Definition of current density in presence of AC electric field

Under time varying AC electric field, the transport problem becomes complicated due to the presence of displacement current. The conventional current density calculated by using the formula $J_{c}=\frac{e}{2m}[((p-eA)\psi)^{*}\psi- \psi^{*} ((p-eA )\psi)]$ is not conserved, which means $\nabla\cdot J_c(r,t)\neq 0$. In order to solve this problem, we will give a new definition of current density by using non-equilibrium Green's function which includes the contributions from the Coulomb interaction in low frequency limit. And we will show that the current calculated from the current density is conserved.