1\cdbalgorithm{projweyl}{}
2
3Projects an expression onto Weyl spinors of positive chirality (this
4algorithm only works in even dimensions). On such a subspace, we have
5\begin{equation}
6\label{e:g10toeps}
7\Gamma^{r_1 \cdots r_{d}}\Big|_{\text{Weyl}} = \frac{1}{\sqrt{-g}}\epsilon^{r_1\cdots
8r_{d}}
9\, ,\quad \epsilon^{0\cdots (d-1)} = +1\, ,
10\end{equation}
11and therefore all gamma matrices with more than $d/2$ indices can be
12converted to their ``dual'' gamma matrices. By repeated contraction
13of~\eqref{e:g10toeps} with gamma matrices on the left one deduces that
14\begin{equation}
15\Gamma^{r_1\cdots r_n}\Big|_{\text{Weyl}} = \frac{1}{\sqrt{-g}} \frac{(-1)^{\frac{1}{2}n(n+1)+1}}{(d-n)!}
16\Gamma_{s_1\cdots s_{d-n}}\Big|_{\text{Weyl}} \epsilon^{s_1\cdots s_{d-n} r_1\cdots r_n}\, .
17\end{equation}
18Here is an example:
19\begin{screen}{1,2}
20{m,n,p,q,r,s,t}::Indices.
21{m,n,p,q,r,s,t}::Integer(0..5).
22\Gamma{#}::GammaMatrix.
23\Gamma_{m n p q};
24@projweyl!(%);
25\end{screen}
26
27\cdbseeprop{GammaMatrix}
28\cdbseealgo{join}
29