# How does higher spin theory evade Weinberg's and the Coleman-Mandula no-go theorem?

Recently I heard some seminar on higher spin gauge theory, and got some interest. I know there are some no-go theorems in quantum field theories:

Weinberg: Massless higher spin amplitudes are forbidden by the general form of the S-mastrix.

Coleman-Mandula: There is no conserved higher spin charge/current, considering nontrivial S-matrix and mass gap formalism.

The speaker says, that by introducing a cosmological constant, i.e. introducing AdS space, one can avoid these no go theorems, but I am not sure how.

Can you give me some explanation for this?

My reference is a talk by Xi Yin, page 5.

• Do you have a link to the talk/a paper by the speaker? Mar 13 '16 at 11:36
• I have no idea what the two theorems you are referring to are supposed to be. The Weinberg-Witten theorem makes a statement about massless conserved currents and stress energies, not about "higher spin amplitudes". The Coleman-Mandula theorem states that there are no non-gauge symmetries except for the Poincaré symmetry, but since spin is essentially the conserved charge of the Lorentz symmetry, I do not see why you say "there is no conserved higher spin". Mar 13 '16 at 12:01
• @ACuriousMind, innisfree, i am refering, talk by Xi Yin. Mar 13 '16 at 12:17
• I'm not an expert on higher spin theories but I've heard similar statements being made. A simple observation that may or may not be relevant is that the theorems you are talking about (Weinberg soft limit for massless spin-s particles, Weinberg-Witten, Coleman Mandela) all assume a Poincaire invariant vacuum state. AdS is not Poincaire invariant, meaning the symmetry group of AdS is not the Poincaire group ISO(1,3). So the speaker may be saying that the theorems don't apply on AdS because the vacuum isn't Poincaire invariant. Again, I'm not an expert so there may be more to it than that. Mar 13 '16 at 13:47
• A detailed answer is at physicsoverflow.org/35557 Mar 14 '16 at 8:17

You can show this assuming only little group invariance and soft limits, without needing a lagrangian description. In a naive fashion, higher spin theories evade this theorem by including an infinite number of massless higher spin fields. This is reminiscent of string theory, in which an infinite number of fields give you a soft behaviour for scattering amplitudes, while a single field of higher spin tend to give divergent contribution going like $\frac{s^J}{s-M^2}$ where $s$ is the usual Mandelstam variable.