THE INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS ...

14 GRAVITY
√ 1
ρΛ −g ∼±
c3¯h 3 E4 Planck = ± 1
¯h 3
√ 4
−gEPlanck . (2.11)
The sign of the vacuum energy is determined by the fermionic and bosonic content
of the quantum field theory.
The vacuum energy was calculated for flat spacetime with the Minkowski
metric in the form gµν = diag(−1,c −2 ,c −2 ,c −2 ), so that √ −g = c −3 . The righthand
side of eqn (2.11) does not depend on the ‘material parameter’ c, and thus
(as we see later) is also applicable to quantum liquids.
2.2.2 Cosmological constant problem
The ‘cosmological constant problem’ refers to a huge disparity between the
naively expected value in eqn (2.11) and the range of actual values. The experimental
observations show that the energy density in our Universe is close to
the critical density ρc, corresponding to a flat Universe. The energy content is
composed of baryonic matter (very few percent), the so-called dark matter which
does not emit or absorb light (about 30%) and vacuum energy (about 70%). Such
a distribution of energy leads to an accelerated expansion of the Universe which
is revealed by recent supernovae Ia observations (Perlmutter et al. 1999; Riess
et al. 2000). The observed value of the vacuum energy ρΛ ∼ 0.7ρc is thus on
the order of 10−123E4 Planck in contradiction with the theoretical estimate in eqn
(2.11). This is probably the largest discrepancy between theory and experiment
in physics.
This huge discrepancy can be cured by supersymmetry, the symmetry between
fermions and bosons. If there is a supersymmetry, the positive contribution
of bosons and negative contribution of fermions exactly cancel each other. However,
it is well known that there is no supersymmetry in our low-energy world.
This means that there must by an energy scale ESuSy below which the supersymmetry
is violated and thus there is no balance between bosons and fermions
in the vacuum energy . This scale is providing the cut-off in eqn (2.9), and the
theoretical estimate of the cosmological constant becomes ρΛ(theor) ∼ E4 SuSy .
Though ESuSy can be much smaller than the Planck scale, the many orders of
magnitude disagreement between theory and reality still persists, and we must
accept the experimental fact that the vacuum energy in eqn (2.9) is not gravitating.
This is the most severe problem in the marriage of gravity and quantum
theory (Weinberg 1989), because it is in apparent contradiction with the general
principle of equivalence, according to which the inertial and gravitating masses
coincide.
One possible way to solve this contradiction is to accept that the theoretical
criteria for setting the absolute zero point of energy are unclear within the
effective theory, i.e. the vacuum energy is by no means the energy of vacuum
fluctuations of effective fields in eqn (2.9). Its estimation requires physics beyond
the Planck scale and thus beyond general relativity. To clarify this issue
we can consider such quantum systems where the elements of the gravitation
are at least partially reproduced, but where the structure of the quantum vacuum
beyond the ‘Planck scale’ is known. Quantum liquids are the right systems,