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

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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.

This post imported from StackExchange Physics at 2016-03-13 14:30 (UTC), posted by SE-user phy_math
Do you have a link to the talk/a paper by the speaker?

This post imported from StackExchange Physics at 2016-03-13 14:30 (UTC), posted by SE-user innisfree
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".

This post imported from StackExchange Physics at 2016-03-13 14:30 (UTC), posted by SE-user ACuriousMind
@ACuriousMind, innisfree, i am refering, talk by Xi Yin.

This post imported from StackExchange Physics at 2016-03-13 14:30 (UTC), posted by SE-user phy_math
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.

This post imported from StackExchange Physics at 2016-03-13 14:30 (UTC), posted by SE-user Andrew
+1 though, if there is a real expert on higher spin on these forums I'd love to hear a fuller explanation.

This post imported from StackExchange Physics at 2016-03-13 14:30 (UTC), posted by SE-user Andrew
I looked into this and found that a) Weinberg's theorem is derived from a factorization property of the S-matrix, not Lorentz invariance itself and b) it's not "higher spin" that can't be conserved, it's a current/charge with higher spin (you did not copy that correctly from the talk). I edited those corrections in.

This post imported from StackExchange Physics at 2016-03-13 14:30 (UTC), posted by SE-user ACuriousMind

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The short answer is that both the theorems are about theories in flat space. To see them fail you need to try to make them work in curved space, but formally they assume flat space. In more detail, the Weinberg low energy theorem studies the soft limit of an amplitude with one spin-s particle attached to an external leg via a cubic vertex. Then one shows that this implies a conservation law that is a polynomial of degree s-1 in momenta.  For s=1 one gets charge conservation. For s=2 we find that gravity couples universally. For s> 2 we have too many conservation laws. If we are in anti-de Sitter we do not know what s-matrix is and the Weinberg does not work. Coleman-Mandula assumes that there are some additional symmetries of the s-matrix that is Poincare invariant. With some mild assumptions one can see more or less the same as in the Weinberg theorem: if symmetries are non trivially mixed with the space time, i.e. Poincare,  they would again lead to some additional conservation laws that trivialize dynamics. This assumes that one can really see higher spin fields, but if they are say confined and for that reason are not observed as asymptotic states then there is no problem. Coleman-Mandula does not apply to ads by construction too.

However, one can define an analog of the s-matrix in ads - ads/cft correlation functions. Then some similar theorems can be proved, check Maldacena-Zhiboedov, but these theorems do not forbid the existence of higher spin theories, right the opposite.

answered Mar 13, 2016 by anonymous

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