Van der Waerden notation

Dotted indices

;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}

Notes

References

• Spinors in physics

References

1. Van der Waerden B.L.. (1929). "Spinoranalyse". Nachr. Ges. Wiss. Göttingen Math.-Phys..
2. Veblen O.. (1933). "Geometry of two-component Spinors". Proc. Natl. Acad. Sci. USA.