Crossed product algebras and generalized entropy for subregions

TL;DR

This paper develops a method to construct type II factors from type III algebras in quantum field theory using crossed products with modular groups, enabling well-defined entropy for arbitrary regions.

Contribution

It generalizes the crossed product construction for algebraic QFT regions, independent of gravity, allowing entropy definitions without UV divergences for arbitrary spacetime subregions.

Findings

Constructed type II factors for various QFT regions

Provided examples in Rindler space, Minkowski space, and AdS/CFT

Offered a holographic argument linking bulk duals to entanglement wedges

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.