Van der Waerden notation

Short description: Notation used for Weyl spinors

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.

## Contents

* 1 Dotted indices
* 2 Hatted indices
* 3 See also
* 4 Notes
* 5 References

## Dotted indices

Undotted indices (chiral indices)

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

[math]\displaystyle{ \Sigma_\mathrm{left} = \begin{pmatrix} \psi_{\alpha}\\\ 0 \end{pmatrix} }[/math]

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.

[math]\displaystyle{ \Sigma_\mathrm{right} = \begin{pmatrix} 0 \\\ \bar{\chi}^{\dot{\alpha}}\\\ \end{pmatrix} }[/math]

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

[math]\displaystyle{ \alpha = 1,2\,,\dot{\alpha} = \dot{1},\dot{2} }[/math]

then a spinor in the chiral basis is represented as

[math]\displaystyle{ \Sigma_\hat{\alpha} = \begin{pmatrix} \psi_{\alpha}\\\ \bar{\chi}^{\dot{\alpha}}\\\ \end{pmatrix} }[/math]

where

[math]\displaystyle{ \hat{\alpha}= (\alpha,\dot{\alpha}) = 1,2,\dot{1},\dot{2} }[/math]

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

[math]\displaystyle{ \Sigma^\hat{\alpha} = \begin{pmatrix} \chi^{\alpha} & \bar{\psi}_{\dot{\alpha}} \end{pmatrix} }[/math]

## 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. doi:10.1073/pnas.19.4.462. PMID 16577541. Bibcode: 1933PNAS...19..462V.

## 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, 1, Cambridge University Press, ISBN 0-521-24527-3
* Budinich, P.; Trautman, A. (1988), The Spinorial Chessboard, Springer-Verlag, ISBN 0-387-19078-3

* v
* t
* e

Tensors

Glossary of tensor theory

Scope|

| Mathematics|

* coordinate system
* multilinear algebra
* Euclidean geometry
* tensor algebra
* dyadic algebra
* differential geometry
* exterior calculus
* tensor calculus

|

* Physics
* Engineering

* continuum mechanics
* electromagnetism
* transport phenomena
* general relativity
* computer vision

Notation|

* index notation
* multi-index notation
* Einstein notation
* Ricci calculus
* Penrose graphical notation
* Voigt notation
* abstract index notation
* tetrad (index notation)
* Van der Waerden notation

Tensor
definitions|

* tensor (intrinsic definition)
* tensor field
* tensor density
* tensors in curvilinear coordinates
* mixed tensor
* antisymmetric tensor
* symmetric tensor
* tensor operator
* tensor bundle
* two-point tensor

Operations|

* tensor product
* exterior product
* tensor contraction
* transpose (2nd-order tensors)
* raising and lowering indices
* Hodge star operator
* covariant derivative
* exterior derivative
* exterior covariant derivative
* Lie derivative

Related
abstractions|

* dimension
* basis
* vector, vector space
* multivector
* covariance and contravariance of vectors
* linear transformation
* matrix
* spinor
* Cartan formalism
* differential form
* exterior form
* connection form
* geodesic
* manifold
* fiber bundle
* Levi-Civita connection
* affine connection

Notable tensors|

| Mathematics|

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

|

Physics|

* moment of inertia
* angular momentum tensor
* spin tensor
* Cauchy stress tensor
* stress–energy tensor
* EM tensor
* gluon field strength tensor
* Einstein tensor
* metric tensor (GR)

Mathematicians|

* Leonhard Euler
* Carl Friedrich Gauss
* Augustin-Louis Cauchy
* Hermann Grassmann
* Gregorio Ricci-Curbastro
* Tullio Levi-Civita
* Jan Arnoldus Schouten
* Bernhard Riemann
* Elwin Bruno Christoffel
* Woldemar Voigt
* Élie Cartan
* Hermann Weyl
* Albert Einstein

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