Polarization-magnetization tensor ($M^{\mu \nu }$)

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 $M^{ij}={ϵ}_{ijk}{M}_k$, together with $F_{i0}=E_i$ and $F_{ij}={ϵ}_{ijk}{B}_k$. 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.