Casimir interaction between normal or superfluid grains in the Fermi sea

We report on a new force that acts on cavities (literally empty regions of space) when they are immersed in a background of non-interacting fermionic matter fields. The interaction follows from the obstructions to the (quantum mechanical) motion of the fermions caused by the presence of bubbles or other (heavy) particles in the Fermi sea, such as, for example, nuclei in the neutron sea in the inner crust of a neutron star or superfluid grains in a normal Fermi liquid. The effect resembles the traditional Casimir interaction between metallic mirrors in the vacuum. However, the fluctuating electromagnetic fields are replaced by fermionic matter fields. We show that the fermionic Casimir problem for a system of spherical cavities can be solved exactly, since the calculation can be mapped onto a quantum mechanical billiard problem of a point-particle scattered off a finite number of non-overlapping spheres or discs. Finally, we generalize the map method to other Casimir systems, especially to the case of a fluctuating massless scalar field between two spheres or a sphere and a plate under Dirichlet boundary conditions.