HKLL bulk reconstruction for small $Δ$

We discuss the extension of the HKLL (Hamilton, Kabat, Lifschytz, and Lowe) bulk reconstruction for non-interacting scalar fields corresponding to conformal weights $Δ$ smaller than the original condition $Δ> d-1$. We give explicit formulas for the cases $d-2<Δ\leq d-1$ and $Δ=d-s$ with integer $s$. In the latter case we show that smearing CFT fields over a region of the boundary consisting of points light-like separated from the bulk point is sufficient for bulk reconstruction, whereas in general smearing over all light-like and space-like separated points is required.

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