The representation of microscopic charge and current densities in terms of polarization and magnetization fields

The microscopic charge and current densities due to an aggregate of charged point particles are shown to be derivable from polarization and magnetization fields defined as sums of line integrals of delta functions along curves joining an arbitrary reference point, which may be moving, to the positions of the particles. The analysis given generalizes previous treatm ents th at deal with a fixed reference point and includes the description of ionic, free electronic and Rontgen currents. The class of ‘admissible’ polarization and magnetization fields, which give rise to a specified charge and current distribution, is shown to be generated by two arbitrary differentiable fields, one scalar and one vector. The class of these latter fields that effect the transformation connecting two given pairs of admissible polarization and magnetization fields is shown in turn to be generated by two arbitrary scalar fields, one a function of position and time and the other a function of time only. The transformation rules are verified for polarization and magnetization fields that are representable as sums of line integrals of delta functions, and the scalar and vector fields that appear in the transformations are identified with integrals over certain moving surfaces and volumes. By means of these identifications it is demonstrated that the class of polarization and magnetization fields that are representable as sums of line integrals of delta functions forms a proper subset of the total class of admissible fields, so that not every admissible polarization or magnetization field is so representable.