Radial Commutativity in AdS/CFT

Question

I am reading Daniel Harlow's Tasi lecture notes on the emergence of bulk physics in AdS/CFT, and the question that I have is regarding the radial commutativity puzzle given in these notes.

First, I will set up a few notations before my question. One can use AdS-Rindler reconstruction to represent a bulk field operator $\phi(x)$ as an integral over local operators in a CFT. The support of the integral is over the boundary domain of dependence $D[R]$ of a spatial region $R$. (The Rindler wedge due to R is colored red in the diagram.) \begin{equation} \label{1} \phi(x)=\int_{D[R]}K(x;Y)\hspace{0.1cm}\mathcal{O}(Y)\hspace{0.1cm} dY. \end{equation} Now consider a CFT local operator $\mathcal{O}(X)$, as shown in the diagram. Since both $\phi(x)$ and $\mathcal{O}(X)$ lie at the same time slice, they should commute. So far, so good. Now according to the time slice axiom, if there is an operator that commutes with all the local operators in a QFT, $[\phi(x),\mathcal{O}(X)]=0$ then, $\phi(x)$ has to be a c-number. This would mean $\phi(x)$ is not an operator at all. This is claimed as a puzzle in the notes. But I don't see what the puzzle is here. Because $\phi(x)$ is just a sum of CFT operators that are spacelike separated from $\mathcal{O}(X)$ and hence would commute with it. \begin{equation} [\phi(x),\mathcal{O}(X)]=\int_{D[R]}K(x;Y)\hspace{0.1cm}[\mathcal{O}(Y),\mathcal{O}(X)]\hspace{0.1cm}dY=0. \end{equation} So it seems to me that it is possible for $\phi(x)$ to not be a c-number and still commute with all the local operators of the boundary CFT. What am I missing here?

Answer 1

The set of local operators form an irreducible representation of the entire algebra of operators of the CFT, and Schur's lemma demands that any operator that commutes with every local operator must be a multiple of the identity.

Harlow's point, if I understand it, is in his statement, "Since $x$ and X are spacelike separated in the bulk, by locality in the radial direction we might expect that, at least at leading order in G, this commutator should indeed vanish". However, while the boundary is a local QFT, the bulk is not, so we need nonlocal boundary operations to capture all of the bulk dynamics. This is what he then explains, that such commutation relations hold only in a subspace of the CFT state space, and therefore need to be replaced with state dependent commutation relations like Equation 4.15 instead. For arbitrary states $|\phi \rangle, |\psi \rangle$, in this subspace (the "code subspace", as described in his next section) we have, $$\langle \phi|[\bar{\mathcal{O}},X_3]|\psi\rangle=\langle \phi|[\mathcal{O}_{12},X_3]|\psi\rangle=0$$ If we want the commutator with $X_2$ to vanish, we have to change the AdS-Rindler representation of the bulk state to get, $$\langle \phi|[\mathcal{O}_{31},X_2]|\psi\rangle=0$$ To summarize, your expectation of how locality should play out here is in contradiction with Schur's lemma, so locality is updated suitably in AdS/CFT.