# 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? Jul 5, 2019 at 16:42
• @CosmasZachos is it not possible to do it using the Casimirs of the Poincaré group directly instead of invoking the stabilisers? Jul 6, 2019 at 8:16
• How?........... Jul 6, 2019 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. Jul 6, 2019 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. Jul 6, 2019 at 15:30