 Solution

ID: past-exam-of-the-groups-and-representations-course-of-the-university-of-oxford/2025/3/b/solution

Choose the chirality operator

Γ = γ^{0} γ^{1} = σ_{3}, Γ^{2} = I, {Γ, γ^{μ}} = 0.

Γ

P_{R} = (I + Γ) /2

and

P_{L} = (I - Γ) /2

. Consequently

ψ_{R} = P_{R} ψ = (ψ_{+} 0), ψ_{L} = P_{L} ψ = (0 ψ_{-}) .

The two independent Weyl spinors each contain one complex component. Since

[S (θ), Γ] = 0

, each chirality is preserved by every connected Lorentz transformation, with

ψ_{R} \mapsto e^{θ /2} ψ_{R}

and

ψ_{L} \mapsto e^{- θ /2} ψ_{L}

The left/right names depend on the sign convention for

Γ

. Choosing

γ_{0} γ_{1} = - σ_{3}

as the chirality operator exchanges the labels

L

and

R

; the two invariant one-component representations are the same.

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