**1.3 Vacuum energy**

The cosmological constant
is a dimensionful
parameter with units of (length)^{-2}. From the point
of view of classical general relativity, there is no
preferred choice for what the length scale defined by
might be. Particle physics,
however, brings
a different perspective to the question. The cosmological
constant turns out to be a measure of the energy density
of the vacuum - the state of lowest energy - and although
we cannot calculate the vacuum energy with any
confidence, this identification allows us to consider the
scales of various contributions to the cosmological
constant [14,
15].

Consider a single scalar field
, with potential
energy *V*(). The action can
be written

(where *g* is the determinant of the metric tensor
*g*_{µ}), and the
corresponding energy-momentum tensor is

In this theory, the configuration with the lowest energy
density (if it exists) will be one in which there is no
contribution from kinetic or gradient energy, implying
ð_{µ} = 0, for which
*T*_{µ} =
- *V*(_{0})
*g*_{µ},
where _{0}
is the value of which minimizes
*V*(). There
is no reason in principle why
*V*(_{0}) should vanish.
The vacuum energy-momentum tensor can thus be written

with _{vac}
in this example given by
*V*(_{0}).
(This form for the vacuum energy-momentum tensor can also be
argued for on the more general grounds that it is the only
Lorentz-invariant form for
*T*^{vac}_{µ}.)
The vacuum can therefore be thought of as a perfect fluid
as in (4), with

The effect of an
energy-momentum tensor of the form (12) is
equivalent to that of a cosmological constant, as can be
seen by moving the
*g*_{µ} term in
(7) to the right-hand side and setting

This equivalence is the origin of the identification of the cosmological constant with the energy of the vacuum. In what follows, I will use the terms ``vacuum energy" and ``cosmological constant" essentially interchangeably.

It is not necessary to introduce scalar fields
to obtain a nonzero vacuum energy. The action for general
relativity in the presence of a ``bare'' cosmological
constant _{0} is

where *R* is the Ricci scalar. Extremizing this action
(augmented by suitable matter terms)
leads to the equations (7). Thus, the cosmological
constant can be thought of as simply a constant term in
the Lagrange density of the theory. Indeed, (15)
is the most general covariant action we can construct out of
the metric and its first and second derivatives, and is
therefore a natural starting point for a theory of gravity.

Classically, then, the effective cosmological constant is
the sum of a bare term
_{0} and the
potential energy
*V*(), where the latter may
change with time as the
universe passes through different phases. Quantum
mechanics adds another contribution, from the
zero-point energies associated with vacuum fluctuations.
Consider a simple harmonic oscillator,
i.e. a particle moving in a one-dimensional potential
of the form *V(x)* = 1/2
^{2} *x*^{2}.
Classically, the ``vacuum'' for this system is the state in
which the particle is motionless and at the minimum of the
potential (*x* = 0), for which the energy in this case vanishes.
Quantum-mechanically, however, the uncertainty principle
forbids us from isolating the particle both in position and
momentum, and we find that the lowest
energy state has an energy *E*_{0} = 1/2
(where I have temporarily re-introduced explicit factors
of for clarity).
Of course, in the absence of gravity either system actually
has a vacuum energy which is completely arbitrary; we could
add any constant to the potential (including, for example,
-1/2
) without changing the theory.
It is important, however, that the zero-point energy
depends on the system, in this case on the frequency
.

A precisely analogous situation holds in field theory.
A (free) quantum field can be thought of as a collection of an
infinite number of harmonic oscillators in momentum space.
Formally, the zero-point energy of such an infinite collection
will be infinite. (See
[10,
3] for further
details.) If, however, we discard the very high-momentum
modes on the grounds that we trust our theory only up to
a certain ultraviolet momentum cutoff *k*_{max}, we
find that the resulting energy density is of the form

This answer could have been guessed by dimensional analysis; the numerical constants which have been neglected will depend on the precise theory under consideration. Again, in the absence of gravity this energy has no effect, and is traditionally discarded (by a process known as ``normal-ordering''). However, gravity does exist, and the actual value of the vacuum energy has important consequences. (And the vacuum fluctuations themselves are very real, as evidenced by the Casimir effect [16].)

The net cosmological constant, from this point of view, is the
sum of a number of apparently disparate contributions,
including potential energies from scalar fields and zero-point
fluctuations of each field theory degree of freedom, as well
as a bare cosmological constant
_{0}. Unlike the last
of these, in the first two cases we can at least make
educated guesses at the magnitudes. In the Weinberg-Salam
electroweak model, the phases of broken and unbroken
symmetry are distinguished by a potential energy
difference of approximately *M*_{EW} ~ 200 GeV
(where 1 GeV = 1.6 x 10^{-3}3 erg); the
universe is in the broken-symmetry phase during our
current low-temperature epoch, and is believed to have
been in the symmetric phase at sufficiently high temperatures
in the early universe. The effective cosmological constant
is therefore different in the two epochs; absent some
form of prearrangement, we would naturally expect a
contribution to the vacuum energy today of order

Similar contributions can arise even without invoking
``fundamental" scalar fields. In the strong interactions,
chiral symmetry is believed to be broken by a nonzero
expectation value of the quark bilinear
*q* (which
is itself a scalar, although constructed from fermions).
In this case the energy difference between the symmetric
and broken phases is of order the QCD scale *M*_{QCD}
~ 0.3 GeV, and we would expect a corresponding
contribution to the vacuum energy of order

These contributions are joined by those from any number
of unknown phase transitions in the early universe,
such as a possible contribution from grand unification
of order *M*_{GUT} ~ 10^{16} GeV.
In the case of vacuum fluctuations, we should choose our
cutoff at the energy past which we no longer trust our
field theory. If we are confident that we can use ordinary
quantum field theory all the way up to the Planck scale
*M*_{Pl} = (8
*G*)^{-1/2} ~
10^{18} GeV, we expect a contribution of order

Field theory may fail earlier, although quantum gravity is the only reason we have to believe it will fail at any specific scale.

As we will discuss later, cosmological observations imply

much smaller than any of the individual effects listed above. The ratio of (19) to (20) is the origin of the famous discrepancy of 120 orders of magnitude between the theoretical and observational values of the cosmological constant. There is no obstacle to imagining that all of the large and apparently unrelated contributions listed add together, with different signs, to produce a net cosmological constant consistent with the limit (20), other than the fact that it seems ridiculous. We know of no special symmetry which could enforce a vanishing vacuum energy while remaining consistent with the known laws of physics; this conundrum is the ``cosmological constant problem''. In section 4 we will discuss a number of issues related to this puzzle, which at this point remains one of the most significant unsolved problems in fundamental physics.