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In scattering theory, dispersion relations of the Kramers-Kronig form are often used to reconstruct the real part of an amplitude from its imaginary part, which is typically easier to measure and/or to compute. It turns out that this technique can be generalized to conformal theories, where it enables to reconstruct four-point correlators from certain ``double commutators’’. One implication of this is that local operators are not independent of each other, but rather organize into analytic functions of spin. In this talk I will review a recent formula which enables this reconstruction, and present some of its applications to the gauge-gravity (AdS/ CFT) correspondence, where it provides a uniquely efficient tool for computing AdS Witten diagrams at tree and loop level. Based on 1703.00278 and 1711.02031.

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