Relativistic kinetic equation for spin-1/2 particles in the long-scale-length approximation

Abstract

In this paper we derive a fully relativistic kinetic theory for spin-1/2 particles and its coupling to Maxwell's equations, valid in the long scale-length limit, where the fields vary on a scale much longer than the localization of the particles; we work to first order in . Our starting point is a Foldy-Wouthuysen (FW) transformation, applicable to this regime, of the Dirac Hamiltonian. We derive the corresponding evolution equation for the Wigner quasi-distribution in an external electromagnetic field. Using a Lagrangian method we find expressions for the charge and current densities, expressed as free and bound parts. It is furthermore found that the velocity is non-trivially related to the momentum variable, with the difference depending on the spin and the external electromagnetic fields. This fact that has previously been discussed as "hidden momentum" and is due to that the FW transformation maps pointlike particles to particle clouds for which the prescription of minimal coupling is incorrect, as they have multipole moments. We express energy and momentum conservation for the system of particles and the electromagnetic field, and discuss our results in the context of the Abraham-Minkowski dilemma.

Relativistic kinetic theory for spin-1/2 particles: Conservation laws, thermodynamics, and linear waves

“…The present quantum kinetic relativistic model for electrons, first derived in Ref. [22], is based on the Dirac equation, where a phenomenological adjustment of the spin g-factor has been made to account for the QEDcontribution. It generalizes the relativistic Vlasov equation to include the spin dynamics, while still allowing for a fully relativistic motion.…”

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

“…The Hamiltonian in (14) includes the spin orbit interaction, see Ref. [12] for more details, but since E B, it is negligible compared to the magnetic interaction. We will therefore neglect the spin-orbit interaction from now on, as the main idea of this paper is to study the effects of a strong magnetic field on the dynamics of a plasma.…”

“…The new effects of this kinetic equation are hiding in ± . First, we have all orders of the spin magnetic moment, compared to previous models [11,12] where only the first-order correction is included. Note also that the equations for W + and W − are decoupled and can be solved separately.…”

“…For this purpose, typically the starting point is the Wigner-Moyal equation [1][2][3][4][5][6], or various generalizations thereof also accounting for physics associated with the electron spin, such as the magnetic dipole force [7,8], spin magnetization [7,9], and spin-orbit interaction [9]. Both weakly [10,11] and strongly [12,13] relativistic treatments have been presented in the recent literature.…”

“…However, most previous theoretical studies starting from the Dirac equation (see also Refs. [11,12]) have been limited to cases where the magnetic field strength is well below the critical field B cr = m 2 c 2 /|q|h.…”