Maybe I am misunderstanding something really stupid, but I am finding it hard to think of local algebras in terms of wedge algebras. One of the claims (see, e.g., Section 3 and 4 of this paper) is that the local algebra $\mathcal{A}(O)$ of quantum field theory for some subregion $O\subseteq \Sigma$ of a spacelike Cauchy surface is type III$_1$ tends to rely on three things:
- Reeh-Schlieder theorem
- Bisognano-Wichmann theorem for wedge algebras
- Some non-trivial scaling limits
I guess I am not fully appreciating the last two conditions. For example, in general curved spacetimes (globally hyperbolic), I don't necessarily have convenient "Rindler wedges" to even think of this relationship. For any practical computation, having to resort to wedge algebras everytime I am working with local subregion seems to be one extra burdening step. Question: How should I think of the double cone algebra (associated to the double commutant $(\mathcal{A}(O))''$ as being related to the wedge algebra?
On a related note, this occurs also when I try to compute relative entropy using Tomita-Takesaki modular theory. Formally, I can always write the expressions in terms of logarithm of modular operators, but if the only way I can ever compute it is when it is in terms of modular Hamiltonian of boosts for the wedge, then isn't that basically the same as saying we hardly could ever compute relative entropy between two subregions unless there's "Rindler wedges" that contain them?
Unless, of course, the "wedge" is not necessarily Rindler and just taken to be basically "almost half" the commutant of $\mathcal{A}(O)$, which I don't know if it is always easy to define in general.
