Annu. Rev. Astron. Astrophys. 1992. 30:
499-542
Copyright © 1992 by . All rights reserved |

In this section we discuss the form that the vacuum energy-momentum
tensor must take, and why the predicted value of
_{vac} is
unreasonably high
(Weinberg 1989;
for nontechnical introductions see
Abbott 1988
and Freedman 1990).

To a particle physicist, the word ``vacuum'' has a different meaning
than to an astronomer. Rather than denoting ``empty space,'' vacuum is
used to mean the ground state (state of lowest energy) of a theory. In
general, this ground state must be Lorentz invariant, that is, must
look the same to all observers. If this is the case, then the
stress-energy- momentum tensor *F*_{µv} of vacuum must be
proportional (in
any locally inertial frame) simply to the diagonal Minkowski metric,
diag(-1, 1, 1, 1), because this is the *only* 4 x 4 matrix that is
invariant
under Lorentz boosts in special relativity (as can easily be checked).
As is well known, a perfect fluid with density
and pressure *P* has
the stress-energy-momentum tensor
diag(, *P, P,
P*). (see, e.g.
Misner et al. 1973;
in this section, we choose units with *c* = 1). Comparing to
the Minkowski metric, it follows that (*a*) ``vacuum'' is a perfect fluid,
and (*b*) it has the equation of state

Not by coincidence, this equation of state is precisely the one that,
under application of the first law of thermodynamics, causes
_{vac} to
remain constant if a volume of vacuum is adiabatically compressed or
expanded: *PdV* work provides exactly the amount of mass-energy to fill
the new volume *dV* to the same density
_{vac}. Thus
_{vac}
remains truly a constant. Its relation to
is simply
= 8*G*
_{vac}.

In nongravitational physics, the energy of the vacuum is irrelevant.
In nongravitational classical mechanics, for example, we speak of
particles with energy *E* = *T* + *V*, where *T* is
the kinetic energy and *V*
the potential energy. The force on a particle is given by the gradient
of *V*; therefore, we may add an arbitrary constant to *V* without
affecting its motion. Often we choose this arbitrary constant so that
the minimum of *V* is zero, and we say that the particle has zero energy
in its vacuum state.

In quantum mechanics the situation is more complicated. Consider,
for example, a simple harmonic oscillator of frequency
; that is, a
particle of mass *m* moving in a one dimensional potential well
*V(x)* =
1/2 *m*^{2}
*x*^{2}. We have chosen the potential such that it has a
minimum *V*(0) =
0. However, the uncertainty principle forbids us from isolating the
particle in a state with zero kinetic energy and zero potential energy (of
Cohen-Tannoudji et
al 1977).
In fact, the vacuum state has a
zero-point energy *E*_{0} =
1/2 . Note that we could have set this energy to
zero, simply by subtracting 1/2
from the definition of the potential;
quantum mechanics does not restrict our freedom to pick the zero point
of energy. However, it does imply that the energy of a vacuum state
will differ from our classical expectation, and that the difference
will depend on the physical system (in this case it is a function of
).

The generalization of this phenomenon to quantum field theory is
straightforward
(Feynman & Hibbs 1965,
Mandl & Shaw 1984).
A relativistic field may be thought of as a collection of harmonic
oscillators of all possible frequencies. A simple example is provided
by a scalar field (i.e. a spinless
boson) of mass *m*. For this
system, the vacuum energy is simply a sum of contributions

where the sum is over all possible modes of the field, i.e. over all
wavevectors **k**. We can do the sum by putting the system in a box of
volume *L*^{3}, and letting *L* go to infinity. If we
impose periodic
boundary conditions, forcing the wavelength (in, say, the *i*th
direction) to be
_{i} = *L /
n _{i}* for some integer

The energy density _{vac} is obtained by letting *L* ->
while
simultaneously dividing both sides by the volume
*L*^{3}. To perform the
integral, we must use _{k}^{2} = *k*^{2} +
*m*^{2} / ^{2}, and
impose a cutoff at a
maximum wavevector *k*_{max} >> *m* /
. Then the integral gives

