22
G. Altarelli and S. Forte
has to be added to the lagrangian (1/λ acts as a lagrangian multiplier). The free
equations of motion are now modified as follows:
[∂
2 g μν − (1 − 1/λ)∂ μ ∂ ν ]V
Aν
= 0.
(2.36)
This operator now has an inverse whose Fourier transform is given by:
D
AB
μν (q) =
i
q 2 + ii
[−g μν + (1 − λ)
q μ q ν
q 2 + ii
] δ
AB
(2.37)
which is the propagator in this class of gauges. The parameter λ can take any value
and it disappears from the final expression of any gauge invariant, physical quantity.
Commonly used particular cases are λ = 1 (Feynman gauge) and λ = 0 (Landau
gauge).
While in an abelian theory the gauge fixing term is all that is needed for a
correct quantization, in a non abelian theory the formulation of complete Feynman
rules involves a further subtlety. This is formally taken into account by introducing
a set of D fictitious ghost fields that must be included as internal lines in
closed loops (Faddeev-Popov ghosts [12]). Given that gauge fields connected by
a gauge transformation describe the same physics, clearly there are less physical
degrees of freedom than gauge field components. Ghosts appear, in the form of
a transformation Jacobian in the functional integral, in the process of elimination
of the redundant variables associated with fields on the same gauge orbit [13].
The correct ghost contributions can be obtained from an additional term in the
lagrangian density. For each choice of the gauge fixing term the ghost langrangian
is obtained by considering the effect of an infinitesimal gauge transformation
V
C
μ = V C
μ − gC ABC θ A V B
μ − ∂ μ θ C on the gauge fixing condition. For ∂ μ V C
μ = 0
one obtains:
∂
μ V
C
μ = ∂
μ V
C
μ − gC ABC ∂
μ (θ
A V
B
μ ) − ∂
2 θ
C
= − [∂
2 δ AC + gC ABC V
B
μ ∂
μ
]θ
A
(2.38)
where the gauge condition ∂ μ V C
μ = 0 has been taken into account in the last step.
The ghost lagrangian is then given by:
Ghost = ¯
η
C
[∂
2 δ AC + gC ABC V
B
μ ∂
μ
]η
A
(2.39)
where η A is the ghost field (one for each index A) which has to be treated as a scalar
field except that a factor (−1) for each closed loop has to be included as for fermion
fields.
Starting from non covariant gauges one can construct ghost-free gauges. An
example, also important in other respects, is provided by the set of “axial” gauges:
n μ V A
μ = 0 where n μ is a fixed reference 4-vector (actually for n μ spacelike one
has an axial gauge proper, for n 2 = 0 one speaks of a light-like gauge and for n μ
G. Altarelli and S. Forte
has to be added to the lagrangian (1/λ acts as a lagrangian multiplier). The free
equations of motion are now modified as follows:
[∂
2 g μν − (1 − 1/λ)∂ μ ∂ ν ]V
Aν
= 0.
(2.36)
This operator now has an inverse whose Fourier transform is given by:
D
AB
μν (q) =
i
q 2 + ii
[−g μν + (1 − λ)
q μ q ν
q 2 + ii
] δ
AB
(2.37)
which is the propagator in this class of gauges. The parameter λ can take any value
and it disappears from the final expression of any gauge invariant, physical quantity.
Commonly used particular cases are λ = 1 (Feynman gauge) and λ = 0 (Landau
gauge).
While in an abelian theory the gauge fixing term is all that is needed for a
correct quantization, in a non abelian theory the formulation of complete Feynman
rules involves a further subtlety. This is formally taken into account by introducing
a set of D fictitious ghost fields that must be included as internal lines in
closed loops (Faddeev-Popov ghosts [12]). Given that gauge fields connected by
a gauge transformation describe the same physics, clearly there are less physical
degrees of freedom than gauge field components. Ghosts appear, in the form of
a transformation Jacobian in the functional integral, in the process of elimination
of the redundant variables associated with fields on the same gauge orbit [13].
The correct ghost contributions can be obtained from an additional term in the
lagrangian density. For each choice of the gauge fixing term the ghost langrangian
is obtained by considering the effect of an infinitesimal gauge transformation
V
C
μ = V C
μ − gC ABC θ A V B
μ − ∂ μ θ C on the gauge fixing condition. For ∂ μ V C
μ = 0
one obtains:
∂
μ V
C
μ = ∂
μ V
C
μ − gC ABC ∂
μ (θ
A V
B
μ ) − ∂
2 θ
C
= − [∂
2 δ AC + gC ABC V
B
μ ∂
μ
]θ
A
(2.38)
where the gauge condition ∂ μ V C
μ = 0 has been taken into account in the last step.
The ghost lagrangian is then given by:
Ghost = ¯
η
C
[∂
2 δ AC + gC ABC V
B
μ ∂
μ
]η
A
(2.39)
where η A is the ghost field (one for each index A) which has to be treated as a scalar
field except that a factor (−1) for each closed loop has to be included as for fermion
fields.
Starting from non covariant gauges one can construct ghost-free gauges. An
example, also important in other respects, is provided by the set of “axial” gauges:
n μ V A
μ = 0 where n μ is a fixed reference 4-vector (actually for n μ spacelike one
has an axial gauge proper, for n 2 = 0 one speaks of a light-like gauge and for n μ