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Van der Waerden notation

{{Short description|Notation used for Weyl spinors}}

In theoretical physics, Van der Waerden notation{{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}}{{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}} 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 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}

See also

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