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.

What the result establishes

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 compares these assumptions and limitations.

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.