Many-body optics I. Dielectric constants and optical dispersion relations

Abstract

This paper is the first of three papers constituting a preliminary mathematical discussion preparatory to the presentation of a series of papers on the observable macroscopic optical properties of a many-body molecular fluid. These properties are viewed wholly as consequences of microscopic many-body interactions, and the theory to be presented is largely concerned with a microscopic theory of the complex refractive index and associated optical response functions: it is nowhere phenomenological. In this paper the classical integral equation, which can be derived from the quantal theory and is fundamental to the optical theory, is presented in limited form (restricted to two-particle interactions) and rigorously solved. Both longitudinal and transverse dispersion relations are obtained and these are generalized to include correlations between sets of particles of any number and to two- (and by implication to multi-) component molecular-fluid systems. All the longitudinal dispersion relations are new results. The theory is strictly a refractive index theory and not a dielectric constant theory: in consequence, the longitudinal dispersion relations admit frequencies close to, but not precisely at, the zeros of the corresponding transverse wave number, and this departure from the conventional picture is associated with the theory of optical scattering. The `local' optical field is due to two- and many-particle complex interaction terms: it mixes `transverse photons' and `longitudinal photons', and both kinds of photon contribute to each dispersion relation. The theory of how this complex local field describes external scattering, Cerenkov radiation and the contribution of virtual photon exchange to the ground-state energy will be developed later. For a theory of external scattering it is necessary to break translational invariance: as a consequence, the optical `extinction theorem', due in the first instance to P. P. Ewald, plays a very significant role in the theory.