Extrapolate Dictionaries versus Interior Reconstruction

The extrapolate map reads the normalizable coefficient of a bulk solution at the conformal boundary. Interior reconstruction is an inverse problem: it must recover every required mode at finite radius from a specified boundary domain and state prescription. The first operation can be valid while the second is nonunique, ill-conditioned, or unavailable. Behind a horizon, analytic continuation and state input are additional assumptions, not consequences of the boundary limit.

Required background. The Bulk Reconstruction Problem supplies the target, region, and error taxonomy. The GKPW Generating-Functional Dictionary supplies the source and normalizable-coefficient convention.

Helpful background. States, Geometries, and Radial Quantization supplies global mode states. Finite N, Horizons, State Dependence, and Reconstruction Limits supplies the later precision boundary.

A boundary limit is not an inverse

For a standard-quantized free scalar,

$$\phi(z,x)=z^{d-\Delta}J(x)+z^\Delta A(x)+\cdots, \qquad \mathcal O(x)=\mathcal N_\Delta A(x).$$

At vanishing source, extrapolation takes a known solution to $A(x)$. It contains no explicit rule for propagating $A$ inward. In global AdS a normalizable solution has

$$\phi(X)=\sum_{n\ell m} \left(a_{n\ell m}f_{n\ell m}(X) +a_{n\ell m}^*f_{n\ell m}^*(X)\right),$$

and its boundary coefficient expands in the corresponding cylinder harmonics. Reconstruction requires extracting every $a_{n\ell m}$ with the correct inner product and then resumming the bulk modes. Completeness, convergence, and access to the required boundary times enter at this second step.

The original boundary-value prescription determines correlators from asymptotic data Witten 1998, §§2–3. The finite-radius smearing inverse is an additional construction Hamilton et al. 2006, §§2–3.

First application: one mode at the boundary and in the bulk

Take a global mode

$$f_{n\ell m}(\tau,\rho,\Omega) =e^{-i\omega_{n\ell}\tau}R_{n\ell}(\rho)Y_{\ell m}(\Omega), \qquad \omega_{n\ell}=\Delta+2n+\ell.$$

Its boundary limit determines

$$A_{n\ell m}(\tau,\Omega) =b_{n\ell}e^{-i\omega_{n\ell}\tau}Y_{\ell m}(\Omega),$$

where $b_{n\ell}=\lim_{\rho\to\infty}z^{-\Delta}R_{n\ell}$. Extracting the Fourier-harmonic coefficient gives $a_{n\ell m}b_{n\ell}$; dividing by the known, nonzero $b_{n\ell}$ and multiplying by $R_{n\ell}(\rho)$ recovers this mode at finite radius. The extrapolate is only the first arrow. The inverse additionally uses the spectrum, normalization, full time dependence, and the assumption that no omitted sector contributes.

Limited boundary data and nonuniqueness

Let $P_D$ project boundary functions onto a limited dataset: a finite collection of time samples, low harmonics, or low-point correlators in a time band. Any normalizable solution with boundary coefficient in $\ker P_D$ is invisible to that dataset but can be nonzero at the bulk point. Thus

$$P_D A_1=P_D A_2 \quad\nRightarrow\quad \phi_1(X)=\phi_2(X).$$

For exact analytic generalized-free fields on a suitable state domain, continuation from an open time interval can restore uniqueness, but that conclusion imports analyticity and potentially severe precision demands. It is not a causal smearing result. A horizon makes the distinction sharper: boundary one-point data plus exterior equations do not choose interior state data or a contour across the horizon.

Adversarial check: identical projected extrapolates

Choose a retained set $S$ of global harmonics and add a mode $q\notin S$. The two solutions have identical $P_SA$ but differ by $a_qf_q(X)$ at finite radius. Formal radial inversion of the retained coefficients still works; it reconstructs only the projected bulk field. Calling it the full operator silently assumes $a_q=0$ or restricts the state sector.

The strongest surviving statement is therefore conditional: within a specified mode-complete sector and with the required time/state data, the finite-radius free solution is reconstructible. No extrapolate identity alone licenses a state-independent operator behind a horizon.

Controlled limits and handoff

The argument is linear and assumes known global-AdS modes. HKLL Reconstruction for Free Bulk Fields packages the inverse as a smearing kernel; Mode Completeness and Smearing-Kernel Domains tests whether that package exists on a chosen patch. Interior black-hole proposals remain outside this result.

Exercises

If a boundary dataset retains only harmonics with $\ell\le L_{\max}$, construct a nonzero bulk perturbation invisible to it.

Solution

Choose any normalizable global mode with $\ell>L_{\max}$. Orthogonality of spherical harmonics makes its projected boundary coefficient vanish, while its radial wavefunction is generally nonzero at finite $\rho$. The dataset therefore fixes only the angularly projected bulk field.

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

• Hamilton, Alex, Daniel Kabat, Gilad Lifschytz, and David A. Lowe. “Holographic Representation of Local Bulk Operators.” Physical Review D 74 (2006): 066009. doi:10.1103/PhysRevD.74.066009. arXiv:hep-th/0606141.
• Witten, Edward. “Anti-de Sitter Space and Holography.” Advances in Theoretical and Mathematical Physics 2 (1998): 253–291. doi:10.4310/ATMP.1998.v2.n2.a2. arXiv:hep-th/9802150.