Normal and lateral Casimir interactions between semi-infinite conductors in the presence of a dispersive medium

Path-integral formalism is employed to study normal and lateral Casimir interactions in a system composed of a dispersive medium surrounded by two semi-infinite ideal conductors. The dispersive medium is modeled by a continuum of harmonic oscillators, and it is shown that for smooth conductors, the normal force at small distances in the presence of a dispersive medium coincides with the original Casimir force, while at large distances, it tends to the original form with a renormalized coefficient. The correction to the normal force because of the roughness on one of the conductors is calculated. When the inner surfaces of both conductors have roughness, the lateral Casimir interaction occurs because of translational symmetry breaking, which is studied. It is shown that both normal and lateral Casimir forces in the presence of a dispersive medium are weaker in comparison with the original one and are proportional to the roughness amplitude squared. The dependence of the normal and lateral interactions on the memory and strength of the dispersive medium is considered.