196
8 BRST Quantization
In this case, the on-shell condition (8.33) reduces to
L 0 = L
⊥
0 − m
2
,L
2
− 1 = 0.
(8.74)
But, additional states can be found in ker B or in a subspace of H on which B
is singular. We have ker B = ker
L
0 such that nothing new can be found there.
However, the operator B is not defined for states with vanishing momentum α
+
0 ∝
p
+
L = 0. In fact, one must also have α
−
0 ∝ p
−
L = 0 (otherwise, the contracting
operator for Q 2 is well-defined and can be used instead). But, these states do not
satisfy the on-shell condition (except for massless states with L ⊥
0 = 1), as it will be
clear later (see [13, sec. 2.2] for more details). For this reason, we assume that states
have a generic non-zero momentum and that there is no pathology.
Full Relative Cohomology
This section aims to construct states in H 0 (
Q B ) from states in H(Q 0 ). We follow
the construction from [5].
Given a state |ψ 0 ∈ H 0 (Q 0 ), the state Q 1 |ψ 0 is Q 0 -closed since Q 0 and Q 1
anti-commute (8.67)
{Q 0 , Q 1 } |ψ 0 = 0 ⇒ Q 0
Q 1 |ψ 0
= 0.
(8.75)
Since Q 1 |ψ 0 is not in ker
L
0 (because Q 1 increases the ghost number by 1), the
state Q 1 |ψ 0 is Q 0 -exact and can be written as Q 0 of another state |ψ 1
Q 1 |ψ 0 =: −Q 0 |ψ 1 ⇒ |ψ 1 = −
B
L
0
Q 1 |ψ 0 .
(8.76)
Computation: Equation (8.76)
Start from the definition and insert (8.70) since
L 0 is invertible
Q 1 |ψ 0 =
Q 0 ,
B
L
0
Q 1 |ψ 0 = Q 0
B
L
0
Q 1 |ψ 0
.
The state |ψ 1 is identified with minus the state inside the parenthesis (up to a
BRST exact state).
As for |ψ 0 , apply {Q 0 , Q 1 } on ψ 1
{Q 0 , Q 1 } |ψ 1 = Q 0
Q 1 |ψ 1 + Q 2 |ψ 0
.
(8.77)
8 BRST Quantization
In this case, the on-shell condition (8.33) reduces to
L 0 = L
⊥
0 − m
2
,L
2
− 1 = 0.
(8.74)
But, additional states can be found in ker B or in a subspace of H on which B
is singular. We have ker B = ker
L
0 such that nothing new can be found there.
However, the operator B is not defined for states with vanishing momentum α
+
0 ∝
p
+
L = 0. In fact, one must also have α
−
0 ∝ p
−
L = 0 (otherwise, the contracting
operator for Q 2 is well-defined and can be used instead). But, these states do not
satisfy the on-shell condition (except for massless states with L ⊥
0 = 1), as it will be
clear later (see [13, sec. 2.2] for more details). For this reason, we assume that states
have a generic non-zero momentum and that there is no pathology.
Full Relative Cohomology
This section aims to construct states in H 0 (
Q B ) from states in H(Q 0 ). We follow
the construction from [5].
Given a state |ψ 0 ∈ H 0 (Q 0 ), the state Q 1 |ψ 0 is Q 0 -closed since Q 0 and Q 1
anti-commute (8.67)
{Q 0 , Q 1 } |ψ 0 = 0 ⇒ Q 0
Q 1 |ψ 0
= 0.
(8.75)
Since Q 1 |ψ 0 is not in ker
L
0 (because Q 1 increases the ghost number by 1), the
state Q 1 |ψ 0 is Q 0 -exact and can be written as Q 0 of another state |ψ 1
Q 1 |ψ 0 =: −Q 0 |ψ 1 ⇒ |ψ 1 = −
B
L
0
Q 1 |ψ 0 .
(8.76)
Computation: Equation (8.76)
Start from the definition and insert (8.70) since
L 0 is invertible
Q 1 |ψ 0 =
Q 0 ,
B
L
0
Q 1 |ψ 0 = Q 0
B
L
0
Q 1 |ψ 0
.
The state |ψ 1 is identified with minus the state inside the parenthesis (up to a
BRST exact state).
As for |ψ 0 , apply {Q 0 , Q 1 } on ψ 1
{Q 0 , Q 1 } |ψ 1 = Q 0
Q 1 |ψ 1 + Q 2 |ψ 0
.
(8.77)