Mode Completeness and Smearing-Kernel Domains

A smearing kernel is an inverse spectral transform. It reconstructs a field only if the chosen modes are complete in the relevant solution space, their boundary limits remain distinguishable, and the inverse converges in a declared topology. Global AdS with reflecting boundary data has a discrete complete scalar basis. A wedge, horizon, missing spectral sector, zero mode, or exponentially ill-conditioned boundary limit can invalidate an ordinary kernel even when formal algebraic inversion remains possible.

Required background. HKLL Reconstruction for Free Bulk Fields supplies the mode inverse. Anti-de Sitter Geometry and the Conformal Boundary supplies the global and patch geometry.

Helpful background. Spectral Decomposition of Two-Point Functions supplies spectral measures. Distributional Kernels and Distributions on Manifolds supplies weak convergence.

The completeness relation

Let $f_\lambda$ be positive-frequency normalizable solutions with Klein–Gordon product

$$(f_\lambda,f_{\lambda'})_{\rm KG}=\delta_{\lambda\lambda'}.$$

Completeness on a global Cauchy slice $\Sigma$ means that the Pauli–Jordan kernel can be expanded as

$$i\Delta(X,Y)=\sum_\lambda \left[f_\lambda(X)f_\lambda^*(Y)-f_\lambda^*(X)f_\lambda(Y)\right],$$

with convergence as a bidistribution. Equivalently, initial data expand in the modes and their normal derivatives. This relation, not merely the list of frequencies, licenses the inverse. Boundary limits $u_\lambda$ must also have nonzero known normalization; otherwise division by their coefficients is singular.

For global AdS with a self-adjoint reflecting scalar boundary condition, regularity at the center and the boundary condition give $\omega_{n\ell}=\Delta+2n+\ell$. The discrete basis is complete in the associated positive-energy solution space, as used in the global smearing construction (Hamilton et al. 2006, §§2–3). It does not include driven nonnormalizable solutions or another self-adjoint extension.

First application: global completeness versus a Rindler restriction

The global completeness test has four steps:

1. normalize $f_{n\ell m}$ in the KG product;
2. verify all allowed $n,\ell,m$ and any exceptional modes are included;
3. take the boundary limits $u_{n\ell m}$ and invert their cylinder inner product;
4. test the mode sum on smooth compactly supported initial data before taking a pointlike distributional limit.

Restricting the resulting field to an AdS-Rindler wedge is valid on the overlap. Rebuilding it solely from wedge modes is a different inverse problem. Large transverse momentum modes can have exponentially small boundary imprint relative to their bulk amplitude, so the inverse kernel grows exponentially and fails as an ordinary distribution on unsmeared boundary data. The obstruction and its relation to trapped or horizon kinematics are exhibited by Leichenauer and Rosenhaus 2013, §§2–4.

Thus global completeness does not imply a well-conditioned real-boundary kernel for every subregion. Wavepacket reconstruction may remain controlled even when a point operator does not.

Spectral truncation and an explicit error

Let $P_S$ retain a spectral set $S$. The reconstructed field is

$$\phi_S(X)=\sum_{\lambda\in S} \left(a_\lambda f_\lambda(X)+a_\lambda^\dagger f_\lambda^*(X)\right),$$

and the omitted vacuum mean-square error is

$$\langle0|\bigl(\phi-\phi_S\bigr)^2|0\rangle =\sum_{\lambda\notin S}|f_\lambda(X)|^2,$$

under the same regulator used to define the coincident field. For a smeared target the sum can converge; for a point field it generally retains the usual ultraviolet divergence. This calculation states exactly what the truncated inverse controls.

Adversarial check: remove a sector or add a horizon

Remove all modes with one angular momentum, or choose a patch whose boundary limits suppress a high-momentum family. Algebraic inversion of the retained coefficients still produces a formula, but the completeness relation acquires a nonzero projector onto the missing sector. The commutator and two-point function then differ by that projector. Near a horizon, exponential inverse weights can amplify boundary errors so that small input uncertainty becomes order-one bulk uncertainty.

The strongest surviving statement is reconstruction of a projected, usually smeared observable in a stated topology. A convergent pointwise kernel or bounded inverse requires its own estimate.

Controlled limits and handoff

This page treats linear spectral reconstruction. Time-Band and Boundary-Diamond Reconstruction restricts boundary time and algebra; interactions later mix single- and multi-trace sectors. Mathematical completeness theorems remain in the spectral-analysis volume.

Exercises

Show that omitting a normalized mode $f_q$ changes the Pauli–Jordan kernel by its antisymmetrized rank-one contribution.

Solution

Subtract the truncated mode sum from the complete one. The missing term is $f_q(X)f_q^*(Y)-f_q^*(X)f_q(Y)$, multiplied by $i$ according to the displayed convention. It is nonzero in general, so the truncated field cannot have the exact free commutator.

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.
• Leichenauer, Stefan, and Vladimir Rosenhaus. “AdS Black Holes, the Bulk-Boundary Dictionary, and Smearing Functions.” Physical Review D 88 (2013): 026003. doi:10.1103/PhysRevD.88.026003. arXiv:1304.6821.