3.2 BRST Quantization
87
The action of the BRST charge on states |ψ ↓ ∈ H ↓ and |ψ ↑ ∈ H ↑ follows from
these relations and from the commutation relation (3.59)
Q B |ψ ↓ = H |ψ ↑ ,
Q B |ψ ↑ = 0,
(3.71)
where H is the worldsheet Hamiltonian defined in (2.26). To prove this relation,
start first with H |ψ ↑ , and then use (3.59)) to get the LHS of the first condition;
then apply Q B to get the second condition (using that Q B commutes with H , and
b + with any other operators building the states). For H = 0, the state |ψ ↓ is not in
the cohomology and |ψ ↑ is exact. Thus, the exact and closed states are
Im Q B =
|ψ ↑ ∈ H ↑ | H |ψ ↑ = 0
,
(3.72a)
ker Q B =
|ψ ↑ ∈ H ↑
∪
|ψ ↓ ∈ H ↓ | H |ψ ↓ = 0
.
(3.72b)
This implies that eigenstates of H in the cohomology satisfy the on-shell condition
H |ψ = 0.
(3.73)
This is consistent with the fact that scattering amplitudes involve on-shell states. In
this case, |ψ ↑ is not exact and is thus a member of the cohomology H(Q B ), as
well as |ψ ↓ since it becomes close. But, the Hilbert space H ↑ must be rejected
for two reasons: there would be an apparent doubling of states and scattering
amplitudes would behave badly. The first problem arises because one can show that
the cohomological subspaces of each space are isomorphic: H ↓ (Q B ) H ↑ (Q B ).
Hence, keeping both subspaces would lead to a doubling of the physical states.
For the second problem, consider an amplitude where one of the external states is
built from |ψ ↑ : the amplitude vanishes if the states are off-shell since the state
|ψ ↑ is exact, but it does not vanish on-shell [16, ch. 4]. This means that it must be
proportional to δ(H ). But, general properties in QFT forbid such dependence in the
amplitude (only poles and cuts are allowed, except if D = 2). Projecting out the
states in H ↑ is equivalent to require
b
+
|ψ = 0
(3.74)
for physical states, which is the second condition in (3.63).
In fact, this condition can be obtained very similarly as the b − = 0 condition:
using the expression of H (2.26) and the commutation relation (3.59), (3.73) is
equivalent to
Q B
dσ b 00 |ψ = 0.
(3.75)
Hence, imposing (3.74) allows to automatically ensure that (3.73) holds.
87
The action of the BRST charge on states |ψ ↓ ∈ H ↓ and |ψ ↑ ∈ H ↑ follows from
these relations and from the commutation relation (3.59)
Q B |ψ ↓ = H |ψ ↑ ,
Q B |ψ ↑ = 0,
(3.71)
where H is the worldsheet Hamiltonian defined in (2.26). To prove this relation,
start first with H |ψ ↑ , and then use (3.59)) to get the LHS of the first condition;
then apply Q B to get the second condition (using that Q B commutes with H , and
b + with any other operators building the states). For H = 0, the state |ψ ↓ is not in
the cohomology and |ψ ↑ is exact. Thus, the exact and closed states are
Im Q B =
|ψ ↑ ∈ H ↑ | H |ψ ↑ = 0
,
(3.72a)
ker Q B =
|ψ ↑ ∈ H ↑
∪
|ψ ↓ ∈ H ↓ | H |ψ ↓ = 0
.
(3.72b)
This implies that eigenstates of H in the cohomology satisfy the on-shell condition
H |ψ = 0.
(3.73)
This is consistent with the fact that scattering amplitudes involve on-shell states. In
this case, |ψ ↑ is not exact and is thus a member of the cohomology H(Q B ), as
well as |ψ ↓ since it becomes close. But, the Hilbert space H ↑ must be rejected
for two reasons: there would be an apparent doubling of states and scattering
amplitudes would behave badly. The first problem arises because one can show that
the cohomological subspaces of each space are isomorphic: H ↓ (Q B ) H ↑ (Q B ).
Hence, keeping both subspaces would lead to a doubling of the physical states.
For the second problem, consider an amplitude where one of the external states is
built from |ψ ↑ : the amplitude vanishes if the states are off-shell since the state
|ψ ↑ is exact, but it does not vanish on-shell [16, ch. 4]. This means that it must be
proportional to δ(H ). But, general properties in QFT forbid such dependence in the
amplitude (only poles and cuts are allowed, except if D = 2). Projecting out the
states in H ↑ is equivalent to require
b
+
|ψ = 0
(3.74)
for physical states, which is the second condition in (3.63).
In fact, this condition can be obtained very similarly as the b − = 0 condition:
using the expression of H (2.26) and the commutation relation (3.59), (3.73) is
equivalent to
Q B
dσ b 00 |ψ = 0.
(3.75)
Hence, imposing (3.74) allows to automatically ensure that (3.73) holds.