# Van der Waerden notation

In theoretical physics, van der Waerden notation 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 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 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

$\Sigma ^{\hat {\alpha }}={\begin{pmatrix}\chi ^{\alpha }&{\bar {\psi }}_{\dot {\alpha }}\end{pmatrix}}$