Multipole theory and the Hehl–Obukhov decomposition of the electromagnetic constitutive tensor

Abstract

The Hehl–Obukhov decomposition expresses the 36 independent components of the electromagnetic constitutive tensor for a local linear anisotropic medium in a useful general form comprising seven macroscopic property tensors: four of second rank, two vectors, and a four-dimensional (pseudo)scalar. We consider homogeneous media and show that in semi-classical multipole theory, the first full realization of this formulation is obtained (in terms of molecular polarizability tensors) at third order (electric octopole–magnetic quadrupole order). The calculations are an extension of a direct method previously used at second order (electric quadrupole–magnetic dipole order). We consider in what sense this theory is independent of the choice of molecular coordinate origins relative to which polarizabilities are evaluated. The pseudoscalar (axion) observable is expressed relative to the crystallographic origin. The other six property tensors are invariant (with respect to an arbitrary choice of each molecular coordinate origin), or zero, at first and second orders. At third order, this invariance has to be imposed (by transformation of the response fields)—an aspect that is required by consideration of isotropic fluids and is consistent with the invariance of transmission phenomena in dielectrics. Alternative derivations of the property tensors are reviewed, with emphasis on the pseudoscalar, constraint-breaking, translational invariance, and uniqueness.