THE INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS ...

26 MICROSCOPIC PHYSICS
structure as the cosmological term in eqn (2.11), with the factor 8/15π2 provided
by the microscopic theory. The Planck mass corresponding to the first Planck
scale EPlanck 1 is the mass m of Bose particles that comprise the vacuum.
The second Planck scale EPlanck 2 marks the border where the discreteness
of the vacuum becomes important: the microscopic parameter which enters this
scale is the number density of particles and thus the distance between the particles
in the vacuum. This scale corresponds to the Debye temperature in solids.
In the model of weakly interacting particles one has EPlanck 1 ≪ EPlanck 2; this
shows that the distance between the particles in the vacuum is so small that
quantum effects are stronger than interactions. This is the limit of strong correlations
and weak interactions. Because of that, the leading term in the vacuum
energy density is
1
2 Un2 = 1
2¯h 3
√ 3
−gEPlanck 2EPlanck 1 . (3.26)
It is much larger than the ‘conventional’ cosmological term in eqn (3.25). This
example clearly shows that the naive estimate of the vacuum energy in eqn (2.9),
as the zero-point energy of relativistic bosonic fields or as the energy of the Dirac
sea in the case of fermionic fields, can be wrong.
3.2.6 Vacuum pressure and cosmological constant
The pressure in the vacuum state is the variation of the vacuum energy over the
vacuum volume at a given number of particles:
P = − d 〈H〉 vac
dV
d(Vɛ(N/V ))
= − = −ɛ(n)+n
dV
dɛ 1
= −
dn V 〈H − µN〉 vac
. (3.27)
In the last equation we used the fact that the quantity dɛ/dn is the chemical
potential µ of the system, which can be obtained from the equilibrium condition
for the vacuum state at T = 0. The equilibrium state of the liquid at T =
0 (equilibrium vacuum) corresponds to the minimum of the energy functional
3 3 d rɛ(n) as a function of n at a given number N = d rn of bare atoms. This
corresponds to the extremum of the following thermodynamic potential
d 3
r˜ɛ(n) ≡ d 3 r(ɛ(n) − µn) =〈H − µN〉 vac , (3.28)
where the chemical potential µ plays the role of the Lagrange multiplier. In
equilibrium one has
d˜ɛ dɛ
=0 or = µ. (3.29)
dn dn
It is important that it is the thermodynamic potential ˜ɛ which provides the
action for effective fields arising in the quantum liquids at low energy including
the effective gravity. That is why ˜ɛ is responsible also for the ‘cosmological
constant’. According to eqn (3.27) the pressure of the vacuum is minus ˜ɛ:
Pvac = −˜ɛvac eq . (3.30)