THE INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS ...

24 MICROSCOPIC PHYSICS
〈H〉 vac – the thermodynamic potential expressed in terms of particle number N.
For the weakly interacting Bose gas these vacuum energies come from eqns (3.8),
(3.15) and (3.16):
and
〈H − µN〉 vac = − µ2 1
V +
2U 2
〈H〉 vac = Evac(N) = 1
2 Nmc2 + 1
2
p
E(p) − p2
2m − mc2 + m3c4 p2
p
E(p) − p2
2m − mc2 + m3c4 p2
, (3.19)
. (3.20)
The last term in both equations m3c4 /p2 is added to take into account the
perturbative correction to the matrix element U; as a result the sum becomes
finite.
Let us compare these equations to the naive estimation of the vacuum energy
(2.9) in RQFT. First of all, the effective theory is unable to resolve between
two kinds of vacuum energy . Second, none of these two vacuum energies is
determined by the zero-point motion of the ‘relativistic’ phonon field. Of course,
eqn (3.19) and eqn (3.20) contain the term 1
2 p E(p), which at low energy is
1
2 p cp, and this can be considered as zero-point energy of relativistic field. But
it represents only a part of the vacuum energy , and its separate treatment is
meaningless.
This divergent ‘zero-point energy’ is balanced by three counterterms in eqn
(3.20) coming from the microscopic physics, that is why they explicitly contain
the microscopic parameter – the mass m of the atom. These counterterms cannot
be properly constructed within the effective theory, which is not aware of the
existence of the microscopic parameter. Actually the theory of weakly interacting
Bose gas can be considered as one of the regularization schemes, which naturally
arise in the microscopic physics. After such ‘regularization’, the contribution of
the ‘zero-point energy’ in eqn (3.20) becomes finite
1
E(p) =
2
p reg
1
E(p) −
2
p
1
2 p
2 2m
p
+ mc2 − m3c4 p2
= 8
15π2 Nmc2 m3c3 3 ,
n¯h
(3.21)
where n = N/V is the particle density in the vacuum. Thus the total vacuum
energy in terms of N is
Evac(N) ≡ d 3 rɛ(n) , (3.22)
ɛ(n) = 1
2 mc2
n + 16
15π2 m3c3 ¯h 3
= 1
2 Un2 8
+
15π2¯h 3 m3/2U 5/2 n 5/2 . (3.23)
Here, we introduce the vacuum energy density ɛ(n) as a function of particle
density n (note that the ‘speed of light’ c(n) also depends on n). This energy
was first calculated by Lee and Yang (1957).