3. Vacuum energy and
Another tradition to consider is the relation between
and the vacuum or dark energy density. In one approach to the
motivation for the Einstein field equation, taken by
McVittie (1956)
and others,
appears as a constant of integration (of the
expression for local conservation of energy and momentum).
McVittie (1956,
p. 35) emphasizes that, as a constant of integration,
"cannot
be assigned any particular value on a priori grounds."
Interesting variants of this line of thought are still under discussion
(Weinberg, 1989;
Unruh, 1989,
and references therein).
The notion of as
a constant of integration may be
related to the issue of the zero point of energy. In laboratory
physics one measures
and computes energy differences. But the net energy matters for
gravity physics, and one can
imagine
represents the difference between the true
energy density and the sum one arrives at by laboratory physics.
Eddington (1939) and
Lemaître (1934,
1949)
make this point.
Bronstein (1933)
(18)
carries the idea further, allowing
for transfer of energy between ordinary matter and that represented
by . In our
notation, Bronstein expresses this picture
by generalizing Eq. (9) to
![]() |
(36) |
where and
p are the energy density and pressure of
ordinary matter and radiation. Bronstein goes on to propose a
violation of local energy conservation, a thought that no longer
seems interesting.
North (1965, p. 81)
finds Eddington's (1939)
interpretation of the zero of energy also somewhat hard to defend.
But for our purpose the important point is that the idea of
as a form of energy has been in the air, in at least some circles, for
many years.
The zero-point energy of fields contributes to the dark energy
density. To make
physical sense the sum over the zero-point mode energies must be
cut off at a short distance or a high
frequency up to which the model under consideration is valid.
The integral of the zero-point energy (k/2) of normal modes (of
wavenumber k) of a massless real bosonic scalar field
(),
up to the wavenumber cutoff kc, gives the vacuum
energy density quantum-mechanical expectation value
(19)
![]() |
(37) |
Nernst (1916)
seems to have been the first to write down this
equation, in connection with the idea that the zero-point energy
of the electromagnetic field fills the vacuum, as a light aether,
that could have physically significant properties.
(20)
This was before Heisenberg and Schrödinger: Nernst's
hypothesis is that each degree of freedom, which classical
statistical mechanics assigns energy kT/2, has "Nullpunktsenergie"
h / 2. This would mean
the ground state energy of a one-dimensional harmonic oscillator is
h
, twice
the correct value. Nernst's expression for the energy density
in the electromagnetic field thus differs from
Eq. (37) by a factor of two (after taking account
of the two polarizations), which is
wonderfully close. For a numerical example, Nernst noted that if
the cutoff frequency were
= 1020 Hz, or
~ 0.4 MeV, the enrgy density of the ``Lichtäther'' (light aether)
would be 1023 erg cm-3, or about 100 g
cm-3.
By the end of the 1920s Nernst's hypothesis was replaced with the
demonstration that in quantum mechanics the zero-point energy of
the vacuum is as real as any other. W. Pauli, in unpublished
work in the 1920s, (21)
repeated Nernst's calculation, with the correct factor of 2, taking
kc to correspond to the
classical electron radius. Pauli knew the value of
is quite unacceptable: the radius of the static Einstein universe
with this value of
"would not even reach to the moon"
(Rugh and Zinkernagel,
2000,
p. 5). (22)
The modern version of this "physicists' cosmological constant problem" is
even more acute, because a natural value for kc is
thought to
be much larger than what Nernst or Pauli used.
(23)
While there was occasional discussion of this issue in the middle of the 20th century (as in the quote from N. Bohr in Rugh and Zinkernagel, 2000, p. 5), the modern era begins with the paper by Zel'dovich (1967) that convinced the community to consider the possible connection between the vacuum energy density of quantum physics and Einstein's cosmological constant. (24)
If the physics of the vacuum looks the same to any inertial
observer its contribution to the stress-energy tensor is the
same as Einstein's cosmological constant (Eq. [19]).
Lemaître (1934)
notes this: "in order that absolute motion, i.e., motion relative to the
vacuum, may not be detected, we must associate a pressure
p = -
c2 to the energy density
c2 of vacuum".
Gliner (1965)
goes further, presenting the relation
between the metric tensor and the stress-energy tensor of a
vacuum that appears the same to any inertial observer. But it was
Zel'dovich (1968)
who presented the argument clearly enough and
at the right time to catch the attention of the community.
With the development of the concept of broken symmetry in the now
standard model for particle physics came the idea that the
expansion and cooling of the universe is accompanied by a
sequence of first-order phase transitions accompanying the
symmetry breaking. Each first-order transition has a latent heat
that appears as a contribution to an effective time-dependent
(t) or dark
energy density.
(25)
The decrease in value of the dark energy density at
each phase transition is much larger than an acceptable present
value (within relativistic cosmology); the natural presumption is
that the dark energy is negligible now. This final
condition seems bizarre, but the picture led to the very
influential concept of inflation. We discussed the basic elements
in connection with Eq. (27); we turn now to some implications.
