Holographic representation of local bulk operators

The Lorentzian anti-de Sitter/conformal field theory correspondence implies a map between local operators in supergravity and nonlocal operators in the CFT. By explicit computation we construct CFT operators which are dual to local bulk fields in the semiclassical limit. The computation is done for general dimension in global, Poincar\'e and Rindler coordinates. We find that the CFT operators can be taken to have compact support in a region of the complexified boundary whose size is set by the bulk radial position. We show that at finite $N$ the number of independent commuting operators localized within a bulk volume saturates the holographic bound.