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Aspects of the bulk flat space limit in AdS/CFT

David Berenstein, Joan Simon

TL;DR

The paper develops a Lorentzian, canonical-quantization framework to study the bulk flat-space limit in AdS/CFT, anchoring the limit in an Inönü–Wigner contraction of AdS isometries to Poincaré and employing the embedding-space formalism to manage scalar primaries and their boosts.Massive states in the flat limit arise from AdS primaries with large conformal dimension Δ, while descendants become null; moving states are generated via AdS isometries (boosts), which become orthogonal in the flat-space limit, yielding the correct massive representation of the Lorentz group.For massless particles, a double scaling limit is required, producing non-plane-wave, Δ-dependent waveforms that remember the AdS operator dimension and admit Mellin-like representations connected to celestial holography concepts; boundary Euclidean preparations underpin the AdS preparation of in/out states and the emergence of the flat S-matrix.Overall, the work provides a coherent, representation-theory–driven route to the flat-space S-matrix from AdS bulk wave functions, clarifying state preparation and boundary-operator roles while outlining key avenues for extending to spinning particles, gravity, and holographic connections.

Abstract

The flat space limit of scalar bulk fields in AdS is discussed within a Lorentzian canonical quantization setup tailored to describe AdS state preparation and to extract the flat S-matrix dynamics. We discuss how the algebraic Ìnönü-Wigner contraction captures the local physics of the equivalence principle in quantum field theory in a fixed background description. 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 non-trivial shape profiles that remember the dimension of the AdS operator. We discuss some of the puzzles attached to these.

Aspects of the bulk flat space limit in AdS/CFT

TL;DR

The paper develops a Lorentzian, canonical-quantization framework to study the bulk flat-space limit in AdS/CFT, anchoring the limit in an Inönü–Wigner contraction of AdS isometries to Poincaré and employing the embedding-space formalism to manage scalar primaries and their boosts.Massive states in the flat limit arise from AdS primaries with large conformal dimension Δ, while descendants become null; moving states are generated via AdS isometries (boosts), which become orthogonal in the flat-space limit, yielding the correct massive representation of the Lorentz group.For massless particles, a double scaling limit is required, producing non-plane-wave, Δ-dependent waveforms that remember the AdS operator dimension and admit Mellin-like representations connected to celestial holography concepts; boundary Euclidean preparations underpin the AdS preparation of in/out states and the emergence of the flat S-matrix.Overall, the work provides a coherent, representation-theory–driven route to the flat-space S-matrix from AdS bulk wave functions, clarifying state preparation and boundary-operator roles while outlining key avenues for extending to spinning particles, gravity, and holographic connections.

Abstract

The flat space limit of scalar bulk fields in AdS is discussed within a Lorentzian canonical quantization setup tailored to describe AdS state preparation and to extract the flat S-matrix dynamics. We discuss how the algebraic Ìnönü-Wigner contraction captures the local physics of the equivalence principle in quantum field theory in a fixed background description. 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 non-trivial shape profiles that remember the dimension of the AdS operator. We discuss some of the puzzles attached to these.

Paper Structure

This paper contains 9 sections, 59 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Ìnönü-Wigner contraction and the flat space limit
  3. Taking the limit
  4. The embedding space formalism
  5. Scalar wave functions
  6. The flat space limit in the embedding space formalism and representation theory
  7. Amplitudes and state preparation
  8. Massless particles and their wave functions
  9. Discussion

Figures (3)

  • Figure 1: Conformal group generators
  • Figure 2: How the CFT generators are mapped to flat space
  • Figure 3: Schematic representation of a scattering event. Particles are prepared in Euclidean time. Evolution then becomes that of real time (from $t=-\frac{\pi}{2}$ to $t=\frac{\pi}{2}$) and becomes Euclidean again for readout.