B. The opportunity for physics
Unless there is some serious and quite unexpected flaw in our understanding of the principles of physics we can be sure the zero-point energy of the electromagnetic field at laboratory wavelengths is real and measurable, as in the Casimir (1948) effect. 5 Like all energy, this zero-point energy has to contribute to the source term in Einstein's gravitational field equation. If, as seems likely, the zero-point energy of the electromagnetic field is close to homogeneous and independent of the velocity of the observer, it manifests itself as a positive contribution to Einstein's , or dark energy. The zero point energies of the fermions make a negative contribution. Other contributions, perhaps including the energy densities of fields that interact only with themselves and gravity, might have either sign. The value of the sum suggested by dimensional analysis is much larger than what is allowed by the relativistic cosmological model. The only other natural value is = 0. If really is tiny but not zero, it adds a most stimulating though enigmatic clue to physics to be discovered.
To illustrate the problem we outline an example of a contribution to . The energy density in the 3 K thermal cosmic microwave background radiation, which amounts to R0 ~ 5 × 10-5 in Eq. (1) (ignoring the neutrinos) peaks at wavelength ~ 2 mm. At this Wien peak the photon occupation number is of order a fifteenth. The zero-point energy amounts to half the energy of a photon at the given frequency. This means the zero-point energy in the electromagnetic field at wavelengths ~ 2 mm amounts to a contribution 0 ~ 4 × 10-4 to the density parameter in or dark energy. The sum over modes scales as -4 (as illustrated in Eq. ). Thus a naive extrapolation to visible wavelengths says the contribution amounts to 0 ~ 5 × 1010, already a ridiculous number.
The situation can be compared to the development of the theory of the weak interactions. The Fermi point-like interaction model is strikingly successful for a considerable range of energies, but it was clear from the start that the model fails at high energy. A fix was discussed -- mediate the interaction by an intermediate boson -- and eventually incorporated into the even more successful electroweak theory. General relativity and quantum mechanics are strikingly successful on a considerable range of length scales, provided we agree not to use the rules of quantum mechanics to count the zero-point energy density in the vacuum, even though we know we have to count the zero-point energies in all other situations. There are thoughts on how to improve the situation, though they seem less focused than was the case for the Fermi model. Maybe a new energy component spontaneously cancels the vacuum energy density; maybe the new component varies slowly with position and here and there happens to cancel the vacuum energy density well enough to allow observers like us to flourish. Whatever the nature of the more perfect theory, it must reproduce the successes of general relativity and quantum mechanics. That includes the method of representing the material content of the observable universe -- all forms of mass and energy -- by the stress-energy tensor, and the relation between the stress-energy tensor and the curvature of macroscopic spacetime. One part has to be adjusted.
The numerical values of the parameters in Eq. (1) also are enigmatic, and possibly trying to tell us something. The evidence is that the parameters have the approximate values
We have written M0 in two parts: B0 measures the density of the baryons we know exist and DM0 that of the hypothetical nonbaryonic cold dark matter we need to fit the cosmological tests. The parameters B0 and DM0 have similar values but represent different things -- baryonic and nonbaryonic matter -- and 0, which is thought to represent something completely different, is not much larger. Also, if the parameters were measured when the universe was one tenth its present size the time-independent parameter would contribute ~ 0.003. That is, we seem to have come on the scene just as has become an important factor in the expansion rate. These curiosities surely are in part accidental, but maybe in part physically significant. In particular, one might imagine that the dark energy density represented by is rolling to its natural value, zero, and is very small now because we measure it when the universe is very old. We will discuss efforts along this line to at least partially rationalize the situation.
5 See Bordag, Mohideen, and Mostepanenko (2001) for a recent review. The attractive Casimir force between two parallel conducting plates results from the boundary condition that suppresses the number of modes of oscillation of the electromagnetic field between the plates, thus suppressing the energy of the system. One can understand the effect at small separation without reference to the quantum behavior of the electromagnetic field, as in the analysis of the Van der Waals interaction in quantum mechanics, by taking account of the term in the particle Hamiltonian for the Coulomb potential energy between the charged particles in the separate neutral objects. But a more complete treatment, as discussed by Cohen-Tannoudji, Dupont-Roc, and Grynberg (1992), replaces the Coulomb interaction with the coupling of the charged particles to the electromagnetic field operator. In this picture the Van der Waals interaction is mediated by the exchange of virtual photons. In either way of looking at the Casimir effect -- the perturbation of the normal modes or the exchange of virtual quanta of the unperturbed modes -- the effect is the same, the suppression of the energy of the system. Back.