Wigner classification of particles vs generic Hamiltonian spectrum Wigner tells us we should associate infinite dimensional unitary irreps of the Poincaré group with particle states. His classification using eigenvectors of the spacetime generators $P^\mu$ and the method of little groups tells us particles are asssociated with two numbers $m^2$ and spin $S$.
However if we look at a generic spectrum of the Hamiltonian for an interacting theory we get something along the lines of

This includes single particle states, bound states, and a continuum of multi-particle and quasi-particle states.
These extra states are parameterised by other discrete and continuous labels. My question is do these extra states form irreps of the Poincaré group? If not, why not? To me, it seems as though his treatment only looks at free theory Hamiltonians, i.e. only looking at the single particle states.
 A: Wigner's classification looks at the quantum numbers that are associated to the Poincaré group. It says nothing about other quantum numbers.
Consider a free theory with a two particle state $|\boldsymbol p_1,\boldsymbol p_2\rangle$. Wigner will tell you that the quantum number associated to the generator of translations, $P^\mu$, is $p^\mu_1+p_2^\mu$, the center-of-mass energy. Wigner will tell you nothing about the other quantum number, $p_1^\mu-p_2^\mu$, the relative momentum. The total momentum is a quantum number associated to a generator of Poincaré; the relative momentum is not. Wigner tells you about the former, not the latter. The total momentum is associated to a spacetime symmetry; the relative momentum is not.
Wigner's classification is nothing but a statement about representation theory: the states of the theory can be organized into representations of a complete set of commuting observables; the Poincaré group gives you a universal set of such operators, but there is no claim that it is complete. In order to fully label the states of the theory, you also need to diagonalize other operators, which in general do not come from spacetime symmetries (for example, you also often find quantum numbers associated to conserved charges for internal symmetries, such as electric charge; these are again not in the Poincaré group).
