The one-particle states in the Hilbert space of a quantized relativistic field theory are said to form irreducible representations of the Poincaré group. Why is that? I mean, popular texts in QFT do not explicitly construct any representation but simply state that one-particle states are representations. Is this so obvious? If not, how can one understand/ensure that they indeed form irreducible representation of the Poincaré group?
EDIT: Moreover, one-particle states are supposed to be the irreducible representations of Poincaré group. Does it mean that any representation which is labelled by unique values of Casimir invariants are irreducible?