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Aspects of the bulk flat space limit in AdS/CFT

David Berenstein1 and Joan Simón2

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Phys. Rev. D 113, 105003 – Published 4 May, 2026

DOI: https://doi.org/10.1103/cwc8-ygdz

Abstract

The flat space limit of bulk scalar fields in anti–de Sitter (AdS) is discussed within a Lorentzian canonical quantization setup. We construct their Lorentzian wave functions and use time dependent perturbation theory to compute amplitudes. These can be computed once the wave functions of the incoming and outgoing states, relevant to set up an scattering problem, have been determined. To get the flat limit we need states tailored so that the flat S-matrix arises naturally. This limit is provided by the algebraic action of the Ìnönü-Wigner contraction on the wave functions we construct. We develop the embedding formalism to describe the bulk AdS scalar primary wave functions as holomorphic functions. Flat space massive particle states are built out of the AdS primary together with AdS boosted wave functions. We compute their inner products and show that these become orthogonal in the flat limit, resulting in the correct continuous spectrum for a standard unitary representation of the Lorentz group. In this same limit the original AdS descendants become null states. We also argue how the flat space S-matrix emerges from standard perturbation theory in the interaction picture. To obtain flat space massless particles requires to consider a double scaled limit in which the boost rapidity is scaled to infinity keeping the average particle energy in the flat space limit fixed. We comment on how this limit generates interesting massless state wave functions with nontrivial shape profiles that remember the dimension of the AdS operator. We discuss some of the puzzles attached to these.

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Phys. Rev. D 113, 105003– Published 4 May, 2026
  • Received 17 January 2026
  • Accepted 25 February 2026
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