**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. [37]). Thus a naive extrapolation to visible
wavelengths says the contribution amounts to
_{0} ~
5 × 10^{10}, 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

(2) |

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.