C Quantum Field Theory
407
The antibracket (graded Poisson bracket) (·, ·) reads
(A, B) =
∂ R A
∂ψ r
∂ L B
∂ψ ∗
r
−
∂ R A
∂ψ ∗
r
∂ L B
∂ψ r ,
(C.83)
where the L and R indices indicate left and right derivatives. It is graded symmetric,
which means that
(A, B) = −(−1)
(|A|+1)(|B|+1) (B, A).
(C.84)
It also satisfies a graded Jacobi identity and the property
N gh ((A, B)) = N gh (A) + N gh (B) + 1,
|(A, B)| = |A| + |B| + 1 mod 2.
(C.85)
Moreover, the antibracket acts as a derivative
(A, BC) = (A, B)C + (−1)
|B|C (A, C)B.
(C.86)
The dynamics of the theory is described by the (classical) master action
S[ψ r , ψ ∗
r ], which satisfies
N gh (S) = 0,
|S| = 0.
(C.87)
In order to reproduce correctly the dynamics of the classical system without ghosts,
this action is required to satisfy the boundary condition
S[ψ
r , ψ
∗
r = 0] = S 0 [φ
i
],
∂ L ∂ R S
∂c ∗
n−1,a n−1
∂c
a n
n
ψ ∗ =0
= R
a n−1
a n .
(C.88)
Indeed, if the antifields are set to zero, the ghost fields cannot appear because they
all have positive ghost numbers, and it is not possible to build terms with vanishing
ghost numbers from them.
In analogy with the Hamiltonian formalism, the master action can be used as
the generator of a global fermionic symmetry, and inspection will show that it
corresponds to a generalization of the BRST symmetry. Writing the generalized
and classical BRST operator as s, the transformations of the fields and antifields
read
δ θ ψ
r
= θ sψ
r
= −θ (S, ψ r ) = θ
∂ R S
∂ψ ∗
r
,
(C.89a)
δ θ ψ
∗
r = θ sψ
∗
r = −θ (S, ψ
∗
r ) = −θ
∂ R S
∂ψ r ,
(C.89b)
407
The antibracket (graded Poisson bracket) (·, ·) reads
(A, B) =
∂ R A
∂ψ r
∂ L B
∂ψ ∗
r
−
∂ R A
∂ψ ∗
r
∂ L B
∂ψ r ,
(C.83)
where the L and R indices indicate left and right derivatives. It is graded symmetric,
which means that
(A, B) = −(−1)
(|A|+1)(|B|+1) (B, A).
(C.84)
It also satisfies a graded Jacobi identity and the property
N gh ((A, B)) = N gh (A) + N gh (B) + 1,
|(A, B)| = |A| + |B| + 1 mod 2.
(C.85)
Moreover, the antibracket acts as a derivative
(A, BC) = (A, B)C + (−1)
|B|C (A, C)B.
(C.86)
The dynamics of the theory is described by the (classical) master action
S[ψ r , ψ ∗
r ], which satisfies
N gh (S) = 0,
|S| = 0.
(C.87)
In order to reproduce correctly the dynamics of the classical system without ghosts,
this action is required to satisfy the boundary condition
S[ψ
r , ψ
∗
r = 0] = S 0 [φ
i
],
∂ L ∂ R S
∂c ∗
n−1,a n−1
∂c
a n
n
ψ ∗ =0
= R
a n−1
a n .
(C.88)
Indeed, if the antifields are set to zero, the ghost fields cannot appear because they
all have positive ghost numbers, and it is not possible to build terms with vanishing
ghost numbers from them.
In analogy with the Hamiltonian formalism, the master action can be used as
the generator of a global fermionic symmetry, and inspection will show that it
corresponds to a generalization of the BRST symmetry. Writing the generalized
and classical BRST operator as s, the transformations of the fields and antifields
read
δ θ ψ
r
= θ sψ
r
= −θ (S, ψ r ) = θ
∂ R S
∂ψ ∗
r
,
(C.89a)
δ θ ψ
∗
r = θ sψ
∗
r = −θ (S, ψ
∗
r ) = −θ
∂ R S
∂ψ r ,
(C.89b)