A theory of the Casimir effect for compact regions (original) (raw)

The Casimir effect for parallel plates revisited

Journal of Mathematical Physics, 2007

The Casimir effect for a massless scalar field with Dirichlet and periodic boundary conditions (b.c.) on infinite parallel plates is revisited in the local quantum field theory (lqft) framework introduced by B.Kay. The model displays a number of more realistic features than the ones he treated. In addition to local observables, as the energy density, we propose to consider intensive variables, such as the energy per unit area ε, as fundamental observables. Adopting this view, lqft rejects Dirichlet (the same result may be proved for Neumann or mixed) b.c., and accepts periodic b.c.: in the former case ε diverges, in the latter it is finite, as is shown by an expression for the local energy density obtained from lqft through the use of the Poisson summation formula. Another way to see this uses methods from the Euler summation formula: in the proof of regularization independence of the energy per unit area, a regularization-dependent surface term arises upon use of Dirichlet b.c., but not periodic b.c.. For the conformally invariant scalar quantum field, this surface term is absent, due to the condition of zero trace of the energy momentum tensor, as remarked by B.De Witt. The latter property does not hold in

The Casimir effect: some aspects

Brazilian Journal of Physics, 2006

We start this paper with a historical survey of the Casimir effect, showing that its origin is related to experiments on colloidal chemistry. We present two methods of computing Casimir forces, namely: the global method introduced by Casimir, based on the idea of zero-point energy of the quantum electromagnetic field, and a local one, which requires the computation of the energy-momentum stress tensor of the corresponding field. As explicit examples, we calculate the (standard) Casimir forces between two parallel and perfectly conducting plates and discuss the more involved problem of a scalar field submitted to Robin boundary conditions at two parallel plates. A few comments are made about recent experiments that undoubtedly confirm the existence of this effect. Finally, we briefly discuss a few topics which are either elaborations of the Casimir effect or topics that are related in some way to this effect as, for example, the influence of a magnetic field on the Casimir effect of charged fields, magnetic properties of a confined vacuum and radiation reaction forces on non-relativistic moving boundaries.

Casimir force: an alternative treatment

2009

The Casimir force between two parallel uncharged closely spaced metallic plates is evaluated in ways alternatives to those usually considered in the literature. In a first approximation we take in account the suppressed quantum numbers of a cubic box, representing a cavity which was cut in a metallic block. We combine these ideas with those of the MIT bag model of hadrons, but adapted to nonrelativistic particles. In a second approximation we consider the particles occupying the energy levels of a Bohr atom, so that the Casimir force depends explicitly on the fine structure constant α. In both treatments, the mean energies which have explicit dependence on the particle mass and on the maximum occupied quantum number (related to the Fermi level of the system) at the beginning of the calculations, have these dependences mutually canceled at the end of them. Finally by comparing the averaged energies computed in both approximations, we are able to make an estimate of the value of the fine structure constant α.

Casimir Effect: The Classical Limit

Annals of Physics, 2001

We analyze the high temperature (or classical) limit of the Casimir effect. A useful quantity which arises naturally in our discussion is the "relative Casimir energy", which we define for a configuration of disjoint conducting boundaries of arbitrary shapes, as the difference of Casimir energies between the given configuration and a configuration with the same boundaries infinitely far apart. Using path integration techniques, we show that the relative Casimir energy vanishes exponentially fast in temperature. This is consistent with a simple physical argument based on Kirchhoff's law. As a result the "relative Casimir entropy", which we define in an obviously analogous manner, tends, in the classical limit, to a finite asymptotic value which depends only on the geometry of the boundaries. Thus the Casimir force between disjoint pieces of the boundary, in the classical limit, is entropy driven and is governed by a dimensionless number characterizing the geometry of the cavity. Contributions to the Casimir thermodynamical quantities due to each individual connected component of the boundary exhibit logarithmic deviations in temperature from the behavior just described. These logarithmic deviations seem to arise due to our difficulty to separate the Casimir energy (and the other thermodynamical quantities) from the "electromagnetic" self-energy of each of the connected components of the boundary in a well defined manner. Our approach to the Casimir effect is not to impose sharp boundary conditions on the fluctuating field, but rather take into consideration its interaction with the plasma of "charge carriers" in the boundary, with the plasma frequency playing the role of a physical UV cutoff. This also allows us to analyze deviations from a perfect conductor behavior.

Casimir forces between arbitrary compact objects

Journal of Physics A: Mathematical and Theoretical, 2008

We develop an exact method for computing the Casimir energy between arbitrary compact objects, both with boundary conditions for a scalar field and dielectrics or perfect conductors for the electromagnetic field. The energy is obtained as an interaction between multipoles, generated by quantum source or current fluctuations. The objects' shape and composition enter only through their scattering matrices. The result is exact when all multipoles are included, and converges rapidly. A low frequency expansion yields the energy as a series in the ratio of the objects' size to their separation. As examples, we obtain this series for two spheres with Robin boundary conditions for a scalar field and dielectric spheres for the electromagnetic field. The full interaction at all separations is obtained for spheres with Robin boundary conditions and for perfectly conducting spheres. Submitted to: J. Phys. A: Math. Gen. ‡ This presentation is based on work performed in collaboration with N. Graham and M. Kardar. For a complete exposition, see Refs. [6] and [7].

