introduces a modified current density that preserves current conservation in commutative phase-space and examines the impact of non-commutativity, highlighting limitations and proposing solutions.

## Key findings

- New current density maintains current in commutative phase-space.
- Non-commutativity violates current conservation unless modified.
- Application to Frahn-Lemmer non-local potential demonstrates method effectiveness.

## Abstract

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.