Crossed product algebras and generalized entropy for subregions

TL;DR

The paper generalizes the crossed product construction of modular automorphisms to arbitrary subregions in quantum field theories, producing type II$_\infty$ algebras that admit a trace and a well-defined generalized entropy. By adjoining the modular Hamiltonian (and its commutant charge) to the local algebra, it yields a UV-finite entropy whose leading term reproduces the Bekenstein–Hawking area contribution and whose subleading piece encodes thermal fluctuations. The authors apply the framework to Rindler space, the domain of dependence of an open ball, and boundary subregions in AdS/CFT, arguing that the bulk dual of a boundary subregion is the entanglement wedge and analyzing excited states and disjoint regions where modular flow can become non-geometric. This provides a gravity-independent, algebraic route to entanglement entropy in QFTs and strengthens the role of modular theory and Haag duality in holography, with implications for the generalized second law and bulk reconstruction. The work broadens the applicability of the crossed product approach beyond prior gravity-centered settings, offering concrete prescriptions for entropy in subregions and clarifying the geometric vs non-geometric nature of modular flow in holographic contexts.

Abstract

An early result of algebraic quantum field theory is that the algebra of any subregion in a QFT is a von Neumann factor of type III$_1$, in which entropy cannot be well-defined because such algebras do not admit a trace or density states. However, associated to the algebra is a modular group of automorphisms characterizing the local dynamics of degrees of freedom in the region, and the crossed product of the algebra with its modular group yields a type II$_\infty$ factor, in which traces and hence von Neumann entropy can be well-defined. In this work, we generalize recent constructions of the crossed product algebra for the TFD to, in principle, arbitrary spacetime regions in arbitrary QFTs, paving the way to the study of entanglement entropy without UV divergences. In contrast to previous works, we emphasize that this construction is independent of gravity. In this sense, the crossed product construction represents a refinement of Haag's assignment of nets of observable algebras to spacetime regions by providing a natural construction of a type II factor. We present several concrete examples: a QFT in Rindler space, a CFT in an open ball of Minkowski space, and arbitrary boundary subregions in AdS/CFT. In the holographic setting, we provide a novel argument for why the bulk dual must be the entanglement wedge, and discuss the distinction arising from boundary modular flow between causal and entanglement wedges for excited states and disjoint regions.

Figures (3)

• Figure 1: The bulk algebra $\frak{A}_b$ is shown in blue, while the commutant $\frak{A}_b'=\frak{A}_{b'}$ is shown in white The dotted lines show two possible sets of entangling surfaces, which exchange dominance when their areas become equal, leading to the switchover effect. Figure taken from Almheiri:2014lwa.
• Figure 2: Restricting to a subregion $B$ on the boundary, the algebras of observables in the bulk are illustrated relative to a particular state in the Hilbert space. Here $\Psi$ represents the global AdS vacuum, and $\Phi$ some non-trivial excited state (e.g., a black hole deep in the bulk, or some collection of matter fields). The backreaction due to the excited state deforms the entangling surface deeper into the bulk, so that the algebra $\frak{A}_\Phi$ is an inclusion of $\frak{A}_\Psi$. Thus, considering the entropy of an excited state in the crossed product construction is formally identical to considering the entropy of the vacuum state, since in either case the expectation value of the modular charge returns the correct area contribution. Only in the case of trivial excitations, e.g., very light insertions of operators in $\frak{A}_\Psi$ or $\frak{A}_\Phi$ which do not alter the location of the corresponding entangling surfaces, does it make sense to consider "excited states" via the relative entropy, in which case we show that this simply gives an additional contribution from the difference in type III entropies.
• Figure 3: Plot of the $\mu\!=\!1$ component of the SCT \ref{['eq:SCT']} with $C^1=-\tfrac{1}{2R}$. We have set $R=1$ for visualization purposes. The horizontal axis is the original Rindler coordinate $X^1$, and the vertical axis is the transformed coordinate $x^1$. Note that the entire right Rindler wedge, represented by the region $X^1>0$, is mapped to the interval $-R<x^1<R$ (indicated by the gray horizonal lines), while the discontinuity at $X^1=-2R$ splits the left Rindler wedge over the complement.