# 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.

• Perhaps you've already seen this physics.SE page physics.stackexchange.com/questions/94644/… (and the physics overflow page linked from its comments)
– j.c.
Oct 22, 2018 at 0:18
• Cf @j.c., with a direct link: see Pelc and Horwitz, arXiv.org/abs/hep-th/9605147, for the most rigorous version I (not an expert) have seen Oct 23, 2018 at 20:51

One possible answer to Q1 is theorem 2.7 in this paper, also see the presentation on YouTube by one of the authors. Roughly, the theorem says that if a quantum field theory

1. is of spacetime dimension at least three;
2. satisfies relativistic invariance;
3. has a compact internal symmetry group;
4. the internal symmetry group stays the same when Wick rotating from Lorentzian to Euclidean signature,

then 'spacetime and internal symmetries must be split'.

This brings us also to Q2: as you say the internal symmetry group and the Lorentz group can indeed combine in some minor way, so we must be careful what we mean by 'split' here. For example, in a system without time-reversing symmetries but with fermions, the (double cover of the Euclidean) Lorentz group often combines with $$G_{int}$$ to give us $$\frac{\operatorname{Spin}(d) \times G_{int}}{\mathbb{Z}_2^F}.$$ Here we quotient by the diagonal $$\mathbb{Z}_2$$-subgroup generated by $$(-1,(-1)^F)$$, where $$-1 \in \operatorname{Spin}(d)$$ is the nontrivial element in the kernel of the double cover to $$SO(d)$$ and $$(-1)^F \in G_{int}$$ is fermion parity.

The situation including time-reversing symmetries is slightly more subtle. I would formulate their formal mathematical definition of 'split' in a language a bit closer to physics as follows. Consider a fermionic symmetry group $$G$$, which I define to be a compact Lie group $$G$$ equipped with a homomorphism $$\theta: G \to \mathbb{Z}_2 = \{\pm 1\}$$ (time-preserving vs time-reversing) and a time-preserving central element $$(-1)^F \in G$$ of square one. The prototypical example of such a group is $$\operatorname{Pin}^+(d)$$, where $$(-1)^F$$ is the nontrivial element of the kernel of the map to $$O(d)$$ and $$\theta_{\operatorname{Pin}}$$ is the composition $$\operatorname{Pin}^+(d) \to O(d) \xrightarrow{\operatorname{det}} \mathbb{Z}_2$$. Every fermionic symmetry group has a canonical automorphism of order at most two: $$g \mapsto \begin{cases} g \quad &\theta(g) = 1, \\ (-1)^F g \quad &\theta(g) = -1. \end{cases}$$ Now according to Freed-Hopkins the most general structure group possible we can endow our Euclidean signature quantum field theories with is $$\ker\left( \theta: \frac{\operatorname{Pin}^+(d) \rtimes G_{int}}{\mathbb{Z}_2^F} \to \mathbb{Z}_2 \right).$$ Here $$(G_{int},(-1)^F,\theta_G)$$ is an arbitrary fermionic symmetry group which acts on $$\operatorname{Pin}^+(d)$$ via the map $$\theta_G: G_{int} \to \mathbb{Z}_2$$ and the canonical automorphism of $$\operatorname{Pin}^+(d)$$. The combined map $$\theta$$ to $$\mathbb{Z}_2$$ is defined by $$(x,g) \mapsto \theta_{\operatorname{Pin}}(x) \theta_G(g)$$.

For example, for $$G_{int} = \operatorname{Pin}^-(1)$$, the internal symmetry group consisting of a single time-reversing symmetry $$T$$ such that $$T^2 = (-1)^F$$, the above group can be shown to be isomorphic to $$\operatorname{Pin}^+(d)$$. For more details on Q2 and other examples, you could consider looking to section 2.1 and 3.2 of my preprint.