Casimir Energies and Pressures for δ\delta-function Potentials

The Casimir energies and pressures for a massless scalar field associated with $\delta$-function potentials in 1+1 and 3+1 dimensions are calculated. For parallel plane surfaces, the results are finite, coincide with the pressures associated with Dirichlet planes in the limit of strong coupling, and for weak coupling do not possess a power-series expansion in 1+1 dimension. The relation between Casimir energies and Casimir pressures is clarified,and the former are shown to involve surface terms. The Casimir energy for a $\delta$-function spherical shell in 3+1 dimensions has an expression that reduces to the familiar result for a Dirichlet shell in the strong-coupling limit. However, the Casimir energy for finite coupling possesses a logarithmic divergence first appearing in third order in the weak-coupling expansion, which seems unremovable. The corresponding energies and pressures for a derivative of a $\delta$-function potential for the same spherical geometry generalizes the TM contributions of electrodynamics. Cancellation of divergences can occur between the TE ($\delta$-function) and TM (derivative of $\delta$-function) Casimir energies. These results clarify recent discussions in the literature.Comment: 16 pages, 1 eps figure, uses REVTeX

Casimir Energy of a Spherical Shell

The Casimir energy for a conducting spherical shell of radius $a$ is computed using a direct mode summation approach. An essential ingredient is the implementation of a recently proposed method based on Cauchy's theorem for an evaluation of the eigenfrequencies of the system. It is shown, however, that this earlier calculation uses an improper set of modes to describe the waves exterior to the sphere. Upon making the necessary corrections and taking care to ensure that no mathematically ill-defined expressions occur, the technique is shown to leave numerical results unaltered while avoiding a longstanding criticism raised against earlier calculations of the Casimir energy.Comment: LaTeX, 14 pages, 1 figur

Vector Casimir effect for a D-dimensional sphere

The Casimir energy or stress due to modes in a D-dimensional volume subject to TM (mixed) boundary conditions on a bounding spherical surface is calculated. Both interior and exterior modes are included. Together with earlier results found for scalar modes (TE modes), this gives the Casimir effect for fluctuating ``electromagnetic'' (vector) fields inside and outside a spherical shell. Known results for three dimensions, first found by Boyer, are reproduced. Qualitatively, the results for TM modes are similar to those for scalar modes: Poles occur in the stress at positive even dimensions, and cusps (logarithmic singularities) occur for integer dimensions $D\le1$. Particular attention is given the interesting case of D=2.Comment: 20 pages, 1 figure, REVTe

Casimir Energy for a Spherical Cavity in a Dielectric: Applications to Sonoluminescence

In the final few years of his life, Julian Schwinger proposed that the ``dynamical Casimir effect'' might provide the driving force behind the puzzling phenomenon of sonoluminescence. Motivated by that exciting suggestion, we have computed the static Casimir energy of a spherical cavity in an otherwise uniform material. As expected the result is divergent; yet a plausible finite answer is extracted, in the leading uniform asymptotic approximation. This result agrees with that found using zeta-function regularization. Numerically, we find far too small an energy to account for the large burst of photons seen in sonoluminescence. If the divergent result is retained, it is of the wrong sign to drive the effect. Dispersion does not resolve this contradiction. In the static approximation, the Fresnel drag term is zero; on the mother hand, electrostriction could be comparable to the Casimir term. It is argued that this adiabatic approximation to the dynamical Casimir effect should be quite accurate.Comment: 23 pages, no figures, REVTe

Observability of the Bulk Casimir Effect: Can the Dynamical Casimir Effect be Relevant to Sonoluminescence?

The experimental observation of intense light emission by acoustically driven, periodically collapsing bubbles of air in water (sonoluminescence) has yet to receive an adequate explanation. One of the most intriguing ideas is that the conversion of acoustic energy into photons occurs quantum mechanically, through a dynamical version of the Casimir effect. We have argued elsewhere that in the adiabatic approximation, which should be reliable here, Casimir or zero-point energies cannot possibly be large enough to be relevant. (About 10 MeV of energy is released per collapse.) However, there are sufficient subtleties involved that others have come to opposite conclusions. In particular, it has been suggested that bulk energy, that is, simply the naive sum of ${1\over2}\hbar\omega$, which is proportional to the volume, could be relevant. We show that this cannot be the case, based on general principles as well as specific calculations. In the process we further illuminate some of the divergence difficulties that plague Casimir calculations, with an example relevant to the bag model of hadrons.Comment: 13 pages, REVTe

Electromagnetic wave scattering by a superconductor

The interaction between radiation and superconductors is explored in this paper. In particular, the calculation of a plane standing wave scattered by an infinite cylindrical superconductor is performed by solving the Helmholtz equation in cylindrical coordinates. Numerical results computed up to $\mathcal{O}(77)$ of Bessel functions are presented for different wavelengths showing the appearance of a diffraction pattern.Comment: 3 pages, 3 figure

Optical BCS conductivity at imaginary frequencies and dispersion energies of superconductors

We present an efficient expression for the analytic continuation to arbitrary complex frequencies of the complex optical and AC conductivity of a homogeneous superconductor with arbitrary mean free path. Knowledge of this quantity is fundamental in the calculation of thermodynamic potentials and dispersion energies involving type-I superconducting bodies. When considered for imaginary frequencies, our formula evaluates faster than previous schemes involving Kramers--Kronig transforms. A number of applications illustrates its efficiency: a simplified low-frequency expansion of the conductivity, the electromagnetic bulk self-energy due to longitudinal plasma oscillations, and the Casimir free energy of a superconducting cavity.Comment: 20 pages, 7 figures, calculation of Casimir energy adde

Casimir effect for the scalar field under Robin boundary conditions: A functional integral approach

In this work we show how to define the action of a scalar field in a such a way that Robin boundary condition is implemented dynamically, i.e., as a consequence of the stationary action principle. We discuss the quantization of that system via functional integration. Using this formalism, we derive an expression for the Casimir energy of a massless scalar field under Robin boundary conditions on a pair of parallel plates, characterized by constants $c_1$ and $c_2$. Some special cases are discussed; in particular, we show that for some values of $c_1$ and $c_2$ the Casimir energy as a function of the distance between the plates presents a minimum. We also discuss the renormalization at one-loop order of the two-point Green function in the $\lambda\phi^4$ theory submitted to Robin boundary condition on a plate.Comment: 16 pages, 2 figures. Version 2: contains a new section on the renormalization of the two-point Green function in the presence of a flat boundary. Accepted for publication in J. Phys.

Identity of the van der Waals Force and the Casimir Effect and the Irrelevance of these Phenomena to Sonoluminescence

We show that the Casimir, or zero-point, energy of a dilute dielectric ball, or of a spherical bubble in a dielectric medium, coincides with the sum of the van der Waals energies between the molecules that make up the medium. That energy, which is finite and repulsive when self-energy and surface effects are removed, may be unambiguously calculated by either dimensional continuation or by zeta function regularization. This physical interpretation of the Casimir energy seems unambiguous evidence that the bulk self-energy cannot be relevant to sonoluminescence.Comment: 7 pages, no figures, REVTe