Traceless multipole moment densities and transformations in macroscopic electromagnetism

We consider whether use of traceless multipole moment densities in macroscopic electromagnetism can yield physically acceptable results. For harmonic plane wave fields it is shown that a traceless electric quadrupole density yields linear constitutive relations for which the dynamical material constants (permittivity and magnetoelectric coefficients) and response fields are unphysical. We further show that, within multipole theory, these constitutive relations cannot be transformed into physically acceptable relations. Specifically, the transformed response field D is unphysical for all orders beyond the electric dipole. This contrasts with use of primitive (traced) moment densities, for which unphysical constitutive relations have been successfully transformed up to electric octopole-magnetic quadrupole order, thereby providing also the leading contribution to the ac permeability.