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AQFT (algebraic quantum field theory) assigns "local algebras of observables" to bounded regions of spacetime, in particular to double-cone ("diamond") regions. These algebras' projection operators are interpreted as possible outcomes of "yes/no" measurements carried out in those regions.

When two double-cone regions X, Y are spacelike-separated, their algebras commute, giving them well-known logical independence properties -- such as the weak property that no nontrivial projection in one algebra is a sub-projection of any nontrivial projection in the other (interpretable as "no outcome in one region guarantees any outcome in the other"), and the strong property that every nonzero projection in X's algebra has a nontrivial intersection with every nonzero projection in Y's algebra (interpretable as "given any pair of desired outcomes in X and Y, a single state could be prepared in a larger region containing X and Y, guaranteeing both outcomes).

My question is: would either or both of these independence properties obtain if X and Y were instead timelike-separated -- say, if every point in X were in the timelike future of every point in Y?

There is lots of literature about the spacelike-separated case, which is understandable since it gives a lot of insight into questions about entanglement, but I've found basically nothing about the timelike-separated case, and I haven't been able to see how to derive an answer to my question from the standard AQFT axioms.

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First of all, you are dealing with von Neumann algebras generated from given GNS representations of C* algebras, otherwise there is no guarantee that projectors associated to observables in an algebra (of their spectral decomposition) are elements of that algebra.

The answer to your question is generally negative, essentially because of the so-called time slice axiom (that turns out to be a theorem for free fields): The algebra in a neighborhood of any fixed Cauchy surface of the spacetime coincides with the whole algebra of the spacetime.

It is also possible to construct explicit counter-examples for free theories. If $f$ has compact support, it holds $\phi(f) = \phi(g)$, where $g$ can be chosen to have compact support in the causal future of the support of $f$. So, pair of elements in timelike separated algebras are strongly correlated. The associated orthogonal projectors must be therefore correlated, too.

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