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Bulk Reconstruction and Gravitational Dressing

Bulk reconstruction asks how boundary data represent an interior observable. Free-field inversion, interacting corrections, and gravitational dressing solve different parts of that problem, each with a specified region and accuracy.

Enter this chapter

Section titled “Enter this chapter”

Begin with The Bulk Reconstruction Problem to identify the bulk observable and boundary data to be reconstructed.

Helpful background. Von Neumann Factors and Type-III Local Algebras supplies the algebraic meaning of a continuum region. The GKPW Generating-Functional Dictionary supplies the boundary extrapolate and source convention. Bulk-to-Boundary and Bulk-to-Bulk Propagators supplies the Green functions used in smearing. CFT Criteria for Approximate Bulk Locality supplies the large-NN and spectral assumptions behind a weakly coupled bulk.

Topics

Section titled “Topics”
  1. The Bulk Reconstruction Problem — Separates extrapolates, smearing maps, relational observables, regions, state domains, and norms.
  2. Extrapolate Dictionaries versus Interior Reconstruction — Shows why a boundary limit is not a finite-radius inverse.
  3. HKLL Reconstruction for Free Bulk Fields — Derives the mode-sum and smearing representation at leading large NN.
  4. Mode Completeness and Smearing-Kernel Domains — Tests spectra, convergence, distributions, and patch dependence.
  5. Time-Band and Boundary-Diamond Reconstruction — Distinguishes causal support from analytic and state-dependent continuation.
  6. Interactions, Gravitational Dressing, and Microcausality — Adds multi-trace corrections and states the order of commutator cancellation.
  7. Gravitational Gauss Laws and Boundary Anchoring — Derives the boundary-visible tail required by a gravitational excitation.
  8. Relational Bulk Observables and Dressing Choices — Compares geodesic, Coulomb, and reference-field definitions.
  9. Causal Wedges and Subregion Reconstruction — Identifies the conservative region fixed by boundary causal access.
  10. Finite N, Horizons, State Dependence, and Reconstruction Limits — Separates perturbative control from exponentially precise and interior claims.
  11. Reconstruction Error Norms and Nonperturbative Precision — Makes the state set and metric part of the conclusion.

How the ideas fit together

Section titled “How the ideas fit together”

HKLL reconstruction inverts a boundary mode expansion to obtain a free bulk field on a controlled patch. Mode completeness and allowed boundary support determine where the smearing representation is valid Hamilton et al. 2006, §§ 2–3. Interactions require additional operator contributions.

Gravity adds constraints: a coordinate-local field is not by itself a gauge-invariant observable. Dressing or a relational definition supplies the missing reference data and can change commutators. Perturbative reconstruction must therefore state its code sector and error criterion before being extended to a subregion or black-hole interior.

The comparison below separates the assumptions and tests needed for these different conclusions.

Claim-domain comparison

Section titled “Claim-domain comparison”
Representative claim domains and validity boundaries for Bulk Reconstruction and Gravitational Dressing
Claim object State, ensemble, and conventions Approximation, status, and evidence timing Uncertainty and counterevidence Falsifier Failure condition Licensed conclusion
HKLL field Declare free equation, patch, and normalizable modes; use the volume conventions unless the page states a local replacement. Model-specific calculation or conditional result. Control chain: boundary region and algebra → bulk equation and smearing → interaction and gravitational dressing → causal and error checks → reconstructable observable. Sources are cited on the destination page; literature checked through 10 August 2026. Track omitted corrections, alternate branches, and competing definitions. A failed “boundary limit and commutator check” check is counterevidence to the promoted claim. Within the stated patch and perturbative order, the reconstructed field has the wrong boundary limit or fails the free-field commutator test. Promoting a perturbative bulk field to an exact finite-N local operator. perturbative bulk field in a wedge
dressed observable Declare anchoring and gauge convention; use the volume conventions unless the page states a local replacement. Model-specific calculation or conditional result. Control chain: boundary region and algebra → bulk equation and smearing → interaction and gravitational dressing → causal and error checks → reconstructable observable. Sources are cited on the destination page; literature checked through 10 August 2026. Track omitted corrections, alternate branches, and competing definitions. A failed “Gauss-law and boundary-charge check” check is counterevidence to the promoted claim. A transformation that is pure gauge at the boundary changes the proposed invariant observable, or its boundary-charge response violates the stated constraint. Assigning strict compact support to a gravitationally dressed observable without its dressing. a gauge-invariant relational observable
subregion reconstruction Declare region, code sector, and norm; use the volume conventions unless the page states a local replacement. Model-specific calculation or conditional result. Control chain: boundary region and algebra → bulk equation and smearing → interaction and gravitational dressing → causal and error checks → reconstructable observable. Sources are cited on the destination page; literature checked through 10 August 2026. Track omitted corrections, alternate branches, and competing definitions. A failed “complement and overlap consistency” check is counterevidence to the promoted claim. Two proposed representatives act differently on the shared code sector beyond the declared error in the chosen norm. Inferring state-independent global locality from region-adapted reconstruction. region-adapted reconstruction

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Where to go next

Section titled “Where to go next”

Continue to Holographic Entropy and Quantum Extremal Geometry to apply the static area prescription with its homology constraint.

References

Section titled “References”