10.1 Classical Action for the Open String
217
was also understood as a way to work with a specific representative of the BRST
cohomology. Since the field is off-shell and the action computes off-shell Green
functions, these arguments cannot be used, which explains why we did not use this
condition earlier.
On the other hand, the condition
b 0 | = 0
(10.30)
can be interpreted as a gauge fixing condition, called Siegel gauge. It can be reached
from any field through a gauge transformation (10.29) with
| = − | ,
,=
b 0
L 0
,
(10.31)
where was defined in (8.39) and will be identified with the propagator. Note that
b 0 = 0 does not imply L 0 = 0 since the string field is not BRST closed.
This gauge choice is well-defined and completely fixes the gauge symmetry offshell, meaning that no solution of the equation of motion is pure gauge after the
gauge fixing. This is shown as follows: assume that |ψ = Q B |χ is an off-shell
pure gauge state with L 0 = 0, and then, because it is also annihilated by b 0 , one
finds
0 = {Q B , b 0 } |ψ = L 0 |ψ ,
(10.32)
which yields a contradiction.
The gauge fixing condition breaks down for L 0 = 0, but this does not pose
any problem when working with Feynman diagrams since they are not physical
by themselves (nor are the off-shell and on-shell Green functions). Only the sum
giving the scattering amplitudes (truncated on-shell Green functions) is physical;
in this case, the singularity L 0 = 0 corresponds to the on-shell condition, and it
is well-known how such infrared divergences for intermediate states are removed
(through the LSZ prescription, mass renormalization and tadpole cancellation).
Computation: Equation (10.31)
Performing a gauge transformation gives
b 0 |
= b 0 | + b 0 Q B | = 0.
(10.33)
Then, one writes
b 0 | = b 0 [Q B , ,] | = b 0 Q B | ,
(10.34)
217
was also understood as a way to work with a specific representative of the BRST
cohomology. Since the field is off-shell and the action computes off-shell Green
functions, these arguments cannot be used, which explains why we did not use this
condition earlier.
On the other hand, the condition
b 0 | = 0
(10.30)
can be interpreted as a gauge fixing condition, called Siegel gauge. It can be reached
from any field through a gauge transformation (10.29) with
| = − | ,
,=
b 0
L 0
,
(10.31)
where was defined in (8.39) and will be identified with the propagator. Note that
b 0 = 0 does not imply L 0 = 0 since the string field is not BRST closed.
This gauge choice is well-defined and completely fixes the gauge symmetry offshell, meaning that no solution of the equation of motion is pure gauge after the
gauge fixing. This is shown as follows: assume that |ψ = Q B |χ is an off-shell
pure gauge state with L 0 = 0, and then, because it is also annihilated by b 0 , one
finds
0 = {Q B , b 0 } |ψ = L 0 |ψ ,
(10.32)
which yields a contradiction.
The gauge fixing condition breaks down for L 0 = 0, but this does not pose
any problem when working with Feynman diagrams since they are not physical
by themselves (nor are the off-shell and on-shell Green functions). Only the sum
giving the scattering amplitudes (truncated on-shell Green functions) is physical;
in this case, the singularity L 0 = 0 corresponds to the on-shell condition, and it
is well-known how such infrared divergences for intermediate states are removed
(through the LSZ prescription, mass renormalization and tadpole cancellation).
Computation: Equation (10.31)
Performing a gauge transformation gives
b 0 |
= b 0 | + b 0 Q B | = 0.
(10.33)
Then, one writes
b 0 | = b 0 [Q B , ,] | = b 0 Q B | ,
(10.34)