32
12. Global Non-Abelian Symmetries
We now recall the matrix γ 5 introduced in section 4.2.1
γ 5 = iγ
0 γ
1 γ
2 γ
3 ,
(12.142)
which takes the form
(
)
1
0
γ 5 =
(12.143)
0 −1
in this representation. The matrix γ 5 plays a prominent role in chiral symmetry, as we shall see. Its defining property is that it anticommutes with the γ
μ
matrices:
{γ 5 , γ
μ
} = 0.
(12.144)
‘Chirality’ means ‘handedness’, from the Greek word for hand, χ∈iρ. Its
use here stems from the fact that, in the limit m → 0 the 2-component spinors
φ, χ become helicity eigenstates (cf problem 9.4), having definite ‘handedness’.
As m → 0 we have E → |p|, and (12.140) and (12.141) reduce to
˜
(σ · p/|p|)φ ˜ = φ
(12.145)
(σ · p/|p|) ˜
χ = − ˜
(12.146)
χ,
so that the limiting spinor φ ˜ has positive helicity, and ˜
χ negative helicity (cf
(3.68) and (3.69)). In this m → 0 limit, the two helicity spinors are decoupled,
reflecting the fact that no Lorentz transformation can reverse the helicity of
a massless particle. Also in this limit, the Dirac energy operator is
(
)
σ · p
0
α · p =
(12.147)
0
−σ · p
which is easily seen to commute with γ 5 . Thus the massless states may equivalently be classified by the eigenvalues of γ 5 , which are clearly ±1 since γ
2 = I.
5
Consider then a massless fermion with positive helicity. It is described
(
)
˜
φ
by the ‘u’-spinor
which is an eigenstate of γ 5 with eigenvalue +1.
0
(
)
0
Similarly, a fermion with negative helicity is described by
which has
χ ˜
γ 5 = −1. Thus for these states chirality equals helicity. We have to be more
careful for antifermions, however. A physical antifermion of energy E and
momentum p is described by a ‘v’- spinor corresponding to −E and −p; but
with m = 0 in (12.140) and (12.141) the equations for φ and χ remain the
same for −E, −p as for E, p. Consider the spin, however. If the physical
1
antiparticle has positive helicity, with p along the z-axis say, then s z = + 2 .
The corresponding v-spinor must then have s z = −
1 (see section 3.4.3) and
2
must therefore be of ˜
χ type (12.146). So the v-spinor for this antifermion of
(
)
0
positive helicity is
which has γ 5 = −1. In summary, for fermions the
χ ˜
γ 5 eigenvalue is equal to the helicity, and for antifermions it is equal to minus
the helicity. It is the γ 5 eigenvalue that is called the ‘chirality’.
12. Global Non-Abelian Symmetries
We now recall the matrix γ 5 introduced in section 4.2.1
γ 5 = iγ
0 γ
1 γ
2 γ
3 ,
(12.142)
which takes the form
(
)
1
0
γ 5 =
(12.143)
0 −1
in this representation. The matrix γ 5 plays a prominent role in chiral symmetry, as we shall see. Its defining property is that it anticommutes with the γ
μ
matrices:
{γ 5 , γ
μ
} = 0.
(12.144)
‘Chirality’ means ‘handedness’, from the Greek word for hand, χ∈iρ. Its
use here stems from the fact that, in the limit m → 0 the 2-component spinors
φ, χ become helicity eigenstates (cf problem 9.4), having definite ‘handedness’.
As m → 0 we have E → |p|, and (12.140) and (12.141) reduce to
˜
(σ · p/|p|)φ ˜ = φ
(12.145)
(σ · p/|p|) ˜
χ = − ˜
(12.146)
χ,
so that the limiting spinor φ ˜ has positive helicity, and ˜
χ negative helicity (cf
(3.68) and (3.69)). In this m → 0 limit, the two helicity spinors are decoupled,
reflecting the fact that no Lorentz transformation can reverse the helicity of
a massless particle. Also in this limit, the Dirac energy operator is
(
)
σ · p
0
α · p =
(12.147)
0
−σ · p
which is easily seen to commute with γ 5 . Thus the massless states may equivalently be classified by the eigenvalues of γ 5 , which are clearly ±1 since γ
2 = I.
5
Consider then a massless fermion with positive helicity. It is described
(
)
˜
φ
by the ‘u’-spinor
which is an eigenstate of γ 5 with eigenvalue +1.
0
(
)
0
Similarly, a fermion with negative helicity is described by
which has
χ ˜
γ 5 = −1. Thus for these states chirality equals helicity. We have to be more
careful for antifermions, however. A physical antifermion of energy E and
momentum p is described by a ‘v’- spinor corresponding to −E and −p; but
with m = 0 in (12.140) and (12.141) the equations for φ and χ remain the
same for −E, −p as for E, p. Consider the spin, however. If the physical
1
antiparticle has positive helicity, with p along the z-axis say, then s z = + 2 .
The corresponding v-spinor must then have s z = −
1 (see section 3.4.3) and
2
must therefore be of ˜
χ type (12.146). So the v-spinor for this antifermion of
(
)
0
positive helicity is
which has γ 5 = −1. In summary, for fermions the
χ ˜
γ 5 eigenvalue is equal to the helicity, and for antifermions it is equal to minus
the helicity. It is the γ 5 eigenvalue that is called the ‘chirality’.