Quantum Mechanics for Mathematicians

where Λ is an element ofSpin(3,1) andS(Λ)∈SL(2,C) is the (^12 ,0) repre-
sentation (see chapter 41, whereS(Λ) = Ω). Λ·xis theSpin(3,1) action on
Minkowski space vectors.
Just as in the Majorana case, parametrizing the space of solutions using
fixed-time fields does not allow one to see the action of boosts on the fields.
In addition, we know that relativistic field quantization requires use of the rel-
ativistic complex structureJr, which is not simply expressed in terms of the
fixed-time fields. To solve both problems we need to study the solutions in
momentum space. To find solutions in momentum space we Fourier transform,
using

ψ(t,x) =

1

(2π)^2

∫

d^4 p ei(−p^0 t+p·x)ψ ̃(p 0 ,p)

and see that the Weyl equations are

(p 0 −σ·p)ψ ̃R= 0

(p 0 +σ·p)ψ ̃L= 0

Since
(p 0 +σ·p)(p 0 −σ·p) =p^20 −(σ·p)^2 =p^20 −|p|^2

bothψ ̃Randψ ̃Lsatisfy
(p^20 −|p|^2 )ψ ̃= 0

so are functions with support on the positive (p 0 =|p|) and negative (p 0 =−|p|)
energy null-cone. These are Fourier transforms of solutions to the massless
Klein-Gordon equation
(
−

∂^2

∂x^20

+

∂^2

∂x^21

+

∂^2

∂x^22

+

∂^2

∂x^23

)

ψ= 0

In the two-component formalism, one can define:

Definition(Helicity).The operator

1

2

σ·p

|p|

(47.18)

on the space of solutions to the Weyl equations is called the helicity operator. It
has eigenvalues±^12 , and its eigenstates are said to have helicity±^12.

The helicity operator is the component of the spin operatorS=^12 σalong the
direction of the momentum of a particle. Single-particle helicity eigenstates of
eigenvalue +^12 are said to have “right-handed helicity”, and described as having
spin in the same direction as their momentum, those with helicity eigenvalue
−^12 are said to have “left-handed helicity” and spin in the opposite direction to
their momentum.
A continuous basis of solutions to the Weyl equation forψLis given by the
wavefunctions
uL(p)ei(−p^0 x^0 +p·x)