There are directions in k-space where the propagator does not fall off at all! This means that when you scatter by spin-1 exchange, these directions can lead to a blow-up in the scattering amplitude at high energies which has to be cancelled somehow. The numerator projects the helicity to be perpendicular to k, and the second term is problematic. 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^$$ This is fixed by Haag–Lopuszanski–Sohnius, who use the Coleman 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. Coleman and Mandula ignore supersymmetries.This is an exercise in Weinberg, which unfortunately I haven't done. The result is extremely plausible, but I am not sure if it is still rigorously true. But Coleman and Madula 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 standard extension of this theorem to the massless case just extends the maximal symmetry from the Poincare group to the conformal group, to allow the space-time part to be bigger. Coleman and Mandula assume a mass gap.But the extended-object exception is the most important one, and must always be kept in the back of the mind. 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. When you have extended fundamental objects, it is not clear that you are doing field theory anymore. There is an excellent complete presentation in Weinberg'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. If you look at Coleman and Mandula's argument (a simple version is presented in Argyres' supersymmetry notes, which gives the flavor. The transformations would become trivial whenever these sheets close in on themselves to make a localized particle. 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. 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. Coleman-Mandula assume that the symmetry is a symmetry of the S-matrix, meaning that it acts nontrivially on some particle state.This theorem is true, given its assumptions, but these assumptions leave out a lot of interesting physics: You should accept the following result of O'Raferteigh, Coleman and Mandula- the continuous symmetries of the particle S-matrix, assuming a mass-gap and Lorentz invariance, are a Lie Group of internal symmetries, plus the Lorentz group. Preliminaries: All possible symmetries of the S-matrix 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. 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 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. Higher spin particles have to be coupled to conserved currents, and there are no conserved currents of high spin in quantum field theories.
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