My teacher said the following:
A particle is a set of states $\{ \vert\Psi\rangle \}$ that transform among themselves under Poincaré transformations. Hence, a particle is a unitary irreducible representation of the Poincaré group.
I understand why it makes sense to talk of particles as in the first sentence. But I don't see how the second sentence follows. Let me try to be mathematically clear:
With the first sentence, we seem to be saying that a particle is an orbit of an element $\vert \Psi \rangle$ of the state space $\mathcal{E}$ (where the Poincaré group $G$ acts) $$ O(\vert\Psi\rangle)=\{ g\vert\Psi\rangle:g\in G \} $$ With the second statement, we're saying that a particle is (the image of an) irreducible unitary representation $\Pi:G\rightarrow M_n(\mathbb{R})$ of $G$.
So I went on looking for a mathematical theorem of representation theorem that said something like "there is a bijection between the sets of all orbits of a set and the set of the images of the irreducible unitary representations of a group that acts on it". I couldn't find it.
Is there such a theorem? Am I interpreting this correctly? Although I am interested in understanding these topics deeply one day, right now I don't have the time, so I just want to see the idea behind this statement about particles.