A Covariant Holographic Entanglement Entropy Proposal (0705.0016v3)

Abstract: With an aim towards understanding the time-dependence of entanglement entropy in generic quantum field theories, we propose a covariant generalization of the holographic entanglement entropy proposal of hep-th/0603001. Apart from providing several examples of possible covariant generalizations, we study a particular construction based on light-sheets, motivated in similar spirit to the covariant entropy bound underlying the holographic principle. In particular, we argue that the entanglement entropy associated with a specified region on the boundary in the context of the AdS/CFT correspondence is given by the area of a co-dimension two bulk surface with vanishing expansions of null geodesics. We demonstrate our construction with several examples to illustrate its reduction to the holographic entanglement entropy proposal in static spacetimes. We further show how this proposal may be used to understand the time evolution of entanglement entropy in a time varying QFT state dual to a collapsing black hole background. Finally, we use our proposal to argue that the Euclidean wormhole geometries with multiple boundaries should be regarded as states in a non-interacting but entangled set of QFTs, one associated to each boundary.

• The paper introduces a covariant formulation of entanglement entropy, defining it via the area of extremal, light-sheet based surfaces in Lorentzian spacetimes.
• It extends the holographic entanglement framework from static to dynamic scenarios, validated by applications to Vaidya-AdS spacetimes reflecting black hole formation and thermalization.
• The work bridges quantum field theory and gravity by offering a robust method to compute time-dependent entanglement, with implications for understanding thermalization and quantum phase transitions.

Covariant Holographic Entanglement Entropy Proposal

In "A Covariant Holographic Entanglement Entropy Proposal," Hubeny, Rangamani, and Takayanagi investigate the concept of entanglement entropy within the field of quantum field theories (QFTs) and propose a covariant generalization geared towards understanding its time dependence, particularly in contexts explored via the AdS/CFT correspondence.

Background and Motivation

The paper of entanglement entropy in QFTs provides crucial insights into the operative degrees of freedom within specific regions of a given spacetime, particularly when intertwined with models of holography inspired by string theory. Within the framework of AdS/CFT duality, a field theory defined on the boundary of an AdS space has a dual gravitational description, allowing for entanglement entropy to be described via geometric constructs. For static spacetimes, this entanglement entropy has been related to the area of a minimal surface within the bulk, but this paper seeks to extend this construct to time-dependent, dynamical scenarios.

Key Proposals and Constructs

1. Covariant Entanglement Entropy and Light-sheets: The authors propose a covariant generalization of entanglement entropy by utilising light-sheets. Light-sheets are surfaces constructed from light-like geodesics, which play a pivotal role in the covariant entropy bounds established by Bousso. The claim is that entanglement entropy in AdS/CFT should be represented by the area of a co-dimension two bulk surface with zero expansion along these null directions.
2. Extremal Surfaces: They identify these surfaces as the solutions to extremization problems where the expansions of orthogonal null vectors vanish, making these surfaces extrema of the area functional in Lorentzian geometries. This results in a universal prescription for calculating entanglement entropy in both static and dynamic spacetimes.
3. Relation to Causal Wedges and Domains of Dependence: To construct these extremal surfaces, the authors utilize the concept of a causal wedge which encompasses the causal domain of dependence related to a boundary region. The boundary conditions required to maintain consistency with notions of causality and holography are further explored.

Results and Implications

The authors verify their proposal through several examples, demonstrating that their covariant construct reproduces known results in static spacetimes. Moreover, it extends naturally to dynamical cases like the Vaidya-AdS spacetimes, illustrative of black hole formation scenarios where entanglement entropy reflects thermalization processes in the dual CFT.

This covariant formulation impacts both theoretical and practical domains:

• Theoretical Implications: It addresses appearent issues in calculating time-dependent entanglement in QFTs encapsulated with gravity holographically. Moreover, it bears on understanding thermalization and quantum phase transitions through a gravitational lens.
• Practical and Computational Considerations: This approach may streamline calculations of entanglement entropy in complex spacetimes, providing a more robust linkage between QFT and gravity domains by making predictions via extremized geometries.

Future Perspectives

The paper suggest avenues for future research that include exploring deeper relationships between entanglement entropy and other thermodynamic quantities or concepts in gravitational theories like the generalized second law. Further, the potential to capture quantum corrections beyond supergravity through this geometric approach poses an exciting frontier for extending the AdS/CFT correspondence.

Given the robustness provided by their proposals, Hubeny, Rangamani, and Takayanagi's work invites expansions into broader frameworks of quantum gravity and non-static spacetimes. Their findings underscore the universality of entanglement entropy and pivotal role geometric constructs play within the rich tapestry of quantum field theories and gravitation. This work may have far-reaching implications in theoretical physics, linking seemingly disparate realms under a cohesive framework.