Emergent Area Operators in the Boundary

TL;DR

This work investigates how an information-theoretic area operator $A$ emerges on the boundary in theories with no bulk local degrees of freedom. It constructs an exact QECC with a central decomposition in holographic 2d CFTs in which the area term vanishes at the operator level, and then shows that coarse-graining under semiclassicality constraints yields a nonzero $A$ that reproduces the entanglement entropy for a class of non-linear states. The approach ties the emergent area to the density of primaries via $ ext{log }n_\a\, ext{roughly}\, ext{log } ho_{ ext{prim}}$, connects to Hawking-Page transitions, and relates the boundary construction to bulk notions such as Virasoro Casimirs and fixed-area states. It also extends the framework to nearly conformal quantum mechanics and single-interval setups, illustrating broad applicability and offering boundary-centric insight into the emergence of bulk geometry.

Abstract

In some cases in two and three bulk dimensions without bulk local degrees of freedom, I look for area operators in a fixed boundary theory. In each case, I define an exact quantum error-correcting code (QECC) and show that it admits a central decomposition. However, the area operator that arises from this central decomposition vanishes. A non-zero area operator, however, emerges after coarse-graining. The expectation value of this operator approximates the actual entanglement entropy for a class of states that do not form a linear subspace. These non-linear constraints can be interpreted as semiclassicality conditions. The coarse-grained area operator is ambiguous, and this ambiguity can be matched with that in defining fixed-area states.

Figures (1)

• Figure 1: Successively more sophisticated coarse-graining schemes, from left to right. In the first, we assume that the fine-grained probability is constant within some bins. Secondly, the data varies but we average over a bin anyway. Finally, the bins become fuzzy also.