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The Coleman-Mandula (CM) theorem states that the Poincar\'e and internalsymmetries of a Minkowski spacetime quantum field theory cannot combinenontrivially in an extended symmetry group. We establish an analogous resultfor quantum field theory in curved spacetimes, assuming local covariance, thetimeslice property, a local dynamical form of Lorentz invariance, andadditivity. Unlike the CM theorem, our result is valid in dimensions $n\ge 2$and for free or interacting theories. It is formulated for theories defined ona category of all globally hyperbolic spacetimes equipped with a globalcoframe, on which the restricted Lorentz group acts, and makes use of a generalanalysis of symmetries induced by the action of a group $G$ on the category ofspacetimes. Such symmetries are shown to be canonically associated with acohomology class in the second degree nonabelian cohomology of $G$ withcoefficients in the global gauge group of the theory. Our main result provesthat the cohomology class is trivial if $G$ is the universal cover $\cal S$ ofthe restricted Lorentz group. Among other consequences, it follows that theextended symmetry group is a direct product of the global gauge group and $\calS$, all fields transform in multiplets of $\cal S$, fields of different spin donot mix under the extended group, and the occurrence of noninteger spin iscontrolled by the centre of the global gauge group. The general analysis isalso applied to rigid scale covariance.

An analogue of the Coleman-Mandula theorem for quantum field theory in curved spacetimes