Having summarised the observational evidence for a cosmological -term let us now turn our attention to some theoretical implicationsi of .

A turning point in our understanding of the cosmological constant
occurred when Zeldovich (1968), intrigued by
-based
cosmological models presented to explain
an excess of quasars near redshift ~ 2, showed
that zero-point vacuum fluctuations must have a
Lorentz invariant form *P*_{vac} = -
_{vac}
*c*^{2}, equivalently *T*_{ik}^{vac} =
*g*_{ik}, *i.e.* the vacuum within the quantum framework
had properties identical to those of a cosmological constant.

Let us review this situation beginning
with an oscillator consisting of a single
particle of mass *m* moving under the influence of a potential
*V* = 1/2 *kx*^{2}.
At the classical level one expects the lowest energy state to be
associated with
the particle at rest at *x* = 0, so that the total energy vanishes:
*E* = *T* + *V* = 0.
Thus, within the classical framework, the vacuum can be viewed as
a state having zero energy and momentum.
However when viewed in terms of quantum mechanics the situation changes,
the uncertainty relation preventing the particle (wave function) from
simultaneously
having a fixed location (*x* = 0) and a fixed velocity (*T* =
0). As a result,
the ground state energy of the oscillator is finite and is given by
*E* = 1/2
, where
= *k*/*m*.
Turning now to quantum theory, it is
well known that after secondary quantization
a classical field can be looked upon as an ensemble of oscillators each with
frequency (*k*).
The net `zero-point energy' of this field is
*E* = _{k}
1/2
(*k*).
Thus the uncertainty relation endows the vacuum with both energy and
pressure!

The existence of zero-point vacuum fluctuations
has been spectacularly demonstrated by the Casimir effect.
^{(10)}
The vacuum energy associated with zero-point fluctuations
is formally infinite and results
in a `cosmological constant problem' for the universe
[205].
Because of the importance of this result we shall
perform a simple calculation aimed at evaluating
the zero-point energy associated with a quantized scalar field in flat
space-time. (The reader is referred to Birrell & Davies 1982 for a
discussion of quantization of higher spin fields.)

Consider the action defined in flat four dimensional space-time
^{(11)}

(53) |

where (*x*) is the
Lagrangian density for a massive scalar field

(54) |

propagating in flat space-time with metric
_{ij}.

The variational principle
_{}
= 0 gives the
Klein-Gordon equation

(55) |

where
^{ik}
_{i}
_{k}.

To quantize the system we treat the field as an operator

(56) |

where *a*_{k}, *a*_{k}^{†} are
annihilation and creation operators
[*a*_{k}, *a*_{k'}^{†}] =
_{kk'}, defining
the vacuum state *a*_{k}| 0> = 0,
*k*.
An orthonormal set of solutions
defined using periodic boundary conditions on a three dimensional
torus of side *L* is given by
[18]

(57) |

where
_{k}^{2}
= *k*^{2} + *m*^{2}, and
the field modes have been normalized using

(58) |

where

(59) |

Consider next, the energy-momentum tensor

(60) |

where *T*_{00} defines the energy density

(61) |

and *T*_{0}
the momentum density

(62) |

Substituting from (56) & (57) into (60) one obtains
for the Hamiltonian *H*

(63) |

which can be further simplified using the commutation relation
[*a*_{k}, *a*_{k'}^{†}] =
_{kk'} to

(64) |

A similar operation on the momentum density yields [18]

(65) |

Inspecting expressions (63) and (65) for
the Hamiltonian *H* and the momentum operator
*P*_{} we
find, for the expectation value of these quantities in the vacuum state
|0>

(66) |

Transforming the sum
_{k} to an integral
we get

(67) |

From (66) & (67) we find that the energy density of
zero-point vacuum fluctuations is dominated by
ultraviolet divergences which diverge as *k*^{4} when
*k*
.
The vacuum state therefore has zero momentum and infinite energy !
(In terms of Feynman diagrams the energy density of zero-point fluctuations
is associated with a one-loop vacuum graph, see
figure (13).)

