Van der Waerden notation

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

Undotted indices (chiral indices)

Spinors with lower undotted indices have a left-handed chirality, and are called chiral indices.

${\displaystyle \Sigma _{\mathrm {left} }={\begin{pmatrix}\psi _{\alpha }\\0\end{pmatrix}}}$

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.

${\displaystyle \Sigma _{\mathrm {right} }={\begin{pmatrix}0\\{\bar {\chi }}^{\dot {\alpha }}\\\end{pmatrix}}}$

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.

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

${\displaystyle \alpha =1,2\,,{\dot {\alpha }}={\dot {1}},{\dot {2}}}$

then a spinor in the chiral basis is represented as

${\displaystyle \Sigma _{\hat {\alpha }}={\begin{pmatrix}\psi _{\alpha }\\{\bar {\chi }}^{\dot {\alpha }}\\\end{pmatrix}}}$

where

${\displaystyle {\hat {\alpha }}=(\alpha ,{\dot {\alpha }})=1,2,{\dot {1}},{\dot {2}}}$

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

${\displaystyle \Sigma ^{\hat {\alpha }}={\begin{pmatrix}\chi ^{\alpha }&{\bar {\psi }}_{\dot {\alpha }}\end{pmatrix}}}$

See also

• Dirac equation
• Infeld–Van der Waerden symbols
• Lorentz transformation
• Pauli equation
• Ricci calculus

Notes

1. Van der Waerden B.L. (1929). "Spinoranalyse". Nachr. Ges. Wiss. Göttingen Math.-Phys. ohne Angabe: 100–109.
2. Veblen O. (1933). "Geometry of two-component Spinors". Proc. Natl. Acad. Sci. USA. 19 (4): 462–474. Bibcode:1933PNAS...19..462V. doi: 10.1073/pnas.19.4.462 . PMC   1086023 . PMID   16577541.

References

• 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