The Bulk Reconstruction Problem

A bulk observable is defined only after specifying what is reconstructed, from which boundary algebra, on which spacetime region and state family, to what perturbative order, and in which norm. A boundary extrapolate may identify a normalizable coefficient without constructing a finite-radius operator. A free HKLL field may be local in a fixed background yet cease to be gauge invariant once gravity is dynamical. This page gives the classification needed before any kernel is written.

Required background. The GKPW Generating-Functional Dictionary supplies the asymptotic field/operator relation. CFT Criteria for Approximate Bulk Locality supplies the large-$N$, sparse-spectrum, and interaction assumptions.

Helpful background. Choosing a Continuum Subsystem: Algebra, Split, or Regulator supplies the region-algebra distinction. CFT-to-Bulk Reconstruction: Uniqueness and Ambiguities supplies the inverse-problem boundary.

Four different reconstruction targets

For a scalar of standard dimension $\Delta$ in asymptotically AdS$_{d+1}$, the extrapolate dictionary is

$$\mathcal O(x)=\mathcal N_\Delta \lim_{z\to0}z^{-\Delta}\phi(z,x),$$

with $\mathcal N_\Delta$ fixed by the kinetic and operator normalization. It maps a bulk solution to boundary data. A free finite-radius field instead requires an inverse such as

$$\phi^{(0)}(X)=\int_{D}d^dx\,K_D(X|x)\mathcal O(x),$$

where $D$, the kernel, and its distributional meaning depend on the patch and boundary conditions. An interacting field adds multi-trace terms order by order. A gravitational observable has the schematic form

$$\Phi(X)=\phi\!\left(X+\kappa V[h;X,\partial M]\right), \qquad \kappa^2=32\pi G_N,$$

where $V$ anchors the point relationally or dresses it to the boundary. These four expressions cannot be interchanged merely because their leading correlators agree.

Domain and error are part of the definition

Let $\mathcal H_{\rm code}$ be a low-energy family around a fixed semiclassical background. A useful state-vector error is

$$\epsilon_{\rm code} =\sup_{\substack{|\psi\rangle\in\mathcal H_{\rm code}\\ \|\psi\|=1}} \left\|\bigl(\widehat\Phi-\Phi_{\rm target}\bigr)|\psi\rangle\right\|.$$

A correlator error tests selected matrix elements and can be much smaller than this supremum. The full operator norm may be infinite for unbounded fields, so an energy cutoff or a bounded functional of the field is often necessary. The region is equally important: a kernel supported on the full boundary cylinder does not establish reconstruction from a proper diamond.

The standard HKLL construction is a leading large-$N$ statement in a generalized-free-field sector Hamilton et al. 2006, §§2–3. Interactions, gravitational constraints, and finite entropy successively narrow its domain rather than converting it into an exact global identity.

First application: classify one global-AdS scalar

Take a free scalar on global AdS with reflecting standard boundary data and frequencies $\omega_{n\ell}=\Delta+2n+\ell$. A complete classification is:

Item	Choice
boundary limit	$\mathcal O=\mathcal N_\Delta\lim z^{-\Delta}\phi$
finite-radius map	full-cylinder global normal-mode smearing
region	universal-cover global AdS, not a black-hole interior
state set	finite occupation of modes below $E_{\max}$ around the global vacuum
dressing	none while gravity is nondynamical
order	free field, leading large $N$
error	equality of projected free-field matrix elements; truncation error stated after a mode cutoff

This record distinguishes a solved linear inverse problem from a claim about exact locality in quantum gravity.

Adversarial check: demand an exact finite-N local observable

Now require an operator that is simultaneously gauge invariant, compactly localized, state independent, exact at finite $N$, and valid on the full Hilbert space. Before constructing a kernel, two assumptions fail. Gravitational Gauss constraints require a nontrivial asymptotic field for an excitation carrying energy, so a gauge-invariant operator needs boundary-anchored or relational dressing. Finite boundary entropy also prevents an unlimited independent algebra of exact local bulk modes.

The strongest surviving claim is perturbative: a dressed observable can approximate local fixed-background behavior on a declared code sector, with commutators and correlators controlled to a stated order in $G_N$ or $1/N$. Different dressings may differ by physical radiative fields. The obstruction to compact localization is developed by Donnelly and Giddings 2016, §§2–4.

Controlled limits and handoff

This taxonomy does not select a unique reconstruction map. Extrapolate Dictionaries versus Interior Reconstruction isolates the radial inverse problem, HKLL Reconstruction for Free Bulk Fields constructs the leading map, and later leaves add dressing and finite-$N$ errors. Entanglement-wedge recovery remains a separate Chapter 15 question.

Exercises

Why does agreement of all two-point functions on a small state family not imply a small full operator norm for $\widehat\Phi-\Phi$?

Solution

Two-point tests sample only selected matrix elements. The difference may act strongly on states outside the tested family or at high energy. The operator norm takes a supremum over the entire declared domain, so it requires either full control or an explicit code/energy restriction.

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

• Donnelly, William, and Steven B. Giddings. “Observables, Gravitational Dressing, and Obstructions to Locality and Subsystems.” Physical Review D 94 (2016): 104038. doi:10.1103/PhysRevD.94.104038. arXiv:1607.01025.
• 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.