Kartik PrabhuSo here’s why what I asked earlier is useful for physics. I don’t think I need to explain why the Lorentz group is useful; relativity and all that. Its double cover is SL(2,C) which is useful for handling spinors (as some would say “fermions”). I’ll be a bit vague though.
In relativity, the kinds of “particles” (really fields) one wants are representations of the Poincare group i.e. spacetime translations and Lorentz. This was solved a long time ago by Wigner with a really ingenious construction. I’ll skip over unitary/faithfulness etc…
The translations are “easily” represented in Fourier space: so instead of a function of spacetime position x you have a function of momentum p (note p and x are Minkowski 4-vectors). Now, we need to represent the Lorentz transformations.
So here’s Wigner’s idea: pick a momentum p and find the largest subgroup of SL(2,C) which leaves this p invariant; that’s the little group L_p of the momentum you chose. Now you can act on this p by Lorentz transformations to get a whole bunch of other momenta.
If you act by anything in L_p it just gives you back p; so you will get something different if you act by SL(2,C)/L_p i.e. Lorentz transformations “not in” L_p. This quotient space is called the orbit space O_p of the chosen p.
In our case O_p is a manifold, and has a measure invariant under SL(2,C). So to construct the representations (wavefunctions/fields) you start with functions on the orbit O_p and integrate them over the orbit. Sounds like fancy nonsense so examples:
1) If you choose p to be a future timelike momentum, then L_p = SU(2) i.e. rotations; and O_p = H3 the Riemannian hyperboloid sitting in the future light cone. And the wavefunctions are functions on H3; these are the usual wavefunctions of massive fields in momentum space.
2,3) If you choose p to be null or spacelike you get a similar story with O_p being future light cone or the one-sheeted Lorentzian hyperboloid. The wavefunctions are then for massless or tachyonic fields.
In each case, we had an “obvious” invariant measure on the orbit space O_p since they all sit nicely as submanifolds of the Minkowski spacetime. This measure is obtained as a “quotient” of the Haar measure on SL(2,C); not sure how explicitly!
So what’s the big deal, we know all the cases we want, right? Well, enter General Relativity, where there is no symmetry. But there is an asymptotic symmetry group, which is BMS not Poincare, if you care about gravitational radiation and memory effect.
The analog of the 4-momentum p is now the supermomentum, which is a function on a 2-sphere not a 4-vector. But you can still play the same little group game as Wigner; with some very important subtleties about topology on infinite-dimensional groups like BMS.
This was worked out a long time ago by P. J. McCarthy e.g. https://doi.org/10.1098/rspa.1972.0157 You get more weird little groups.
1) One little group you get is a simple double cover of U(1), whose orbit space is H3 × S2
2) Another one is little group Z2, whose orbit space is H3 × RP3, this is just the Lorentz group (no double cover)
In each case an invariant measure on the orbit space is guaranteed to exist due to math (McCarthy does this in an appendix) but is there a nice formula for these measures?