How to code \gamma^{5}

Question

Suppose you want to do supersymmetry calculations in 4d with standard gamma matrices. What is the best way to code $\Gamma^{5}$ ? One can make it explicit with $$\begin{equation} \Gamma^{5}\sim \Gamma^{0}\Gamma^{1}\Gamma^{2}\Gamma^{3}, \end{equation}$$ but is there a way to cope with a suitable new symbol, with its simple properties like $\left\{\Gamma^{5}, \Gamma^{\mu}\right\}=0$ ?

Answer 1

You could use something like this

rules := { \gamma^{\star} -> 1/ 24 \epsilon_{m n p q} \Gamma^{m n p q} };

and then compute, say $(\gamma^5)^2$ as follows

gs_gs := \gamma^{\star} \gamma^{\star};
substitute(gs_gs, rules, repeat=True);
epsilon_to_delta(gs_gs);
expand_delta(gs_gs)
distribute(_)
eliminate_kronecker(_)
canonicalise(_)
collect_terms(_)
join_gamma(_);

Cheers.

Comments

Thanks a lot !