Van Der Waerden Notation - Dotted Indices

Dotted Indices

Undotted indices (chiral indices)

Spinors with lower undotted indices have a left-handed chiralty, 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 chiralty when no index is indicated.

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