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.

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} $$