Quantum electrodynamics in the presence of dielectrics and conductors. II. Theory of dispersion forces

The different kinds of response functions introduced in a previous paper are used to calculate the dispersion forces. An exact expression for the interaction energy in a system of harmonic oscillators interacting with a second-quantized radiation field is obtained in terms of appropriate response functions. The radiation field may be either the field appropriate to entire free space or the field altered by the presence of the dielectric. The result is valid for arbitrary geometries involving dielectric and conducting surfaces. An expansion of our result in powers of $e^2$ leads to the results of other authors. The calculation of the dispersion force for the case of excited states is also briefly discussed. Next the problem of the dispersion force between macroscopic bodies is considered. Lifshitz's expression for the dispersion force is rederived using the response functions and the two methods are compared. The role of surface modes in the determination of dispersion force is discussed. Finally the dispersion force between a spatially dispersive and spatially nondispersive dielectric is calculated exactly. When the spatial dispersion is weak, then it is found that the first-order term is repulsive in nature, in contrast to the zeroth-order term which is attractive. As a by-product of our analysis, surface-polariton dispersion relations are obtained.