Hilbert space in the flat limit of AdS/CFT. I. Massive particles
Abstract
The flat limit of AdS4 spacetime corresponds, at the level of the symmetry algebra, to the Inönü--Wigner contraction so(2,3) → iso(1,3). We describe the emergence of the Hilbert space of massive scattering states of arbitrary integer spin s from unitary lowest-weight representations 𝒟(Δ,s) of the conformal group, using a basis of `pseudo-momentum' states first introduced by Fronsdal. These states nicely reduce to Wigner's momentum eigenstates in the limit of infinite AdS curvature radius ℓ → ∞, at fixed m=Δ/ℓ. A simple conformal map relates this description to conformal fields in ℝ3 familiar from radial quantization, with the interior and exterior of the unit ball respectively corresponding to outgoing and ingoing momenta. Under the contraction, and upon appropriate rescaling, the inner product on 𝒟(Δ,s) converges to the standard Lorentz-invariant inner product of Wigner's massive particles, so that unitarity is preserved throughout. The contraction to massless particles is briefly discussed here, and will be treated in detail in a forthcoming paper.
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BibTeX
@misc{nguyen2026hilbert,
title = {{Hilbert space in the flat limit of AdS/CFT. I. Massive particles}},
author = {Kevin Nguyen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.31055},
eprint = {2609.31055},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31055},
}
RIS
TY - UNPB
TI - Hilbert space in the flat limit of AdS/CFT. I. Massive particles
AU - Kevin Nguyen
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2609.31055
UR - https://arxiv.org/abs/2609.31055
AB - The flat limit of AdS_4 spacetime corresponds, at the level of the symmetry algebra, to the Inönü--Wigner contraction so(2,3) → iso(1,3). We describe the emergence of the Hilbert space of massive scattering states of arbitrary integer spin s from unitary lowest-weight representations 𝒟(Δ,s) of the conformal group, using a basis of `pseudo-momentum' states first introduced by Fronsdal. These states nicely reduce to Wigner's momentum eigenstates in the limit of infinite AdS curvature radius ℓ → ∞, at fixed m=Δ/ℓ. A simple conformal map relates this description to conformal fields in ℝ^3 familiar from radial quantization, with the interior and exterior of the unit ball respectively corresponding to outgoing and ingoing momenta. Under the contraction, and upon appropriate rescaling, the inner product on 𝒟(Δ,s) converges to the standard Lorentz-invariant inner product of Wigner's massive particles, so that unitarity is preserved throughout. The contraction to massless particles is briefly discussed here, and will be treated in detail in a forthcoming paper.
ER -