Within the framework of Newtonian gravity and either classical or quantum
mechanics, an infinite (or very large)
vacuum energy does not cause serious problems since interaction
between particles is governed not by the absolute value of the potential
energy *V*, but by its gradient
*V*.
As a result one can always redefine
*V'*
*V* + *V*_{0}
so that the minimum of *V'* has zero net energy. The situation changes
dramatically
when we view the vacuum within the framework of general relativity.
A central tenet of the general theory of relativity is that the
gravitational force couples to *all* forms of energy through the
Einstein equations *G*_{ik} =
(8*G* /
*c*^{4})*T*_{ik}.
Therefore if the vacuum has energy then it also
gravitates ! In order to probe this effect further one needs to know the
equation of state possessed by the vacuum energy, equivalently the form
of its energy momentum tensor *T*^{ik}_{vac}.
This question was answered by
Zeldovich (1968) who showed that the vacuum state
had to have a Lorentz-invariant form, one that was left unchanged by
a velocity transformation and hence appeared the same to all observers.
This requirement is exactly satisfied by the equation of
state *P* = -
possessed by the cosmological constant, since the relation
*T*_{ik} =
*g*_{ik} is manifestly Lorentz-invariant.
^{(12)}

All fields occurring in nature contribute an energy density to the vacuum and expressions analogous to (66) for bosons can also be derived for fermions. Since fermionic creation, annihilation operators anti-commute this leads to

(68) |

Comparing (68) with (66) we find that the zero-point energy of fermions is equal and opposite to that of bosons (having identical mass).

The advent of Supersymmetry in the 1980s, incorporating a
fundamental symmetry between bosons and fermions, led to the hope that the
cosmological constant problem would finally be resolved, since the
one-to-one correspondence between bosons and fermions in such theories
was expected to lead to cancellation between bosonic and fermionic
infinities
[216].
However Supersymmetry
is expected to exist only at very high energies/temperatures.
At low temperatures
such as those existing in the universe today, Supersymmetry is broken.
One might therefore expect the cosmological constant to vanish in the early
universe only to reappear later, when the universe has cooled sufficiently
so that *T* << *T*_{SUSY} > 10^{3} GeV.
Thus the
cosmological constant problem re-emerges to haunt the present epoch !

Although the cosmological constant problem remains unresolved, an important aspect of Zeldovich's work was that it demonstrated a firm physical mechanism for the generation of a cosmological constant. Later work, mostly associated with Inflationary model-building, further strengthened this idea by showing that an effective cosmological constant could arise due to diverse physical processes including symmetry breaking, vacuum polarization in curved space-time, higher dimensional `Kaluza-Klein' theories etc. Some of these developments have been reviewed in Einstein is quoted as saying [132].

^{10} The Casimir
effect arises because vacuum fluctuations satisfy the
quantum mechanical wave equation and hence are sensitive to boundary
conditions.
As shown by Casimir (1948) the presence of two flat parallel conducting plates
at a separation *l*, alters the distribution of electromagnetic
field modes existing in the vacuum,
resulting in an attractive force per unit area between
the plates:
*F* = -
*c* ^{2} /
240*l*^{4} which is of vacuum origin.
The Casimir effect has been experimentally measured by Spaarnay (1957)
and others
Einstein is quoted as saying
[18,
189,
205,
139].
Back.

^{11} Following Landau & Lifshitz (1975)
we use Latin
indices to describe space-time coordinates, so that i,j,k... = 0,1,2,3;
and Greek indices to describe spatial coordinates:
,
,
.. = 1,2,3.
Back.

^{12} Zero-point fluctuations
are usually regularized by `normal ordering' - a rather ad hoc procedure
which involves the substitution
*a*_{k} *a*_{k}^{†}
*a*_{k}^{†} *a*_{k} in
[63].
In curved space-time a single regularization is not
enough to rid <*T*_{ik}> of all its divergences.
Three remaining `infinities' must be
regularized, leading to the renormalization of additional terms in
the one-loop effective Lagrangian for the gravitational field, which, in
an FRW universe becomes:
_{eff} =
-*g*
[_{} + *R* /
16
*G*_{}
+ _{} *R*^{2} +
_{} *R*_{ij}
*R*^{ij}]. Renormalization of the first term
_{}
0 corresponds to
normal ordering. The presence of the second term
*R* / 16
*G*_{}, led
Sakharo
v to postulate that the gravitational field might be `induced' by one-loop
quantum effects in a curved background geometry, since one could recover
the ordinary Einstein action by renormalizing the `bare' value
*G*_{} to its
observed value:
*G*_{}
*G*_{obs}
[173].
Thus *both* the cosmological constant
and
the gravitational constant *G* may be induced by quantum effects.
The remaining two terms in
_{eff} give
rise to vacuum polarization effects and have been
extensively discussed in the literature
[18,
81].
Back.