Null quantization, shadows and boost eigenfunctions in Lorentzian AdS

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.

💡 Research Summary

The paper revisits the canonical quantization of a free scalar field in four‑dimensional Lorentzian Anti‑de Sitter space (AdS₄) and introduces a novel “null quantization” framework. By rotating the global embedding coordinates, the authors construct a foliation of AdS₄ by null cones, leading to Bondi‑type coordinates (τ, r, Ω) where the metric reads ds² = −(ℓ² + r²)dτ² − 2ℓ dτ dr + r²dΩ². Solving the Klein‑Gordon equation in these coordinates yields mode functions Ψ_Δ(τ, r, Ω) = Y_ℓm(Ω) e^{−iωτ} R(r) with R(r) expressed as a linear combination of two hypergeometric functions. Each term is individually invariant under the shadow transformation Δ → 3 − Δ, a property absent in the usual global‑AdS basis.

A key observation is that for generic frequency ω the radial part R(r) contains both the normalizable fall‑off r^{−Δ} and the non‑normalizable fall‑off r^{Δ−3} as r → ∞ (the AdS boundary). Thus, null‑cone modes inevitably mix source‑like and response‑like behaviours, reminiscent of the mixing observed on null infinity in asymptotically flat spacetimes. Only when ω takes the discrete values ω = Δ + ℓ + 2n does the non‑normalizable component vanish, reproducing the standard highest‑weight representations. Conversely, imposing normalizability along a null direction forces the solution to be singular at r = 0 unless Δ belongs to the principal series Δ = 3/2 + iλ, in which case the modes are normalizable but correspond to an imaginary bulk mass.

The authors then turn to bulk‑to‑boundary propagators G_Δ(P;X) = (2Δ − 1)Γ(Δ)π^{3/2}Γ(Δ − ½)(−P·X + iε)^{−Δ}. By evaluating the Klein‑Gordon inner product on a spacelike slice Σ of constant τ, they find that the inner product ⟨G_Δ(P₁;X), G_Δ(P₂;X)⟩Σ is non‑zero only when both boundary insertion points lie on the same side of Σ (Eq. 16). This establishes a positive‑frequency (in‑) and negative‑frequency (out‑) basis of solutions, directly analogous to the decomposition of flat‑space fields into plane waves. When one of the propagators is replaced by its shadow partner G{3−Δ}, the inner product acquires a δ‑function support on null‑separated boundary points (Eq. 17), reproducing the characteristic two‑point functions of Carrollian and celestial CFTs.

Motivated by these observations, the paper proposes a new bulk reconstruction formula (Eq. 14): \