24
12. Global Non-Abelian Symmetries
minus the corresponding structure constant. Such a representation is always
possible for a Lie group, and is called the adjoint, or regular, representation
(see appendix M, section M.5). These representations are of particular importance in gauge theories, as we will see, since gauge quanta always belong
to the adjoint representation of the gauged group (for example, the 8 gluons
in SU(3) c ).
Further flavours c, b and t of course exist, but the mass differences are now
so large that it is generally not useful to think about higher flavour groups
such as SU(4) f etc. Instead, we now move on to consider the field-theoretic
formulation of global SU(2) f and SU(3) f .
12.4 Non-Abelian global symmetries in Lagrangian
quantum field theory
12.4.1 SU(2) f and SU(3) f
As may already have begun to be apparent in chapter 7, Lagrangian quantum
field theory is a formalism which is especially well adapted for the description
of symmetries. Without going into any elaborate general theory, we shall now
give a few examples showing how global flavour symmetry is very easily built
into a Lagrangian, generalizing in a simple way the global U(1) symmetries
considered in section 7.1 and section 7.2. This will also prepare the way for
the (local) gauge case, to be considered in the following chapter.
Consider, for example, the Lagrangian
ˆ ¯
¯ ˆ
L = u ˆ(i / ∂ − m)ˆ u + d(i / ∂ − m)d ˆ
(12.85)
describing two free fermions ‘u’ and ‘d’ of equal mass m, with the overbar
now meaning the Dirac conjugate for the four-component spinor fields. Note
carefully that we are suppressing the space-time arguments of the quantum
fields ˆ
u(x), d ˆ (x). As in (12.50), we are using the convenient shorthand ψ ˆ u = ˆ
u
and ψ ˆ d = d ˆ . Let us introduce
(
)
u ˆ
q ˆ =
(12.86)
ˆ
d
so that L ˆ can be compactly written as
ˆ ¯
L = q ˆ(i / ∂ − m)ˆ q.
(12.87)
In this form it is obvious that L ˆ – and hence the associated Hamiltonian H ˆ –
is invariant under the global U(1) transformation
′
iα ˆ
q ˆ = e q
(12.88)
12. Global Non-Abelian Symmetries
minus the corresponding structure constant. Such a representation is always
possible for a Lie group, and is called the adjoint, or regular, representation
(see appendix M, section M.5). These representations are of particular importance in gauge theories, as we will see, since gauge quanta always belong
to the adjoint representation of the gauged group (for example, the 8 gluons
in SU(3) c ).
Further flavours c, b and t of course exist, but the mass differences are now
so large that it is generally not useful to think about higher flavour groups
such as SU(4) f etc. Instead, we now move on to consider the field-theoretic
formulation of global SU(2) f and SU(3) f .
12.4 Non-Abelian global symmetries in Lagrangian
quantum field theory
12.4.1 SU(2) f and SU(3) f
As may already have begun to be apparent in chapter 7, Lagrangian quantum
field theory is a formalism which is especially well adapted for the description
of symmetries. Without going into any elaborate general theory, we shall now
give a few examples showing how global flavour symmetry is very easily built
into a Lagrangian, generalizing in a simple way the global U(1) symmetries
considered in section 7.1 and section 7.2. This will also prepare the way for
the (local) gauge case, to be considered in the following chapter.
Consider, for example, the Lagrangian
ˆ ¯
¯ ˆ
L = u ˆ(i / ∂ − m)ˆ u + d(i / ∂ − m)d ˆ
(12.85)
describing two free fermions ‘u’ and ‘d’ of equal mass m, with the overbar
now meaning the Dirac conjugate for the four-component spinor fields. Note
carefully that we are suppressing the space-time arguments of the quantum
fields ˆ
u(x), d ˆ (x). As in (12.50), we are using the convenient shorthand ψ ˆ u = ˆ
u
and ψ ˆ d = d ˆ . Let us introduce
(
)
u ˆ
q ˆ =
(12.86)
ˆ
d
so that L ˆ can be compactly written as
ˆ ¯
L = q ˆ(i / ∂ − m)ˆ q.
(12.87)
In this form it is obvious that L ˆ – and hence the associated Hamiltonian H ˆ –
is invariant under the global U(1) transformation
′
iα ˆ
q ˆ = e q
(12.88)