Path-integral approach to the dynamic Casimir effect with fluctuating boundaries

A path-integral formulation is developed for the dynamic Casimir effect. It allows us to study small deformations in space and time of the perfectly reflecting (conducting) boundaries of a cavity. The mechanical response of the intervening vacuum is calculated to linear order in the frequency-wave-vector plane, using which a plethora of interesting phenomena can be studied. For a single corrugated plate we find a correction to mass at low frequencies and an effective shear viscosity at high frequencies that are both anisotropic. The anisotropy is set by the wave vector of the corrugation. For two plates, the mass renormalization is modified by a function of the ratio between thr: separation of the plates and the wavelength of corrugations. The dissipation rate is not modified for frequencies below the lowest optical mode of the cavity and there is a resonant dissipation for all frequencies greater than that. In this regime, a divergence in the response function implies that such high-frequency deformation modes of the cavity cannot be excited by any macroscopic external forces. This phenomenon is intimately related to resonant particle creation. For particular examples of two corrugated plates that are stationary, or moving uniformly in the lateral directions, Josephson-like effects are observed. For capillary waves on the surface of mercury a renormalization to surface tension and sound velocity is obtained.