Appendix
317
The superoperator formulation of the electron propagator reads
G
S
pq (ω) = (c q |(ω ˆ ˆ
I − ˆ ˆ
H )
−1
|c p )
(A.8.7)
where
ˆ ˆ
R(ω) = (ω ˆ ˆ
I − ˆ ˆ
H )
−1
(A.8.8)
is referred to as the superoperator resolvent. Expanding the binary matrix element
yields
G
S
pq (ω) = =0|c
†
q
ˆ ˆ
R(ω)c p |0 + +0|( ˆ ˆ
R(ω)c p )c
†
q |0
(A.8.9)
which makes explicit that
G
S
pq (ω) = G
S(−)
pq (ω) + G
S(+)
pq (ω)
(A.8.10)
is composed of an (N −1)- and an (N +1)-electron part.
Equivalence with the Original Definition
It is not obvious that the superoperator form of the electron propagator is equivalent
to the usual definition, as given by Eqs. (3.24), (3.25). To verify that equivalence,
one may expand the superoperator resolvent
(ω ˆ ˆ
I − ˆ ˆ
H )
−1
=
1
ω
∞
ν=0
ˆ ˆ
H
ν
ω ν
(A.8.11)
and compare the resulting nested commutator expressions with the terms obtained
by applying the Baker–Hausdorff expansion
c p [t] = e
i ˆ
Ht c p e
−i ˆ
Ht
= c p + it[ ˆ
H , c p ] +
(it)
2
2!
[ ˆ
H , [ ˆ
H , c p ]] + . . .
(A.8.12)
in the definition (3.3) of the electron propagator and then performing the time integration of the Fourier transform (3.10).
A simpler proof of the equivalence is as follows [7]. According to Eq. (A.8.8), the
superoperator resolvent ˆ ˆ
R(ω) is defined as the inverse of the superoperator (ω ˆ ˆ
I − ˆ ˆ
H ).
Thus, the action of ˆ ˆ
R(ω) on an arbitrary operator ˆ
A
ˆ ˆ
R(ω) ˆ
A = ˆ
X
(A.8.13)
generates an ordinary operator ˆ
X , such that
(ω ˆ ˆ
I − ˆ ˆ
H ) ˆ
X = ˆ
A
(A.8.14)