Analog of spin VS helicity for internal symmetries This might be more of a soft question, since I don't While learning about representations of the Lorentz group, I found in Maggiore's book (Chapter 2) that massive particles of spin $j$ have $2j+1$ degrees of freedom, whereas massless particles only have one degree of freedom, the helicity. That's why a photon, which we often call a massless spin-1 particle, has two polarizations and its "spin" (helicity really) can only take two values.
The argument follows from having two Casmir operators for the Lorentz group, $P^\mu P_\mu, W^\mu W_\mu$, with $W^\mu$ being the Pauli-Lubanski 4-vector, which commute. After that, we make the distiction between $P^\mu P_\mu=m=0$ and $m\neq 0$. So, then $m=0$ reduces the degrees of freedom of the field that transforms under the Lorentz group. Please, do correct me if I'm wrong.
This might be a bit generic but, in the case of an internal symmetry (like a gauge symmetry), can there be an analogous restriction to the degrees of freedom? If so, are there any examples of models that do that?
As an example, I'm thinking about a theory with internal symmetry under a group $G$, with its Casimirs, $C_1, C_2, .. C_n$, with one $C_j$ (analogous to $P^\mu P_\mu$ for the Lorentz group) such that $C_j=0$ (with $C_j$ not being trivial).
 A: What you should really be thinking of is a group and representations.
In the massive case, the little group is $SU(2)$ whose representations are labelled by a half-integer $j$ and has dimension $2j+1$.
In the massless case, the little group os $U(1)$ whose representations are labelled by a half-integer $h$ and have dimension $1$. However, we want our Hilbert space to be CPT invariant so for every helicity $h$ representation, we also include a helicity $-h$ representation giving 2 d.o.f. for each $|h|$.
There is no "restriction" of any sense happening anywhere. The two types of particles are described by two completely different groups so their structures are different.
Internal symmetry groups can also have such properties. It's definitely possible. For instance $SL(2,{\mathbb R})$ has the usual highest weight representations (which has real and discrete scaling dimensions), but they also have a continuous series representation (which has complex and continuous scaling dimensions) which are totally different in structure.
We do not generically see such things in QFT since internal symmetry groups are usually compact (The Poincare group and $SL(2,{\mathbb R})$ are not!)
