Subregions, Minimal Surfaces, and Entropy in Semiclassical Gravity

TL;DR

The work develops a semiclassical framework to understand the entanglement entropy of a subregion in gravity by analyzing the reduced density matrix through Euclidean path integrals and covariant phase-space methods. It shows that, in the semiclassical limit, the reduced density matrix is dominated by configurations that minimize a boost-charge associated with the entangling surface, leading to a leading entropy term proportional to the minimal entangling-surface functional: in GR this is the area, giving S = ⟨𝒜_min⟩/(4Għ), and in higher-derivative theories a dynamical Wald-type entropy S_dyn,min. These results hold without replicas and generalize beyond AdS/CFT, tying quantum information measures to geometric functionals of the entangling surface and suggesting a broader entangling-surface duality. The paper also clarifies gauge ambiguities at the entangling surface and outlines future work on regularization, Lorentzian mapping, and potential connections to a string-like interpretation of Υ.

Abstract

For a large class of density matrices in semiclassical gravity, it is shown that the reduced density matrix which corresponds to tracing over the degrees of freedom in a spatial subregion is dominated by states for which the area of the boundary of the subregion is minimised. In the semiclassical limit, the entropy of the reduced density matrix is found to have a leading order contribution equal to one quarter of the minimal area in natural units. This is consistent with the Ryu-Takayanagi conjecture. An extension to higher derivative theories of gravity is established, for which the area is replaced by a dynamical generalisation of the Wald entropy.