Diploma thesis - Fachbereich Physik

3.3. GROUND-STATE ENERGY WITH EXTERNAL CURRENT 47
(2.138),
Z =
∮ { ∫ ¯hβ M
Dδx exp − A(X) − dτ[
1¯h
1¯h 0 2 δẋ2 (τ) (3.44)
+ M (
2 ω2 1 + 6gAX )
Mω + 12g2 BX 2
δx 2 + ( gA + 4g 2 BX ) ]}
δx 3 + g 2 Bδx 4 ,
2 Mω 2
where, according to the background method, the first order term has been omitted, and
the integration runs over all fluctuations δx that are periodic in the imaginary time, i.e.
δx(0) = δx(¯hβ). Due to (2.37) and (2.127), one has the following relation between the
partition function Z and the effective potential:
Z = exp [−βV eff (X)] . (3.45)
And by comparing (3.44) and (3.45), one obtains
∮ {
exp [−βV eff (X) + βV (X)] = Dδx exp − 1¯h
∫ ¯hβ
dτ (3.46)
0
[ M
×
2 δẋ2 (τ) + M (
2 ω2 1 + 6gAX )
Mω + 12g2 BX 2
δx 2 + ( gA + 4g 2 BX ) ]}
δx 3 + g 2 Bδx 4 .
2 Mω 2
In Section 2.7 and 2.8, we have evaluated the expression on the right-hand side of the
last identity in the saddle-point approximation, i.e. we have performed an expansion in ¯h.
However, in (3.46), the powers of the fluctuations δx are not related to the powers of the
coupling constant g. Taking into account that the potential (3.4) vanishes at the origin, one
obtains from (3.46)
exp[−βV eff (X = 0)] = (3.47)
∮ {
Dδx exp − 1¯h
∫ ¯hβ
[ M
dτ
2 δẋ2 (τ) + M ]}
2 ω2 δx 2 + gAδx 3 + g 2 Bδx 4 .
Thus, the effective potential can be transformed according to
V eff (X) = V (X) + V eff (X = 0) ∣
0
∣
ω 2 =Ω 2 , A=A+4gBX
, (3.48)
where Ω is defined by (2.157). It is important to note that in contrast to (3.46), there
is a fixed relation between the powers of the fluctuations and the coupling constant in
(3.47). Therefore, except for the tree-level, an expansion for the effective potential in the
coupling constant g can be transformed into the corresponding ¯h-expansion by performing
the substitutions
X → 0 , ω → Ω , A → A + 4gBX (3.49)
in (3.42) and reintroducing ¯h (and M) into Tab. 3.3. Evaluating (3.42), Tab. 3.3 in the
second order of the coupling constant according to (3.49) reproduces (2.165). In the fourth
order, one reobtains (2.170). In general, when the ¯h-expansion is given in the form
V eff (X) =
N∑
¯h l V (l) (X) + O(¯h N+1 ) , (3.50)
l=0