\psi=\left(\begin{array}{c} \psi_{L}\\ \psi_{R} \end{array}\right)=\left(\begin{array}{c} {\psi_{L}}_{\alpha}\\ {\psi_{R}}^{\dot{\alpha}} \end{array}\right)

left-handed spinorとright-handed spinorのLorentz変換は次のようにかける。

{\psi_{L}}_{\alpha} \rightarrow{\left(\exp\left[-i\theta_{k}\frac{\sigma^{k}}{2}-\beta_{k}\frac{\sigma^{k}}{2}\right]\right)_{\alpha}}^{\beta}{\psi_{L}}_{\beta}

{\psi_{R}}^{\dot{\alpha}} \rightarrow{\left(\exp\left[-i\theta_{k}\frac{\sigma^{k}}{2}+\beta_{k}\frac{\sigma^{k}}{2}\right]\right)^{\dot{\alpha}}}_{\dot{\beta}}{\psi_{R}}^{\dot{\beta}}

\sigma^{2}\psi_{L}^{*}はright-handed spinorのような変換則を持つ。

{\psi_{L}^{*}}_{\dot{\alpha}} \rightarrow{\left(\exp\left[i\theta_{k}\frac{\sigma^{*k}}{2}-\beta_{k}\frac{\sigma^{*k}}{2}\right]\right)_{\dot{\alpha}}}^{\dot{\beta}}{\psi_{L}^{*}}_{\dot{\beta}}

\left(i\sigma^{2}\right)^{\dot{\gamma}\dot{\alpha}}{\psi_{L}^{*}}_{\dot{\alpha}} \rightarrow\left(i\sigma^{2}\right)^{\dot{\gamma}\dot{\alpha}}{\left(\exp\left[i\theta_{k}\frac{\sigma^{*k}}{2}-\beta_{k}\frac{\sigma^{*k}}{2}\right]\right)_{\dot{\alpha}}}^{\dot{\beta}}\left(-i\sigma^{2}\right)_{\dot{\beta}\dot{\delta}}\left(i\sigma^{2}\right)^{\dot{\delta}\dot{\kappa}}{\psi_{L}^{*}}_{\dot{\kappa}}

\varepsilon^{\dot{\gamma}\dot{\alpha}}{\psi_{L}^{*}}_{\dot{\alpha}} \rightarrow\varepsilon^{\dot{\gamma}\dot{\alpha}}{\left(\exp\left[i\theta_{k}\frac{\sigma^{*k}}{2}-\beta_{k}\frac{\sigma^{*k}}{2}\right]\right)_{\dot{\alpha}}}^{\dot{\beta}}\varepsilon_{\dot{\beta}\dot{\delta}}\varepsilon^{\dot{\delta}\dot{\kappa}}{\psi_{L}^{*}}_{\dot{\kappa}}

\left(i\sigma^{2}\psi_{L}^{*}\right)^{\dot{\gamma}} \rightarrow{\left(\exp\left[i\theta_{k}\frac{\sigma^{*k}}{2}-\beta_{k}\frac{\sigma^{*k}}{2}\right]\right)^{\dot{\gamma}}}_{\dot{\delta}}\left(i\sigma^{2}\psi_{L}^{*}\right)^{\dot{\delta}}