Translational invariance, the Post constraint and uniqueness in macroscopic electrodynamics

Abstract

We consider semi-classical multipole theory for non-magnetic molecules interacting with harmonic plane electromagnetic waves, to electric octopole–magnetic quadrupole order and relative to an arbitrary set of molecular coordinate origins {On}. Spatial averaging of expectation values of induced molecular multipole moments produces a macroscopic theory for linear, homogeneous, anisotropic media that has three shortcomings: it is only partially invariant with respect to {On}, it is ambivalent on the Post constraint (equality of the traces of the magnetoelectric tensors), and it yields non-unique dynamic response fields D and H. To remedy these, we present a fully invariant theory that is consistent (affirmative) on the Post constraint, and is based on five time-even, invariant molecular polarizability tensors (one each of electric dipole and electric quadrupole–magnetic dipole order, and three of electric octopole–magnetic quadrupole order). As in previous work on linear phenomena, translational invariance is achieved through the Van Vleck–Buckingham condition. Uniqueness of the invariant response fields is demonstrated, based on linear independence of molecular polarizability tensors at each multipole order above electric dipole. Our results are compared with previously published expressions for two invariant polarizabilities.