Higher spin particles have to be coupled to conserved currents, and there are no conserved currents of high spin in quantum field theories. The only conserved currents are vector currents associated with internal symmetries, the stress-energy tensor current, the angular momentum tensor current, and the spin-3/2 supercurrent, for a supersymmetric theory.
This restriction on the currents constrains the spins to 0,1/2 (which do not need to be coupled to currents), spin 1 (which must be coupled to the vector currents), spin 3/2 (which must be coupled to a supercurrent) and spin 2 (which must be coupled to the stress-energy tensor). The argument is heuristic, and I do not think it rises to the level of a mathematical proof, but it is plausible enough to be a good guide.
Preliminaries: All possible symmetries of the S-matrix
You should accept the following result of <PERSON>, <PERSON> and <PERSON> the continuous symmetries of the particle S-matrix, assuming a mass-gap and <PERSON> invariance, are a Lie Group of internal symmetries, plus the <PERSON> group. This theorem is true, given its assumptions, but these assumptions leave out a lot of interesting physics:
* Coleman-Mandula assume that the symmetry is a symmetry of the S-matrix, meaning that it acts nontrivially on some particle state. This seems innocuous, until you realize that you can have a symmetry which doesn't touch particle states, but only acts nontrivially on objects like strings and membranes. Such symmetries would only be relevant for the scattering of infinitely extended infinite energy objects, so it doesn't show up in the S-matrix. The transformations would become trivial whenever these sheets close in on themselves to make a localized particle. If you look at <PERSON> and <PERSON>'s argument (a simple version is presented in <PERSON>' supersymmetry notes, which gives the flavor. 976
There is an excellent complete presentation in <PERSON>'s quantum field theory book, and the original article is accessible and clear), it almost begs for the objects which are charged under the higher symmetry to be spatially extended. When you have extended fundamental objects, it is not clear that you are doing field theory anymore. If the extended objects are solitons in a renormalizable field theory, you can zoom in on ultra-short distance scattering, and consider the ultra-violet fixed point theory as the field theory you are studying, and this is sufficient to understand most examples. But the extended-object exception is the most important one, and must always be kept in the back of the mind.
* <PERSON> and <PERSON> assume a mass gap. The standard extension of this theorem to the massless case just extends the maximal symmetry from the <PERSON> group to the conformal group, to allow the space-time part to be bigger. But <PERSON> and <PERSON> use analyticity properties which I am not sure can be used in a conformal theory with all the branch-cuts which are not controlled by mass-gaps. The result is extremely plausible, but I am not sure if it is still rigorously true. This is an exercise in <PERSON>, which unfortunately I haven't done.
* <PERSON> and <PERSON> ignore supersymmetries. This is fixed by <PERSON>–Lopuszanski–Sohnius, who use the <PERSON> mandula theorem to argue that the maximal symmetry structure of a quantum field theory is a superconformal group plus internal symmetries, and that the supersymmetry must close on the stress-energy tensor.
What the Coleman Mandula theorem means in practice is that whenever you have a conserved current in a quantum field theory, and this current acts nontrivially on particles, then it must not carry any space-time indices other than the vector index, with the only exceptions being the geometric currents: a spinor supersymmetry current, $J^{\alpha\mu}$, the (<PERSON> symmetric) stress-energy tensor $T^{\mu\nu}$, the (Belinfante) angular momentum tensor $S^{\mu\nu\lambda} = x^{\mu} T^{\nu\lambda} - x^\nu T^{\mu\lambda}$, and sometimes the dilation current $D^\mu = x^\mu T^\alpha_\alpha$ and conformal and superconformal currents too. 800