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 0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Van der Waerden notation. Read more | Retrieved from "https://handwiki.org/wiki/index.php?title=Van_der_Waerden_notation&oldid=2234906" *[v]: View this template *[t]: Discuss this template *[e]: Edit this template