6. Applications

We discuss in this section several potential consequences of Casimir energies mostly for microscopic phenomena.

Sections 3 and 4 showed that the total Casimir energies of finite cavities may be negative. (An example is a sufficiently elongated conducting cavity containing electromagnetic field in three dimensions.) Such negative energies violate the dominant energy condition commonly postulated in classical general relativity (e.g. [38]). Failure of the dominant energy conditions renders the Hawking-Penrose singularity theorems impotent. (Theorems are also invalidated by violations of the strong energy condition [39].) The total energy (or mass) of a cavity is negative only if the energy of its walls is sufficiently small. The calculation of the wall energies can be performed only with an explicit physical model for their construction. One suspects that the result will show that an isolated system with negative total mass cannot exist. (14) If such a system were possible, then the usual vacuum should be unstable and decay into such systems. (15)

In simple models, composite particles are often taken to consist of a bag containing approximately free fields. An early semiclassical model for the electron involved a charged spherical conductor, whose positive electrostatic energy was cancelled by negative Casimir energy [40,41]. The requirement of cancellation implied a definite value for the electromagnetic coupling constant . In practice, a spherical (zero angular momentum) conducting cavity or shell has positive [5] rather than negative electromagnetic Casimir energy, and no value of yields a stable system. The QCD bag model takes hadrons to consist of a bag containing approximately free quark and gluon fields, yielding again a positive Casimir energy [42]. However, in a model with scalar constituents, the results of Section 4 show that binding of a composite particle by Casimir energy is potentially possible.

Casimir energy may also be important in determining the structure of the vacuum state in interacting field theories. According to a simple model (e.g. [43]), the vacuum state in QCD consists of a ``foam-like'' collection of regions (``bubbles'') of low field strength, separated by walls of high field strength. The size of the bubbles is governed by the characteristic distance at which the effective QCD coupling constant becomes strong. Similarly, in QED and quantum gravity [44], a foam structure associated with the increasing strength of the effective coupling at very short distances might be expected. Many phenomena may contribute in determining the structure of the foam. The properties of the walls are crucial, but can presumably be found only by consideration of the complete interacting field theory. The bubbles may contain for example magnetic fields whose energy density determines their size. However, a universal feature is the presence of Casimir energy arising solely from the confinement of the fields in bubbles. Section 4 shows that in, for example, the case of QCD or QED in three space dimensions, this Casimir energy leads to a force which tends to deform bubbles of fixed volume into long tubes. (In a foam, the total volume of each bubble should remain fixed.) If Casimir energy is dominant, this suggests that the vacuum in QCD should consist of a foam of long tubes rather than approximately spherical bags.

For certain dimensionalities and boundary conditions, the Casimir energy is minimized when some of the dimensions of a system are very long compared to the others. In such cases, bubbles of a certain dimensionality should become very thin in some directions and very long in others, so that the dimensionality of their interior is effectively reduced. The bubbles in the vacuum state for an interacting field theory in space dimensions could thus deform as a result of Casimir forces into bubbles which are long in, say, 3 directions but short in . Excitations along the three long directions could be approximately massless, but those along the short directions could be unobservable because of their large masses. Casimir energy could thus potentially lead to spontaneous reduction in the effective dimensionality of a field theory.

A complete spatially finite universe also exhibits a non-zero Casimir energy. The resulting pressure may well be anisotropic, as in the case of physical boundaries. An intriguing possibility is that in a high dimensional universe, Casimir energy could cause collapse along some directions, and expansion along others (cf. [45]), thereby reducing the apparent dimensionality of spacetime. A quantitative investigation of this possibility would require evaluation of the Casimir energy in anisotropic universes (e.g., governed by static mixmaster metrics, as recently considered in [46]).

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