Appendix
291
(iii) m ≥ r > s: Here,
τ r − τ s = −x s − x s+1 − · · · − x r −1
which leads to the factors
e
i p x j e
ηx j , for j = s, s + 1, . . . , r − 1
in the integrands of the x-integrals.
To summarize: A particle line crossing two successive vertices contributes to the corresponding x- or y-integration. The orbital-energy factors in the respective integrals
are
e
−i p y i for y-integrals
e
+i p x i for x-integrals
(A.4.10)
The convergence factors contribute as follows
e
−η y i for y-integrals
e
ηx i for x-integrals
(A.4.11)
In the same way, we can analyze a hole line, G
0
h (τ s , τ r ), running from τ r to τ s . As
above, we may distinguish the following three cases
(i) r > s ≥ m + 1;
(ii) r > m ≥ s;
(iii) m ≥ r > s, (or τ r = 0);
and analyze them separately. The finding is that any hole line crossing a pair of
successive vertices contributes to the associated x- or y-integration, the orbital-energy
factors being
e
i h y i for y-integrals
e
−i h x i for x-integrals
(A.4.12)
Note that the orbital energies enter the exponential factors with signs being opposite
to those in the particle case (A.4.10). By contrast, the convergence factors deriving
from the hole lines agree with those from the particle lines, as given by (A.4.11).
Finally, we have to consider the exponential factor e
iωt associated with the Fourier
transform. As assumed in Fig. A.1, the vertex τ v , v ≥ m + 1, is assigned to the
original time argument t. This is consistent with the class (I) of time-orderings
(t > 0) considered so far. Expressing τ v in terms of new variables,
τ v = y m+1 + y m+2 + · · · + y v
(A.4.13)
shows that the factors e
iω y i enter the integrations over y i , i = m + 1, . . . , v. Obviously, those y i variables correspond to pairs of successive τ vertices crossed by the