7.2 Performing the Inner Time Integrations
103
Here, the time-ordering t > t 1 > t
is assured by the product θ(t − t 1 )θ(t 1 − t
) of
step functions in the integrand. Recalling that G
0
p (t, t 1 ) is given by
G
0
p (t, t 1 ) = −iθ(t − t 1 )e
−i( p −iη)(t−t 1 )
¯
n p + iθ(t 1 − t)e
−i( p +iη)(t−t 1 ) n p
we see that the first part is compatible with time-ordering (a), whereas the second
part is projected out according to θ(t 1 − t)θ(t − t 1 ) ≡ 0:
G
0
p (t, t 1 ) → −iθ(t − t 1 )e
−i( p −iη)(t−t 1 )
¯
n p
Analogously, the second part of G
0
q (t 1 , t
) is to be discarded, that is,
G
0
q (t 1 , t
) → −iθ(t 1 − t
)e
−i( q −iη)(t 1 −t
)
¯
n q
This finding may readily be generalized. In a time-ordered diagram, the direction (up
or down) of a free fermion line has a distinct meaning: In upwards directed lines, the
first time argument is larger than the second one, which means that only the particle
part ( ¯
n r = 1) of the free electron propagator comes into play; in downwards directed
lines, the first time argument is smaller than the second one, and therefore, only the
hole part (n r = 1) survives. Accordingly, we will denote upwards and downwards
directed lines in time-ordered diagrams as particle and hole lines, respectively.
We may now write Eq. (7.23) more explicitly as
D
(a)
(ω 1 , ω 2 ) =(−i)
2
¯
n p ¯
n q w pq
∞
−∞
dte
i(ω 1 − p +iη)t
t
−∞
dt 1 e
it 1 ( p − q )
t 1
−∞
dt
e
−i(ω 2 − q +iη)t
(7.24)
where the various factors have been reordered according to their dependence on t, t 1 ,
and t
. Note that the convergence factors for the t 1 -integration have cancelled each
other out. Now, the three integrations can be performed successively in the order
t
, t 1 , and t. The t
-integration yields
t 1
−∞
dt
e
−i(ω 2 − q +iη)t
=
e
it 1 ( q −ω 2 −iη)
i( q − ω 2 − iη)
(7.25)
While the numerator enters the ensuing t 1 -integration, the denominator constitutes a
factorial part of the final result (see Eq. 7.29 below). As is stated more specifically in
the diagram rules below, this factor can directly be related to a horizontal line (“cut”)
above the t
-vertex. The horizontal line crosses the particle line with the index q and
an auxiliary ω-line connecting the outer vertices t and t
, which allows one to specify
the entries ( q , ω) and their signs in the denominator.
103
Here, the time-ordering t > t 1 > t
is assured by the product θ(t − t 1 )θ(t 1 − t
) of
step functions in the integrand. Recalling that G
0
p (t, t 1 ) is given by
G
0
p (t, t 1 ) = −iθ(t − t 1 )e
−i( p −iη)(t−t 1 )
¯
n p + iθ(t 1 − t)e
−i( p +iη)(t−t 1 ) n p
we see that the first part is compatible with time-ordering (a), whereas the second
part is projected out according to θ(t 1 − t)θ(t − t 1 ) ≡ 0:
G
0
p (t, t 1 ) → −iθ(t − t 1 )e
−i( p −iη)(t−t 1 )
¯
n p
Analogously, the second part of G
0
q (t 1 , t
) is to be discarded, that is,
G
0
q (t 1 , t
) → −iθ(t 1 − t
)e
−i( q −iη)(t 1 −t
)
¯
n q
This finding may readily be generalized. In a time-ordered diagram, the direction (up
or down) of a free fermion line has a distinct meaning: In upwards directed lines, the
first time argument is larger than the second one, which means that only the particle
part ( ¯
n r = 1) of the free electron propagator comes into play; in downwards directed
lines, the first time argument is smaller than the second one, and therefore, only the
hole part (n r = 1) survives. Accordingly, we will denote upwards and downwards
directed lines in time-ordered diagrams as particle and hole lines, respectively.
We may now write Eq. (7.23) more explicitly as
D
(a)
(ω 1 , ω 2 ) =(−i)
2
¯
n p ¯
n q w pq
∞
−∞
dte
i(ω 1 − p +iη)t
t
−∞
dt 1 e
it 1 ( p − q )
t 1
−∞
dt
e
−i(ω 2 − q +iη)t
(7.24)
where the various factors have been reordered according to their dependence on t, t 1 ,
and t
. Note that the convergence factors for the t 1 -integration have cancelled each
other out. Now, the three integrations can be performed successively in the order
t
, t 1 , and t. The t
-integration yields
t 1
−∞
dt
e
−i(ω 2 − q +iη)t
=
e
it 1 ( q −ω 2 −iη)
i( q − ω 2 − iη)
(7.25)
While the numerator enters the ensuing t 1 -integration, the denominator constitutes a
factorial part of the final result (see Eq. 7.29 below). As is stated more specifically in
the diagram rules below, this factor can directly be related to a horizontal line (“cut”)
above the t
-vertex. The horizontal line crosses the particle line with the index q and
an auxiliary ω-line connecting the outer vertices t and t
, which allows one to specify
the entries ( q , ω) and their signs in the denominator.