# Coleman-Mandula theorem in mathematical language

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Every supersymmetry text starts off mentioning the Coleman-Mandula theorem. Often it is introduced using rather colloquial terminology. I was wondering if anyone knew a precise mathematical formulation of the theorem.

This post imported from StackExchange Physics at 2014-08-11 14:58 (UCT), posted by SE-user SWV
For everyone interested, here's a link to the original paper: prola.aps.org/abstract/PR/v159/i5/p1251_1

This post imported from StackExchange Physics at 2014-08-11 14:58 (UCT), posted by SE-user Vibert
Weinberg Volume 3 has a derivation of the theorem in the first (or maybe second) chapter.

This post imported from StackExchange Physics at 2014-08-11 14:58 (UCT), posted by SE-user Prahar

This post imported from StackExchange Physics at 2014-08-11 14:58 (UCT), posted by SE-user Sanath Devalapurkar
I find that this paper is particularly accessible and constitutes a nice derivation of the individual elements of the proof. ippp.dur.ac.uk/~busbridge/files/documents/CMtheorem.pdf

This post imported from StackExchange Physics at 2014-08-11 14:58 (UCT), posted by SE-user Autolatry

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See http://arxiv.org/pdf/hep-th/9605147v1.pdf. Most of the following is verbatim from there.

Theorem: With the following assumptions: $G$ is a group of symmetries of the $S$-matrix $\mathscr{S}$$G$ contains a subgroup $\mathcal{P}^\prime_0$ that is locally isomorphic to $\mathcal{P}(r,1)$ for $r\geq3$; all particle types correspond to positive-energy time-like representations of the universal covering group of $\mathcal{P}^\prime_0$; the number of particle types is finite; at least locally, $G$ is generated by generators represented in the one-particle space $\mathcal{H}^{(1)}$ by (generalized) integral operators in momentum space, with distributions as kernels; the amplitudes for elastic scattering of two particles do not vanish identically; and the scattering amplitudes are regular functions of the momenta, and the amplitudes for scattering between two-particle states are analytic functions in some neighborhood of the physical region, we can state that $G$ is isomorphic to the direct product of $\mathcal{P}^\prime_0$ and the direct product of some internal symmetry group.

See the linked paper for the proof of the statement, if needed.

answered Sep 12, 2014 by (285 points)

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