7.2 First-Order bc Ghost System
175
=
n>0
n b −n c n +
n>0
n c −n b n + a λ ,
=
n
n
b −n c n
+ a λ ,
using that
0
n=−λ+1
c −n b n = −
λ−1
n=0
n c n b −n = −
λ−1
n=0
n (− b −n c n +1) =
λ−1
n=0
n b −n c n +a λ .
The result also follows from (6.163).
Computation: Equation (7.162)
j 0 = −
n
: b −n c n : = −
n≥λ
b −n c n +
n>−λ
c −n b n
= −
n≥λ
b −n c n +
n>0
c −n b n +
λ−1
n=1
c n b −n + c 0 b 0
= −
n≥λ
b −n c n +
n>0
c −n b n −
λ−1
n=1
b −n c n + (λ − 1) + c 0 b 0
= −
n>0
b −n c n +
n>0
c −n b n + (λ − 1) + c 0 b 0 .
Finally, one can write
(λ − 1) = −
q λ
2
−
2
.
(7.168)
The result also follows from (6.163). The second expression is obtained by
symmetrizing the last term such that
c 0 b 0 + − 1) =
2
c 0 b 0 +
1
2
(−b 0 c 0 + ) + (λ − 1)
=
1
2
(( c 0 b 0 − b 0 c 0 ) +
λ −
1
2
.