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

TL;DR

This paper derives a kinetic equation for spin-1/2 electrons in ultra-strong magnetic fields near magnetars, incorporating Landau quantization and relativistic effects, with implications for plasma wave dispersion.

Contribution

It introduces a novel kinetic framework starting from the Dirac equation that accounts for Landau quantization and relativistic spin effects in ultra-strong magnetic fields.

Findings

Derived a kinetic equation with an energy operator depending on spin and momentum derivatives.

Computed energy eigenstates as eigenfunctions of the new operator.

Analyzed dispersion relations for electrostatic waves in magnetar-like conditions.

Abstract

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 ultra-strong 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.