Does boundary quantum mechanics imply quantum mechanics in the bulk?

TL;DR

This paper argues that boundary quantum mechanics in AdS/CFT is only well-defined perturbatively, as the free bulk field operator suffers from ambiguities and non-associativity, implying bulk quantum mechanics is inherently perturbative.

Contribution

It introduces a new perspective on bulk reconstruction, linking the well-definedness of bulk operators to perturbative quantum mechanics validity.

Findings

Correlation functions with $^{(0)}$ have ambiguities due to analytic continuation.

$^{(0)}$ cannot be a well-defined linear operator in the CFT.

Bulk quantum mechanics is likely only valid perturbatively.

Abstract

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.