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?

  • 2
    $\begingroup$ The answer to my question physics.stackexchange.com/q/497456/122952 (and the references therein) give a good explanation. $\endgroup$
    – NDewolf
    May 4, 2020 at 16:46
  • 2
    $\begingroup$ Linked. $\endgroup$ May 4, 2020 at 17:54
  • $\begingroup$ @NDewolf Many thanks; very helpful and well explained. (I would never have found your answer without your comment.) $\endgroup$
    – iSeeker
    May 4, 2020 at 17:56
  • $\begingroup$ @CosmasZachos Useful refs therein - together with other replies to date that'll keep me busy for a while. Thanks $\endgroup$
    – iSeeker
    May 4, 2020 at 18:17
  • $\begingroup$ 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? $\endgroup$
    – iSeeker
    May 5, 2020 at 13:10

1 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.

  • $\begingroup$ That's a direction that's quite new to me - it looks to be, as you say, not an easy topic, but all grist to the (mental) mill. $\endgroup$
    – iSeeker
    May 4, 2020 at 18:15

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