Surprises in theoretical Casimir physics : quantum forces in inhomogeneous media

Abstract

This thesis considers the problem of determining Casimir-Lifshitz forces in inhomogeneous media. The ground-state energy of the electromagnetic field in a piston-geometry is discussed. When the cavity is empty, the Casimir pressure on the piston is finite and independent of the small-scale physics of the media that compose the mirrors. However, it is demonstrated that, when the cavity is filled with an inhomogeneous dielectric medium, the Casimir energy is cut-off dependent. The local behavior of the stress tensor commonly used in calculations of Casimir forces is also determined. It is shown that the usual expression for the stress tensor is not finite anywhere within such a medium, whatever the temporal dispersion or index profile, and that this divergence is unlikely to be removed by modifying the regularisation. These findings suggest that the value of the Casimir pressure may be inextricably dependent on the detailed behavior of the mirror and the medium at large wave vectors. This thesis also examines two exceptions to this rule: first, the case of an idealised metamaterial is considered which, when introduced into a cavity, reduces the magnitude of the Casimir force. It is shown that, although the medium is inhomogeneous, it does not contribute additional scattering events but simply modifies the effective length of the cavity, so the predicted force is finite and can be stated exactly. Secondly, a geometric argument is presented for determining a Casimir stress in a spherical mirror filled with the inhomogeneous medium of Maxwell’s fish-eye. This solution questions the idea that the Casimir force of a spherical mirror is repulsive, but prompts additional questions concerning regularisation and the role of non-local effects in determining Casimir forces.