96
7 Time-Ordered or Goldstone Diagrams
G pq (t, t
) =
1
2π
∞
−∞
dω e
−iω(t−t
) G pq (ω)
(7.1)
using here the fact that G pq (t, t
) depends only on the difference t − t
of the time
arguments.
In a similar way, any diagram D pq (t, t
) contributing to G pq (t, t
) can be transformed to the ω-representation,
D pq (ω) =
∞
−∞
d(t − t
)e
iω(t−t
) D pq (t, t
)
(7.2)
Let us consider the second-order contribution, represented by the second-order
Abrikosov diagram (Fig. 6.7, Eq. 6.15):
G
(2)
pq (ω) =
1
2
r,u,v
V pr[uv] V uv[qr]
∞
−∞
d(t − t
)e
iω(t−t
)
∞
−∞
dt 1
∞
−∞
dt 2 G
0
p (t, t 1 )G
0
u (t 1 , t 2 )G
0
v (t 1 , t 2 )G
0
r (t 2 , t 1 )G
0
q (t 2 , t
)
We may replace the time-dependent free Green’s functions by their Fourier transforms,
G
0
r (t, t
) =
1
2π
∞
−∞
dωe
−iω(t−t
) G
0
r (ω)
(7.3)
where
G
0
r (ω) =
n r
ω − ω r + iη
+
n r
ω − ω r − iη
(7.4)
This leads to the following expression,
G
(2)
pq (ω) =
1
2
r,u,v
V pr[uv] V uv[qr]
∞
−∞
d(t − t
)e
iω(t−t
)
∞
−∞
dt 1
∞
−∞
dt 2
∞
−∞
dω 1
2π
. . .
∞
−∞
dω 5
2π
e
−iω 1 (t−t 1 ) e
−iω 2 (t 1 −t 2 ) e
−iω 3 (t 1 −t 2 ) e
−iω 4 (t 2 −t 1 ) e
−iω 5 (t 2 −t
)
G
0
p (ω 1 )G
0
u (ω 2 )G
0
v (ω 3 )G
0
r (ω 4 )G
0
q (ω 5 )