The description of the Coleman–Mandula theorem on Wikipedia starts with the following assumptions:

Every quantum field theory satisfying the assumptions,

  1. Below any mass M, there is only a finite number of particle types.

  2. Any two-particle state undergoes some reaction at almost all energies

  3. The amplitude for elastic two-body scattering are analytic functions of scattering angle at almost all energies,

Then one has non-trivial interactions if the theory has a Lie group symmetry which is always a direct product of the Poincaré group and an internal group if there is a mass gap: no mixing between these two is possible. As the authors say in the introduction to the 1967 publication, "We prove a new theorem on the impossibility of combining space-time and internal symmetries in any but a trivial way."


  • Why are assumptions 2. and 3. taken for granted? Why are they necessary or sufficient?

  • Why is the presence of a mass gap an issue to establish a direct product of the Poincaré group and an internal group? If it is massless, can we establish instead a direct product of the conformal group and an internal group?


The 2001 SUSY notes by Argyres at https://homepages.uc.edu/~argyrepc/cu661-gr-SUSY/index.html have a good summary at the beginning. Essentially Poincare symmetry in a scattering process guarantees momentum conservation \begin{equation} p_1^\mu + p_2^\mu = q_1^\mu + q_2^\mu \end{equation} but spacetime symmetries beyond Poincare would lead to tensorial generalizations like \begin{equation} p_1^\mu p_1^\nu + p_2^\mu p_2^\nu = q_1^\mu q_1^\nu + q_2^\mu q_2^\nu. \end{equation} For this to be a problem, assumption 2 sounds necessary because otherwise you could use the trivial solution $p_i = q_i$. Also, assumption 3 is necessary because otherwise you could use the non-analytic solution $p_1 = q_2, p_2 = q_1$ which is like the one that shows up in $1 + 1d$ integrable QFTs. But as soon as you disallow these possibilities, the higher spin symmetry is ruled out because there are no other solutions.

A mass gap is implicit in the above arguments because this is usually needed to define scattering states. But in CFTs, it is indeed the case that a similar result holds ruling out spacetime symmetries beyond the conformal group. This was proven in https://arxiv.org/abs/1112.1016 by Maldacena and Zhiboedov.


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