Covariant representation of microscopic charge and current densities in terms of polarisation and magnetisation fields

Abstract

It is shown that a previously developed formalism (Healy 1977) for representing microscopic charge and current densities in terms of polarisation and magnetisation fields can be written in a manifestly Lorentz covariant manner. The charge-current density four-vector associated with an aggregate of charged point particles is first constructed and its behaviour under the time reversal transformation is discussed. Particular polarisation-magnetisation tensors defined as sums of line integrals of delta functions are then shown to reproduce the charge and current densities in the required fashion; it is noted that the components of these tensors, although involving instantaneous integrals along spatial curves, are defined in the same way by all observers related by homogeneous Lorentz transformations, provided only that the speed of any point of the curves is less than c. The general polarisation-magnetisation tensor is derived through a transformation generated by an arbitrary pseudovector field. The explicit form of this field, as well as of a subsidiary pseudoscalar field, is obtained for those transformations that interrelate polarisation-magnetisation tensors of the line integral kind.