AdS Wavepackets and Boundary Extraction of Flat-Space Scattering

Boundary sources can create normalizable AdS wavepackets that meet once in a region much smaller than the curvature radius. After subtracting disconnected propagation and dividing by external norms, the $L\to\infty$ limit yields a flat-space scattering element. Sources that are not localized allow repeated AdS reflections and do not isolate one collision.

Required background. Lorentzian Witten diagrams supplies state preparation and ordering. S- and T-matrix normalization supplies the target observable.

Helpful background. The Mellin flat limit supplies an alternative extraction. Wavepackets, modes, and frames supplies localization. Amplitude/background matching supplies finite-curvature errors.

Boundary preparation of normalizable packets

Choose compact boundary sources $f_i(t,\Omega)$ and form

$$\psi_i(X)=\int_{\partial\mathrm{AdS}}dt\,d\Omega\, K_{\mathrm{Lor}}(X;t,\Omega)f_i(t,\Omega).$$

The frequency support is restricted to positive-energy normalizable modes after the source is switched off. Sources centered at appropriate boundary times and angles focus incoming packets near the AdS center. Their proper width $\sigma$ must satisfy

$$\Lambda_{\mathrm{bulk}}^{-1}\ll\sigma\ll L,$$

so the packets resolve the desired momentum but see negligible curvature. Outgoing sources project onto packets before the first return from the timelike boundary, on a time scale $O(L)$ Gary, Giddings, and Penedones 2009.

First application: one central 2-to-2 collision

Prepare two incoming scalar packets with central momenta $p_1,p_2$ and two outgoing projectors $p_3,p_4$. The connected boundary functional gives

$$\mathcal A_L[f_i]= \langle\Psi^{\mathrm{out}}_3\Psi^{\mathrm{out}}_4 \mid\Psi^{\mathrm{in}}_1\Psi^{\mathrm{in}}_2\rangle_{\mathrm{conn}}.$$

Subtract the identity contractions, divide by each packet’s Klein–Gordon norm, and take $L\to\infty$ with physical $p_i$ and $\sigma$ fixed. The central region becomes Minkowski space, AdS energy spacing tends to zero, and $\mathcal A_L$ approaches $(2\pi)^{d+1}\delta^{d+1}(\sum p_i)i\mathcal T(S,T)$ smeared by the packet profiles. Finite-$L$ errors scale with powers of $\sigma/L$ and the probability of secondary interactions.

Adversarial control: broad sources and repeated reflections

Use boundary sources of duration $O(L)$ with angular support over the whole sphere. The resulting states overlap throughout AdS and refocus after boundary reflections. The connected correlator sums several collisions and curvature propagation; no division by an external norm turns it into a single LSZ event. Narrowing the packets and restricting the observation window must stabilize the extracted amplitude.

The evidence ceiling is a normalized flat-space S-matrix element for wavepackets, species, and perturbative orders possessing this limit. It does not establish asymptotic completeness of the full bulk or locality at energies near the gap. Finite-gap errors bound the derivative expansion used inside the collision.

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

• Gary, M., Giddings, S. B., and Penedones, J. (2009), “Local Bulk S-Matrix Elements and CFT Singularities,” Physical Review D 80, 085005. arXiv:0903.4437.
• Penedones, J. (2011), “Writing CFT Correlation Functions as AdS Scattering Amplitudes,” Journal of High Energy Physics 2011(03), 025. arXiv:1011.1485.