C Quantum Field Theory
401
These transformations are generated by a (fermionic) charge Q B called the BRST
charge
δ φ
i
= i [Q B , φ
i
]
(C.47)
and similarly for the other fields (stripping the outside the commutator turns it
to an anti-commutator if the field is fermionic). Taking the ghosts to be Hermitian
leads to a Hermitian charge.
An important consequence is that the two additional terms of the action can be
rewritten as the BRST exact terms:
S gf + S gh = {Q B , b A F
A
}.
(C.48)
A small change in the gauge fixing condition δF leads to a variation of the action
δS = {Q B , b A δF
A
}.
(C.49)
The BRST charge should commute with the Hamiltonian in order to be conserved:
this should hold in particular when changing the gauge fixing condition
[Q B , {Q B , b A δF
A
}] = 0 ⇒ Q
2
B = 0.
(C.50)
Some vocabulary is needed before proceeding further. A state |ψ is said to be
BRST closed if it is annihilated by the BRST charge
|ψ closed ⇐⇒ |ψ ∈ ker Q B ⇐⇒ Q B |ψ = 0.
(C.51)
States that are in the image of Q B (i.e. they can be written as Q B applied on some
other states) are said to be exact
|ψ exact ⇐⇒ |ψ ∈ Im Q B ⇐⇒ ∃ |χ : |ψ = Q B |χ .
(C.52)
The cohomology H(Q B ) of Q B is the set of closed states that are not exact
|ψ ∈ H(Q B ) ⇐⇒ |ψ ∈ ker Q B , |χ : |ψ = Q B |χ .
(C.53)
Hence, the cohomology corresponds to
H(Q B ) =
ker Q B
Im Q B
.
(C.54)
Two elements of the cohomology differing by an exact state are in the same
equivalence class
|ψ |ψ + Q B |χ .
(C.55)
401
These transformations are generated by a (fermionic) charge Q B called the BRST
charge
δ φ
i
= i [Q B , φ
i
]
(C.47)
and similarly for the other fields (stripping the outside the commutator turns it
to an anti-commutator if the field is fermionic). Taking the ghosts to be Hermitian
leads to a Hermitian charge.
An important consequence is that the two additional terms of the action can be
rewritten as the BRST exact terms:
S gf + S gh = {Q B , b A F
A
}.
(C.48)
A small change in the gauge fixing condition δF leads to a variation of the action
δS = {Q B , b A δF
A
}.
(C.49)
The BRST charge should commute with the Hamiltonian in order to be conserved:
this should hold in particular when changing the gauge fixing condition
[Q B , {Q B , b A δF
A
}] = 0 ⇒ Q
2
B = 0.
(C.50)
Some vocabulary is needed before proceeding further. A state |ψ is said to be
BRST closed if it is annihilated by the BRST charge
|ψ closed ⇐⇒ |ψ ∈ ker Q B ⇐⇒ Q B |ψ = 0.
(C.51)
States that are in the image of Q B (i.e. they can be written as Q B applied on some
other states) are said to be exact
|ψ exact ⇐⇒ |ψ ∈ Im Q B ⇐⇒ ∃ |χ : |ψ = Q B |χ .
(C.52)
The cohomology H(Q B ) of Q B is the set of closed states that are not exact
|ψ ∈ H(Q B ) ⇐⇒ |ψ ∈ ker Q B , |χ : |ψ = Q B |χ .
(C.53)
Hence, the cohomology corresponds to
H(Q B ) =
ker Q B
Im Q B
.
(C.54)
Two elements of the cohomology differing by an exact state are in the same
equivalence class
|ψ |ψ + Q B |χ .
(C.55)