Semiclassical algebraic reconstruction for type III algebras

TL;DR

This paper extends the algebraic reconstruction theorem to type III algebras using crossed product algebras and semiclassical methods, providing a framework for bulk-boundary duality in holography.

Contribution

It introduces a semiclassical algebraic reconstruction approach for type III algebras, incorporating the algebraic RT formula within a holographic framework.

Findings

Relative entropy factorizes in crossed product algebras

Constructs holographic crossed product algebras for bulk and boundary

Extends algebraic reconstruction theorem to include algebraic RT formula

Abstract

In this work, we address the unresolved type III cases of the algebraic reconstruction theorem by integrating crossed product algebras and semiclassical approximations. We first derive that the relative entropy in crossed product algebras factorizes into contributions from the original algebra and observer wavefunctions. By constructing ``holographic'' crossed product algebras for ``bulk'' and ``boundary'' type III factors, we extend the algebraic reconstruction theorem to include the algebraic Ryu-Takayanagi (RT) formula semiclassically, which provides a complete algebraic description of the reconstruction theorem, as an intrinsic framework for the algebraic version of bulk-boundary correspondences in holographic duality.