Van der Waerden notation

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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[edit]

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[edit]

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}

See also[edit]

Notes[edit]

  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. 

References[edit]