A charge $q$ in an electric potential $\phi$ acquires potential energy $q\phi$.
Likewise, a charge density $\rho(x)$ aquires potential energy density $\rho(x)\phi(x)$.
A moving charge $q$ in an electromagnetic 4potential $A$ ``acquires'' an additional momentum $qA$ so that the canonical momentum equals $qA$ plus the kinetic momentum.
Q2. Is there a physically meaningful notion of the energymomentum tensor $T(x)$ which a 4current density $j(x)$ ``acquires'' in a 4potential $A(x)$?
Q3. Is there a tensor field $T(x)$ sush that

$T(x)$ depends only on $j(x)$, the electromagnetic 4potential $A(x)$, and maybe their derivatives;

$T(x)=0$ identically, once $j(x)=0$ identically;

$T(x)+T_{field}(x)$ is conserved, where $T_{field}(x)$ is the (BelinfanteRosenfeld) energymomentum tensor of the field itself.
One expects the same tensor field $T(x)$ for Q2 and Q3.
An immediate idea could be to apply the Noether theorem. But the theorem is not applicable here because the Lagrangian explicitly depends on $x$ through the term $A_\mu(x)j^\mu(x)$, hence is NOT translational invariant. Notice that the field $j(x)$ is given, it is not dynamical.
A guess for the required tensor could be $T^\mu_\nu=A_\nu j^\mu\delta^\mu_\nu A_\lambda j^\lambda$. But this does not work because the divergence $\partial_\mu T^\mu_\nu=F_{\mu\nu}j^\muA_\mu\partial_\nu j^\mu$ contains an extra term in addition to the Lorentz 4force, which contradicts to point 3 above.
Setting formally $T(x)=T_{field}(x)$ does not work because it contradicts to point 2.
In the literature known to me (e.g., LandauLifshitz, sections 3233 and 94 of volume 2) such tensor $T(x)$ is never constructed. Instead, a particular type of particles producing the current $j(x)$ is chosen and the energymomentum tensor of the particles is added to $T_{field}(x)$. This results in a conserved tensor. But the result depends on the particular type of the particles, not just on $j(x)$, which contradicts to point 1.
I would be very grateful to you for any insight, optimally including an explicit YES/NO answer for either Q2 or Q3.
EDIT Qestion Q1 has been removed (Q1:Is the quantity $\rho(x)\phi(x)$ a part of some Lorentz covariant tensor field $T(x)$? The answer is trivially yes: just take $T(x)=A_\nu (x)j^\mu(x)$.)
This post imported from StackExchange Physics at 20170405 15:37 (UTC), posted by SEuser Mikhail Skopenkov