Kinetic theory for spin-1/2 particles in ultrastrong magnetic fields

When the Zeeman energy approaches the characteristic kinetic energy of electrons, Landau quantization becomes important. In the vicinity of magnetars, the Zeeman energy can even be relativistic. We start from the Dirac equation and derive a kinetic equation for electrons, focusing on the phenomenon of Landau quantization in such ultrastrong but constant magnetic fields, neglecting short-scale quantum phenomena. It turns out that the usual relativistic $\gamma$ factor of the Vlasov equation is replaced by an energy operator, depending on the spin state, and also containing momentum derivatives. Furthermore, we show that the energy eigenstates in a magnetic field can be computed as eigenfunctions of this operator. The dispersion relation for electrostatic waves in a plasma is computed, and the significance of our results is discussed.

Figure 1

The normalized number density for different energy states $E_m$ at different values of the parameters $\beta ={\mu }_B{B}_0/m$ and $\tau =k_{BT}/m$.