Appendix
319
Operator-Based Resolution of the Identity for Binary Products
The use of operator-based resolution of the identity in binary products, being a
basic ingredient in the superoperator approach to the electron propagator, is not selfexplanatory. To better understand the mathematical structure of the ensuing operator
algebra, one may first visit the Excursus at the end of this Appendix, reviewing
analogous procedures in the simpler case of state representations.
Consider two operators, ˆ
A, ˆ
B, generating (N −1)-electron states, ˆ
A|0 and ˆ
B|0.
(Likewise, one could proceed with operators for (N +1)-electron states.) What we
want to show is that a RI in terms of basis operators can be applied to the binary
product (A.8.5) in the following form:
( ˆ
A| ˆ
B) = ( ˆ
A|h)(h|h)
−1
(h| ˆ
B)
(A.8.20)
Here,
h ≡ { ˆ
O I }
(A.8.21)
is a short notation for the set of the basis operators (16.6),
{ ˆ
O I } = {c k ; c
†
a c k c l ; . . . } ∪ {c a ; c
†
k c b c a ; . . . }
(A.8.22)
which were introduced in Sect. 16.1. The binary product can be rewritten as
( ˆ
A| ˆ
B) ==0|{ ˆ
A
†
, ˆ
B}|0
==0| ˆ
A
† ˆ
B|0 + +0| ˆ
B ˆ
A
†
|0
==0| ˆ
A
† ˆ
B|0 + +0| ˆ
A ˆ
B
†
|0
==0|( ˆ
A
†
+ ˆ
A)( ˆ
B
†
+ ˆ
B)|0
(A.8.23)
Scalar products of states with different particle numbers vanish, so that
0| ˆ
A ˆ
B|0 = =0| ˆ
A
† ˆ
B
†
|0 = 0
(A.8.24)
which has been used to arrive at the last equation. In the third equation, it is assumed
that ˆ
A, ˆ
B are real-valued operators and the ground-state wave function is real-valued
too, so that
0| ˆ
B ˆ
A
†
|0 = =0| ˆ
B ˆ
A
†
|0
∗
= =0| ˆ
A ˆ
B
†
|0
(A.8.25)
In the case of complex operators, one may instead resort to the identity ( ˆ
A| ˆ
B) =
0|( ˆ
A
∗†
+ ˆ
A)( ˆ
B
∗†
+ ˆ
B)|0 (see Sect. 16.2 and Exercises 16.2 and 16.3).
Now we may bring in the (N ± 1)-electron (Fock space) states (16.27) discussed
in Sect. 16.2,
| I = ( ˆ
O
†
I + ˆ
O I )|0
(A.8.26)
Obviously, the overlap matrix elements (16.29) are just the binary products, ( ˆ
O I | ˆ
O J ):