Cadabra
a field theory motivated approach to computer algebra

sort_spinors

Sort Majorana spinor bilinears
Sorts Majorana spinor bilinears using the Majorana flip property, which for anti-commuting spinors takes the form \begin{equation} \bar\psi_1 \Gamma_{r_1\cdots r_n}\psi_2 = \alpha \beta^n (-)^{\frac{1}{2}n(n-1)}\, \bar\psi_1 \Gamma_{r_1\cdots r_n}\psi_2\, . \end{equation} Here $\alpha$ and $\beta$ determine the properties of the charge conjugation matrix, \begin{equation} {\cal C}^T = \alpha {\cal C}\,,\quad {\cal C}\Gamma_r {\cal C}^{-1} = \beta \Gamma_r^T\, . \end{equation} Here is an example.
{\chi, \psi, \psi_{m}}::Spinor(dimension=10, type=MajoranaWeyl). {\chi, \psi, \psi_{m}}::AntiCommuting. \bar{#}::DiracBar. \Gamma{#}::GammaMatrix. {\psi_{m}, \psi, \chi}::SortOrder. ex:=\bar{\chi} \Gamma_{m n} \psi;
\(\displaystyle{}\bar{\chi} \Gamma_{m n} \psi\)
sort_spinors(_);
\(\displaystyle{}-\bar{\psi} \Gamma_{m n} \chi\)
Copyright © 2001-2017 Kasper Peeters
Questions? info@cadabra.science