40
13. Local Non-Abelian (Gauge) Symmetries
indicated. Notice that we have inserted a parameter g in the exponent to
make the analogy with the electromagnetic U(1) case
ψ
′ (x) = exp[iqχ(x)]ψ(x)
(13.3)
even stronger: g will be a coupling strength, analogous to the electromagnetic
charge q. The consideration of theories based on (13.2) was the fundamental
step taken by Yang and Mills (1954); see also Shaw (1955).
Global symmetries and their associated (possibly approximate) conservation laws are certainly important, but they do not have the dynamical significance of local symmetries. We saw in section 7.4 how the ‘requirement’ of
local U(1) phase invariance led almost automatically to the local gauge theory
¯ ˆ
of QED, in which the conserved current ψγ
μ ψ ˆ of the global U(1) symmetry is
‘promoted’ to the role of dynamical current which, when dotted into the gauge
field A ˆ μ , gave the interaction term in L ˆ QED . A similar link between symmetry and dynamics appears if, following Yang and Mills, we generalize the
non-Abelian global symmetries of the preceding chapter to local non-Abelian
symmetries, which are the subject of the present one.
However, as mentioned in the introduction to chapter 12, the original
Yang-Mills attempt to get a theory of hadronic interactions by ‘localizing’
the flavour symmetry group SU(2) turned out not to be phenomenologically
viable (although a remarkable attempt was made to push the idea further
by Sakurai (1960)). In the event, the successful application of a local SU(2)
symmetry was to the weak interactions. But this is complicated by the fact
that the symmetry is ‘spontaneously broken’, and consequently we shall delay
the discussion of this application until after QCD – which is the theory of
strong interactions, but at the quark, rather than the composite (hadronic)
level. QCD is based on the local form of an SU(3) symmetry; once again,
however, it is not the flavour SU(3) of section 12.2, but a symmetry with
respect to a totally new degree of freedom, colour. This will be introduced in
the following chapter.
Although the application of local SU(2) symmetry to the weak interactions
will follow that of local SU(3) to the strong, we shall begin our discussion
of local non-Abelian symmetries with the local SU(2) case, since the group
theory is more familiar. We shall also start with the ‘wavefunction’ formalism,
deferring the field theory treatment until section 13.3.
13.1 Local SU(2) symmetry
13.1.1 The covariant derivative and interactions with matter
In this section we shall introduce the main ideas of the non-Abelian SU(2)
gauge theory which results from the demand of invariance, or covariance,
13. Local Non-Abelian (Gauge) Symmetries
indicated. Notice that we have inserted a parameter g in the exponent to
make the analogy with the electromagnetic U(1) case
ψ
′ (x) = exp[iqχ(x)]ψ(x)
(13.3)
even stronger: g will be a coupling strength, analogous to the electromagnetic
charge q. The consideration of theories based on (13.2) was the fundamental
step taken by Yang and Mills (1954); see also Shaw (1955).
Global symmetries and their associated (possibly approximate) conservation laws are certainly important, but they do not have the dynamical significance of local symmetries. We saw in section 7.4 how the ‘requirement’ of
local U(1) phase invariance led almost automatically to the local gauge theory
¯ ˆ
of QED, in which the conserved current ψγ
μ ψ ˆ of the global U(1) symmetry is
‘promoted’ to the role of dynamical current which, when dotted into the gauge
field A ˆ μ , gave the interaction term in L ˆ QED . A similar link between symmetry and dynamics appears if, following Yang and Mills, we generalize the
non-Abelian global symmetries of the preceding chapter to local non-Abelian
symmetries, which are the subject of the present one.
However, as mentioned in the introduction to chapter 12, the original
Yang-Mills attempt to get a theory of hadronic interactions by ‘localizing’
the flavour symmetry group SU(2) turned out not to be phenomenologically
viable (although a remarkable attempt was made to push the idea further
by Sakurai (1960)). In the event, the successful application of a local SU(2)
symmetry was to the weak interactions. But this is complicated by the fact
that the symmetry is ‘spontaneously broken’, and consequently we shall delay
the discussion of this application until after QCD – which is the theory of
strong interactions, but at the quark, rather than the composite (hadronic)
level. QCD is based on the local form of an SU(3) symmetry; once again,
however, it is not the flavour SU(3) of section 12.2, but a symmetry with
respect to a totally new degree of freedom, colour. This will be introduced in
the following chapter.
Although the application of local SU(2) symmetry to the weak interactions
will follow that of local SU(3) to the strong, we shall begin our discussion
of local non-Abelian symmetries with the local SU(2) case, since the group
theory is more familiar. We shall also start with the ‘wavefunction’ formalism,
deferring the field theory treatment until section 13.3.
13.1 Local SU(2) symmetry
13.1.1 The covariant derivative and interactions with matter
In this section we shall introduce the main ideas of the non-Abelian SU(2)
gauge theory which results from the demand of invariance, or covariance,