1

I am trying to study and understand QFT from the perspective of symmetries. I was referred to this super helpful answer by @ACuriousMind : https://physics.stackexchange.com/a/174908/50583. I still have a lingering question though.

I was going through these set of notes : https://www.physics.uci.edu/~tanedo/files/notes/FlipSUSY.pdf. Here the author says :

The key, however, is that one must look at full Poincaré group (incorporating translations as well as Lorentz transformations) to develop a consistent picture. Adding the translation generator P to the algebra surprisingly turns out to cure the ills of non-unitarity (i.e. of non-compactness). The cost for these features, as mentioned above, is that the representations will become infinite dimensional, but this infinite dimensionality is well-understood physically: we can boost into any of a continuum of frames where the particle has arbitrarily boosted four-momentum.

Now as I understand it, the non-compactness of both the Lorentz and the Poincare groups yields infinite dimensional unitary representations only. This is fine, because, the elements of these infinite dimensional representations are not the classical fields themselves (i.e, scalars, 4-vectors, 4-spinors etc.), but rather fields as seen as operator-valued distributions on some infinite dimensional Hilbert space.

But then, why don't the unitary, infinite dimensional representations of the Lorentz group work? I mean, I get that the universe respects more than just Lorentzian symmetry; it also respects translations, so it makes sense to look for representations of the full Poincare group. But the aforementioned quote from the author seems to suggest that we somehow need translation symmetry (and the full Poincare group) to get any unitary representations at all. Is the author incorrect here or am I missing something?

qavidfostertollace
  • 31
  • it is a perfectly well-defined problem to study the unitary representations of the Lorentz group in a standalone way, i.e., without considering translations. In fact, the Lorentz group is isomorphic to the global conformal group in two-dimensions less. In four-dimensional spacetime, the 4D Lorentz group is isomorphic to the 2D global conformal group and therefore studying just its unitary representations is interesting for 2D CFT for example. The important point, though, is that for a 4D relativistic theory translations are a symmetry and you can't dispense with them. – Gold Nov 11 '23 at 20:24
  • This means that the Hilbert space of a 4D relativistic system will carry a unitary representation of the full Poincaré group. In the standard Wigner classification approach explained in Weinberg's chapter 2, one starts by studying the translations and then finding out how the Lorentz transformations act. One could do the opposite: start studying the Lorentz representations and finding a basis of states that simplifies it, and then finding out how translations act (this is done for example in Celestial Holography). In either way, you need to have full Poincaré. – Gold Nov 11 '23 at 20:26
  • Aah okay, perfect, thanks! And yes I've been meaning to properly go through Weinberg. I've only skimmed through it, but if I remember correctly, he doesn't study the unitary representation of the Lorentz group alone. He studies the full Poincare group only. But many thanks again! – qavidfostertollace Nov 11 '23 at 20:40
  • He studies full Poincaré because his objective is to study quantum relativistic theories in 4D. His method is to diagonalize translations first, then understand how Lorentz transformations get represented. For standalone unitary reps of the Lorentz group see https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group and the discussion of the principal and complementary series. – Gold Nov 11 '23 at 20:54

0 Answers0