Van Der Waerden Notation - Hatted Indices

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

then a spinor in the chiral basis is represented as

\Sigma_\hat{\alpha} = \begin{pmatrix} \psi_{\ell}\\ \bar{\chi}^{\dot{r}}\\ \end{pmatrix}

where

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}

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