72
5 Introducing Diagrams
to the external operators. Since there are
n
ν
ways to choose ν interaction parts out of
n ones, there will be altogether
n
ν
contraction schemes differing from the original
one only by the choice of the interaction parts. Considering the observations 3 and
4, those
n
ν
contraction schemes will all result in the same contribution. This allows
us to rewrite the nth-order term according to
i ˜
G
(n)
pq (t, t
) =
(−i)
n
n!
n
ν=0
n!
ν!μ!
∞
−∞
dt 1 . . .
∞
−∞
dt ν 0 | ˆ
T T T
ˆ
H I (t 1 ) . . . ˆ
H I (t ν )c p (t)c
†
q (t
)
| 0 C
linked contributions only
×
∞
−∞
dt ν+1 . . .
∞
−∞
dt n 0 | ˆ
T T T
ˆ
H I (t ν+1 ) . . . ˆ
H I (t n )
| 0
μ=n−ν interaction operators
Here the switching functions have been omitted for brevity. Using the usual resummation technique for the exponential series,
∞
n=0
(−i)
n
n!
n
ν=0
n!
ν!(n − ν)!
→
∞
ν=0
(−i)
ν
ν!
∞
μ=0
(−i)
μ
μ!
(5.41)
the numerator i ˜
G pq (t, t
) can be written as
i ˜
G pq (t, t
) =
∞
ν=0
(−i)
ν
ν!
∞
−∞
dt 1 . . .
∞
−∞
dt ν 0 | ˆ
T T T
ˆ
H I (t 1 ) . . . ˆ
H I (t ν )c p (t)c
†
q (t
)
| 0 C
×
∞
μ=0
(−i)
μ
μ!
∞
−∞
dt 1 . . .
∞
−∞
dt μ 0 | ˆ
T T T
ˆ
H I (t 1 ) . . . ˆ
H I (t μ )
| 0
(5.42)
The second factor is seen to be the perturbation expansion of 0 | ˆ
U (∞, −∞)| 0 ,
which concludes the proof.
The linked-cluster theorem guarantees the existence (and triviality) of the adiabatic limit. For any linked contribution to the right-hand side of Eq. (5.39), the limit
→ 0 can be safely performed within the integrand of the time integration, reducing
the switching functions to unity. Accordingly, Eq. (5.39) becomes