# What are the physical manifestations of the finite-dimensional irreducible non-unitary representations of the inhomogeneous 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.

• The answer to my question physics.stackexchange.com/q/497456/122952 (and the references therein) give a good explanation. – NDewolf May 4 at 16:46
• – Cosmas Zachos May 4 at 17:54
• @NDewolf Many thanks; very helpful and well explained. (I would never have found your answer without your comment.) – iSeeker May 4 at 17:56
• @CosmasZachos Useful refs therein - together with other replies to date that'll keep me busy for a while. Thanks – iSeeker May 4 at 18:17
• NDewolf’s link leads to the following concerning the Poincare Group (PG): "The PG appears in two different ways in QFT: • Particles, described by unitary (and hence infinite-dimensional) reps of PG, and • Fields, described by finite-dimensional (and hence non-unitary) reps of PG." Can this be taken (from an ignorant chemist’s viewpoint) as meaning that with PG including translations, particle states include continuous (i.e. unbound) states requiring infinite-D reps, whereas fields described by Fock space have discrete states and finite-D reps? – iSeeker May 5 at 13:10