Aspects of the bulk flat space limit in AdS/CFT

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 \`{I}n\"on\"u-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.