7.2 Performing the Inner Time Integrations
105
The three successive time integrations can easily be performed, yielding D
(b)
(ω 1 , ω 2 ),
from which the final result
D
(b)
(ω) = −w pq
¯
n q
(ω − q + iη)
n p
( p − q )
(7.31)
can be deduced. As above, D
(b)
(ω) is given essentially by a product of two denominators, which can be related to cuts above the vertex t
and t. The latter cut does not
cross the auxiliary ω-line, and accordingly, the second denominator is ω-independent.
Note that p = q because of the restriction n p ¯
n q = 1. Accordingly, the infinitesimal
in the ω-independent denominator (being originally of the form ( p − q − 2iη)
−1 )
is dispensable and has been omitted in Eq. (7.31).
In a similar way, the remaining time-ordered diagrams D
(c)
, . . . , D
( f ) can be
evaluated (see Exercise 7.1). One may observe that whenever t > t
, such as in
D
(a)
, D
(b)
, D
(c) , the ω-dependent denominators are of the form (ω · · · + iη)
−1 (representing poles in the lower complex ω-plane). By contrast, in the time-orderings
D
(d)
, D
(e)
, D
( f ) with t < t
, the poles are located in the upper complex ω-plane.
It is interesting to contrast the compact result of Eq. (7.13) with the fragmented
form associated with the use of time-ordered diagrams (see Exercise 7.3):
D(ω) = D
(a)
(ω) + D
(b)
(ω) + · · · + D
( f )
(ω)
(7.32)
Obviously, the product G
0
p (ω)G
0
q (ω) can be expanded in a sum of four products
of each two simple poles, two of which can readily be identified with D
(a)
(ω) and
D
(d)
(ω). The other two products are of mixed type combining poles in the upper and
lower complex plane. Here, the expansion into partial fractions generates each two
contributions of the type D
(b,c)
(ω) and D
(e, f )
(ω).
Rules for Time-Ordered or Goldstone Diagrams
A general treatment of the inner time and Fourier integrations in the Feynman (or
Abrikosov) diagrams is given in Appendix A.4, which establishes the following diagram rules:
(G1) A Feynman (or Abrikosov) diagram of nth order gives rise to (n + 2)! timeordered or Goldstone diagrams corresponding to the (n + 2)! permutations of
the two outer vertices t, t
and n inner vertices t 1 , . . . , t n . Draw an auxiliary
line (ω-line) from vertex t to vertex t
.
(G2) In the time-ordered diagrams, the direction of the G
0 -lines has the following
meaning: upwards and downwards directed lines are associated with unoccupied (particle) and occupied (hole) orbitals, respectively. Introduce the corresponding restrictions in the one-particle indices.
(G3) Each (horizontal) ‘cut’ between two successive vertices gives rise to a denominator of the kind
(σω + ε k + ε l + · · · − ε a − ε b − · · · + iη)
−1
105
The three successive time integrations can easily be performed, yielding D
(b)
(ω 1 , ω 2 ),
from which the final result
D
(b)
(ω) = −w pq
¯
n q
(ω − q + iη)
n p
( p − q )
(7.31)
can be deduced. As above, D
(b)
(ω) is given essentially by a product of two denominators, which can be related to cuts above the vertex t
and t. The latter cut does not
cross the auxiliary ω-line, and accordingly, the second denominator is ω-independent.
Note that p = q because of the restriction n p ¯
n q = 1. Accordingly, the infinitesimal
in the ω-independent denominator (being originally of the form ( p − q − 2iη)
−1 )
is dispensable and has been omitted in Eq. (7.31).
In a similar way, the remaining time-ordered diagrams D
(c)
, . . . , D
( f ) can be
evaluated (see Exercise 7.1). One may observe that whenever t > t
, such as in
D
(a)
, D
(b)
, D
(c) , the ω-dependent denominators are of the form (ω · · · + iη)
−1 (representing poles in the lower complex ω-plane). By contrast, in the time-orderings
D
(d)
, D
(e)
, D
( f ) with t < t
, the poles are located in the upper complex ω-plane.
It is interesting to contrast the compact result of Eq. (7.13) with the fragmented
form associated with the use of time-ordered diagrams (see Exercise 7.3):
D(ω) = D
(a)
(ω) + D
(b)
(ω) + · · · + D
( f )
(ω)
(7.32)
Obviously, the product G
0
p (ω)G
0
q (ω) can be expanded in a sum of four products
of each two simple poles, two of which can readily be identified with D
(a)
(ω) and
D
(d)
(ω). The other two products are of mixed type combining poles in the upper and
lower complex plane. Here, the expansion into partial fractions generates each two
contributions of the type D
(b,c)
(ω) and D
(e, f )
(ω).
Rules for Time-Ordered or Goldstone Diagrams
A general treatment of the inner time and Fourier integrations in the Feynman (or
Abrikosov) diagrams is given in Appendix A.4, which establishes the following diagram rules:
(G1) A Feynman (or Abrikosov) diagram of nth order gives rise to (n + 2)! timeordered or Goldstone diagrams corresponding to the (n + 2)! permutations of
the two outer vertices t, t
and n inner vertices t 1 , . . . , t n . Draw an auxiliary
line (ω-line) from vertex t to vertex t
.
(G2) In the time-ordered diagrams, the direction of the G
0 -lines has the following
meaning: upwards and downwards directed lines are associated with unoccupied (particle) and occupied (hole) orbitals, respectively. Introduce the corresponding restrictions in the one-particle indices.
(G3) Each (horizontal) ‘cut’ between two successive vertices gives rise to a denominator of the kind
(σω + ε k + ε l + · · · − ε a − ε b − · · · + iη)
−1