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 at 16:46
  • 2
    $\begingroup$ Linked. $\endgroup$ – Cosmas Zachos May 4 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 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 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 at 13:10

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.

| cite | improve this answer | |
  • $\begingroup$ 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 at 18:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.