7.1 Energy Representation of Diagrams
97
comprising altogether three time and five ω-integrations. The three time integrations
and three of the ω-integrations can be successively performed as described in the
following.
1. Integrate over inner time arguments t 1 , t 2 :
1
2π
∞
−∞
dt 1 e
it 1 (ω 1 −ω 2 −ω 3 +ω 4 )
= δ(ω 1 − ω 2 − ω 3 + ω 4 )
1
2π
∞
−∞
dt 2 e
it 2 (ω 2 +ω 3 −ω 4 −ω 5 )
= δ(ω 2 + ω 3 − ω 4 − ω 5 )
The delta functions resulting here impose energy conservation for the ω-variables
at each inner vertex:
ω 1 + ω 4 = ω 2 + ω 3
(7.5)
ω 2 + ω 3 = ω 4 + ω 5
(7.6)
This, in turn, implies
ω 1 = ω 5
(7.7)
2. The two delta functions obtained in the first step allow one to perform two ωintegrations. Choosing ω 4 and ω 5 results in replacing ω 4 by ω 2 + ω 3 − ω 1 and ω 5
by ω 1 .
3. Integrate over t − t
:
1
2π
∞
−∞
d(t − t
)e
i(ω−ω 1 )(t−t
)
= δ(ω − ω 1 )
4. Integrate over ω 1 , using the delta function arising in the preceding step, which
means to replace ω 1 by ω.
The final result of the foregoing “integral algebra” reads
G
(2)
pq (ω) =
1
2
r,u,v
V pr[uv] V uv[qr] I uvr (ω)G
0
p (ω)G
0
q (ω)
(7.8)
where
I uvr (ω) =
dω 1
2π
dω 2
2π
G
0
u (ω 1 )G
0
v (ω 2 )G
0
r (ω 1 + ω 2 − ω)
(7.9)
Note that here the original integration variables ω 2 , ω 3 have been changed to ω 1 , ω 2 .