Constraints on a fine-grained AdS/CFT correspondence

For a boundary CFT to give a good approximation to the bulk flat-space S-matrix, a number of conditions need to be satisfied: some of those are investigated here. In particular, one would like to identify an appropriate set of approximate asymptotic scattering states, constructed purely via boundary data. We overview, elaborate, and simplify obstacles encountered with existing proposals for these. Those corresponding to normalizable wavefunctions undergo multiple interactions; we contrast this situation with that needed for a flat-space LSZ treatment. Non-normalizable wavefunctions can have spurious interactions, due either to power-law tails of wavepackets or to their non-normalizable behavior, which obscure S-matrix amplitudes we wish to extract; although in the latter case we show that such gravitational interactions can be finite, as a result of gravitational red shift. We outline an illustrative construction of arbitrary normalizable wavepackets from boundary data, that also yields such spurious interactions. Another set of non-trivial questions regard the form of unitarity relations for the bulk S-matrix, and in particular its normalization and multi-particle cuts. These combined constraints, together with those found earlier on boundary singularity structure needed for bulk momentum conservation and other physical/analytic properties, are a non-trivial collection of obstacles to surmount if a fine-grained S-matrix, as opposed to a coarse-grained construction, is to be defined purely from boundary data.