Null quantization, shadows and boost eigenfunctions in Lorentzian AdS

TL;DR

This work analyzes the canonical quantization of a free scalar in Lorentzian AdS4 using a null-cone foliation, uncovering that time-translation eigenmodes with generic energy mix normalizable and non-normalizable fall-offs near the boundary. It develops scattering bases built from bulk-to-boundary propagators, defines a bulk reconstruction formula in terms of boundary primaries and their shadow, and demonstrates how iε prescriptions encode time-ordering and shadow relations, connecting AdS holography to Carrollian and celestial structures. The authors construct boost-eigenstate wavefunctions in both hyperbolic and null foliations, showing these reduce in the flat-space limit to massive and massless conformal primary waves, respectively, thereby linking AdS dynamics to flat-space holography and suggesting a universal bulk origin for observed symmetry enhancements. Overall, the paper provides a concrete framework for AdS bulk reconstruction, null quantization, and boost-diagonal bases that illuminate the AdS–flat space correspondence and offer tools for exploring holography beyond standard AdS/CFT.

Abstract

We revisit the quantization of a free scalar in 4-dimensional (4d) Lorentzian Anti-de-Sitter spacetime (AdS$_4$). We derive solutions to the wave equation that diagonalize time translations in a foliation of AdS$_4$ with null cones. We show that time-translation eigenmodes of arbitrary mass fields that admit a flat space limit must contain both normalizable and non-normalizable fall-offs as one approaches the boundary along a null leaf. We then show that AdS bulk-to-boundary propagators with suitable time orderings provide alternative bases of solutions to the wave equation. We propose an AdS bulk reconstruction formula relating an on-shell free scalar at a spacetime point to CFT primary operators and their shadow transforms. In the flat space limit, this formula reduces to the Carrollian expansion of a free field in flat space. We finally construct Lorentz boost eigenfunctions in AdS in both hyperbolic and null foliations and show that they respectively become massive and massless conformal primary wavefunctions in the flat space limit.

Figures (2)

• Figure 1: Time-ordered bulk-to-boundary propagators provide a decomposition of the space of solutions to the KG equation into positive/negative frequency components for boundary points in the in/out regions. The in/out sectors are orthogonal with respect to the KG inner product on $\Sigma$, in analogy with the plane wave decomposition of solutions to the KG equation in flat space.
• Figure 2: Schematic representation of AdS$_4$ foliated into EAdS$_3$ (left) and dS$_3$ (right) leaves.