THE INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS ...

30 MICROSCOPIC PHYSICS
P = 0. Thus the relevant vacuum energy density ˜ɛ ≡ V −1 〈H − µN〉 vac = −P
in eqn (3.30) is also zero: ˜ɛ = 0. This can now be compared to the cosmological
term in RQFT, eqn (3.30). Vanishing of both the energy density and the pressure
of the vacuum, PΛ = −ρΛ = 0, means that, if the effective gravity arises
in the liquid, the cosmological constant would be identically zero without any
fine tuning. The only condition for this vanishing is that the liquid must be in
complete equilibrium at T = 0 and isolated from the environment.
Note that no supersymmetry is needed for exact cancellation. The symmetry
between the fermions and bosons is simply impossible in 4 He, since there are no
fermionic fields in this Bose liquid.
This scenario of vanishing vacuum energy survives even if the vacuum undergoes
a phase transition. According to conventional wisdom, the phase transition,
say to the broken-symmetry vacuum state, is accompanied by a change of the
vacuum energy , which must decrease in a phase transition. This is what usually
follows from the Ginzburg–Landau description of phase transitions. However,
if the liquid is isolated from the environment, its chemical potential µ will be
automatically adjusted to preserve the zero external pressure and thus the zero
energy ˜ɛ of the vacuum. Thus the relevant vacuum energy is zero above the transition
and (after some transient period) below the transition, meaning that T =0
phase transitions do not disturb the zero value of the cosmological constant. We
shall see this in the example of phase transitions between two superfluid states
at T = 0 in Sec. 29.2.
The vacuum energy vanishes in liquid-like vacua only. For gas-like states,
the chemical potential is positive, µ>0, and thus these states cannot exist
without an external pressure. That is why one might expect that the solution
of the cosmological constant problem can be provided by the mere assumption
that the vacuum of RQFT is liquid-like rather than gas-like. However, as we
shall see later, the gas-like states suggest their own solution of the cosmological
constant problem. One finds that, though the vacuum energy is not zero in the
gas-like vacuum, in the effective gravity arising there the vacuum energy is not
gravitating if the vacuum is in equilibrium. It follows from the local stability of
the vacuum state (see Sec. 7.3.6).
Thus in both cases it is the principle of vacuum stability which leads to the
(almost) complete cancellation of the cosmological constant in condensed matter.
It is possible that the same arguement of vacuum stability can be applied to the
‘cosmological fluid’ – the quantum vacuum of the Standard Model. If so, then
this generates the principle of non-gravitating vacuum, irrespective of what are
the internal variables of this fluid. According to Brout (2001) the role of the
variable n in a quantum liquid can be played by the inflaton field – the filed
which causes inflation.
Another important lesson from the quantum liquid is that the naive estimate
of the vacuum energy density from the zero-point fluctuations of the effective
bosonic or fermionic modes in eqn (2.9) never gives the correct magnitude or
even sign. The effective theory suggesting such an estimate is valid only in the
sub-Planckian region of energies, in other words for fermionic and bosonic zero