On bulk reconstruction in Lorentzian AdS and its flat space limit

TL;DR

This paper develops a method to reconstruct free scalar fields in Lorentzian AdS$_4$ using boundary operators, connecting it to flat space limits and different basis decompositions, including plane wave and Carrollian bases.

Contribution

It introduces a new bulk reconstruction framework in Lorentzian AdS$_4$ that relates boundary operators to bulk fields via time-ordered propagators and explores flat space limits.

Findings

Bulk scalar fields can be expressed in terms of boundary operators with time orderings.

The construction generalizes to different bases, including plane wave and Carrollian.

In the flat space limit, AdS scalar wavefunctions become flat space conformal primary wavefunctions.

Abstract

We revisit the reconstruction of a free quantum field in 4-dimensional Lorentzian Anti-de-Sitter (AdS$_4$) spacetime in terms of primary operators in the boundary 3d CFT (CFT$_3$). We show that the positive and negative energy subspaces of solutions to the Klein-Gordon equation in AdS can be spanned with bulk-to-boundary propagators with appropriate time orderings. As a result, free scalar fields on a codimension-1 bulk hypersurface $\Sigma_{\tau}$ can be expressed in terms of operators integrated over boundary regions in the past or future of $\Sigma_{\tau}$ with kernels given by time-ordered or anti-time-ordered propagators. We present various equivalent representations for the bulk field in terms of either CFT primaries or their shadows. We show from both a representation theoretic perspective and by direct computation of various flat space limits that our construction is the AdS analog of the canonical quantization of a free scalar in flat space. Depending on the choice of $\Delta$ and the location of the boundary insertions one obtains a decomposition of the scalar in either a plane wave basis ($\Delta \rightarrow \infty$) or a Carrollian basis (fixed $\Delta$). Finally, we show that the free scalar in AdS$_4$ can be alternatively decomposed in terms of wavefunctions associated with principal series representations of dimension $\delta = 1 + i\lambda$ of an $\mathfrak{so}(3,1)$ subalgebra of the AdS$_4$ isometry algebra. We demonstrate that the latter become, in the limit of large AdS radius and for fixed $\delta$, scalar conformal primary wavefunctions in flat space.