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Crossed Products, Extended Phase Spaces and the Resolution of Entanglement Singularities

Marc S. Klinger, Robert G. Leigh

TL;DR

The paper addresses entanglement entropy in subregions of quantum field theories where the naive subregion algebra is Type III and lacks a finite trace. It proposes a deep correspondence between the crossed product construction for von Neumann algebras and the extended phase space (EPS) treatment of corner symmetries, supported by a commutative diagram linking geometric quantization with operator-algebraic extensions. By incorporating universal corner symmetry (UCS) generators through the crossed product, the authors show that the subregion algebra becomes semi-finite, enabling well-defined entanglement measures and a unified understanding of modular flow and surface deformations. They connect these ideas to the extended Hilbert space formalism, argue for the necessity of the full UCS in gluing subregions, and outline a universal regularization mechanism for entanglement in gauge theories, independent of holographic considerations. The framework suggests a robust, regulator-free route to finite entanglement data in general QFTs and lays groundwork for extensions to dynamical geometries and quantum gravity contexts.

Abstract

We identify a direct correspondence between the crossed product construction which plays a crucial role in the theory of Type III von Neumann algebras, and the extended phase space construction which restores the integrability of non-zero charges generated by gauge symmetries in the presence of spatial substructures. This correspondence provides a blue-print for resolving singularities which are encountered in the computation of entanglement entropy for subregions in quantum field theories. The extended phase space encodes quantities that would be regarded as `pure gauge' from the perspective of the full theory, but are nevertheless necessary for gluing together, in a path integral sense, physics in different subregions. These quantities are required in order to maintain gauge covariance under such gluings. The crossed product provides a consistent method for incorporating these necessary degrees of freedom into the operator algebra associated with a given subregion. In this way, the extended phase space completes the subregion algebra and subsequently allows for the assignment of a meaningful, finite entropy to states therein.

Crossed Products, Extended Phase Spaces and the Resolution of Entanglement Singularities

TL;DR

The paper addresses entanglement entropy in subregions of quantum field theories where the naive subregion algebra is Type III and lacks a finite trace. It proposes a deep correspondence between the crossed product construction for von Neumann algebras and the extended phase space (EPS) treatment of corner symmetries, supported by a commutative diagram linking geometric quantization with operator-algebraic extensions. By incorporating universal corner symmetry (UCS) generators through the crossed product, the authors show that the subregion algebra becomes semi-finite, enabling well-defined entanglement measures and a unified understanding of modular flow and surface deformations. They connect these ideas to the extended Hilbert space formalism, argue for the necessity of the full UCS in gluing subregions, and outline a universal regularization mechanism for entanglement in gauge theories, independent of holographic considerations. The framework suggests a robust, regulator-free route to finite entanglement data in general QFTs and lays groundwork for extensions to dynamical geometries and quantum gravity contexts.

Abstract

We identify a direct correspondence between the crossed product construction which plays a crucial role in the theory of Type III von Neumann algebras, and the extended phase space construction which restores the integrability of non-zero charges generated by gauge symmetries in the presence of spatial substructures. This correspondence provides a blue-print for resolving singularities which are encountered in the computation of entanglement entropy for subregions in quantum field theories. The extended phase space encodes quantities that would be regarded as `pure gauge' from the perspective of the full theory, but are nevertheless necessary for gluing together, in a path integral sense, physics in different subregions. These quantities are required in order to maintain gauge covariance under such gluings. The crossed product provides a consistent method for incorporating these necessary degrees of freedom into the operator algebra associated with a given subregion. In this way, the extended phase space completes the subregion algebra and subsequently allows for the assignment of a meaningful, finite entropy to states therein.
Paper Structure (20 sections, 125 equations, 1 figure)

This paper contains 20 sections, 125 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Operator Algebras and the Crossed Product
  3. The Crossed Product Algebra
  4. The Dual Weight Theorem
  5. Comparison with Extended Hilbert Space
  6. The Crossed Product and the Extended Phase Space
  7. Preliminaries
  8. G-actions on symplectic manifolds
  9. The Extended Phase Space
  10. Geometric Quantization
  11. Subregions and Entangling Surfaces in Quantum Field Theory
  12. The Diffeomorphism Extended Phase Space
  13. The Modular Automorphism Group of a Subregion Algebra
  14. A synthesis
  15. Discussion
  16. ...and 5 more sections

Figures (1)

  • Figure 1: The crossed product of the naive subregion algebra with the UCS may be understood as the algebra of ultra-local operators in $\mathcal{R}$ along with operators that have support in the tubular neighborhood of the entangling surface, an infinitesimal region generated by the UCS generators.

Theorems & Definitions (13)

  • Definition 3.1: Field of $C^*$ algebras
  • Definition 3.2: Strict deformation quantization of a Poisson manifold
  • Definition E.1: Groupoid
  • Example E.1: Lie group and Bundle of Lie groups
  • Example E.2: Pair groupoid
  • Example E.3: Action groupoid
  • Example E.4: General linear groupoid
  • Example E.5: Gauge groupoid
  • Definition E.2: Transitivity
  • Definition E.3: Lie groupoid representations
  • ...and 3 more