Significance of the Little group

My current understanding of the Little group is that it is the symmetry of a given state in the Fock space. This means that given a massive or massless particle in n dimensions, I can tell the number of degrees of freedom by looking at its little group. Or given the little group of a particle, I can classify it as a massive or massless particle.

Now, my question is: I don’t understand why the little group is necessary for any of the above. Anything I can do with the Little group, I can also do with the Poincaré group. So, why study the little group? Or is there some other way that it’s useful that I’m missing?

• Yes, the stabilizer subgroups of the Poincaré group. That's how Wigner does it. Can you detail how you do it instead? – Cosmas Zachos Jul 5 at 16:42
• @CosmasZachos is it not possible to do it using the Casimirs of the Poincaré group directly instead of invoking the stabilisers? – adithya Jul 6 at 8:16
• How?........... – Cosmas Zachos Jul 6 at 10:19
• The eigenvalue of $P^{\mu}P_{\mu}$ will give the mass of the particle and the eigenvalue of the Pauli-Lubanski vector can be used for the spin, I guess? I am not sure though. I am not very clear how the classification works or how it needs the Little group. – adithya Jul 6 at 10:34
• Both the WP article and most good books review that. You go to a rest frame; or a momentum aligned frame for massless particles; and the respective little groups are different, so their representation theories differ, etc... These are the ones determining the respective particles' properties, as they fit into the Casimir above. – Cosmas Zachos Jul 6 at 15:30