Van der Waerden notation

;Undotted indices (chiral indices) Spinors with lower undotted indices have a left-handed chirality, and are called chiral indices. :\Sigma_\mathrm{left} = \begin{pmatrix} \psi_{\alpha}\\ 0 \end{pmatrix} ;Dotted indices (anti-chiral indices) Spinors with raised dotted indices, plus an overbar on the symbol (not index), are right-handed, and called anti-chiral indices. :\Sigma_\mathrm{right} = \begin{pmatrix} 0 \\ \bar{\chi}^{\dot{\alpha}}\\ \end{pmatrix} Without the indices, i.e. "index free notation", an overbar is retained on right-handed spinor, since ambiguity arises between chirality when no index is indicated. ==Hatted indices==

Indices which have hats are called Dirac indices, and are the set of dotted and undotted, or chiral and anti-chiral, indices. For example, if : \alpha = 1,2\,,\dot{\alpha} = \dot{1},\dot{2} then a spinor in the chiral basis is represented as :\Sigma_\hat{\alpha} = \begin{pmatrix} \psi_{\alpha}\\ \bar{\chi}^{\dot{\alpha}}\\ \end{pmatrix} where : \hat{\alpha}= (\alpha,\dot{\alpha}) = 1,2,\dot{1},\dot{2} In this notation the Dirac adjoint (also called the Dirac conjugate) is :\Sigma^\hat{\alpha} = \begin{pmatrix} \chi^{\alpha} & \bar{\psi}_{\dot{\alpha}} \end{pmatrix} ==See also==