Scalar Casimir Energies of Tetrahedra and Prisms

Journal of Physics A Mathematical and Theoretical (Impact Factor: 1.77). 02/2012; DOI: 10.1088/1751-8113/45/42/425401
Source: arXiv

ABSTRACT New results for scalar Casimir self-energies arising from interior modes are
presented for the three integrable tetrahedral cavities. Since the eigenmodes
are all known, the energies can be directly evaluated by mode summation, with a
point-splitting regulator, which amounts to evaluation of the cylinder kernel.
The correct Weyl divergences, depending on the volume, surface area, and the
edges, are obtained, which is strong evidence that the counting of modes is
correct. Because there is no curvature, the finite part of the quantum energy
may be unambiguously extracted. Cubic, rectangular parallelepipedal, triangular
prismatic, and spherical geometries are also revisited. Dirichlet and Neumann
boundary conditions are considered for all geometries. Systematic behavior of
the energy in terms of geometric invariants for these different cavities is
explored. Smooth interpolation between short and long prisms is further
demonstrated. When scaled by the ratio of the volume to the surface area, the
energies for the tetrahedra and the prisms of maximal isoareal quotient lie
very close to a universal curve. The physical significance of these results is

  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We derive boundary conditions for electromagnetic fields on a $\delta$-function plate. The optical properties of such a plate are shown to necessarily be anisotropic in that they only depend on the transverse properties of the plate. We unambiguously obtain the boundary conditions for a perfectly conducting $\delta$-function plate in the limit of infinite dielectric response. We show that a material does not "optically vanish" in the thin-plate limit. The thin-plate limit of a plasma slab of thickness $d$ with plasma frequency $\omega_p^2=\zeta_p/d$ reduces to a $\delta$-function plate for frequencies ($\omega=i\zeta$) satisfying $\zeta d \ll \sqrt{\zeta_p d} \ll 1$. We show that the Casimir interaction energy between two parallel perfectly conducting $\delta$-function plates is the same as that for parallel perfectly conducting slabs. Similarly, we show that the interaction energy between an atom and a perfect electrically conducting $\delta$-function plate is the usual Casimir-Polder energy, which is verified by considering the thin-plate limit of dielectric slabs. The "thick" and "thin" boundary conditions considered by Bordag are found to be identical in the sense that they lead to the same electromagnetic fields.
    Physical review D: Particles and fields 06/2012; 86(8).
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: In an attempt to understand a recently discovered torque anomaly in quantum field theory with boundaries, we calculate the Casimir energy and torque of a scalar field subject to Dirichlet boundary conditions on an annular sector defined by two coaxial cylinders intercut by two planes through the axis. In this model the particularly troublesome divergence at the cylinder axis does not appear, but new divergences associated with the curved boundaries are introduced. All the divergences associated with the volume, the surface area, the corners, and the curvature are regulated by point separation either in the direction of the axis of the cylinder or in the (Euclidean) time; the full divergence structure is isolated, and the remaining finite energy and torque are extracted. Formally, only the regulator based on axis splitting yields the expected balance between energy and torque. Because of the logarithmic curvature divergences, there is an ambiguity in the linear dependence of the energy on the wedge angle; if the terms constant and linear in this angle are removed by a process of renormalization, the expected torque-energy balance is preserved.
    Physical review D: Particles and fields 06/2013; 88(2).
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Based on calculations involving an idealized boundary condition, it has long been assumed that the stress on a spherical conducting shell is repulsive. We use the more realistic case of a Drude dielectric to show that the stress is attractive, matching the generic behavior of Casimir forces in electromagnetism. We trace the discrepancy between these two cases to interactions between the electromagnetic quantum fluctuations and the dielectric material.
    Physics Letters B 05/2013; · 4.57 Impact Factor

Full-text (3 Sources)

Available from
May 26, 2014

E. K. Abalo