The equations of motion are derived here with the help of the Heisenberg EoM. Starting in the Heisenberg picture the time evolution of an operator is \begin{equation} \frac{\partial}{\partial t} \hat{O}_\text{H} (x,t) = \left[ \hat{O}_\text{H} (x,t) , \hat{H}_\text{H} (t) \right] + \left( \frac{\partial}{\partial t} \hat{O}_\text{S} (x) \right)_\text{H} \quad . \end{equation}

The EoM for the electronic creation and annihilation operator is \begin{equation} \frac{\partial}{\partial t} \hat{\Psi}_\text{H} (x,t) = \dots \end{equation} according to the calculation here.

Condition: \begin{equation} \nu (\vec{r},\vec{r}^\prime) = \nu (\vec{r}^\prime,\vec{r}) \end{equation}

The EoM of the particle number operator \begin{equation} \frac{\partial}{\partial t} \hat{n}_\text{H}= \dots \end{equation} is of the form of a continuity equation. The calculation is done here.

Wtih the help of the EoM of the fermionic field operator, the EoM for the current density operator $J(x,t)$ becomes \begin{equation} \frac{\partial}{\partial t} \hat{J}_\text{H} (x,t) = \frac{\partial}{\partial t} \hat{j}_\text{H} (x,t) + \frac{\partial}{\partial t} \hat{j}_\text{d,H} (x,t) = \dots + \dots \quad . \end{equation} Insert Link to para and diramagnetic definitions This holds for symmetric correlation potentials \begin{equation} \nu (\vec{r},\vec{r}^\prime) = \nu (\vec{r}^\prime,\vec{r}) \end{equation} and (Do a page where I discuss where this occurs) \begin{equation} \frac{\partial}{\partial t} \frac{\partial}{\partial k} = \frac{\partial}{\partial k} \frac{\partial}{\partial t} \quad . \end{equation} The calculation can be found here.