# Coleman–Mandula theorem and a mathematical proof

Coleman–Mandula theorem (by Sidney Coleman and Jeffrey Mandula) [1] is a no-go theorem in theoretical physics. It states that "space-time and internal symmetries cannot be combined in any but a trivial way". Since "realistic" theories contain a mass gap, the only conserved quantities, apart from the generators of the Poincaré group, must be Lorentz scalars.

Question 1: Since this statement is only a theorem to physicists, I wonder whether there is a mathematical version and a mathematical proof of it?

Question 2: Stating "space-time and internal symmetries cannot be combined in any but a trivial way" seems to be too restricted. For example, we are allowed to have twisted symmetries combining the spacetime bundle $$G_{\text{spacetime}}$$-bundle (like Spin, SO, Pin-groups) and the gauge $$G_{\text{gauge}}$$-bundle (like a compact Lie group SU(N)), and their connections in a nontrivial way (instead of being the product of bundles). Can we clarify this statement of Coleman–Mandula further --- what is a precise rigorous restriction?

[1] Sidney Coleman, Jeffrey Mandula, "All Possible Symmetries of the S Matrix, "Physical Review, 159(5), 1967, pp. 1251–1256.