L_{D} = i \overset{ˉ}{ψ} γ^{μ} \partial_{μ} ψ - m \overset{ˉ}{ψ} ψ .

ψ_{M} = (i σ ^{2} u _{L}^{*} u _{L}) .

Substitution shows that its two chiral kinetic terms agree after integration by parts and that

- m \overset{ˉ}{ψ}_{M} ψ_{M} = im {u_{L}^{T} σ^{2} u_{L} - u_{L}^{†} σ^{2} u_{L}^{*}} .

Thus

L_{D} [ψ_{M}]

equals twice the displayed one-Weyl-field Lagrangian, up to a total derivative.

The lower chiral component of the Dirac equation is

i \overset{σ}{ˉ}^{μ} \partial_{μ} u_{L} - m u_{R} = 0.

Putting

u_{R} = i σ^{2} u_{L}^{*}

gives exactly

i \overset{σ}{ˉ}^{μ} \partial_{μ} u_{L} - im σ^{2} u_{L}^{*} = 0

; the upper component is its complex conjugate.

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