HKLL bulk reconstruction for small $\Delta$

TL;DR

This paper extends the HKLL bulk reconstruction method to scalar fields with smaller conformal weights, providing explicit formulas and demonstrating the conditions under which boundary smearing suffices for bulk reconstruction.

Contribution

It introduces explicit formulas for HKLL reconstruction for small elta and shows when boundary smearing over light-like separated points is sufficient.

Findings

Explicit formulas for elta in (d-2, d-1] and integer elta=d-s cases.

Reconstruction with smearing over light-like separated points for elta=d-s.

Smearing over all light-like and space-like points needed in general.

Abstract

We discuss the extension of the HKLL (Hamilton, Kabat, Lifschytz, and Lowe) bulk reconstruction for non-interacting scalar fields corresponding to conformal weights $\Delta$ smaller than the original condition $\Delta > d-1$. We give explicit formulas for the cases $d-2<\Delta\leq d-1$ and $\Delta=d-s$ 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.