Quantum Mechanics for Mathematicians

47.3 Weyl spinors

For the casem= 0 of the Dirac equation, it turns out that there is an interesting
operator acting on the space of solutions:

Definition(Chirality).The operator

γ 5 =iγ 0 γ 1 γ 2 γ 3

is called the chirality operator. It has eigenvalues± 1 and its eigenstates are said
to have chirality± 1. States with chirality+1are called “right-handed”, those
with chirality− 1 are called “left-handed”.

Note that the operatorJW =−iγ 5 =γ 0 γ 1 γ 2 γ 3 satisfiesJW^2 =−1 and
provides a choice of complex structure on the spaceVof real-valued solutions
of the Dirac equation. We can complexify such solutions and write

V ⊗C=VL⊕VR (47.15)

whereVL is the +i eigenspace ofJW (the negative or left-handed chirality
solutions), andVRis the−ieigenspace ofJW (the positive or right-handed
chirality solutions).
To work withJWeigenvectors, it is convenient to adopt a choice ofγ-matrices
in whichγ 5 is diagonal. This cannot be done with real matrices, but requires
complexification. One such choice was already described in 41.2, the chiral or
Weyl representation. In this choice, theγmatrices can be written in 2 by 2
block form as

γ 0 =−i

(

0 1

1 0

)

,γ 1 =−i

(

0 σ 1

−σ 1 0

)

,γ 2 =−i

(

0 σ 2

−σ 2 0

)

,γ 3 =−i

(

0 σ 3

−σ 3 0

)

and the chirality operator is diagonal

γ 5 =

(

−1 0

0 1

)

We can thus write (complexified) solutions in terms of chiral eigenstates as

ψ=

(

ψL

ψR

)

whereψLandψRare two-component wavefunctions, of left and right chirality
respectively.
The Dirac equation 47.2 is then

−i

(

−im ∂t∂ +σ·∇

∂

∂t−σ·∇ −im

)(

ψL

ψR

)

= 0

or, in terms of two-component functions
(
∂
∂t

+σ·∇

)

ψR=imψL