Pseudo- symmetric Dirac equation: effect of a new mean spin angular momentum operator on Gilbert damping

Abstract

The pseudo- symmetric Dirac equation is proposed and analyzed by using a non-unitary Foldy–Wouthuysen transformations. A new spin operator symmetric expectation value (called the mean spin operator) for an electron interacting with a time-dependent electromagnetic field is obtained. We show that spin magnetization—which is the quantity usually measured experimentally—is not described by the standard spin operator but by this new mean spin operator to properly describe magnetization dynamics in ferromagnetic materials and the corresponding equation of motion is compatible with the phenomenological model of the Landau–Lifshitz–Gilbert equation (LLG).