Question

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 (P² and W²). 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?

Answer 1

Composite particles form reducible representations of the Poincare group that combine the irreducible representations of their constituents. Particles with the same mass and spin, such as electrons and positrons, belong to the same irreducible representation. To incorporate CPT transformation, the combined effect of charge conjugation, parity, and time reversal transformations should be considered.

Step-by-step explanation:

Composite particles, such as a proton consisting of quarks, behave differently under the transformation of the Poincare group compared to one-particle states.

While one-particle states transform under irreducible unitary representations, composite particles form reducible representations that combine the irreducible representations of their constituents.

The irreducible unitary representations of the Poincare group are characterized by mass and spin, so particles with the same mass and spin, like electrons and positrons, do belong to the same irreducible representation.

To incorporate CPT transformation in the original argument, one would need to consider the combined effect of charge conjugation (C), parity (P), and time reversal (T) transformations.