62
13. Local Non-Abelian (Gauge) Symmetries
It is possible to verify that
ˆ μ
ˆ ν ˆ
(D ) ki (D ) ij F jμν = 0
(13.114)
where i, j, k are the SU(2) matrix indices, which implies that
ˆ
(D
μ
ˆ
) ki ∂ μ B i = 0.
(13.115)
ˆ
This is the crucial result: it implies that the auxiliary field B is not a free
ˆ
μ
field in this non-Abelian case, and so neither (from (13.113)) is ∂ μ W . In
consequence, the obvious generalizations of (7.108) or (13.110) cannot be used
to define the physical (transverse) states. The reason is that a condition like
(13.110) must hold for all times, and only if the field is free is its time variation
known (and essentially trivial).
Let us press ahead nevertheless, and assume that the rules we have derived
so far are the correct Feynman rules for this gauge theory. We will see that
this leads to physically unacceptable consequences, namely to the violation of
unitarity.
In fact, this is a problem which threatens all gauge theories if the gauge
field is treated covariantly, i.e. as a 4-vector. As we saw in section 7.3.2, this
introduces unphysical degrees of freedom which must somehow be eliminated
from the theory, or at least prevented from affecting physical processes. In
QED we do this by imposing the condition (7.111), or (13.110), but as we
have seen the analogous conditions will not work in the non-Abelian case, and
so unphysical states may make their presence felt, for example in the ‘sum
over intermediate states’ which arises in the unitarity relation. This relation
determines the imaginary part of an amplitude via an equation of the form
(cf (11.65))
∫ ∑
2 Im ‹f | M | i› =
n
‹f | M | n›‹n | M
†
| i›dρ n
(13.116)
where ‹f | M | i› is the (Feynman) amplitude for the process i → f, and
the sum is over a complete set of physical intermediate states | n›, which
can enter at the given energy; dρ n represents the phase space element for
the general intermediate state | n›. Consider now the possibility of gauge
quanta appearing in the states | n›. Since unitarity deals only with physical
states, such quanta can have only the two degrees of freedom (polarizations)
allowed for a physical massless gauge field (cf section 7.3.1). Now part of the
power of the ‘Feynman rules’ approach to perturbation theory is that it is
manifestly covariant. But there is no completely covariant way of selecting
out just the two physical components of a massless polarization vector ∈ μ ,
from the four originally introduced precisely for reasons of covariance. In
fact, when gauge quanta appear as virtual particles in intermediate states in
Feynman graphs, they will not be restricted to having only two polarization
states (as we shall see explicitly in a moment). Hence there is a real chance
13. Local Non-Abelian (Gauge) Symmetries
It is possible to verify that
ˆ μ
ˆ ν ˆ
(D ) ki (D ) ij F jμν = 0
(13.114)
where i, j, k are the SU(2) matrix indices, which implies that
ˆ
(D
μ
ˆ
) ki ∂ μ B i = 0.
(13.115)
ˆ
This is the crucial result: it implies that the auxiliary field B is not a free
ˆ
μ
field in this non-Abelian case, and so neither (from (13.113)) is ∂ μ W . In
consequence, the obvious generalizations of (7.108) or (13.110) cannot be used
to define the physical (transverse) states. The reason is that a condition like
(13.110) must hold for all times, and only if the field is free is its time variation
known (and essentially trivial).
Let us press ahead nevertheless, and assume that the rules we have derived
so far are the correct Feynman rules for this gauge theory. We will see that
this leads to physically unacceptable consequences, namely to the violation of
unitarity.
In fact, this is a problem which threatens all gauge theories if the gauge
field is treated covariantly, i.e. as a 4-vector. As we saw in section 7.3.2, this
introduces unphysical degrees of freedom which must somehow be eliminated
from the theory, or at least prevented from affecting physical processes. In
QED we do this by imposing the condition (7.111), or (13.110), but as we
have seen the analogous conditions will not work in the non-Abelian case, and
so unphysical states may make their presence felt, for example in the ‘sum
over intermediate states’ which arises in the unitarity relation. This relation
determines the imaginary part of an amplitude via an equation of the form
(cf (11.65))
∫ ∑
2 Im ‹f | M | i› =
n
‹f | M | n›‹n | M
†
| i›dρ n
(13.116)
where ‹f | M | i› is the (Feynman) amplitude for the process i → f, and
the sum is over a complete set of physical intermediate states | n›, which
can enter at the given energy; dρ n represents the phase space element for
the general intermediate state | n›. Consider now the possibility of gauge
quanta appearing in the states | n›. Since unitarity deals only with physical
states, such quanta can have only the two degrees of freedom (polarizations)
allowed for a physical massless gauge field (cf section 7.3.1). Now part of the
power of the ‘Feynman rules’ approach to perturbation theory is that it is
manifestly covariant. But there is no completely covariant way of selecting
out just the two physical components of a massless polarization vector ∈ μ ,
from the four originally introduced precisely for reasons of covariance. In
fact, when gauge quanta appear as virtual particles in intermediate states in
Feynman graphs, they will not be restricted to having only two polarization
states (as we shall see explicitly in a moment). Hence there is a real chance