Lectures on entanglement, von Neumann algebras, and emergence of spacetime

TL;DR

The work develops an operator-algebraic framework for quantum gravity, arguing that spacetime and locality emerge from entanglement encoded in von Neumann algebras rather than Hilbert-space factorization. It introduces modular theory, GNS construction, and crossed products to characterize entanglement types (I, II, III) and emergent time, then applies these tools to AdS/CFT, subregion-subalgebra duality, and islands, providing algebraic reformulations of entanglement wedges, causal wedges, and firewall/information- island diagnostics. The paper also ventures into the stringy regime and gravitational dressing, suggesting how Type II structures and generalized entropies arise nonperturbatively and how bulk causal structure can be read from boundary algebras, with implications for holographic quantum error correction and ER=EPR. Overall, it offers a comprehensive algebraic blueprint for bulk reconstruction, entropy, and spacetime emergence in the semiclassical to quantum-gravity transition. The approach promises a robust, nonperturbative language for understanding quantum gravity, holography, and the entanglement-geometry nexus across regimes.

Abstract

We review recent developments in the use of von Neumann algebras to analyze the entanglement structure of quantum gravity and the emergence of spacetime in the semi-classical limit. Von Neumann algebras provide a natural framework for describing quantum subsystems when standard tensor factorizations are unavailable, capturing both kinematic and dynamical aspects of entanglement. The first part of the review introduces the fundamentals of von Neumann algebras, including their classification, and explains how they can be applied to characterize entanglement. Topics covered include modular and half-sided modular flows and their role in the emergence of time, as well as the crossed-product construction of von Neumann algebras. The second part turns to applications in quantum gravity, including an algebraic formulation of AdS/CFT in the large-$N$ limit, the emergence of bulk spacetime structure through subregion-subalgebra duality, and an operator-algebraic perspective on gravitational entropy. It also discusses simple operator-algebraic models of quantum gravity, which provide concrete settings in which to explore these ideas. In addition, several original conceptual contributions are presented, including a diagnostic of firewalls and an algebraic formulation of entanglement islands. The review concludes with some speculative remarks on the mathematical structures underlying quantum gravity.

Figures (35)

• Figure 1: Cartoon of entanglement structure of a CFT for a subregion $A$. There are two types of entanglement: short-range entanglement near the boundary $\partial A$ (represented by short blue lines) due to local interactions there and long-range entanglement (represented by long red dashed lines). We assume that the CFT is put on a lattice, so that the former is regularized. In the $N \to \infty$ limit, both types of entanglement become infinite as they are proportional to the number of field theoretical degrees of freedom, leading to new entanglement structures. In particular, as we will discuss in detail in this review, the divergence of the long-range entanglement is responsible for the emergence of local bulk physics and bulk causal structure in the $G_N \to 0$ limit.
• Figure 2: (a) Cartoon of subregion-subalgebra duality. A bulk subregion $O_1$ is dual to a boundary subalgebra ${{\mathcal{A}}}_1$, which captures all bulk physical operations in $O_1$. Similarly bulk subregions $O_2$ and $O_3$ correspond to boundary algebras ${{\mathcal{A}}}_2$ and ${{\mathcal{A}}}_3$, respectively. Bulk geometric and causal relations among these regions translate into algebraic relations among the corresponding algebras. For example, $O_1 \subset O_2$ becomes ${{\mathcal{A}}}_1 \subset {{\mathcal{A}}}_2$. That $O_1$ and $O_3$ are spacelike separated is captured by $[{{\mathcal{A}}}_1 , {{\mathcal{A}}}_3] =0$, i.e., the bulk causal structure in one higher dimension is captured by the emergent boundary commutant structure in the large $N$ limit. That these regions can be sharply defined in the $G_N \to 0$ limit requires a specific bulk entanglement structure. This structure is captured by the type III$_1$ nature of the ${{\mathcal{A}}}_i$'s, which arises from the infinite long-range entanglement described in Fig. \ref{['fig:newSt']}. (b) A causal diamond in the global AdS that does not touch the boundary is dual to an emergent subalgebra in the boundary system that does not have a geometric description.
• Figure 3: (a) A one-dimensional lattice gauge theory: dynamical variables $U_{ij} \in G$ are assigned to each oriented link $ij$ joining lattice points $i$ and $j$, and gauge transformations are $U_{ij} \to V_i U_{ij} V_j^{\dagger}$ with $V_i, V_j \in G$, where $G$ is the gauge group. (b) Two copies of a system of $N$ spins (blue dots) in an entangled state. Red dashed lines represent entanglement between pairs of spins. (c) A spatial slice of a relativistic quantum field theory is separated into $R$ and $L$ regions by surface $x=0$. Dashed blue lines represent short-range entanglement near $x=0$.
• Figure 4: Rindler regions of $(1+1)$-dimensional Minkowski spacetime, whose metric can be written as $ds^2 = - dt^2 +d x^2 = - \rho^2 d \eta^2 + d \rho^2$. A Rindler observer's worldline is a line of constant $\rho$, corresponding to a hyperbola parameterized by Rindler time $\eta$, which is the worldline of a Rindler observer. The proper time of a Rindler observer is thus given by $d \tau = \rho d \eta$. A boost generates a translation in $\eta$, and maps the right Rindler region $\hat{R}$ to itself.
• Figure 5: Slightly separated $R$ and $L$ regions on a spatial slice. We denote the green region as $I_\epsilon$. $R_\epsilon = R \cup I_\epsilon$ denotes the union of $R$ and the green region while $L_\epsilon = L \cup I_\epsilon$ denotes the union of $L$ and the green region.

Theorems & Definitions (9)

• Proposition 2.1
• Proposition 2.2
• Lemma 4.1
• Definition 8.1
• Definition 8.2
• Definition 8.3
• Proposition 8.1
• Definition 8.4
• Proposition 8.2