# Why are one-particle states called irreducible representations of Poincaré group?

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?

• The general idea is that any representation of the Poincare' group can be labelled with a pair $(m,s)$, with each of the two variables undergoing some particular conditions that lead us to read them as mass and spin. The complete argument is more complex, but that is somehow where one starts from. Dec 18, 2015 at 14:51
• Possible duplicates: physics.stackexchange.com/q/73593/2451 , physics.stackexchange.com/q/65839/2451 and links therein. Dec 18, 2015 at 17:02
• @Gennaro Tedesco- do you mean that since the one-particle states are labeled by eigenvalues of the casimir operators, they belong to the representation of the Poincare group?
– SRS
Dec 18, 2015 at 17:15
• Yes, I do (the entire argument calls in the Casimir operators, as you pointed out). Dec 18, 2015 at 17:21
• Note that in some sources fields which represents explicitly irreducible representations of the Poincare group are constucted explicitly. Dec 20, 2015 at 8:36