7.2 Feynman Transform
179
The repeated use of (6.53) and (7.18) gives
• uv • ab F( ˚
C )(ρ)ιC (ρ) ◦
ab
◦
uv
= • uv F( ˚
C )
ρ| S{a,b}
• uv ι ◦
uv ˚
C
ρ| S{a,b}
◦
uv
= F( ˚
C )
ρ| S
• ab • uv ι ◦
uv
◦
ab ˚
C
ρ| S ) = • ab • uv ι ◦
uv
◦
ab ,
so indeed
• uv • ab ι ◦
ab
◦
uv
= • ab • uv ι ◦
uv
◦
ab .
Let us attend to 2 . There are four possibilities.
1. Case u, v ∈ A. Then S = S 1 S 2 , A = S 1 {u, v} and B = S 2 for some finite
sets S 1 , S 2 , and 2 equals
S=S 1 2
• uv a • b (ι ⊗ ι)
a
S 1 {u, v}
◦
b
S 2
◦
uv .
We rewrite this expression, using (6.83) and (7.23), as
−
S=S 1 2
a • b (◦ uv ⊗ 1)(ι ⊗ ι)(◦
uv
⊗ 1)
a
S 1
◦
b
S 2
,
which is 3 with the minus sign.
2. Case u, v ∈ B. The mirror image of the previous one. By precisely the same
arguments we obtain 5 with the minus sign.
3. Case u ∈ A, v ∈ B. There clearly exist finite sets S 1 , S 2 such that A = S 1 {u}
and B = S 2 {v}, so that 2 can be rewritten as
S=S 1 2
• uv a • b (ι ⊗ ι)
a
S 1 {u}
◦
b
S 2 ◦
uv
(7.29)
which, by (6.82) and (7.22), equals
−
S=S 1 2
• ab u • v (ι ⊗ ι)
u
S 1 {a}
◦
v
S 2 ◦
ab .