Casimir Effect for a Perfectly Conducting Wedge in Terms of Local Zeta Function

Abstract

The vacuum energy density of electromagnetic field inside a perfectly conducting wedge is calculated by making use of the local zeta function technique. This regularization completely eliminates divergent expressions in the course of calculations and gives rise to a finite expression for the energy density in question without any subtractions. Employment of the Hertz potentials for constructing the general solution to the Maxwell equations results in a considerable simplification of the calculations. Transition to the global zeta function is carried out by introducing a cutoff nearby the cusp at the origin. Proceeding from this the heat kernel coefficients are calculated and the high temperature asymptotics of the Helmholtz free energy and of the torque of the Casimir forces are found. The wedge singularity gives rise to a specific high temperature behaviour ∼ T 2 of the quantities under consideration. The obtained results are directly applicable to the free energy of a scalar massless field and electromagnetic field on the background of a cosmic string.

Scalar Casimir Energies for Separable Coordinate Systems: Application to Semi-Transparent Planes in an Annulus

“…We will enforce Dirichlet boundary condition on the inner and outer cylinder. This is similar to situations studied by Nesterenko et al 15,16 for global Casimir energies for the case of one circular boundary and by Saharian et al 17,18 for the local properties of the stress energy tensor for the case of both one and two circular boundaries. The radial potentials will be semi-transparent deltafunction potentials in the angular coordinates, λ 2 δ(θ − α).…”

Local Zeta Regularization and the Scalar Casimir Effect

“…We also consider a variation of this configuration, which corresponds essentially to identify the sides of the wedge; this is the so-called case of the "cosmic string" (see subsection 5.9). Some of these cases have already been treated by Dowker et al [12,13], Deutsch and Candelas [11] (also discussing the electromagnetic case) and, more recently, by Saharian et al [29,31] and by Fulling et al [23] (see also the citations in these works and [6,7,26]); nearly all of these authors use the point splitting approach, or some variant of it. More in detail, in [12] and [11] attention is restricted to the conformal part of the stress-energy VEV for either Dirichlet or Neumann boundary conditions, while in [29,31] also the non-conformal part is considered, but in the Dirichlet case only; in [23], instead, the authors only show the graphs of the energy density and of the pressure components (for which no explicit expression is given), derived via a point-splitting approach for several configurations and various choices of the parameters describing the theory.…”

Electrodynamic Casimir effect in a medium-filled wedge

“…The Casimir energy and stress in a wedge geometry was approached already in the 1970s [2,3]. Since that time, various embodiments of the wedge with perfectly conducting walls have been treated by Brevik and co-workers [4,5,6] and others [7]. More recently a wedge intercut by a cylindrical shell was considered by Nesterenko and collaborators, first for a semicircular wedge [8], then for arbitrary dihedral angle [9].…”