User:Maschen/spinor index notation

In mathematics and mathematical physics, spinor index notation is the index notation analogue of tensor index notation for spinors. Many of the tensor index notation conventions carry over to spinor index notation.

Overlap with tensor index notation

Differences

Either capital Latin letters (instead of lower case Latin letters), or Greek lower case, are used for indices. For example

$\psi ^{A}\,,\quad \phi _{B}\,,\quad \chi ^{A}{}_{B}\,,\ldots$

other conventions...

Similarities

Summation convention for twice-repeated indices

$\psi ^{A}\phi _{A}\equiv \sum _{A}\psi ^{A}\phi _{A}$

Covariance

Lower indices, ψ_μν...

Contravariance

Upper indices, ψ^αβ...

Mixed variance

Upper and lower indices ψ^α_μ.

Raising and lowering indices via the metric spinor also applies, e.g.

$\psi ^{\alpha }=g^{\alpha \beta }\psi _{\beta }$

Symmetry

Antisymmetry

van der Waerden notation

In theoretical physics, van der Waerden notation 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.

Dots on 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.

Hats on 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

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

See also

• Dirac equation
• bra–ket notation
• Infeld–van der Waerden symbols
• Lorentz transformation
• Pauli equation
• Ricci calculus

References

• M. Carmeli, S. Malin (2000). Theory of Spinors. World Scientific. ISBN 9789812564726.

• Landau, L.D.; Lifshitz, E. M. (1977). Quantum Mechanics: Non-Relativistic Theory. Vol. Vol. 3 (3rd ed.). Pergamon Press. ISBN 978-0-08-020940-1. {{cite book}}: |volume= has extra text (help) Online copy

• R. M. Wald (1984). General Relativity. Chicago University Press. ISBN 9780226870335. (spinor/twistor chapter, spinors of type (a,b;c,d)?)

• 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. Vol. 1, Cambridge University Press, ISBN 0-521-24527-3 {{citation}}: |volume= has extra text (help)
• Budinich, P.; Trautman, A. (1988), The Spinorial Chessboard, Springer-Verlag, ISBN 0-387-19078-3