Perturbative bulk reconstruction in AdS/CFT starts by representing a free bulk field $\phi^{ (0)}$ as a smeared operator in the CFT. A series of $1/N$ corrections must be added to $\phi^{ (0)}$ to represent an interacting bulk field $\phi$. These corrections have been determined in the literature from several points of view. Here we develop a new perspective. We show that correlation functions involving $\phi^{ (0)}$ suffer from ambiguities due to analytic continuation. As a result $\phi^{ (0)}$ fails to be a well-defined linear operator in the CFT. This means bulk reconstruction can be understood as a procedure for building up well-defined operators in the CFT which thereby singles out the interacting field $\phi$. We further propose that the difficulty with defining $\phi^{ (0)}$ as a linear operator can be re-interpreted as a breakdown of associativity. Presumably $\phi^{ (0)}$ can only be corrected to become an associative operator in perturbation theory. This suggests that quantum mechanics in the bulk is only valid in perturbation theory around a semiclassical bulk geometry.
Categories: hep-th