Mage's Rhizome

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======Equation of motion======
The equations of motion are derived here with the help of the [[QM representations|Heisenberg EoM]].
Starting in the [[QM representations|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}

=====Fermionic Field Operator=====
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 [[forcebalance:calculation:EoM fermionic field operator|here]].

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

=====Particle Number Operator=====
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 [[forcebalance:continuity equation]].
The calculation is done [[forcebalance:calculation:EoM particle number operator|here]].

=====Current Density Operator=====
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}
FIXME 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 [[forcebalance:calculation:EoM current density operator|here]].