10.1 Classical Action for the Open String
215
Since the Hilbert space splits as H = H 0 ⊕ c 0 H 0 with H 0 = H ∩ ker b 0 ,
see (8.31), it is natural to split the field as (this is discussed further in Sect. 10.2)
| = | ↓ + c 0 |
↓ ,
(10.19)
where
↓ ,
↓ ∈ H 0 ⇒ b 0 | ↓ = b 0 |
↓ = 0.
(10.20)
The ghost number of each component is
N gh (( ↓ ) = 1,
N gh (
↓ ) = 0.
(10.21)
Remembering the decomposition (8.14a) of the BRST operator
Q B = c 0 L 0 − b 0 M +
Q B ,
(10.22)
inserting the decomposition (10.19) in the action (10.14) gives
S =
1
2
↓ |c 0 L 0 | ↓ +
1
2
↓ |c 0 M |
↓
+
↓ |c 0
Q B | ↓
.
(10.23)
The equations of motion are obtained by varying the different fields:
0 = −M |
↓ +
Q B | ↓ ,
0 = c 0 L 0 | ↓ + c 0
Q B |
↓ .
(10.24)
Computation: Equation (10.23)
Let us introduce the projector s = b 0 c 0 on the space H 0 = H ∩ ker b 0 and
the orthogonal projector ¯
s = c 0 b 0 such that
| = | ↓ + | ↑ ,
| ↓ = s | ,
| ↑ = ¯
s | .
(10.25)
We then have
s Q B | = −b 0 M | ↑ +
Q B | ↓ ,
¯
s Q B | = c 0 L 0 | ↓ +
Q B | ↑ ,
(10.26)
using
[ s ,
Q B ] = [ s , M] = [ s , L 0 ] = 0.
(10.27)