Semiclassical estimates of electromagnetic Casimir self-energies of spherical and cylindrical metallic shells

The leading semiclassical estimates of the electromagnetic Casimir stresses on a spherical and a cylindrical metallic shell are within $1\%$ of the field theoretical values. The electromagnetic Casimir energy for both geometries is given by two decoupled massless scalars that satisfy conformally covariant boundary conditions. Surface contributions vanish for smooth metallic boundaries, and the finite electromagnetic Casimir energy in leading semiclassical approximation is due to quadratic fluctuations about periodic rays in the interior of the cavity only. Semiclassically, the nonvanishing Casimir energy of a metallic cylindrical shell is almost entirely due to Fresnel diffraction.

Figure 1

Classical periodic rays of a spherical and a cylindrical cavity. (a) The shortest primitive rays in sectors $(n,w)\in \{(2,1),(4,1),(6,2)\}$ that contribute to the electromagnetic SCE. (b) Closed paths in sectors $(n,w)\in \{(1,0),(5,1),(5,2)\}$ that reflect an odd number of times off the surface and whose contribution to the electromagnetic SCE vanishes. Caustic surfaces are indicated as thin circles. The dashed part of any trajectory is on one sheet, and its solid part is on the other sheet of a two-sheeted covering space. The phase space of the (5,2) sector is indicated by the hatched area. Note that the caustics are of second order for a spherical cavity but are of first order for a cylindrical one.

Figure 2

Integration over the longitudinal momentum fraction. The upper solid curve shows the exact integral in Eq. (22) over the longitudinal momentum fraction $\alpha$; the lower solid line is its semiclassical approximation to quadratic order in the fluctuations. The dashed curve gives the simple exponential approximation $e^{-\pi x/4}$.