# Representations of the Poincare group

Which type of states carry the irreducible unitary representations of the Poincare group? Multi-particle states or Single-particle states?

• Related: physics.stackexchange.com/q/21801/2451 , physics.stackexchange.com/q/22449/2451 and links therein. Feb 25, 2014 at 16:58
• by definition, single particle states are irreps of Poincare or any other group of interest. Multiparticle states belong to symemtrized (bosons) or antisymmetrized (fermions) products thereof
– John
Feb 25, 2014 at 17:54

Proof. A multi-particle representation is the tensor product of the representations in each factor one-particle subspace. If $P_\mu$ denotes the total four-momentum operator of the system of particles, the bounded unitary operator $e^{i a P_\mu P^\mu}$ ($a\in \mathbb R$) commutes with all the unitary operators of the tensor representations and it is not proportional to the identity operator (as it instead happens for a one-particle space). In view of Schur's lemma the representation cannot be irreducible.
An invariant closed subspace is, evidently, the subspace of state vectors where the squared mass $M^2= P_\mu P^\mu$ assumes values (in the sense of spectral decomposition) inside a fixed interval $[a,b]$.
• Thanks for your answer. By "The said representation", do you mean the multiparticle Hilbert space? Furthermore, is $P^\mu P_\mu$ a Casimir operator? Feb 26, 2014 at 8:03