Bulk Ignorance vs Boundary Algebra

 Most of the time in AdS/CFT the bulk is the more complicated story, although in the general discussion of holography the bulk and the boundary both are somewhat mischievous. A result that I am working with is that the type III and type II attribution of the boundary algebras reflect the algebraic-ification of algebras $\mathcal{A}(\mathcal{U})$ associated to a bulk bounded region $\mathcal{U}$, in the following sense. One works from the basis of Haag-Kastler setting, where we attribute to these $\mathcal{U}\subset \Sigma $ in a globally hyperbolic manifold $(M, g)$. We do not wish to work in a non-holographic theory for the sake of this article, although being in a non-holographic theory allows some amount of ease with things like the split property. It should be clear immediately that this can be extended to bulk subregions, for which the algebra is type III as Liu-Leutheusser showed. In this framework, when $\mathcal{A}(\widetilde{R})$ (for some boundary subregion $\widetilde{R}$) is a type II algebra, one can associate the notion of density matrices and entanglement entropy; the bulk dual $\mathcal{A}(R)$ ($R$ being the bulk subregion dual to $\widetilde{R}$) would then have a plausible definition of generalized entropy. This is a very general argument; one can define entropy for $\widetilde{R}$, $\mathcal{S}(\widetilde{R})$ and attribute to it some generalized entropy, and establish a neat relation with the relative entropy. Formally, in type III von Neumann entropy is not well-defined, but nonetheless one can describe the relative entropy of some semiclassical state $|\hat{\Phi }\rangle $ with $|\hat{\Psi }\rangle $ a cyclic and separating state, set with $\delta A_{X}=0$, which looks something like 

\[S(\hat{\Phi }|\hat{\Psi })=-S(\hat{\Phi })-\langle \ln \rho _{\hat{\Psi }}\rangle _{\hat{\Phi }}\;,\]

which can be simplified into a better form by identifying the modular Hamiltonian. In type II, this is no longer merely ``formal", and allows us to elaborately identify the generalized entropy of the subregion. Of course, to work with bulk algebras more formally is certainly interesting, since usually the bulk is the more complicated half of AdS/CFT. A very vague idea of what I had initially is to essentially do this in terms of LCQFTs to see what happens to the split property established at the bifurcation surface $X$. While I did make some good-ish progress, I never got to formalizing it. However, in between, I am working on some set of notes on LCQFTs, which might be a good read once completed. 

P.S. More Hans Niemann debate in the chess world from the St. Louis club. Apparently SLCC did not invite him for their tournaments because of some ``inappropriate behaviour", which Niemann apologized for but is now claiming is just more drama. Catch up with Niemann's video on their letter to him here: I have fully addressed all of the claims made by the STL Chess club in their letter, addressed their private letter to me and given context to my history with the Club. Let's make one detail absolutely clear, I received 0 invitations to STL Events, before I regretably caused..."