As we let the cutoff *k*_{max} approach infinity,
_{vac}
becomes divergent.
In the venerable rhetoric of quantum field theory, this is known as an
``ultraviolet divergence,'' since it comes about due to the contribution
from modes with very high *k*. Such divergences are only modestly
worrisome. We know that no simple low-energy theory is likely to be
exactly true at high energies, where other particles, and possibly new
kinds of forces, become important. Therefore, we can estimate
*k*_{max} as
the energy scale at which our confidence in the formalism no longer
holds. For example, it is widely believed that the Planck energy
*E*^{*}
10^{19} GeV
10^{16} erg marks a point where conventional field theory
breaks down due to quantum gravitational effects. Choosing
*k*_{max} = *E*^{*} /
, we obtain

This, as we will see later, is approximately 120 orders of magnitude larger than is allowed by observation.

We might boldly ignore a discrepancy this large, if it were not for
gravity. As in classical mechanics, the absolute value of the vacuum
energy has no measurable effect in (nongravitational) quantum field
theory. However, one of the postulates of general relativity is that
gravitation couples universally to all energy and momentum; this must
include the energy of the vacuum. Since gravity is the only force for
which this is true, the only manifestation of vacuum energy will be
through its gravitational influence. For a density as high as given by
Equation 8, this manifestation is dramatic: if
_{vac} =
10^{92} g/cm^{3}, the
cosmic microwave background would have cooled below 3 K in the first
10^{-41} s after the Big Bang.

One may object that we have simply chosen an unrealistically high
value for *k*_{max}. However, to satisfy cosmologically observed
constraints, we would need *k*_{max} < 10^{-3}
cm^{-1}; in other words, we must
neglect effects at energies higher than 10^{-14} erg
10^{-2} eV. This is not
very high at all; the binding energy of the electron in a hydrogen
atom is much larger, and is experimentally tested to very high
precision. Moreover, there is direct experimental evidence for the
reality of a vacuum energy density in the
Casimir (1948)
effect: The
vacuum energy between two parallel plate conductors depends on the
separation between the plates. This leads to a force between the
plates, experimentally measured by Sparnaay
(1957; also
Tabor & Winterton
1969),
who found agreement with Casimir's prediction.
Fulling (1989),
in a lucid discussion of the Casimir
effect, notes that ``No worker in the field of overlap of quantum
theory and general relativity can fail to point this fact out in tones
of awe and reverence.''

One can postulate an additional ``bare'' cosmological constant,
opposite in sign and exactly equal in magnitude to
8*G
*_{vac}, so that
the ``net'' cosmological constant is exactly zero. However, the vacuum
energy of quantum field theory does not simply result from the
fluctuations of a single scalar field. In the real world there are
many different particles, each with its own somewhat different
contribution, and with additional contributions derived from their
interactions. Given the large number of elementary fields in the
standard model of particle physics, it is most unlikely that they
conspire to produce a vanishing vacuum energy.

We should finally note that possible solutions to the cosmological constant problem are particularly constrained if they are to be compatible with the inflationary universe scenario (Guth 1981, Linde 1982, Albrecht & Steinhardt 1982; recent reviews are by Narlikar & Padmanabhan 1991, Linde 1990, and Kolb & Turner 1990). Inflationary cosmology postulates an early, exponential expansion driven by the vacuum energy density of a scalar field trapped in a ``false vacuum,'' away from the true minimum of its potential. During the exponential phase, this vacuum energy density is, in fact, a non-zero (and quite large) cosmological constant. Thus, to be compatible with inflation, whatever physical process enforces = 0 today must also allow it to have had a large value in the past.

For physicists, then, the cosmological constant problem is this: There are independent contributions to the vacuum energy density from the virtual fluctuations of each field, from the potential energy of each field, and possibly from a bare cosmological constant itself. Each of these contributions should be much larger than the observational bound; yet, in the real world, they seem to combine to be zero to an uncanny degree of accuracy. Most particle theorists take this situation as an indication that new, unknown physics must play a decisive role. The quest to solve this puzzle has led to a number of intriguing speculations, some of which we will review in Section 5.