Time-Band and Boundary-Diamond Reconstruction

A finite boundary time band contains ordinary causal access only to the bulk region that can both send signals to and receive signals from that band. It can nevertheless determine more in a restricted theory and state sector: discrete global-AdS frequencies, analyticity, and a code-subspace assumption can turn short-time data into an ill-conditioned continuation problem. The crucial distinction is therefore between causal reconstruction from a boundary diamond and analytic reconstruction from exact, sector-restricted data. Here AdS has unit radius, Lorentzian signature, global time $\tau$, and the standard $i0$ inherited from the vacuum Wightman function.

Required background. HKLL reconstruction supplies the free-field mode map, and the AdS conformal boundary supplies global causal times and boundary domains of dependence.

Helpful background. Localized algebraic operations separate an operation’s algebra from a tensor factor; nonlocal modular generators warn against assuming geometric flow; and communication through fields distinguishes pre-existing correlations from controllable signaling.

Short time bands do not enlarge the causal wedge

Write global $\mathrm{AdS}_{d+1}$ as

$$ds^2=\frac{1}{\cos^2\rho} \left(d\tau^2-d\rho^2-\sin^2\rho\,d\Omega_{d-1}^2\right), \qquad 0\leq \rho<\frac{\pi}{2}.$$

A radial null ray obeys $d\tau=\pm d\rho$, so light takes $\pi/2$ to travel between the center and the boundary. Consider the full-sphere boundary time band

$$\mathcal T_T=\{(\tau,\Omega):|\tau|<T\}, \qquad T<\frac{\pi}{2}.$$

No point at $(\tau,\rho)=(0,0)$ can both exchange causal signals with $\mathcal T_T$: its earliest future boundary arrival and latest past boundary departure occur at $\tau=\pm\pi/2$, outside the band. A smearing formula supported in $\mathcal T_T$ therefore cannot be justified merely by a retarded or advanced Green function. This causal statement is state-independent.

A spatial subregion $A$ with boundary domain of dependence $D[A]$ has the same logic. Operations in its algebra can causally control the bulk causal wedge $J^+(D[A])\cap J^-(D[A])$, subject to the field equations and boundary conditions. A larger reconstruction claim needs input other than causal propagation; correlations alone do not provide a signaling channel.

The analytic route and its price

For a generalized-free scalar on the boundary cylinder, the positive-frequency part has the discrete expansion

$$\mathcal O^+(\tau,\Omega) =\sum_{n,\ell,m}a_{n\ell m}\,N_{n\ell} e^{-i\omega_{n\ell}\tau}Y_{\ell m}(\Omega), \qquad \omega_{n\ell}=\Delta+2n+\ell.$$

If this series is known exactly, belongs to the assumed positive-energy class, and converges in the relevant distribution topology, its values on any open time interval determine its analytic continuation. One may then extract the mode coefficients and insert them into the global HKLL expansion, even when the target is the center and $T<\pi/2$. The extra assumptions are doing the work: exact continuum data, the global spectral condition, the generalized-free code sector, and a specified topology of convergence. The construction is a precursor representation, not a new causal path.

The inverse is badly conditioned. For a finite spectral cutoff $0\leq n\leq M$, mode extraction amounts to inverting the Gram matrix

$$G_{nn'}(T)=\int_{-T}^{T}d\tau\, e^{i(\omega_n-\omega_{n'})\tau} =2T\,\operatorname{sinc}\!\big[(\omega_n-\omega_{n'})T\big].$$

On the full orthogonality interval this matrix is well separated. On a short band, high-frequency combinations can nearly cancel throughout the interval, and the smallest singular value falls rapidly as $M$ grows. If measured time-band data have error $\delta f$, the inferred coefficients satisfy only

$$\lVert\delta a\rVert_2 \leq \sigma_{\min}(G)^{-1}\lVert\delta f\rVert_2.$$

Thus exact uniqueness can coexist with operational uselessness. Hamilton, Kabat, Lifschytz, and Lowe make the global mode reconstruction explicit, while later time-band constructions emphasize the restricted-sector assumptions behind such precursors (Hamilton et al. 2006, §§2–3; Banerjee et al. 2016, §§2–4).

Central-mode application

Take a spherically symmetric free mode,

$$\phi_{n00}(\tau,0)=a_{n00}f_{n0}(0)e^{-i\omega_{n0}\tau}+\text{h.c.}$$

and suppose boundary data are supplied only on $|\tau|<T<\pi/2$. There are three distinct answers.

1. Causal answer: the central insertion is not in the causal wedge of the band, so causal Green-function evolution from the band does not reconstruct it.
2. Band-limited answer: if frequencies through $M$ are assumed and $G(T)$ is inverted, the truncated central field is reconstructed with an error amplified by $\sigma_{\min}^{-1}$.
3. Exact analytic answer: in the ideal generalized-free, positive-energy sector, the open interval fixes all modes by analytic continuation. No uniform finite-precision bound follows from this statement.

This calculation records the missing premise instead of hiding it in a formal kernel. The relevant object is also an operator in a chosen time-band algebra or its suitable unbounded extension; equality of a few correlators is weaker than operator equality.

An indistinguishable-state test

Now retain only correlators of at most $k$ insertions and frequencies below $\Lambda$. Choose two states $\rho$ and $\rho'$ with identical reduced moment data in that finite set but differing in the occupation of a mode above $\Lambda$. Every supplied low-point time-band datum agrees, yet a central operator containing the omitted mode has

$$\operatorname{Tr}\!\left[(\rho-\rho')\phi(0,0)^2\right]\neq0.$$

The proposed reconstruction is therefore nonunique on the enlarged state set. It becomes unique only after specifying enough data or restricting the state/code sector. Reeh–Schlieder-type density statements do not remove the problem: the approximating operators can be unbounded in norm and hypersensitive to errors, and density in a vacuum Hilbert-space topology is not a uniform recovery guarantee.

The strongest controlled statement is consequently conditional: a boundary diamond causally reconstructs its causal wedge; a shorter time band may represent additional free-field operators in a declared positive-energy code sector, but the representation is analytic, state-sensitive, and generally ill-conditioned.

Controlled limits and handoff

As $T\to\pi/2$, the center enters causal contact and ordinary global smearing becomes available. At fixed $T<\pi/2$, keeping a finite frequency cutoff makes the matrix problem finite but leaves a cutoff and noise error. Taking $M\to\infty$ before sending the data error to zero is not controlled. Interactions add multi-trace corrections, gravity adds boundary-anchored dressing, and finite $N$ invalidates exact generalized-free spectra. Those issues are developed in interacting reconstruction and finite-$N$ limits; genuinely larger subregion recovery belongs to the later entanglement-wedge chapter.

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

• Banerjee, S., Bryan, J.-W., Papadodimas, K., and Raju, S. (2016). “A toy model of black hole complementarity.” Journal of High Energy Physics 2016(11), 144. DOI.
• Hamilton, A., Kabat, D., Lifschytz, G., and Lowe, D. A. (2006). “Local bulk operators in AdS/CFT: A boundary view of horizons and locality.” Physical Review D 74, 066009. DOI.