Appendix
305
BCC Order Relations
In the derivation of the canonical order relations (17.45) for the LL part of M
cc ,
we shall essentially follow the presentation in App. 1 of Ref. [13], which itself is
based on the general proof given in Ref. [12]. An alternative approach was pursued
by Christiansen et al. [14] and Hald et al. [15].
Consider an LL matrix element M
cc
I J with [I ] > [J ]. What is to be shown is
M
cc
I J ∼ O([I ] − [J ])
(A.6.16)
which means that in the PT expansion of M
cc
I J the non-vanishing contributions do not
begin before order n = [I ] − [J ]. Let us write the matrix element somewhat more
conveniently as
M
cc
I J == 0 | ˆ
C
†
I e
− ˆ
T
[ ˆ
H , ˆ
C J ] e
ˆ
T
| 0
== 0 | ˆ
C
†
I e
− ˆ
T ˆ
K J e
ˆ
T
| 0
(A.6.17)
where
ˆ
K J = [ ˆ
H , ˆ
C J ]
(A.6.18)
denotes the commutator of the hamiltonian and the excitation operator ˆ
C J . Obviously,
ˆ
K J can be partitioned according to
ˆ
K J = ˆ
K
(0)
J + ˆ
K
(1)
J
(A.6.19)
into a zeroth-order contribution, ˆ
K
(0)
J = [ ˆ
H 0 , ˆ
C J ], and a first-order contribution,
ˆ
K
(1)
J = [ ˆ
H I , ˆ
C J ]. The considered matrix element is of the form
M
cc
I J ∼ ∼ 0 | ˆ
C
†
I
ˆ
O J | 0
(A.6.20)
where the operator
ˆ
O J = e
− ˆ
T ˆ
K J e
ˆ
T
(A.6.21)
is subject to a PT expansion, ˆ
O J = ˆ
O
(0)
J + ˆ
O
(1)
J + . . . . What we have to show is
that the matrix element vanishes for the orders ν = 0, . . . , [I ] − [J ] − 1, of this
series. To this end, the concept of the rank of an operator is essential. For a (chargeconserving) operator given by a product of fermion operators, the rank is the number
of creation operators (c
† ) in the product. If an operator is a sum of such fermion
operator products, the rank is defined as the maximal rank of its constituents. For
example, the rank of ˆ
H 0 and ˆ
H I is 1 and 2, respectively. The rank of ˆ
C I is its
excitation class number, r = [I ], and the class-specific ˆ
T μ operators are of rank μ.
Why is the operator rank of interest? Because the nth order contribution to the matrix
element (A.6.20) necessarily vanishes,
305
BCC Order Relations
In the derivation of the canonical order relations (17.45) for the LL part of M
cc ,
we shall essentially follow the presentation in App. 1 of Ref. [13], which itself is
based on the general proof given in Ref. [12]. An alternative approach was pursued
by Christiansen et al. [14] and Hald et al. [15].
Consider an LL matrix element M
cc
I J with [I ] > [J ]. What is to be shown is
M
cc
I J ∼ O([I ] − [J ])
(A.6.16)
which means that in the PT expansion of M
cc
I J the non-vanishing contributions do not
begin before order n = [I ] − [J ]. Let us write the matrix element somewhat more
conveniently as
M
cc
I J == 0 | ˆ
C
†
I e
− ˆ
T
[ ˆ
H , ˆ
C J ] e
ˆ
T
| 0
== 0 | ˆ
C
†
I e
− ˆ
T ˆ
K J e
ˆ
T
| 0
(A.6.17)
where
ˆ
K J = [ ˆ
H , ˆ
C J ]
(A.6.18)
denotes the commutator of the hamiltonian and the excitation operator ˆ
C J . Obviously,
ˆ
K J can be partitioned according to
ˆ
K J = ˆ
K
(0)
J + ˆ
K
(1)
J
(A.6.19)
into a zeroth-order contribution, ˆ
K
(0)
J = [ ˆ
H 0 , ˆ
C J ], and a first-order contribution,
ˆ
K
(1)
J = [ ˆ
H I , ˆ
C J ]. The considered matrix element is of the form
M
cc
I J ∼ ∼ 0 | ˆ
C
†
I
ˆ
O J | 0
(A.6.20)
where the operator
ˆ
O J = e
− ˆ
T ˆ
K J e
ˆ
T
(A.6.21)
is subject to a PT expansion, ˆ
O J = ˆ
O
(0)
J + ˆ
O
(1)
J + . . . . What we have to show is
that the matrix element vanishes for the orders ν = 0, . . . , [I ] − [J ] − 1, of this
series. To this end, the concept of the rank of an operator is essential. For a (chargeconserving) operator given by a product of fermion operators, the rank is the number
of creation operators (c
† ) in the product. If an operator is a sum of such fermion
operator products, the rank is defined as the maximal rank of its constituents. For
example, the rank of ˆ
H 0 and ˆ
H I is 1 and 2, respectively. The rank of ˆ
C I is its
excitation class number, r = [I ], and the class-specific ˆ
T μ operators are of rank μ.
Why is the operator rank of interest? Because the nth order contribution to the matrix
element (A.6.20) necessarily vanishes,