AdS Flat Limits as Candidate Flat-Holography Dictionaries

A controlled AdS/CFT pair can encode a flat-space S-matrix when the AdS radius is taken large while boundary sources create normalizable wavepackets localized in a nearly Minkowski interaction region. This extraction is a stringent dictionary test. It produces scattering data from an existing boundary theory; it does not by itself identify a theory living intrinsically at null infinity.

Required background. AdS Wavepackets and Boundary Extraction of Flat-Space Scattering develops the bulk construction, while S-Matrix and T-Matrix Normalization fixes the target distribution.

Helpful background. Mellin Amplitudes and Penedones-Type Flat-Space Limits supplies the Mellin route, and Finite-Gap Corrections and Locality Error Budgets quantifies contamination by nonlocal bulk physics.

Localized scattering inside AdS

Global AdS acts like a cavity. A generic boundary source produces waves that refocus and scatter repeatedly, so it does not isolate one flat-space event. Instead choose boundary smearing functions $K_{p_i}^{(\ell)}(b_i)$ whose normalizable bulk wavepackets overlap in a region of size $L\ll\ell$ and separate before returning from the boundary. A schematic extraction is

$$\mathcal A(p_1,\ldots,p_n) =\lim_{\ell\to\infty}\mathcal N(\ell,p_i) \int\prod_i db_i\;K_{p_i}^{(\ell)}(b_i) \langle\mathcal O_1(b_1)\cdots\mathcal O_n(b_n)\rangle_{\rm conn}.$$

The normalization $\mathcal N$ is fixed by matching boundary two-point functions to unit-normalized one-particle states. The connected correlator removes disconnected propagation. In the large-radius limit, approximate AdS energy selection sharpens into the flat momentum-conserving distribution.

Contact interaction as a normalization test

For a local bulk vertex $\lambda\phi^4/4!$, four localized wavepackets give, at tree level,

$$\langle p_3p_4|iT|p_1p_2\rangle =i(2\pi)^{d+1}\delta^{(d+1)} \!\left(p_1+p_2-p_3-p_4\right)\mathcal M, \qquad \mathcal M=-\lambda$$

under the stated action and S-matrix convention. Recovering both the constant amplitude and the delta-function coefficient tests the wavepacket normalization. Finite-$\ell$ corrections broaden energy conservation and probe curvature or repeated images.

First application. Extract a tree-level scalar contact amplitude from an AdS correlator using localized wavepackets and verify its normalization and momentum-conserving distribution. Normalize each leg with the CFT two-point function, keep the interaction region fixed while $\ell\to\infty$, and compare the result with the chosen LSZ convention rather than only its momentum dependence.

Order of limits

The flat-radius, large-$N$, and large-gap limits control different errors. Large $N$ suppresses bulk loops; a large higher-spin gap supports a local EFT; large $\ell$ removes curvature. Reversing limits can retain loop or string thresholds at a different physical scale. Boundary sources with long time support can also excite multiple AdS passages, contaminating a single-collision interpretation.

Adversarial control. Use a nonlocalized source and find the repeated-collision terms, or take $N\to\infty$ while holding dimensions rather than physical masses fixed as $\ell\to\infty$. If the extracted distribution or pole spectrum changes, the proposed flat limit lacks a declared scaling prescription. The existence of one well-controlled limit survives; uniqueness does not.

Evidence ceiling

AdS wavepacket and Mellin limits recover perturbative flat-space amplitudes from suitable CFT data and provide powerful checks of bulk locality. They do not construct an autonomous flat-boundary Hilbert space, solve gravitational infrared dressing, or show that every asymptotically flat observable is obtainable from one limit.

The Mellin-space large-radius relation to flat scattering is formulated by Penedones 2011, while normalizable scattering-state preparation in AdS requires the wavepacket construction of Fitzpatrick and Kaplan 2011.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Fitzpatrick, A. Liam, and Jared Kaplan. “Scattering States in AdS/CFT.” (2011). arXiv:1104.2597.
• Giddings, Steven B. “The Boundary S-Matrix and the AdS to CFT Dictionary.” Physical Review Letters 83 (1999): 2707–2710. DOI; Open PDF.
• Penedones, João. “Writing CFT Correlation Functions as AdS Scattering Amplitudes.” Journal of High Energy Physics 2011, no. 3 (2011): 025. DOI; Open PDF.