Phase space formulation of Classical Electrodynamics Is there a Lorentz invariant formulation of classical electrodynamics whose source terms are the components of a phase space distribution of current densities?  I assume that from one perspective it would provide a net current density equal to an integral over all current velocities (and corresponding current densities) at each point in configuration space.  I've done a lot of searching online, but have only found articles that discuss plasma dynamics and quantum mechanics.
 A: If we are working with potentials (developing fomulations for fields is not much more cumbersome) then the wave equation for the four-potential ($A$) is:
$\square^2 A^\nu = \mu_0 J^\nu$
With Lorentz gauge:
$\partial_\nu A^\nu =0$
Where $J^\nu$ is four-current. Let the fourier tranforms be:
$\tilde{J}^\nu \left(\mathbf{k},\omega\right) = \iiint d^3r \int cdt \exp\left(i\left(\frac{\omega}{c}\cdot ct - \mathbf{k}.\mathbf{r}\right)\right) J^\mu\left(\mathbf{r}, t\right)$
Where $c$ is the speed of light. It makes sense to switch to four-position $x^\nu = \left(ct,\mathbf{r}\right)^\mu$ and four-wavevector $k^\nu=\left(\frac{\omega}{c},\mathbf{k}\right)^\mu$. Then:
$\tilde{J}^\nu \left(k\right) = \int d^4x \exp\left(i k_\sigma x^\sigma\right) J^\mu\left(x\right)$
Same for the four-potential. Then the equations become:
$-k_\sigma k^\sigma\tilde{A}^\mu = \mu_0 \tilde{J}^\nu$
$k_\nu \tilde{A}^\nu = 0$
Is this what you were looking for? For four current we have $\partial_\nu J^\nu=0$, so $k_\sigma \tilde{J}^\sigma = 0$, thus one of the four components (e.g. charge density) can be determined from the other ones in most cases.
