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In this section we discuss the form that the vacuum energy-momentum tensor must take, and why the predicted value of rhovac 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 Fµv 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 rho and pressure P has the stress-energy-momentum tensor diag(rho, 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

Equation 4 4.

Not by coincidence, this equation of state is precisely the one that, under application of the first law of thermodynamics, causes rhovac 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 rhovac. Thus rhovac remains truly a constant. Its relation to Lambda is simply Lambda = 8piG rhovac.

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 omega; that is, a particle of mass m moving in a one dimensional potential well V(x) = 1/2 momega2 x2. 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 E0 = 1/2 hbaromega. Note that we could have set this energy to zero, simply by subtracting 1/2 hbaromega 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 omega).

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 phi (i.e. a spinless boson) of mass m. For this system, the vacuum energy is simply a sum of contributions

Equation 5 5.

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 L3, and letting L go to infinity. If we impose periodic boundary conditions, forcing the wavelength (in, say, the ith direction) to be Lambdai = L / ni for some integer ni, then, since ki = 2pi / lambdai, there are dkiL / 2pi discrete values of ki in the range (ki, ki + dki). Therefore expression 5 becomes

Equation 6 6.

The energy density rhovac is obtained by letting L -> infty while simultaneously dividing both sides by the volume L3. To perform the integral, we must use omegak2 = k2 + m2 / hbar2, and impose a cutoff at a maximum wavevector kmax >> m / hbar. Then the integral gives

Equation 7 7.

As we let the cutoff kmax approach infinity, rhovac 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 kmax 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* approx 1019 GeV approx 1016 erg marks a point where conventional field theory breaks down due to quantum gravitational effects. Choosing kmax = E* / hbar, we obtain

Equation 8 8.

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 rhovac = 1092 g/cm3, 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 kmax. However, to satisfy cosmologically observed constraints, we would need kmax < 10-3 cm-1; in other words, we must neglect effects at energies higher than 10-14 erg approx 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 8piG rhovac, 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 Lambda = 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.

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