190
8 BRST Quantization
This shows that = b 0 /L 0 given by (8.39) is not a contracting homotopy
operator. A proper definition involves the projector P 0 on the kernel of L 0
|ψ ∈ ker L 0 : P 0 |ψ = |ψ ,
|ψ ∈ (ker L 0 )
⊥
: P 0 |ψ = 0.
(8.44)
Then, the appropriate contracting homotopy operator reads (1 − P 0 ), and (8.37)
is changed to
{Q B , ,(1 − P 0 )} = (1 − P 0 ).
(8.45)
This parallels completely the definition of the Green function in the presence
of zero-modes, see (B.3). By abuse of language, we will also say that is a
contracting homotopy operator, remembering that this statement is correct only
when multiplying with (1 − P 0 ).
We will revisit these aspects later from the SFT perspective. In fact, we will
find that Q B is the kinetic operator of the gauge invariant theory, while is the
gauge fixed propagator in the Siegel gauge. This is expected from experience with
standard gauge theories: the inverse of the kinetic operator (Green function) is not
defined when the gauge invariance is not fixed.
The on-shell condition (8.35) is already a good starting point. In order to simplify
the analysis further, one can restrict the question of computing the cohomology on
the subspace
H 0 := H ∩ ker b 0 = H m ⊗ H gh,0 ,
(8.46)
where H gh,0 = H gh ∩ ker b 0 was defined in (7.2.6). This subspace contains all states
|ψ such that
|ψ ∈ H 0 ⇒ b 0 |ψ = 0.
(8.47)
In this subspace, there is no exact state |ψ with L 0 |ψ = 0 such that b 0 |ψ =
Q B |ψ = 0. Indeed, assuming these conditions, (8.42) leads to a contraction
b 0 |ψ = Q B |ψ = 0, L 0 |ψ = 0 ⇒ |ψ = 0.
(8.48)
Note that the converse statement is not true: there are on-shell states such that
b 0 |ψ = 0. This also makes sense because the ghost Hilbert space can be
decomposed with respect to the ghost zero-modes. The cohomology of Q B in the
subspace H 0 is called the relative cohomology
H rel (Q B ) := H 0 (Q B ) =
|ψ ∈ H 0
Q B |ψ = 0, |χ ∈ H
|ψ = Q B |χ
.
(8.49)
8 BRST Quantization
This shows that = b 0 /L 0 given by (8.39) is not a contracting homotopy
operator. A proper definition involves the projector P 0 on the kernel of L 0
|ψ ∈ ker L 0 : P 0 |ψ = |ψ ,
|ψ ∈ (ker L 0 )
⊥
: P 0 |ψ = 0.
(8.44)
Then, the appropriate contracting homotopy operator reads (1 − P 0 ), and (8.37)
is changed to
{Q B , ,(1 − P 0 )} = (1 − P 0 ).
(8.45)
This parallels completely the definition of the Green function in the presence
of zero-modes, see (B.3). By abuse of language, we will also say that is a
contracting homotopy operator, remembering that this statement is correct only
when multiplying with (1 − P 0 ).
We will revisit these aspects later from the SFT perspective. In fact, we will
find that Q B is the kinetic operator of the gauge invariant theory, while is the
gauge fixed propagator in the Siegel gauge. This is expected from experience with
standard gauge theories: the inverse of the kinetic operator (Green function) is not
defined when the gauge invariance is not fixed.
The on-shell condition (8.35) is already a good starting point. In order to simplify
the analysis further, one can restrict the question of computing the cohomology on
the subspace
H 0 := H ∩ ker b 0 = H m ⊗ H gh,0 ,
(8.46)
where H gh,0 = H gh ∩ ker b 0 was defined in (7.2.6). This subspace contains all states
|ψ such that
|ψ ∈ H 0 ⇒ b 0 |ψ = 0.
(8.47)
In this subspace, there is no exact state |ψ with L 0 |ψ = 0 such that b 0 |ψ =
Q B |ψ = 0. Indeed, assuming these conditions, (8.42) leads to a contraction
b 0 |ψ = Q B |ψ = 0, L 0 |ψ = 0 ⇒ |ψ = 0.
(8.48)
Note that the converse statement is not true: there are on-shell states such that
b 0 |ψ = 0. This also makes sense because the ghost Hilbert space can be
decomposed with respect to the ghost zero-modes. The cohomology of Q B in the
subspace H 0 is called the relative cohomology
H rel (Q B ) := H 0 (Q B ) =
|ψ ∈ H 0
Q B |ψ = 0, |χ ∈ H
|ψ = Q B |χ
.
(8.49)