Executive Summary : | The AdS/CFT correspondence suggests that quantum gravity in a d-dimensional AdS spacetime is equivalent to a conformal field theory in a d-1 dimensional flat spacetime. However, the dictionary between these theories is still incomplete. The bulk reconstruction program aims to complete this by constructing operators in the boundary theory that reproduce the correlation functions of bulk fields. These 'boundary representations' can be obtained by solving a non-standard Cauchy problem where data is specified on the boundary instead of an initial value hypersurface. This yields nonlocal operators in the bulk consisting of boundary primaries smeared by a'smearing function'. One can also obtain boundary representations from data specified on a limited portion of the boundary, such as the AdS-Rindler patch, which covers only part of the full AdS spacetime. However, this results in a boundary operator that is apparently inequivalent to the one obtained from the global patch. Several open questions remain: 1) Are the different boundary representations really inequivalent? In the global calculation, redundant modes can be identified and eliminated, but not for the AdS-Rindler representation. 2) The non-standard Cauchy problem has not received much attention, and general results are missing. 3) Previous work has established that smearing function diverges when evanescent modes appear in the bulk spacetime, but it is unclear why this should be so. 4) What are the implications of inequivalent representations on the connection between quantum error correction and holography? |