Bulk-Point Singularities and Locality Diagnostics

A bulk-point singularity occurs after Lorentzian continuation when boundary insertions can all be connected by null geodesics to one bulk interaction point with conserved local momenta. Its strength can resemble a flat-space scattering amplitude. Finite string length, finite higher-spin gap, wavepacket smearing, and finite $N$ soften the signal, so absence in Euclidean or finite-precision data is not evidence against locality.

Required background. The GKPW generating functional supplies the correlator. Regge and eikonal causality supplies the Lorentzian sheet. Bulk-point and flat-space limits supplies the imported result.

Helpful background. Lorentzian causal orderings supplies analytic continuation.

The common-null-point configuration

Choose boundary points $P_i$ and seek a bulk point $X$ with $P_i\cdot X=0$ for all $i$, together with signs assigning incoming and outgoing null rays and momenta satisfying $\sum_i k_i=0$. This is the AdS analogue of a Landau configuration. In cross-ratio variables the relevant approach lies on a Lorentzian sheet near $z=\bar z$, not on the Euclidean line by itself.

For a local contact vertex, saddle-point integration focuses all external propagators near $X$ and produces a power singularity whose exponent depends on $d$, the external dimensions, and derivative order. Its coefficient is related to a high-energy bulk amplitude Maldacena, Simmons-Duffin, and Zhiboedov 2017.

First application: local contact versus string softening

Construct four boundary wavepackets whose null rays meet at the AdS center. A pointlike $\phi^4$ vertex gives coherent support down to transverse resolution $\delta x$. If the microscopic interaction is spread over $\ell_s$, the singular growth stops when the inferred momentum satisfies $p\ell_s\sim1$, equivalently when $EL$ approaches the string-scale gap $L/\ell_s$.

At finite $N$, loops and nonperturbative level spacing also regulate an ideal large-$N$ singularity. Smearing the boundary insertions replaces a distributional divergence by a finite wavepacket amplitude. The comparison is therefore made at fixed smearing and then extrapolated in gap and $N$, not by counting pixels in an unsmeared plot.

Adversarial control: infer from Euclidean precision

Sample a Euclidean correlator away from its cuts and observe no divergence. The bulk-point configuration lies on another sheet, and analytic continuation from noisy finite data is ill conditioned. Without controlled Lorentzian continuation, error bounds, and sufficient resolution relative to $\Delta_{\mathrm{gap}}^{-1}$, the absence has no locality content.

The evidence ceiling is a kinematic diagnostic compatible with approximately local bulk propagation in a specified large-$N$, large-gap regime. It is neither necessary at finite resolution nor sufficient for a complete local quantum gravity. AdS wavepackets turn the same geometric focusing into a normalized collision protocol.

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.
• Maldacena, J., Simmons-Duffin, D., and Zhiboedov, A. (2017), “Looking for a Bulk Point,” Journal of High Energy Physics 2017(01), 013. arXiv:1509.03612.