  A quantum theory of dispersion for an inhomogeneous solid is obtained, from a
starting point of multipolar coupled atoms interacting with an electromagnetic
field. The dispersion relations obtained are equivalent to the standard
classical Sellmeir equations obtained from the Drude-Lorentz model. In the
homogeneous (plane-wave) case, we obtain the detailed quantum mode structure of
the coupled polariton fields, and show that the mode expansion in all branches
of the dispersion relation is completely defined by the refractive index and
the group-velocity for the polaritons. We demonstrate a straightforward
procedure for exactly diagonalizing the Hamiltonian in one, two or
three-dimensional environments, even in the presence of longitudinal
phonon-exciton dispersion, and an arbitrary number of resonant transitions with
different frequencies. This is essential, since it is necessary to include at
least one phonon (I.R.) and one exciton (U.V.) mode, in order to accurately
represent dispersion in transparent solid media. Our method of diagonalization
does not require an explicit solution of the dispersion relation, but relies
instead on the analytic properties of Cauchy contour integrals over all
possible mode frequencies. When there is longitudinal phonon dispersion, the
relevant group-velocity term is modified so that it only includes the purely
electromagnetic part of the group velocity.
