We define a perturbatively calculable quantity—the on-shell correlator—which furnishes a unified description of particle dynamics in curved spacetime. Specializing to the case of flat and anti-de Sitter space, on-shell correlators coincide precisely with on-shell scattering amplitudes and boundary correlators, respectively. Remarkably, we find that symmetric manifolds admit a generalization of on-shell kinematics in which the corresponding momenta are literally the isometry generators of the spacetime acting on the external kinematic data. These isometric momenta are intrinsically non-commutative but exhibit on-shell conditions that are identical to those of flat space, thus providing a common language for computing and representing on-shell correlators which is agnostic about the underlying geometry.
As applications of these tools, we compute n-point scalar correlators in AdS in terms of isometric momenta. In many cases, the results are direct lifts of flat-space expressions. We provide field-theoretic proofs of color-kinematics duality and BCJ relations in AdS at n-points in biadjoint scalar theory and the nonlinear sigma model. We discuss possible extensions to generic curved spacetimes without symmetry.
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