Casimir effect for a semitransparent wedge and an annular piston

Abstract

We consider the Casimir energy due to a massless scalar field in a geometry of an infinite wedge closed by a Dirichlet circular cylinder, where the wedge is formed by δ-function potentials, so-called semitransparent boundaries. A finite expression for the Casimir energy corresponding to the arc and the presence of both semitransparent potentials is obtained, from which the torque on the sidewalls can be derived. The most interesting part of the calculation is the nontrivial nature of the angular mode functions. Numerical results are obtained which are closely analogous to those recently found for a magnetodielectric wedge, with the same speed of light on both sides of the wedge boundaries.Alternative methods are developed for annular regions with radial semitransparent potentials, based on reduced Green's functions for the angular dependence, which allows calculations using the multiple-scattering formalism. Numerical results corresponding to the torque on the radial plates are likewise computed, which generalize those for the wedge geometry. Generally useful formulas for calculating Casimir energies in separable geometries are derived.

Investigations of the torque anomaly in an annular sector. I. Global calculations, scalar case

“…Indeed, when a smaller plot range is specified the results agree rather closely with those found in Ref. [8]. There, only three-body (wedge-inner cylinderouter cylinder) effects were considered from the outset.…”

“…These curves are very similar to those found for an annular piston in Ref. [8], the difference there being that both sides of the radial walls are considered, so E (α) → E (α)+E (2π−α), so a similar attraction appears near α = 2π. Indeed, when a smaller plot range is specified the results agree rather closely with those found in Ref.…”

“…[7]. However, because of the logarithmic curvature divergences, there is an ambiguity in the linear dependence on the wedge angle in both the divergent and finite parts of the energy; if this arbitrary linear dependence is removed by a process of "renormalization," all divergences and anomalies disappear, and the physical torque is the negative derivative with respect to the wedge angle of the physical energy, in agreement with previous considerations of the annular piston [8].…”

“…Let us simplify the following discussion by considering Dirichlet wedge surfaces, where the eigenvalues are explicit: 8) and the eigenfunctions are explicitly…”

“…Although not classically convergent, the z integral has a well-defined meaning: 8) and then, approximating the sum on m by an integral:…”