4Spinors
III Quantum Field Theory
4.5 Chiral/Weyl spinors and γ
5
Recall that if we picked the chiral representation of the Clifford algebra, the
corresponding representation of the spin group is
S[Λ] =
e
1
2
χ·σ
0
0 e
−
1
2
χ·σ
!
for boosts
e
i
2
φ·σ
0
0 e
i
2
φ·σ
!
for rotations
.
It is pretty clear from the presentation that this is actually just two independent
representations put together, i.e. the representation is reducible. We can then
write our spinor ψ as
ψ =
U
+
U
−
,
where
U
+
and
U
−
are 2-complex-component objects. These objects are called
Weyl spinors or chiral spinors.
Definition
(Weyl/chiral spinor)
.
A left (right)-handed chiral spinor is a 2-
component complex vector
U
+
and
U
−
respectively that transform under the
action of the Lorentz/spin group as follows:
Under a rotation with rotation parameters φ, both of them transform as
U
±
7→ e
iφ·σ/2
U
±
,
Under a boost χ, they transform as
U
±
7→ e
±χ·σ/2
U
±
.
So these are two two-dimensional representations of the spin group.
We have thus discovered
Proposition.
A Dirac spinor is the direct sum of a left-handed chiral spinor
and a right-handed one.
In group theory language,
U
+
is the (0
,
1
2
) representation of the Lorentz
group, and U
−
is the (
1
2
, 0) representation, and ψ is in (
1
2
, 0) ⊕ (0,
1
2
).
As an element of the representation space, the left-handed part and right-
handed part are indeed completely independent. However, we know that the
evolution of spinors is governed by the Dirac equation. So a natural question to
ask is if the Weyl spinors are coupled in the Dirac Lagrangian.
Decomposing the Lagrangian in terms of our Weyl spinors, we have
L =
¯
ψ(i
/
∂ − m)ψ
=
U
†
+
U
†
−
0 1
1 0
i
0 ∂
t
+ σ
i
∂
i
∂
t
− σ
i
∂
i
0
− m
1 0
0 1
U
+
U
−
= iU
†
−
σ
µ
∂
µ
U
−
+ iU
†
+
¯σ
µ
∂
µ
U
+
− m(U
†
+
U
−
+ U
†
−
U
+
),
where
σ
µ
= (1, σ), ¯σ
µ
= (1, −σ).
So the left and right-handed fermions are coupled if and only if the particle is
massive. If the particle is massless, then we have two particles satisfying the
Weyl equation:
Definition (Weyl equation). The Weyl equation is
i¯σ
µ
∂
µ
U
+
= 0.
This is all good, but we produced these Weyl spinors by noticing that in our
particular chiral basis, the matrices
S
[Λ] looked good. Can we produce a more
intrinsic definition of these Weyl spinors that do not depend on a particular
representation of the Dirac spinors?
The solution is to introduce the magic quantity γ
5
:
Definition (γ
5
).
γ
5
= −iγ
0
γ
1
γ
2
γ
3
.
Proposition. We have
{γ
µ
, γ
5
} = 0, (γ
5
)
2
= 1
for all γ
µ
, and
[S
µν
, γ
5
] = 0.
Since (γ
5
)
2
= 1, we can define projection operators
P
±
=
1
2
(1 ± γ
5
).
Example. In the chiral representation, we have
γ
5
=
1 0
0 −1
.
Then we have
P
+
=
1 0
0 0
, P
−
=
0 0
0 1
.
We can prove, in general, that these are indeed projections:
Proposition.
P
2
±
= P
±
, P
+
P
−
= P
−
P
+
= 0.
Proof. We have
P
2
±
=
1
4
(1 ± γ
5
)
2
=
1
4
(1 + (γ
5
)
2
± 2γ
5
) =
1
2
(1 ± γ
5
),
and
P
+
P
−
=
1
4
(1 + γ
5
)(1 − γ
5
) =
1
4
(1 − (γ
5
)
2
) = 0.
We can think of these
P
±
as projection operators to two orthogonal subspaces
of the vector space
V
of spinors. We claim that these are indeed representations
of the spin group. We define
V
±
= {P
±
ψ : ψ ∈ V }.
We claim that
S
[Λ] maps
V
±
to itself. To show this, we only have to compute
infinitesimally, i.e. that
S
µν
maps
V
±
to itself. But it follows immediately from
the fact that S
µν
commutes with γ
5
that
S
µν
P
±
ψ = P
±
S
µν
ψ.
We can then define the chiral spinors as
ψ
±
= P
±
ψ.
It is clear from our previous computation of the
P
±
in the chiral basis that these
agree with what we’ve defined before.