18 Kragh (1996, p. 36) describes Bronstein's motivation and history. We discuss this model in more detail in Sec. III.E, and comment on why decay of dark energy into ordinary matter or radiation would be hard to reconcile with the thermal spectrum of the 3 K cosmic microwave background radiation. Decay into the dark sector may be interesting. Back.
19 Eq. (37), which usually figures in
discussions of the vacuum energy puzzle, gives a helpful indication of the
situation: the zero-point energy of each mode is real and the
sum is large. The physics is seriously incomplete,
however. The elimination of spatial momenta with magnitudes k
> kc
only makes sense if there is a preferred reference frame in which
kc is defined. That is implicitly taken to be the rest
frame for the large-scale distribution of matter and radiation. It
seems strange to think the microphysics cares about large-scale
structure, but maybe it happens in a sea of interacting
fields. The cutoff in Eq. (37) might be
applied at fixed comoving wavenumber
kc
a(t)-1, or
at a fixed physical value of kc. The first
prescription can be described by an action written as a sum of terms
2 / 2 +
k2
2 /
(2a(t)2) for the
allowed modes. The zero-point energy of each mode scales with the
expansion of the universe as a(t)-1, and the
sum over modes scales as
a(t)-4, consistent with
kc
a(t)-1. In the limit of exact spatial
homogeneity, an equivalent approach uses the spatial average of
the standard expression for the field stress-energy tensor. Indeed,
Akhmedov (2002)
shows that the vacuum expectation value of the stress-energy tensor,
expressed as an integral cut off at k = kc, and
computed in the preferred coordinate frame, is diagonal with space part
p
=
/ 3, for
the massless field we are
considering. That is, in this prescription the vacuum zero point
energy acts like a homogeneous sea of radiation. This defines a
preferred frame of motion, where the stress-energy tensor is
diagonal, which is not unexpected because we need a preferred
frame to define kc. It is unacceptable as a model for
the properties of dark energy, of course. For example, if the
dark energy density were normalized to the value now under
discussion, and varied as
a(t)-4, it
would quite mess up the standard model for the origin of the
light elements. We get a more acceptable model for the behavior of
from the second prescription, with the cutoff
at a fixed physical momentum. If we also want to satisfy local
energy conservation we must take the pressure to be
p
= -
. This
does not contradict the derivation of
p
in the first prescription, because the second
situation
cannot be described by an action: the pressure must be
stipulated, not derived. What is worse, the known fields at
laboratory momenta certainly do not allow this stipulation; they
are well described by analogs of the action in the first
prescription. This quite unsatisfactory situation illustrates how
far we are from a theory of the vacuum energy.
Back.
20 A helpful discussion of Nernst's ideas on cosmology is in Kragh (1996, pp. 151-7). Back.
21 This is discussed in Enz and Thellung (1960), Enz (1974), Rugh and Zinkernagel (2000, pp. 4-5), and Straumann (2002). Back.
22 In an unpublished letter in 1930, G. Gamow considered the gravitational consequences of the Dirac sea (Dolgov, 1989, p. 230). We thank A. Dolgov for helpful correspondence on this point. Back.
23 In terms of an energy scale
defined by
=
4, the Planck energy
G-1/2 is
about 30 orders of magnitude larger than the "observed" value of
.
This is, of course, an extreme case, since a lot of the theories of
interest break down well below the Planck scale. Furthermore, in
addition to other contributions, one is allowed to add a counterterm
to Eq. (37) to predict any value of
.
With reference to this point, it is interesting to note that while Pauli
did not publish his computation of
, he
remarks in his
famous 1933 Handbuch der Physik review on quantum mechanics that
it is more consistent to "exclude a zero-point energy for each degree
of freedom as this energy, evidently from experience, does not interact
with the gravitational field"
(Rugh and Zinkernagel,
2000,
p. 5). Pauli was fully aware that one must take account of zero-point
energies in the binding energies of molecular structure, for
example (and we expect he was aware that what contributes to the
energy contributes to the gravitational mass). He chose to drop
the section with the above comment from the second (1958) edition of the
review
(Pauli, 1980, pp.
iv-v). In a globally supersymmetric field theory there are equal
numbers of bosonic and fermionic degrees of freedom, and the net
zero-point vacuum energy density
vanishes
(Iliopoulos and Zumino,
1974;
Zumino, 1975).
However, supersymmetry is not a
symmetry of low energy physics, or even at the electroweak unification
scale. It must be broken at low energies, and the proper setting for
a discussion of the zero-point
in
this case is locally supersymmetric supergravity.
Weinberg (1989,
p. 6) notes "it is very
hard to see how any property of supergravity or superstring theory
could make the effective cosmological constant sufficiently small".
Witten (2001) and
Ellwanger (2002)
review more recent developments
on this issue in the superstring/M theory/branes scenario.
Back.
24 For subsequent more detailed discussions of this issue, see Zel'dovich (1981), Weinberg (1989), Carroll, Press, and Turner (1992), Sahni and Starobinsky (2000), Carroll (2001), and Rugh and Zinkernagel (2000). Back.
25 Early references to this point are Linde (1974), Dreitlein (1974), Kirzhnitz and Linde (1974), Veltman (1975), Bludman and Ruderman (1977), Canuto and Lee (1977), and Kolb and Wolfram (1980). Back.