Local bulk operators in AdS/CFT correspondence: A boundary view of horizons and locality

We develop the representation of local bulk fields in anti-de Sitter (AdS) space by nonlocal operators on the boundary, working in the semiclassical limit and using AdS₂ as our main example. In global coordinates we show that the boundary operator has support only at points which are spacelike separated from the bulk point. We construct boundary operators that represent local bulk operators inserted behind the horizon of the Poincar\'e patch and inside the Rindler horizon of a two-dimensional black hole. We show that these operators respect bulk locality and comment on the generalization of our construction to higher dimensional AdS black holes.