On bulk reconstruction in Lorentzian AdS and its flat space limit

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.