12
Global Non-Abelian Symmetries
12.1 The Standard Model
In the preceding volume, a very successful dynamical theory – QED – has been
introduced, based on the remarkably simple gauge principle: namely, that the
theory should be invariant under local phase transformations on the wavefunctions (chapter 2) or field operators (chapter 7) of charged particles. Such
transformations were characterized as Abelian in section 2.6, since the phase
factors commuted. The second volume of this book will be largely concerned
with the formulation and elementary application of the remaining two dynamical theories within the Standard Model – that is, QCD and the electroweak
theory. They are built on a generalization of the gauge principle, in which the
transformations involve more than one state, or field, at a time. In that case,
the ‘phase factors’ become matrices, which generally do not commute with
each other, and the associated symmetry is called a ‘non-Abelian’ one. When
the phase factors are independent of the space-time coordinate x, the symmetry is a ‘global non-Abelian’ one; when they are allowed to depend on x, one
is led to a non-Abelian gauge theory. Both QCD and the electroweak theory
are of the latter type, providing generalizations of the Abelian U(1) gauge
theory which is QED. It is a striking fact that all three dynamical theories in
the Standard Model are based on a gauge principle of local phase invariance.
In this chapter we shall be mainly concerned with two global non-Abelian
symmetries, which lead to useful conservation laws but not to any specific
dynamical theory. We begin in section 12.1 with the first non-Abelian symmetry to be used in particle physics, the hadronic isospin ‘SU(2) symmetry’
proposed by Heisenberg (1932) in the context of nuclear physics, and now
understood as following from QCD and the smallness of the u and d quark
masses as compared with the QCD scale parameter Λ
(see section 18.3.3).
MS
In section 12.2 we extend this to SU(3) f flavour symmetry, as was first done
by Gell-Mann (1961) and Ne’eman (1961) – an extension seen, in its turn, as
reflecting the smallness of the u, d and s quark masses as compared with Λ MS .
The ‘wavefunction’ approach of sections 12.1 and 12.2 is then reformulated in
field-theoretic language in section 12.3.
In the last section of this chapter, we shall introduce the idea of a global
chiral symmetry, which is a symmetry of theories with massless fermions. This
may be expected to be a good approximate symmetry for the u and d quarks.
3
Global Non-Abelian Symmetries
12.1 The Standard Model
In the preceding volume, a very successful dynamical theory – QED – has been
introduced, based on the remarkably simple gauge principle: namely, that the
theory should be invariant under local phase transformations on the wavefunctions (chapter 2) or field operators (chapter 7) of charged particles. Such
transformations were characterized as Abelian in section 2.6, since the phase
factors commuted. The second volume of this book will be largely concerned
with the formulation and elementary application of the remaining two dynamical theories within the Standard Model – that is, QCD and the electroweak
theory. They are built on a generalization of the gauge principle, in which the
transformations involve more than one state, or field, at a time. In that case,
the ‘phase factors’ become matrices, which generally do not commute with
each other, and the associated symmetry is called a ‘non-Abelian’ one. When
the phase factors are independent of the space-time coordinate x, the symmetry is a ‘global non-Abelian’ one; when they are allowed to depend on x, one
is led to a non-Abelian gauge theory. Both QCD and the electroweak theory
are of the latter type, providing generalizations of the Abelian U(1) gauge
theory which is QED. It is a striking fact that all three dynamical theories in
the Standard Model are based on a gauge principle of local phase invariance.
In this chapter we shall be mainly concerned with two global non-Abelian
symmetries, which lead to useful conservation laws but not to any specific
dynamical theory. We begin in section 12.1 with the first non-Abelian symmetry to be used in particle physics, the hadronic isospin ‘SU(2) symmetry’
proposed by Heisenberg (1932) in the context of nuclear physics, and now
understood as following from QCD and the smallness of the u and d quark
masses as compared with the QCD scale parameter Λ
(see section 18.3.3).
MS
In section 12.2 we extend this to SU(3) f flavour symmetry, as was first done
by Gell-Mann (1961) and Ne’eman (1961) – an extension seen, in its turn, as
reflecting the smallness of the u, d and s quark masses as compared with Λ MS .
The ‘wavefunction’ approach of sections 12.1 and 12.2 is then reformulated in
field-theoretic language in section 12.3.
In the last section of this chapter, we shall introduce the idea of a global
chiral symmetry, which is a symmetry of theories with massless fermions. This
may be expected to be a good approximate symmetry for the u and d quarks.
3