4
12. Global Non-Abelian Symmetries
But the anticipated observable consequences of this symmetry (for example,
nucleon parity doublets) appear to be absent. This puzzle will be resolved
in Part VII, via the profoundly important concept of ‘spontaneous symmetry
breaking’.
The formalism introduced in this chapter for SU(2) and SU(3) will be
required again in the following one, when we consider the local versions of
these non-Abelian symmetries and the associated dynamical gauge theories.
The whole modern development of non-Abelian gauge theories began with
the attempt by Yang and Mills (1954) (see also Shaw 1955) to make hadronic
isospin into a local symmetry. However, the beautiful formalism developed
by these authors turned out not to describe interactions between hadrons.
Instead, it describes the interactions between the constituents of the hadrons,
namely quarks – and this in two respects. First, a local SU(3) symmetry
(called SU(3) c ) governs the strong interactions of quarks, binding them into
hadrons (see Part VI). Secondly, a local SU(2) symmetry (called weak isospin)
governs the weak interactions of quarks (and leptons); together with QED, this
constitutes the electroweak theory (see Part VIII). It is important to realize
that, despite the fact that each of these two local symmetries is based on
the same group as one of the earlier global (flavour) symmetries, the physics
involved is completely different. In the case of the strong quark interactions,
the SU(3) c group refers to a new degree of freedom (‘colour’) which is quite
distinct from flavour u, d, s (see chapter 14). In the weak interaction case,
since the group is an SU(2), it is natural to use ‘isospin language’ in talking
about it, particularly since flavour degrees of freedom are involved. But we
must always remember that it is weak isospin, which (as we shall see in chapter
20) is an attribute of leptons as well as of quarks, and hence physically quite
distinct from hadronic isospin. Furthermore, it is a parity-violating chiral
gauge theory.
Despite the attractive conceptual unity associated with the gauge principle, the way in which each of QCD and the electroweak theory ‘works’ is
actually quite different from QED, and from each other. Indeed it is worth
emphasizing very strongly that it is, a priori, far from obvious why either the
strong interactions between quarks, or the weak interactions, should have anything to do with gauge theories at all. Just as in the U(1) (electromagnetic)
case, gauge invariance forbids a mass term in the Lagrangian for non-Abelian
gauge fields, as we shall see in chapter 13. Thus it would seem that gauge
field quanta are necessarily massless. But this, in turn, would imply that the
associated forces must have a long-range (Coulombic) part, due to exchange of
these massless quanta – and of course in neither the strong nor the weak interaction case is that what is observed.
1 As regards the former, the gluon quanta
are indeed massless, but the contradiction is resolved by non-perturbative effects which lead to confinement, as we indicated in chapter 1. We shall discuss
1 Pauli had independently developed the theory of non-Abelian gauge fields during 1953,
but did not publish any of this work because of the seeming physical irrelevancy associated
with the masslessness problem (Enz 2002, pages 474-82; Pais 2000, pages 242-5).
12. Global Non-Abelian Symmetries
But the anticipated observable consequences of this symmetry (for example,
nucleon parity doublets) appear to be absent. This puzzle will be resolved
in Part VII, via the profoundly important concept of ‘spontaneous symmetry
breaking’.
The formalism introduced in this chapter for SU(2) and SU(3) will be
required again in the following one, when we consider the local versions of
these non-Abelian symmetries and the associated dynamical gauge theories.
The whole modern development of non-Abelian gauge theories began with
the attempt by Yang and Mills (1954) (see also Shaw 1955) to make hadronic
isospin into a local symmetry. However, the beautiful formalism developed
by these authors turned out not to describe interactions between hadrons.
Instead, it describes the interactions between the constituents of the hadrons,
namely quarks – and this in two respects. First, a local SU(3) symmetry
(called SU(3) c ) governs the strong interactions of quarks, binding them into
hadrons (see Part VI). Secondly, a local SU(2) symmetry (called weak isospin)
governs the weak interactions of quarks (and leptons); together with QED, this
constitutes the electroweak theory (see Part VIII). It is important to realize
that, despite the fact that each of these two local symmetries is based on
the same group as one of the earlier global (flavour) symmetries, the physics
involved is completely different. In the case of the strong quark interactions,
the SU(3) c group refers to a new degree of freedom (‘colour’) which is quite
distinct from flavour u, d, s (see chapter 14). In the weak interaction case,
since the group is an SU(2), it is natural to use ‘isospin language’ in talking
about it, particularly since flavour degrees of freedom are involved. But we
must always remember that it is weak isospin, which (as we shall see in chapter
20) is an attribute of leptons as well as of quarks, and hence physically quite
distinct from hadronic isospin. Furthermore, it is a parity-violating chiral
gauge theory.
Despite the attractive conceptual unity associated with the gauge principle, the way in which each of QCD and the electroweak theory ‘works’ is
actually quite different from QED, and from each other. Indeed it is worth
emphasizing very strongly that it is, a priori, far from obvious why either the
strong interactions between quarks, or the weak interactions, should have anything to do with gauge theories at all. Just as in the U(1) (electromagnetic)
case, gauge invariance forbids a mass term in the Lagrangian for non-Abelian
gauge fields, as we shall see in chapter 13. Thus it would seem that gauge
field quanta are necessarily massless. But this, in turn, would imply that the
associated forces must have a long-range (Coulombic) part, due to exchange of
these massless quanta – and of course in neither the strong nor the weak interaction case is that what is observed.
1 As regards the former, the gluon quanta
are indeed massless, but the contradiction is resolved by non-perturbative effects which lead to confinement, as we indicated in chapter 1. We shall discuss
1 Pauli had independently developed the theory of non-Abelian gauge fields during 1953,
but did not publish any of this work because of the seeming physical irrelevancy associated
with the masslessness problem (Enz 2002, pages 474-82; Pais 2000, pages 242-5).