218
10 Free BRST String Field Theory
using the relation (8.37), the expression (8.39) and the fact that b 2
0 = 0.
Plugging this back in the first equation gives
b 0 Q B ( | + |) = 0.
(10.35)
The factor of b 0 can be removed by multiplying with c 0 , and the parentheses
should vanish (since it is not identically closed), which means that (10.31)
holds up to a BRST exact state.
Example 10.1: Gauge Fixing and Singularity
In Maxwell’s theory, the gauge transformation
A
μ = A μ + ∂ μ λ
(10.36)
is used to impose the Lorentz condition
∂
μ A
μ = 0 ⇒ λ = −∂
μ A μ .
(10.37)
In momentum space, the parameter reads
λ = −
k μ
k 2 A μ .
(10.38)
It is singular when k is on-shell, k 2 = 0. However, this does not prevent from
computing Feynman diagrams.
To understand the effect of the gauge fixing on the string field components,
decompose the field as (10.19) | = | ↓ + c 0 |
↓ . Then, imposing the
condition (10.30) yields
|
↓ = 0 ⇒ | = | ↓ .
(10.39)
This has the expected effect of dividing by two the number of states and shows that
they are not physical.
Plugging this condition in the action (10.23) leads to gauge fixed action
S =
1
2
|c 0 L 0 | ,
(10.40)
for which the equation of motion is
L 0 | = 0.
(10.41)