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
    Commented May 4, 2020 at 16:46
  • 2
    $\begingroup$ Linked. $\endgroup$ Commented 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
    Commented 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
    Commented May 4, 2020 at 18:17
  • 1
    $\begingroup$ @iSeeker: the Fock spaces used to quantize free or nearly-free fields correspond precisely to particle Hilbert spaces. Because fields can take on arbitrary values on arbitrary distinct points, these Hilbert spaces are infinite dimensional (moreso for 'normal' commuting fields than for Grassmann-algebra-valued ones). The 'sections' (i.e. values that fields take over points on the invariant manifold or base space) of classical fields transform under the non-unitary representations, just like how the value of a 1-form rotates covariantly with the coordinate system (under spatial rotations.) $\endgroup$
    – TLDR
    Commented May 22 at 4:16

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
    Commented May 4, 2020 at 18:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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