A Computation of the Casimir Energy on a Parallelepiped

The original computations deriving the Casimir energy and force consists of first taking limits of the spectral zeta function and afterwards analytically extending the result. This process of computation presents a weakness in Hendrik Casimir's original argument since limit and analytic continuation do not commute. A case of the Laplacian on a parallelepiped box representing the space as the vacuum between two plates modelled with Dirichlet and periodic Neumann boundary conditions is constructed to address this anomaly. It involves the derivation of the regularised zeta function in terms of the Riemann zeta function on the parallelepiped. The values of the Casimir energy and Casimir force obtained from our derivation agree with those of Hendrik Casimir.

An intuitive picture of the Casimir effect

The Casimir effect, which predicts the emergence of an attractive force between two parallel, highly reflecting plates in vacuum, plays a vital role in various fields of physics, from quantum field theory and cosmology to nanophotonics and condensed matter physics. Nevertheless, Casimir forces still lack an intuitive explanation and current derivations rely on regularisation procedures to remove infinities. Starting from special relativity and treating space and time coordinates equivalently, this paper overcomes no-go theorems of quantum electrodynamics and obtains a local relativistic quantum description of the electromagnetic field in free space. When extended to cavities, our approach can be used to calculate Casimir forces directly in position space without the introduction of cut-off frequencies.

The Casimir effect in a solid ball when εμ = 1

Annals of Physics, 1982

The Casimir surface force density on a solid ball is calculated assuming that the medium satisfies the relationship y = 1, E being the permittivity and p the permeability. Remarkably enough, the cutoff problems which otherwise plague calculations of the Casimir stress on dielectric nonmagnetic balls disappear, and one arrives at a cutoff independent, finite, and repulsive result for the force. When p + 0 or p-t a, one finds exactly the same result as in the case of a perfectly conducting shell. Another virtue of the theory is that one may avoid the subtraction of contact terms, as the contact terms are simply vanishing. Finally, the theory is immediately applicable to a gluonic (nonquark) bag in QCD, assuming no gluonic interaction. The surface force in QCD becomes repulsive, just as in QED. I. INTRODUCTION The old semiclassical electron model of Casimir [ 11, according to which the "electron" is pictured as a perfectly conducting spherical shell, has recently been subject to renewed interest. Casimir's idea was that the electromagnetic zero-point fluctuations would give rise to forces stabilizing the electron against the Coulomb repulsion. After specific calculations had been carried out it became clear, however, that the Casimir stress is repulsive, thus contrary to Casimir's expectation. Boyer [2] was the first to show this fact. The calculation has later on been improved by several others 13-51, and the result is now known to great accuracy. Due to delicate cancellations between interior and exterior energy contributions the result turns out to be independent of the cutoff parameter. An interesting generalization of Casimir's electron model is obtained if one regards the electron not as a perfectly conducting shell but instead as a dielectric compact spherical bull. Milton [6] has recently investigated such a possibility, assuming the interior medium to be nonmagnetic. He managed to isolate finite terms describing an attactive force caused by the zero-point fluctuations in accordance with Casimir's idea, although a characteristic difficulty in the formalism was the presence of 179

Casimir forces between compact objects: The scalar case

Physical Review D - Particles, Fields, Gravitation and Cosmology, 2008

We have developed an exact, general method to compute Casimir interactions between a finite number of compact objects of arbitrary shape and separation. Here, we present details of the method for a scalar field to illustrate our approach in its most simple form; the generalization to electromagnetic fields is outlined in Ref. . The interaction between the objects is attributed to quantum fluctuations of source distributions on their surfaces, which we decompose in terms of multipoles. A functional integral over the effective action of multipoles gives the resulting interaction. Each object's shape and boundary conditions enter the effective action only through its scattering matrix. Their relative positions enter through universal translation matrices that depend only on field type and spatial dimension. The distinction of our method from the pairwise summation of two-body potentials is elucidated in terms of the scattering processes between three objects. To illustrate the power of the technique, we consider Robin boundary conditions φ − λ∂nφ = 0, which interpolate between Dirichlet and Neumann cases as λ is varied. We obtain the interaction between two such spheres analytically in a large separation expansion, and numerically for all separations. The cases of unequal radii and unequal λ are studied. We find sign changes in the force as a function of separation in certain ranges of λ and see deviations from the proximity force approximation even at short separations, most notably for Neumann boundary conditions.

Global Versus Local Casimir Effect

Annales Henri Poincaré, 2010

This paper continues the investigation of the Casimir effect with the use of the algebraic formulation of quantum field theory in the initial value setting. Basing on earlier papers by one of us (AH), we approximate the Dirichlet and Neumann boundary conditions by simple interaction models whose nonlocality in physical space is under strict control, but which at the same time are admissible from the point of view of algebraic restrictions imposed on models in the context of Casimir backreaction. The geometrical setting is that of the original parallel plates. By scaling our models and taking appropriate limit, we approach the sharp boundary conditions in the limit. The global force is analyzed in that limit. One finds in Neumann case that although the sharp boundary interaction is recovered in the norm resolvent sense for each model considered, the total force per area depends substantially on its choice and diverges in the sharp boundary conditions limit. On the other hand the local energy density outside the interaction region, which in the limit includes any compact set outside the strict position of the plates, has a universal limit corresponding to sharp conditions. This is what one should expect in general, and the lack of this discrepancy in Dirichlet case is rather accidental. Our discussion pins down its precise origin: the difference in the order in which scaling limit and integration over the whole space is carried out.