# Van der Waerden notation

In theoretical physics, van der Waerden notation [1][2] refers to the usage of two-component spinors (Weyl spinors) in four spacetime dimensions. This is standard in twistor theory and supersymmetry. It is named after Bartel Leendert van der Waerden.

## 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.

## 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

1. ^ Van der Waerden B.L. (1929). "Spinoranalyse". Nachr. Ges. Wiss. Göttingen Math.-Phys. 1929: 100–109.
2. ^ Veblen O. (1933). "Geometry of two-component Spinors". Proc. Natl. Acad. Sci. USA 19: 462–474.