View source for Van der Waerden notation

{{Short description|Notation used for Weyl spinors}}
In [[theoretical physics]], '''Van der Waerden notation'''<ref>{{cite journal|title=Spinoranalyse|author=Van der Waerden B.L.|journal=Nachr. Ges. Wiss. Göttingen Math.-Phys.|volume=ohne Angabe|year=1929|pages=100–109}}</ref><ref>{{cite journal|title=Geometry of two-component Spinors|author=Veblen O.|journal=Proc. Natl. Acad. Sci. USA|volume=19|year=1933|issue=4|pages=462–474|doi=10.1073/pnas.19.4.462 |bibcode = 1933PNAS...19..462V |pmc=1086023|pmid=16577541|doi-access=free}}</ref> refers to the usage of two-component [[spinor]]s ([[Weyl spinor]]s) 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.

:<math>\Sigma_\mathrm{left} = 
\begin{pmatrix}
\psi_{\alpha}\\
0
\end{pmatrix}
</math>

;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.

:<math>\Sigma_\mathrm{right} = 
\begin{pmatrix}
0 \\
\bar{\chi}^{\dot{\alpha}}\\
\end{pmatrix}
</math>

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

:<math> \alpha = 1,2\,,\dot{\alpha} = \dot{1},\dot{2}</math>

then a spinor in the chiral basis is represented as

:<math>\Sigma_\hat{\alpha} = 
\begin{pmatrix}
\psi_{\alpha}\\
\bar{\chi}^{\dot{\alpha}}\\
\end{pmatrix}
</math>

where

:<math> \hat{\alpha}= (\alpha,\dot{\alpha}) = 1,2,\dot{1},\dot{2}</math>

In this notation the [[Dirac adjoint]] (also called the '''Dirac conjugate''') is

:<math>\Sigma^\hat{\alpha} = 
\begin{pmatrix}
\chi^{\alpha} & \bar{\psi}_{\dot{\alpha}}
\end{pmatrix}
</math>

==See also==
* [[Dirac equation]]
* [[Infeld–Van der Waerden symbols]]
* [[Lorentz transformation]]
* [[Pauli equation]]
* [[Ricci calculus]]

==Notes==
{{Reflist}}

==References==
* [https://web.archive.org/web/20160920000629/https://terpconnect.umd.edu/~wdlinch3/MyNotes/07.04.30.14.30.RTGSpin.pdf Spinors in physics]
*{{citation|author= P. Labelle|title=Supersymmetry|year = 2010|publisher = Demystified series, McGraw-Hill (USA)|isbn=978-0-07-163641-4}}
* {{citation|last1 = Hurley|first1=D.J.|last2 = Vandyck|first2=M.A.|title=Geometry, Spinors and Applications|year = 2000|publisher = Springer |isbn=1-85233-223-9}}
* {{citation|last1 = Penrose|first1=R.|last2 = Rindler|first2=W.|title=Spinors and Space–Time|year = 1984|publisher = Cambridge University Press|volume=1|isbn=0-521-24527-3}}
* {{citation|last1 = Budinich|first1=P.|last2 = Trautman|first2=A.|title=The Spinorial Chessboard|year = 1988|publisher = Springer-Verlag|isbn=0-387-19078-3}}

{{tensors}}

[[Category:Spinors]]
[[Category:Mathematical notation]]