294
Appendix
τ m+1 + · · · + τ n+1 = (n − m + 1)y m+1 + (n − m)y m+2 + · · · + 3y n−1 + 2y n + y n+1
(A.4.20)
As noted above, the external time argument τ v , being
τ v = y m+1 + · · · + y v−1 + y v
(A.4.21)
in terms of the y arguments, has to be discarded. Altogether, for each y i integration,
there is an additional convergence factor, e
−λ j y j , with λ j being a positive integer
for j = m + 1, . . . , n, and λ n+1 ≥ 0. In the limit → 0, the -functions act as convergence factors together with the original η convergence functions in eliminating
the upper y-integration limits. As above, there is no difference between taking the
adiabatic limit a priori or a posteriori.
Remark:
From the analysis given above, it follows that the argument of the redundancy of the
-functions might also be reversed: One could keep the -functions and discard the η
convergence factors in the G
0 -expressions (A.4.9) as redundant. Here it is required,
though, to introduce a convergence factor in the Fourier transform (A.4.1):
D(ω) =
∞
−∞
dt e
iωt e
−η|t| D(t, 0)
(A.4.22)
Then, the evaluation of the time integrations for a given Feynman diagram can be
based on the expression
D(ω) = lim
→0
∞
−∞
dt
∞
−∞
dt n
∞
−∞
dt n−1 . . .
∞
−∞
dt 1 e
iωt e
−η|t| D(t, 0; t 1 , . . . , t n )e
−(|t1|+···+|tn |)
(A.4.23)
where all η factors in D(t, 0; t 1 , . . . , t n ) have been omitted. The general procedure
used to establish the Goldstone rules (G1)–(G4) can be applied in a completely
analogous way, leading to identical results. Here, the infinitesimal and the remaining
infinitesimal η ensure that the respective upper and lower integration limits vanish. In
the denominators resulting from the x- and y-integrals, the limit → 0 is trivial, while
the η infinitesimal enters the ω dependent denominators in the form (ω · · · ± iη) for
time-orderings of class (I) or (II), respectively.
Appendix
τ m+1 + · · · + τ n+1 = (n − m + 1)y m+1 + (n − m)y m+2 + · · · + 3y n−1 + 2y n + y n+1
(A.4.20)
As noted above, the external time argument τ v , being
τ v = y m+1 + · · · + y v−1 + y v
(A.4.21)
in terms of the y arguments, has to be discarded. Altogether, for each y i integration,
there is an additional convergence factor, e
−λ j y j , with λ j being a positive integer
for j = m + 1, . . . , n, and λ n+1 ≥ 0. In the limit → 0, the -functions act as convergence factors together with the original η convergence functions in eliminating
the upper y-integration limits. As above, there is no difference between taking the
adiabatic limit a priori or a posteriori.
Remark:
From the analysis given above, it follows that the argument of the redundancy of the
-functions might also be reversed: One could keep the -functions and discard the η
convergence factors in the G
0 -expressions (A.4.9) as redundant. Here it is required,
though, to introduce a convergence factor in the Fourier transform (A.4.1):
D(ω) =
∞
−∞
dt e
iωt e
−η|t| D(t, 0)
(A.4.22)
Then, the evaluation of the time integrations for a given Feynman diagram can be
based on the expression
D(ω) = lim
→0
∞
−∞
dt
∞
−∞
dt n
∞
−∞
dt n−1 . . .
∞
−∞
dt 1 e
iωt e
−η|t| D(t, 0; t 1 , . . . , t n )e
−(|t1|+···+|tn |)
(A.4.23)
where all η factors in D(t, 0; t 1 , . . . , t n ) have been omitted. The general procedure
used to establish the Goldstone rules (G1)–(G4) can be applied in a completely
analogous way, leading to identical results. Here, the infinitesimal and the remaining
infinitesimal η ensure that the respective upper and lower integration limits vanish. In
the denominators resulting from the x- and y-integrals, the limit → 0 is trivial, while
the η infinitesimal enters the ω dependent denominators in the form (ω · · · ± iη) for
time-orderings of class (I) or (II), respectively.