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| 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=1929|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|pages=462–474}}</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]].
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| ==Dotted indices==
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| ;Undotted indices (chiral indices)
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| Spinors with lower undotted indices have a left-handed chiralty, and are called chiral indices.
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| :<math>\Sigma_\mathrm{left} =
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| \begin{pmatrix}
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| \psi_{\alpha}\\
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| 0
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| \end{pmatrix}
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| </math> | |
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| ;Dotted indices (anti-chiral indices)
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| Spinors with raised dotted indices, plus an overbar on the symbol (not index), are right-handed, and called anti-chiral indices.
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| :<math>\Sigma_\mathrm{right} = | |
| \begin{pmatrix}
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| 0 \\
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| \bar{\chi}^{\dot{\alpha}}\\
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| \end{pmatrix}
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| </math>
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| 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.
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| ==Hatted indices==
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| 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
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| :<math> \alpha = 1,2\,,\dot{\alpha} = \dot{1},\dot{2}</math>
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| then a spinor in the chiral basis is represented as
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| :<math>\Sigma_\hat{\alpha} =
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| \begin{pmatrix}
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| \psi_{\alpha}\\
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| \bar{\chi}^{\dot{\alpha}}\\
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| \end{pmatrix}
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| </math>
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| where
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| :<math> \hat{\alpha}= (\alpha,\dot{\alpha}) = 1,2,\dot{1},\dot{2}</math>
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| In this notation the [[Dirac adjoint]] (also called the '''Dirac conjugate''') is
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| :<math>\Sigma^\hat{\alpha} =
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| \begin{pmatrix}
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| \chi^{\alpha} & \bar{\psi}_{\dot{\alpha}}
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| \end{pmatrix}
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| </math>
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| ==See also==
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| * [[Dirac equation]]
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| * [[bra-ket notation]]
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| * [[Infeld–van der Waerden symbols]]
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| * [[Lorentz transformation]]
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| * [[Pauli equation]]
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| * [[Ricci calculus]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * [http://www.math.sunysb.edu/rtg/Images/07.04.30.14.30.RTGSpin.pdf Spinors in physics]
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| *{{citation|author= P. Labelle|title=Supersymmetry|year = 2010|publisher = Demystified series, McGraw-Hill (USA)|isbn=978-0-07-163641-4}}
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| * {{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}}
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| * {{citation|last1 = Penrose|first1=R.|last2 = Rindler|first2=W.|title=Spinors and Space–Time|year = 1984|publisher = Cambridge University Press|volume=Vol. 1|isbn=0-521-24527-3}}
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| * {{citation|last1 = Budinich|first1=P.|last2 = Trautman|first2=A.|title=The Spinorial Chessboard|year = 1988|publisher = Spinger-Verlag|isbn=0-387-19078-3}}
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| {{DEFAULTSORT:Van Der Waerden Notation}}
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| [[Category:Quantum field theory]]
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| [[Category:Spinors]]
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| [[Category:Mathematical notation]]
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| {{Phys-stub}}
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