Non-relativistic approximation of the dirac current

Abstract

By making use of theFoldy-Wouthuysen transformation the non-relativistic charge and current density is derived from the Dirac equation up to the second order inp/mc. The terms obtained in addition to non-relativistic Pauli charge and current density are interpreted as follows. The moving magnetic dipole corresponds to an electric one interacting with the electric field like a polarization charge ϱπ div π. The current corresponding to this polarization charge appears in the usual form\(\frac{{\partial \pi }}{{\partial t}}\). The Thomas precession gives the contributions\(\rho r = \frac{1}{2}div \pi \) and\(jr = \frac{1}{2}\frac{{\partial \pi }}{{\partial t}}\) to the charge and current density, respectively, and, thus, only half of the polarization charge and current is effective. An E×σ type term arises in the current siuce the mechanical and canonical momenta are not now simply related through\(m\gamma v = p - \frac{e}{c}A\). An additional Darwin type term arises in the charge density which cannot be interpreted classically.