Electromagnetic Response Theory with Relativistic Corrections: Selfconsistency and Validity of Variables

TL;DR

This paper develops a relativistic correction-based electromagnetic response theory from the Schrödinger-Pauli equation, ensuring consistency with classical relations and identifying minimal variables for describing EM responses.

Contribution

It formulates a second order weakly relativistic response theory with explicit charge, current, and polarization variables, clarifying the role of spin current in this framework.

Findings

Charge and current densities satisfy classical continuity and polarization relations.

The minimal set of variables for the theory is identified as φ, A, ρ, J.

The theory encompasses all EM responses within the second order weakly relativistic approximation.

Abstract

Schr\"odinger-Pauli equation (SP-eq) derived from weakly relativistic approximation (WRA) of Dirac eq, combined with Electromagnetic (EM) field Lagrangian for variational principle, is expected to give a new level of EM response theory. A complete process of this formulation within the second order WRA is given, with explicit forms of charge and current densities, $\rho , \vec{J}$, and electric and magnetic polarizations, $\vec{P}$, $\vec{M}$ containing correction terms. They fulfill, not only the continuity equation, but also the relations $\nabla \cdot \vec{P}=-\rho, \ \partial \vec{P}/\partial t + c \nabla \times \vec{M} = \vec{J}$, known in the classical EM theory for the corresponding macroscopic variables. This theory should be able to describe all the EM responses within the second order WRA, and the least necessary variables are ${\phi, \vec{A}, \rho, \vec{J}}$ (six independent components). From this viewpoint, there emerges a problem about the use of "spin current" popularly discussed in spintronics, because it does not belong to the group of least necessary variables.