292
Appendix
Fig. A.2 Particle and hole
lines crossing the cut (dashed
line) between vertex levels
τ j , τ j+1
τ j+1
τ j
x j
p 1
p 2 p 3
h 1
h 2 h 3
auxiliary ω line starting at the vertex τ v = t and ending at the vertex t
= 0 (between
τ m and τ m+1 ).
Now the x i - and y i -integrations can successively be performed. Let us consider a
particular x-integration, say, over x j (1 ≤ j ≤ m). Let there be n p particle lines with
indices p 1 , p 2 , . . . running between the vertices τ j and τ j+1 (or likewise crossing
the “cut” between τ j and τ j+1 ), as well as n h hole lines, h 1 , h 2 , . . . . A schematic
depiction of the situation is given in Fig. A.2. Each of the particle and hole lines
contributes a factor to the integrand, yielding altogether
f (x j ) = e
i( p 1 + p 2 +···− h 1 − h 2 −... )x j e
(n p +n h )ηx j
(A.4.14)
Note that the overall convergence factor e
(n p +n h )ηx j guarantees that the integrand
vanishes in the limit x j → −∞, as n p + n h > 0 is a non-vanishing integer. The
integral becomes
0
−∞
dx j f (x j ) = i
h 1 + h 2 + · · · − p 1 − p 2 + · · · + i(n p + n h )η
−1
(A.4.15)
Obviously, the infinitesimal i(n p + n h )η in the denominator is no longer relevant
and can be omitted.
A similar procedure applies to the y integrals. For the integration over y j , j =
m + 1, . . . , n + 1, the integrand is of the form
f (y j ) = e
i(ωσ j − p
1
− p
2
+···+ h
1
+ h
2
+... )y j e
−(n
p +n
h )η y j
(A.4.16)
assuming here the presence of n
p particle and n
h hole lines. In addition to the contributions from the particle and hole lines crossing the cut between the τ j and τ j−1
vertices, the factor e
iω y j comes into play if the auxiliary ω-line crosses the cut. The
parameter σ j = 1, 0 accounts for the two possibilities.
The integral can be evaluated to give
∞
0
dy j f (y j ) = i
ωσ j + h
1
+ h
2
+ · · · − p
1
− p
2
− · · · + i(n
p + n
h )η
−1
(A.4.17)
where the upper limit of the integral vanishes due to the cumulated convergence
factor for the y j -integration, e
−(n
p +n
h )η y j . As above, the infinitesimal i(n
p + n
h )η
Appendix
Fig. A.2 Particle and hole
lines crossing the cut (dashed
line) between vertex levels
τ j , τ j+1
τ j+1
τ j
x j
p 1
p 2 p 3
h 1
h 2 h 3
auxiliary ω line starting at the vertex τ v = t and ending at the vertex t
= 0 (between
τ m and τ m+1 ).
Now the x i - and y i -integrations can successively be performed. Let us consider a
particular x-integration, say, over x j (1 ≤ j ≤ m). Let there be n p particle lines with
indices p 1 , p 2 , . . . running between the vertices τ j and τ j+1 (or likewise crossing
the “cut” between τ j and τ j+1 ), as well as n h hole lines, h 1 , h 2 , . . . . A schematic
depiction of the situation is given in Fig. A.2. Each of the particle and hole lines
contributes a factor to the integrand, yielding altogether
f (x j ) = e
i( p 1 + p 2 +···− h 1 − h 2 −... )x j e
(n p +n h )ηx j
(A.4.14)
Note that the overall convergence factor e
(n p +n h )ηx j guarantees that the integrand
vanishes in the limit x j → −∞, as n p + n h > 0 is a non-vanishing integer. The
integral becomes
0
−∞
dx j f (x j ) = i
h 1 + h 2 + · · · − p 1 − p 2 + · · · + i(n p + n h )η
−1
(A.4.15)
Obviously, the infinitesimal i(n p + n h )η in the denominator is no longer relevant
and can be omitted.
A similar procedure applies to the y integrals. For the integration over y j , j =
m + 1, . . . , n + 1, the integrand is of the form
f (y j ) = e
i(ωσ j − p
1
− p
2
+···+ h
1
+ h
2
+... )y j e
−(n
p +n
h )η y j
(A.4.16)
assuming here the presence of n
p particle and n
h hole lines. In addition to the contributions from the particle and hole lines crossing the cut between the τ j and τ j−1
vertices, the factor e
iω y j comes into play if the auxiliary ω-line crosses the cut. The
parameter σ j = 1, 0 accounts for the two possibilities.
The integral can be evaluated to give
∞
0
dy j f (y j ) = i
ωσ j + h
1
+ h
2
+ · · · − p
1
− p
2
− · · · + i(n
p + n
h )η
−1
(A.4.17)
where the upper limit of the integral vanishes due to the cumulated convergence
factor for the y j -integration, e
−(n
p +n
h )η y j . As above, the infinitesimal i(n
p + n
h )η