Appendix
293
in the denominator can be discarded if the denominator is a constant (σ j = 0); for
ω-dependent denominators (σ j = 1), the infinitesimal i(n
p + n
h )η may be replaced
by iη.
For the case t > 0 considered so far, the expressions (A.4.15), (A.4.17) establish
rule (G3); note, moreover, that each integral introduces a factor i, as accounted for by
rule (G4). The time-orderings of class (II), where t < 0, can be treated in a completely
analogous way. The ω-dependent denominators in the time-ordered diagrams of class
(I) and (II) are of the form ω + · · · + iη and ω + · · · − iη, respectively.
Revisiting the Adiabatic Limit
After having established the rules for performing the time integrations in the Feynman
diagrams, we now may revisit the issue of the adiabatic limit → 0 underlying the
Gell-Mann and Low theorem (see Sect. 4.2). Here, is the parameter of the adiabatic
switching function e
−|t| augmenting the interaction part of the hamiltonian (cf.
Eq. 4.36).
As stated in Sects. 5.3 and 7.2, the adiabatic limit exists (and is trivial) for the
linked Feynman diagrams. How can this claim be justified? Just take that limit a
priori by skipping the n switching functions e
−|t i |
, i = 1, . . . , n, for the internal
vertices of an nth order diagram and verify that a well-defined analytical expression
can be assigned to that diagram according the Goldstone rules of Sect. 7.2. However,
the analysis given above can easily be extended to take into account the adiabatic
switching functions as well, allowing us to take the adiabatic limit a posteriori. Let
us consider again a specific time-ordering, such as (A.4.3) belonging to class (I).
Using the τ arguments, the product of the n switching functions can be written as
e
−(|t 1 |+···+|t n |)
= e
(τ 1 +τ 2 +···+τ m ) e
−(τ m+1 +...τ m+2 +···+τ n+1 )+τ v
(A.4.18)
Note that τ v = t is the time argument of an external vertex and has to be excluded
from the product on the right-hand side, more specifically, from the second factor
(supposing the time-ordering t > 0). This is the achieved by the final term, +τ v , in
the exponent.
Using the transformations (A.4.5), (A.4.7), the τ -variables are replaced by the xand y-variables. According to
τ 1 + τ 2 + · · · + τ m = mx m + (m − 1)x m−1 + · · · + x 1
(A.4.19)
the first factor on the right-hand side of Eq. (A.4.18) can be expressed by a product
of factors e
j x j , j = 1, . . . , m. Thus, for each x j integral, the original convergence
factor, e
η(n
( j)
p +n
( j)
h )x j , due to the hole and particle lines crossing the cut j is augmented
by the factor e
j x j , acting as a “supporting” convergence factor in the limit → 0.
This means that the -factors are redundant and can safely be omitted: There is no
difference between taking the limit → 0 a priori or a posteriori.
In a similar way, the -functions of the second factor in (A.4.18) can be introduced
in the y-integrations. Here, the relation between the sums in the exponent is
293
in the denominator can be discarded if the denominator is a constant (σ j = 0); for
ω-dependent denominators (σ j = 1), the infinitesimal i(n
p + n
h )η may be replaced
by iη.
For the case t > 0 considered so far, the expressions (A.4.15), (A.4.17) establish
rule (G3); note, moreover, that each integral introduces a factor i, as accounted for by
rule (G4). The time-orderings of class (II), where t < 0, can be treated in a completely
analogous way. The ω-dependent denominators in the time-ordered diagrams of class
(I) and (II) are of the form ω + · · · + iη and ω + · · · − iη, respectively.
Revisiting the Adiabatic Limit
After having established the rules for performing the time integrations in the Feynman
diagrams, we now may revisit the issue of the adiabatic limit → 0 underlying the
Gell-Mann and Low theorem (see Sect. 4.2). Here, is the parameter of the adiabatic
switching function e
−|t| augmenting the interaction part of the hamiltonian (cf.
Eq. 4.36).
As stated in Sects. 5.3 and 7.2, the adiabatic limit exists (and is trivial) for the
linked Feynman diagrams. How can this claim be justified? Just take that limit a
priori by skipping the n switching functions e
−|t i |
, i = 1, . . . , n, for the internal
vertices of an nth order diagram and verify that a well-defined analytical expression
can be assigned to that diagram according the Goldstone rules of Sect. 7.2. However,
the analysis given above can easily be extended to take into account the adiabatic
switching functions as well, allowing us to take the adiabatic limit a posteriori. Let
us consider again a specific time-ordering, such as (A.4.3) belonging to class (I).
Using the τ arguments, the product of the n switching functions can be written as
e
−(|t 1 |+···+|t n |)
= e
(τ 1 +τ 2 +···+τ m ) e
−(τ m+1 +...τ m+2 +···+τ n+1 )+τ v
(A.4.18)
Note that τ v = t is the time argument of an external vertex and has to be excluded
from the product on the right-hand side, more specifically, from the second factor
(supposing the time-ordering t > 0). This is the achieved by the final term, +τ v , in
the exponent.
Using the transformations (A.4.5), (A.4.7), the τ -variables are replaced by the xand y-variables. According to
τ 1 + τ 2 + · · · + τ m = mx m + (m − 1)x m−1 + · · · + x 1
(A.4.19)
the first factor on the right-hand side of Eq. (A.4.18) can be expressed by a product
of factors e
j x j , j = 1, . . . , m. Thus, for each x j integral, the original convergence
factor, e
η(n
( j)
p +n
( j)
h )x j , due to the hole and particle lines crossing the cut j is augmented
by the factor e
j x j , acting as a “supporting” convergence factor in the limit → 0.
This means that the -factors are redundant and can safely be omitted: There is no
difference between taking the limit → 0 a priori or a posteriori.
In a similar way, the -functions of the second factor in (A.4.18) can be introduced
in the y-integrations. Here, the relation between the sums in the exponent is