We begin by checking that the free Dirac equation is indeed Lorentz invariant. We shall show
that
(
γμ∂′μ−m
)
ψ′(x′) =D(Λ) (γμ∂μ−m)ψ(x) (22.34)To establish this relation, it suffices to compute the left hand side, using the following transformation
results,
ψ′(x′) = D(Λ)ψ(x)
∂μ′ = Λμν∂ν
γμΛμν = D(Λ)γνD(Λ)−^1 (22.35)If now (γμ∂μ−m)ψ(x) = 0, then it follows that also
(
γμ∂μ′−m)
ψ′(x′) = 0, and the equation is
covariant.
Group theoretically, the structure of the Dirac equation isorganized as follows,ψ ∼(
1
2, 0
)
⊕(
0 ,1
2
)∂ ∼
(
1
2,
1
2
)∂ψ ∼(
1
2,
1
2
)
⊗[(
1
2, 0
)
⊕(
0 ,1
2
)]=
(
0 ,1
2
)
⊕(
1 ,1
2
)
⊕(
1
2, 0
)
⊕(
1
2, 1
)
(22.36)Theγ-matrix is responsible for projecting out the representations (1,^12 )⊕(^12 ,1) and retaining only
(^12 ,0)⊕(0,^12 ). In the Weyl basis, we have
γ^0 =(
0 I 2
−I 2 0)
γi=(
0 σi
σi 0)
γ^5 =(
I 2 0
0 −I 2)
(22.37)It is customary to introduce the Pauli matrices with a Lorentz index, as follows,
σμ ≡ (+I 2 ,σi)
σ ̄μ ≡ (−I 2 ,σi) (22.38)The Dirac matrices may then be expressed in a manifestly Lorentz covariant way,
γμ=(
0 σμ
̄σμ 0)
(22.39)Decomposing now also the Dirac fieldψinto 2-component spinorsψLandψR, referred to as the
left and the right Weyl spinors,
ψ=(
ψL
ψR)
(22.40)