5. Discussion and Extensions

The calculations of Section 3 and 4 were performed only for the case of cuboidal cavities. The Casimir energy is, however, not expected to be sensitive to the detailed shape of the cavity (cf. [6]). Comparison of the electromagnetic Casimir energies for three-dimensional spheres and cubes bears out this expectation. Table I gives the Casimir energy of a cubical cavity with side length as , yielding an energy divided by volume of . The corresponding energy for a spherical cavity of radius is [5,6,8] yielding , very close to that for a cube.

A further simplification in the calculations of Sections 2, 3 and 4 was to assume that the field existed only inside the cavity. In some physical situations, the field may also exist outside the cavity, satisfying appropriate boundary conditions on the exterior as well as interior surface of the cavity wall. Nevertheless, as mentioned in Section 2, fields exterior to a pair of parallel plates do not affect the Casimir energy associated with the plates. In systems with , however, exterior fields are expected to affect the Casimir energy. Any divergences associated with external mode sums should cancel in the computation of physical Casimir energy differences. (13) The contribution of exterior fields for cuboidal regions is difficult to calculate (the necessary modes are discussed in [28]). For a spherical region, inclusion of exterior modes reduces the Casimir energy per unit volume from 0.100 to 0.073 [5,6,8]. The energy per unit area for a circle with both interior and exterior modes included is [29] while the Casimir energy for the interior of a unit square is .

In Sections 2, 3 and 4 we considered only the total Casimir energy of a cavity, and not the energy density as a function of position. Forces exerted on walls confining a field depend only on this total energy. However, gravitational effects may depend on the local energy density.

The energy density for fields with periodicity constraints is always independent of position. For massless fields, this energy density is finite. For massive fields, a single divergence proportional to the total volume appears. As in Section 2, this may be cancelled by addition of a constant to the Hamiltonian density (a ``cosmological term'' or ``bag constant'').

In the presence of explicit boundaries the possible divergences and ``surface'' Lagrangian counterterms become more complicated. For example, for a scalar field the energy-momentum tensor with minimal coupling diverges like when the distance from a boundary tends to zero [30]. With conformal invariant coupling, the divergences in the energy-momentum tensor are proportional to times the curvature of the boundary [30,31,32]. With dimensional regularization, the formal integral of such pure power divergences is zero, and thus does not affect the total Casimir energy. The origin of divergences in the local energy density is presumably the unphysical assumption of a precisely localized boundary. Such a boundary could be maintained only with an infinite binding energy which must compensate the infinite local Casimir energy at the boundary. The binding energy of the boundary depends only on the form of the boundary itself. It is, for example, independent of the separation between two boundary planes, and thus does not contribute to the Casimir force.

Sections 2, 3 and 4 considered Casimir energy resulting from the modification of modes in quantized fields associated with the introduction of boundaries. The modes are also modified by the presence of localized ``particles.'' A polarizable particle perturbs the modes of the electromagnetic field in a large box. The perturbations associated with the polarizable electrically-neutral particles lead to ``van der Waals forces'' between the particles [33,34]. For positive polarizability, these forces are always attractive.

There are two other common interpretations of van der Waals forces. In the first (e.g. [35]), the forces arise from two-photon exchange. The corresponding Feynman diagram contains a closed photon loop, which represents vacuum fluctuations in the electromagnetic field, perturbed by two insertions representing the two particles. An integration is performed over the possible momenta of the virtual intermediate photons. In the second interpretation [36,37], the form (2.5) of the zero-point fluctuations in the electromagnetic field is assumed, and the resulting force of attraction on macroscopic surfaces is calculated.

Given the van der Waals forces between a pair of particles, a simple scheme would take the Casimir forces between surfaces as a superposition of forces between their constituent particles. A conducting surface may be taken to consist of a collection of polarizable particles. According to the simple scheme, the Casimir forces between conducting surfaces would then always be attractive. The results of Section 4 show that in practice they are often repulsive (as in the case of a cubical cavity). The simple scheme fails for essentially two reasons. First, the presence of the boundaries modifies the modes of the electromagnetic field, and thus changes the virtual photon propagator and the spectrum of the zero-point fluctuations in the electromagnetic field. Second, whenever the boundary is connected (as when ), it is not possible to separate the ``self-energy'' of the boundary from the true Casimir energy of the confined field. In the case of two parallel planes, the simple scheme nevertheless leads to a correct Casimir energy. This result is probably fortuitous: the scheme is known to fail for and probably fails with non-planar boundaries even with .

The failure of superposition for van der Waals forces is in principle amenable to experimental investigation. The repulsive nature of Casimir forces in a spherical cavity could perhaps be seen in the behaviour of small bubbles (possibly in liquid ).

In physical systems, the assumption of perfectly conducting cavity walls holds at best only for frequencies much lower than those characteristic of the interatomic spacing. The Casimir energies of dielectric systems may be given as integrals over the dielectric constant [8,36,37]. The decrease in conductivity at high frequencies (leading to free transmission of high modes by the cavity walls) appears to affect only details of Casimir energies: their approximate magnitude and sign is left unchanged.

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