# Coleman–Mandula theorem and its assumptions on QFT

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."

## Questions

• 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 $$$$p_1^\mu + p_2^\mu = q_1^\mu + q_2^\mu$$$$ but spacetime symmetries beyond Poincare would lead to tensorial generalizations like $$$$p_1^\mu p_1^\nu + p_2^\mu p_2^\nu = q_1^\mu q_1^\nu + q_2^\mu q_2^\nu.$$$$ For this to be a problem, assumption 2 sounds necessary because otherwise you could use the trivial solution $$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.