Isotropically averaged polarisation and magnetisation fields

Abstract

In the minimal-coupling Lagrangian for the interaction of the electromagnetic field and nonrelativistic charged particles, the charge and current densities are coupled to scalar and vector potentials. In the multipolar Lagrangian, on the other hand, the aggregate or particles is particularly (through sometimes completely) described by polarisation and magnetisation fields and these are coupled to the electric and magnetic induction fields. It is shown that if isotropically averaged polarisation and magnetisation fields are introduced, the minimal-coupling and multipolar Lagrangians are identical, provided also the potentials used are those of the Coulomb gauge. The associated canonical transformation of the Hamiltonian is the identity transformation. Thus the minimal-coupling Hamiltonian can be written directly in multipolar form, without any change in the canonical dynamical variables.