C Quantum Field Theory
399
The third equality follows by developing the product and ordering the θ 0i :
minus signs result from anti-commuting the θ 0i such that one gets the
determinant of the basis elements.
C.2
BRST Quantization
Consider an action S m [φ i ] that depends on some fields φ i subject to a gauge
symmetry:
δφ
i
=
a δ a φ
i
=
a R
i
a (φ),
(C.34)
where a are the (local) bosonic parameters, such that the action is invariant
a δ a S m = 0.
(C.35)
The gauge transformations form a Lie algebra with structure coefficients f c
ab
[δ a , δ b ] = f
c
ab δ c .
(C.36)
It is important that (1) the algebra closes off-shell (without using the equations
of motion), (2) the structure coefficients are field independent and (3) the gauge
symmetry is irreducible (each gauge parameter is independent).
Remark C.3 (Interpretation of the R i
a Matrices) If the φ i transforms in a representation R of the gauge group, then the transformation is linear in the field
R
i
a (φ) = (T
R
a )
i
j φ
j ,
(C.37)
with T R
a the generators in the representation R. But, in full generality, this is not the
case: for example, the gauge fields A a
μ do not transform in the adjoint representation
even if they carry an adjoint index (only the field strength does), and in this case,
R
b
aμ = δ
b
a ∂ μ + f
c
ab A
b
μ .
(C.38)
When the fields φ i form a non-linear sigma models, the R i
a (φ)’s correspond to
Killing vectors of the target manifold.
In order to fix the gauge symmetry in the path integral
Z =
−1
gauge
dφ
i e
−S m ,
(C.39)
399
The third equality follows by developing the product and ordering the θ 0i :
minus signs result from anti-commuting the θ 0i such that one gets the
determinant of the basis elements.
C.2
BRST Quantization
Consider an action S m [φ i ] that depends on some fields φ i subject to a gauge
symmetry:
δφ
i
=
a δ a φ
i
=
a R
i
a (φ),
(C.34)
where a are the (local) bosonic parameters, such that the action is invariant
a δ a S m = 0.
(C.35)
The gauge transformations form a Lie algebra with structure coefficients f c
ab
[δ a , δ b ] = f
c
ab δ c .
(C.36)
It is important that (1) the algebra closes off-shell (without using the equations
of motion), (2) the structure coefficients are field independent and (3) the gauge
symmetry is irreducible (each gauge parameter is independent).
Remark C.3 (Interpretation of the R i
a Matrices) If the φ i transforms in a representation R of the gauge group, then the transformation is linear in the field
R
i
a (φ) = (T
R
a )
i
j φ
j ,
(C.37)
with T R
a the generators in the representation R. But, in full generality, this is not the
case: for example, the gauge fields A a
μ do not transform in the adjoint representation
even if they carry an adjoint index (only the field strength does), and in this case,
R
b
aμ = δ
b
a ∂ μ + f
c
ab A
b
μ .
(C.38)
When the fields φ i form a non-linear sigma models, the R i
a (φ)’s correspond to
Killing vectors of the target manifold.
In order to fix the gauge symmetry in the path integral
Z =
−1
gauge
dφ
i e
−S m ,
(C.39)