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 chiralty, 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 chiralty 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}}

Notes

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

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

Tensors

Glossary of tensor theory

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Physics Engineering	continuum mechanics electromagnetism transport phenomena general relativity computer vision

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Mathematics	Kronecker delta Levi-Civita symbol metric tensor nonmetricity tensor Christoffel symbols Ricci curvature Riemann curvature tensor Weyl tensor torsion tensor

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Mathematicians

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Van der Waerden notation

Dotted indices

Hatted indices

See also

Notes

References