What are the physical manifestations of the finite-dimensional irreducible non-unitary representations of the inhomogeneous Lorentz group?

Question

According to https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group:

"The finite-dimensional irreducible non-unitary representations together with the irreducible infinite-dimensional unitary representations of the inhomogeneous Lorentz group, the Poincare group, are the representations that have direct physical relevance."

While the meaning of unitary representations seems clear (over-simplistically, perhaps) reps of symmetry operations corresponding to rotations in the Hilbert space of states (OK?), I can find no elementary description of the physical manifestations ("direct physical relevance") of the finite-dimensional irreducible non-unitary representations.

Can someone please enlighten me?

Answer 1

Indecomposable representations (which are non-unitary and finite dimensional) of Poincaré appear in theories of unstable particles. It's not an easy topic but was explored in this paper:

Raczka, R. "A theory of relativistic unstable particles." Annales de l'IHP Physique théorique. Vol. 19. No. 4. 1973.