Appendix
287
A.4 Time-Ordered Diagrams: Derivation of Goldstone
Rules
To establish the rules (G1)–(G4) of Sect. 7.2 for assigning analytic expressions to the
time-ordered Feynman diagrams, one has to perform the required time integrations
for an arbitrary nth order diagram in a systematic and generic way. The procedure
presented in the following is based on notes by F. Mertins [7].
Let us consider an nth order Feynman diagram, D = D(t, t
), where t, t
denote
the two external time arguments. The Fourier transformed diagram, D(ω), can be
obtained according to
D(ω) =
∞
−∞
dt e
iωt D(t, 0)
(A.4.1)
using here the legitimate choice t
= 0. This means that altogether the computation
of D(ω) requires n + 1 time integrations:
D(ω) =
∞
−∞
dt
∞
−∞
dt n
∞
−∞
dt n−1 . . .
∞
−∞
dt 1 e
iωt D(t, 0; t 1 , . . . t n )
(A.4.2)
Here, D(t, 0; t 1 , . . . , t n ) denotes the full time-dependent form of D (prior to the n
internal time integrations).
The n + 1 time integrations in Eq. (A.4.2) can be decomposed with respect to
distinct time-orderings of the time arguments. Let τ 1 , τ 2 , . . . , τ n+1 denote a particular
permutation P of the time arguments t, t 1 , . . . , t n such that
τ 1 ≤ τ 2 ≤ · · · ≤ τ m ≤ 0 ≤ τ m+1 ≤ · · · ≤ τ n+1
(A.4.3)
The corresponding contribution to the full time integral (A.4.2) is given by
D
P (ω) =
∞
0
dτ n+1
τn+1
0
dτ n . . .
τm+2
0
dτ m+1
(B)
0
−∞
dτ m
τm
−∞
dτ m−1 . . .
τ2
−∞
dτ 1
(A)
D(τ 1 , . . . )
(A.4.4)
where D(τ 1 , . . . ) is the integrand of Eq. (A.4.2) as a function of the τ -variables. The
original Feynman diagram may be redrawn such that the order of the vertices (both
internal and external) reflects the particular permutation of the time arguments. This
is referred to as a time-ordered or Goldstone diagram (rule G1).
Let us first consider part (A) of the integral, comprising the variables τ i ≤ 0. In
analogy to the treatment of the time-evolution operator in Sect. 4.4, we introduce
new variables, x 1 , . . . , x m :