Quantum-Mechanical Relation for a Magnetized Electron Gas

The following theorem is proven: The $\mathrm{xx}$ and $\mathrm{zz}$ components of the electron energy-momentum tensor $t_{\mu \nu }$ and the magnetization $M$ of an electron gas in a constant magnetic field $B$ satisfy the relation $t_{\mathrm{zz}}-t_{\mathrm{xx}}=BM$. This relation is valid at any density, temperature, and magnetic field strength, if the system is in thermal equilibrium. Since the electromagnetic energy-momentum tensor $t_{\mu \nu }$ is anisotropic itself, this relation makes the total energy-momentum tensor $T_{\mu \nu }=t_{\mu \nu }+{\tau }_{\mu \nu }$ become a scalar. Because $t_{\mathrm{xx}}$ and $t_{\mathrm{zz}}$ can simply describe the electron pressure in these directions, the above theorem states that the difference $p_{\mathrm{zz}}-p_{\mathrm{xx}}$ equals twice the magnetic field energy, since $M=4\mathrm{}\pi B$.