Quantum Mechanics for Mathematicians

(

∂

∂t

−σ·∇

)

ψL=imψR

Whenm= 0 the equations decouple and one can consistently restrict atten-
tion to just right-handed or left-handed solutions, giving:

Definition(Weyl equations). The Weyl wave equations for two-component
spinors are (
∂
∂t

+σ·∇

)

ψR= 0 (47.16)

(

∂

∂t

−σ·∇

)

ψL= 0 (47.17)

Also in the massless case, the chirality operator satisfies

[γ 5 ,HD] = 0

(HDis the Dirac Hamiltonian 47.5) since for eachj

[γ 5 ,γ 0 γj] = 0

This follows from the fact that commutingγ 0 throughγ 5 gives three minus
signs, commutingγjthroughγ 5 gives another three. In this case chirality is a
conserved quantity, and the complex structureJW=−iγ 5 commutes withHD.
JW then takes positive energy solutions to positive energy solutions, negative
energy to negative energy solutions, and thus commutes with the relativistic
complex structureJr.
We now have two commuting complex structuresJWandJronV, and they
can be simultaneously diagonalized (much like the situation in section 44.1.2).
We get a decomposition

VJ+r=H 1 =H 1 ,L⊕H 1 ,R

of the positive energy solutions into +i(H 1 ,L) and−i(H 1 ,R) eigenspaces of
JW. Restricting to the solutions of equation 47.17 we get a decomposition

VL=H 1 ,L⊕H 1 ,L

into positive and negative energy left-handed solutions. We can then take Weyl
spinor fields to be two-component objects

ΨL(x),ΨL(x)

that are continuous basis elements ofH 1 ,LandH 1 ,Lrespectively. The action of
the (double cover of the) Poincar ́e group on space-time dependent Weyl fields
will be given by

ΨL(x)→(a,Λ)ΨL(x) =S(Λ)−^1 ΨL(Λ·x+a)