What do you mean by "which Pin group Nature favors" ?
If I understood correctly it is possible that some particles are described by Pin(1,3) and others by Pin(3,1), and so Nature could choose one group or the other for differents particles (?).
Hello and thank you all for your answers.
For my confusion about the Pin groups I understood that in reality you can fix the signature and simply change the definition of the generators of the group.
So that's it, everything is clear to me thank you.
Hello,
I've always heard that the choice of signature for the metric was just a matter of convention, i.e. taking (+---) or (-+++) had no physical impact. The groups O(1,3) and O(3,1) being isomorphic it made sense to me.
However, I came across an article discussing the Pin(1,3) and Pin(3,1)...