I am writing a note about the Poincare group and I am trying to explain that argument that one-particle state transforms under irreducible unitary representations of the Poincare group. However, there are some ambiguities which confuse me.
- Is a composite particle transformation under an irreducible unitary representation of the Poincare group? For example, a proton consists of some quarks. It seems they can't be separated by just rotating or translating or boosting, therefore it does form an irreducible unitary representation of the Poincare group, even though it's not a fundamental particle.
- The irreducible unitary representation of the Poincare group is characterised by mass and spin, which are given by two Casimir operators ($P^2$ and $W^2$). However, the electron and positron have the same mass and spin, but definitely different particles. So it seems the original argument has a loophole and we should somehow incorporate CPT transformation to modify the original argument. If I'm right, how to incorporate CPT in the original argument?
Edit: I just found that both questions are explained in Winberg Chapter Two, but I would still appreciate it if someone would like to explain it again.