It is maybe simpler to consider all the generators as representations of SL(2,C), so, using spinor indices, you will have : Mα˙αβ˙β,Pβ˙β,Qα,ˉQ˙β
Indices are raised and lowered with the Levi-Civita symbols ϵαβ,ϵαβ,ϵ˙α˙β,ϵ˙α˙β
Now, what is [Pβ˙β,Qα] ?
We see that there is no generator with the form Gβ˙βα.
Levi-Civita symbols are not useful too, because they have 2 lower or upper indices of same kind, so we cannot write something like [Pβ˙β,Qα]=ϵαβQ˙β (there would be an obvious problem with the β indice).
So the only solution is a contraction on indices α and β, that is :
[Pβ˙β,Qα]=δβαˉQ˙β
With Pμ=σμβ˙βPβ˙β, (which means simply that the (12,12) representation of SL(2,C) is equivalent to the fundamental representation of SO(3,1) ) we get finally :
[Pμ,Qα]=σμβ˙βδβαˉQ˙β=σμα˙βˉQ˙β
This post imported from StackExchange Physics at 2014-08-12 09:38 (UCT), posted by SE-user Trimok