= Polarization-magnetization tensor
{title2=$M^{\mu\nu}$}
An <antisymmetric tensor> combines the <electric polarization> and <magnetization> of a medium. For <Minkowski metric> $(-,+,+,+)$ and units with light speed one, choose $M^{i0}=P_i$ and $M^{ij}=\epsilon_{ijk}M_k$, together with $F_{i0}=E_i$ and $F_{ij}=\epsilon_{ijk}B_k$. Then the <Lorentz scalar> interaction is $M^{\mu\nu}F_{\mu\nu}/2=\mathbf P\cdot\mathbf E+\mathbf M\cdot\mathbf B$. The <bound charge and bound current> have four-current $j_b^\mu=\partial_\nu M^{\mu\nu}$, giving $\rho_b=-\nabla\cdot\mathbf P$ and $\mathbf J_b=\partial_t\mathbf P+\nabla\times\mathbf M$. This current is identically conserved because partial derivatives commute and $M$ is antisymmetric.
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