Does boundary quantum mechanics imply quantum mechanics in the bulk?

Perturbative bulk reconstruction in AdS/CFT starts by representing a free bulk field ϕ⁽⁰⁾ as a smeared operator in the CFT. A series of 1/N corrections must be added to ϕ⁽⁰⁾ to represent an interacting bulk field ϕ. 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 ϕ⁽⁰⁾ suffer from ambiguities due to analytic continuation. As a result ϕ⁽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 ϕ. We further propose that the difficulty with defining ϕ⁽⁰⁾ as a linear operator can be re-interpreted as a breakdown of associativity. Presumably ϕ⁽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.