Spin-dependent relativistic quantum magnetized electron gas

Abstract

The covariant form of the response 4-tensor is derived for a spin-dependent, relativistic magnetized quantum electron gas. The electron gas is described by its occupation number, nϵns(pz), where ϵ = ±1 labels electron and positron states, n the Landau level, pz the parallel momentum, and s = ±1 the spin, which corresponds to the parallel component of the magnetic moment operator. A spin-dependent electron gas corresponds to nϵn +(pz) ≠ nϵn −(pz). We evaluate the spin-dependent contribution to the response tensor and show that it can be written such that its tensorial form is independent of the occupation number, which appears only in relativistic plasma dispersion functions that are independent of the perpendicular wave vector, k⊥. We discuss the special cases of parallel propagation, complete degeneracy, the synchrotron-emitting limit, and nϵns(pz)∝δ(pz). We expand the exact quantum result in powers of ℏ and find that the correction of order ℏ is nonzero for the spin-dependent part, with the lowest order correction for the spin-independent part being of order ℏ2. We find inconsistencies when the result is compared with quasi-classical calculations of the spin-dependent response. We conclude that until these inconsistencies are understood and resolved, the validity of the quasi-classically derived, spin-dependent results is uncertain.