For arbitrary spin and to all orders in an external source, the boundary limit of conformally smeared bulk operators equals the CFT generating functional, so the GKP-Witten relation follows from symmetry and OPE alone.
Flow equation, conformal symmetry and AdS geometry
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abstract
We argue that the Anti-de-Sitter (AdS) geometry in d+1 dimensions naturally emerges from an arbitrary conformal field theory in d dimensions using the free flow equation. We first show that an induced metric defined from the flowed field generally corresponds to the quantum information metric, called the Bures or Helstrom metric, if the flowed field is normalized appropriately. We next verify that the induced metric computed explicitly with the free flow equation always becomes the AdS metric when the theory is conformal. We finally prove that the conformal symmetry in d dimensions converts to the AdS isometry in d+1 dimensions after d dimensional quantum averaging. This guarantees the emergence of AdS geometry without explicit calculation.
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Derivation of the GKP-Witten relation by symmetry without Lagrangian
For arbitrary spin and to all orders in an external source, the boundary limit of conformally smeared bulk operators equals the CFT generating functional, so the GKP-Witten relation follows from symmetry and OPE alone.