Notation used for Weyl spinors
Intheoretical physics ,Van der Waerden notation [ 1] [ 2] spinors (Weyl spinors ) in four spacetime dimensions. This is standard intwistor theory andsupersymmetry . It is named afterBartel Leendert van der Waerden .
Undotted indices (chiral indices) Spinors with lower undotted indices have a left-handedchirality , and are called chiral indices.
Σ l e f t = ( ψ α 0 ) {\displaystyle \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.
Σ r i g h t = ( 0 χ ¯ α ˙ ) {\displaystyle \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.
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
α = 1 , 2 , α ˙ = 1 ˙ , 2 ˙ {\displaystyle \alpha =1,2\,,{\dot {\alpha }}={\dot {1}},{\dot {2}}} then a spinor in the chiral basis is represented as
Σ α ^ = ( ψ α χ ¯ α ˙ ) {\displaystyle \Sigma _{\hat {\alpha }}={\begin{pmatrix}\psi _{\alpha }\\{\bar {\chi }}^{\dot {\alpha }}\\\end{pmatrix}}} where
α ^ = ( α , α ˙ ) = 1 , 2 , 1 ˙ , 2 ˙ {\displaystyle {\hat {\alpha }}=(\alpha ,{\dot {\alpha }})=1,2,{\dot {1}},{\dot {2}}} In this notation theDirac adjoint (also called theDirac conjugate ) is
Σ α ^ = ( χ α ψ ¯ α ˙ ) {\displaystyle \Sigma ^{\hat {\alpha }}={\begin{pmatrix}\chi ^{\alpha }&{\bar {\psi }}_{\dot {\alpha }}\end{pmatrix}}} Spinors in physics P. Labelle (2010),Supersymmetry , Demystified series, McGraw-Hill (USA),ISBN 978-0-07-163641-4 Hurley, D.J.; Vandyck, M.A. (2000),Geometry, Spinors and Applications , Springer,ISBN 1-85233-223-9 Penrose, R.; Rindler, W. (1984),Spinors and Space–Time , vol. 1, Cambridge University Press,ISBN 0-521-24527-3 Budinich, P.; Trautman, A. (1988),The Spinorial Chessboard , Springer-Verlag,ISBN 0-387-19078-3
