HKLL Reconstruction for Free Bulk Fields

HKLL reconstructs a free normalizable bulk field by inverting its boundary mode expansion. The resulting smearing kernel depends on the AdS patch, scalar quantization branch, boundary condition, and convergence prescription. At leading large $N$, the boundary operator behaves as a generalized free field, so the smeared two-point function equals the bulk Wightman function and the commutator vanishes at bulk-spacelike separation. These statements hold on the declared Fock/code sector, not as exact finite-$N$ gravitational identities.

Required background. Bulk-to-Boundary and Bulk-to-Bulk Propagators supplies the Green functions. The Bulk Reconstruction Problem supplies the region, state, and norm requirements.

Helpful background. Bulk Fields and Boundary Operators supplies the scalar branch. Distributional Kernels and Distributions on Manifolds supplies the weak kernel interpretation. Bounded, Compact, and Integral Operators supplies the operator-domain distinction. Wavepackets, Modes, Frames, and Localization supplies mode localization.

Global mode inversion

Use Lorentzian global AdS$_{d+1}$ with reflecting standard boundary data and signature $(+,-,\ldots,-)$. For $m^2L^2=\Delta(\Delta-d)$ on the $\Delta_+$ branch,

$$\phi^{(0)}(X)=\sum_\lambda \left(a_\lambda f_\lambda(X)+a_\lambda^\dagger f_\lambda^*(X)\right), \qquad \lambda=(n,\ell,m),$$

with $\omega_{n\ell}=\Delta+2n+\ell$. The boundary limit has the same creation and annihilation operators,

$$\mathcal O(\tau,\Omega) =\sum_\lambda\left(a_\lambda u_\lambda(\tau,\Omega) +a_\lambda^\dagger u_\lambda^*(\tau,\Omega)\right).$$

Orthogonality of $u_\lambda$ extracts $a_\lambda$. Substitution yields

$$\phi^{(0)}(X)=\int_{\partial\mathrm{AdS}}d\tau'\,d\Omega' \,K(X|\tau',\Omega')\mathcal O(\tau',\Omega'),$$

where $K$ is the resummed inverse-mode kernel. Equivalent kernels can differ by functions that integrate to zero against the allowed boundary spectrum. The global construction and its patch-dependent support are derived by Hamilton et al. 2006, §§2–3.

First application: a solvable AdS2 kernel

For a massless scalar in Poincaré AdS$_2$,

$$ds^2=\frac{L^2}{z^2}(dt^2-dz^2), \qquad \Delta=1,$$

normalizable Dirichlet solutions obey the flat two-dimensional wave equation. If $\mathcal O(t)=\lim_{z\to0}z^{-1}\phi(t,z)$, then

$$\phi(t,z)=\frac12\int_{t-z}^{t+z}dt'\,\mathcal O(t').$$

Indeed, differentiating with respect to $z$ and taking $z\to0$ returns $\mathcal O(t)$. Smearing twice gives

$$\langle\phi(t,z)\phi(t',z')\rangle =\frac14\int_{t-z}^{t+z}du \int_{t'-z'}^{t'+z'}dv\,\langle\mathcal O(u)\mathcal O(v)\rangle.$$

With the vacuum boundary distribution $\langle\mathcal O(u)\mathcal O(v)\rangle=C/(u-v-i0)^2$, the integrals reproduce the normalizable AdS$_2$ Wightman function, including its $i0$. The same calculation shows that the commutator vanishes when the two bulk points are spacelike. This is an operator statement at generalized-free order.

Kernel support and state data

The displayed AdS$_2$ kernel has compact real support on the boundary interval spacelike related to the point. In higher dimensions or other patches, a real compact kernel may require analytic continuation of boundary coordinates or exist only as a distribution. A choice of Wightman, time-ordered, or retarded correlator fixes the contour; changing $i0$ changes the reconstructed boundary value even when the formal kernel is unchanged.

Normalizable reconstruction uses the operator spectrum of a chosen state representation. Turning on a nonnormalizable source adds a driven solution determined by the bulk-to-boundary propagator; it is not part of the same normalizable operator map.

Adversarial check: leave the naive kernel domain

For some dimensions and AdS-Rindler kinematics, the formal mode sum grows too rapidly to define an ordinary real-boundary smearing function. Declaring the divergent sum to be a function hides the failure. One must instead give a distributional prescription, use complexified boundary support, smear the target into a wavepacket, or weaken the norm. Black-hole and wedge examples where a conventional kernel fails are analyzed by Leichenauer and Rosenhaus 2013, §§2–4.

The strongest surviving claim is mode-by-mode reconstruction on the stated spectral domain. Pointwise kernel existence is a stronger result and must be demonstrated separately.

Controlled limits and handoff

HKLL here is free, leading large $N$, and fixed-background. Mode Completeness and Smearing-Kernel Domains tests the spectral inverse, while Interactions, Gravitational Dressing, and Microcausality adds $1/N$ corrections and gauge invariance. No exact finite-$N$ local operator follows.

Exercises

Verify the AdS$_2$ boundary limit of the smearing formula.

Solution

For continuous $\mathcal O$, the integral is $\frac12(2z\mathcal O(t)+O(z^3))=z\mathcal O(t)+O(z^3)$. Hence $z^{-1}\phi(t,z)\to\mathcal O(t)$.

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.