Effective entropy of quantum fields coupled with gravity (2007.02987v1)

Abstract: Entanglement entropy quantifies the amount of uncertainty of a quantum state. For quantum fields in curved space, entanglement entropy of the quantum field theory degrees of freedom is well-defined for a fixed background geometry. In this paper, we propose a generalization of the quantum field theory entanglement entropy by including dynamical gravity. The generalized quantity named effective entropy, and its Renyi entropy generalizations, are defined by analytic continuation of a gravitational path integral on replica geometry with a co-dimension-$2$ brane at the boundary of region we are studying. We discuss different approaches to define the region in a gauge invariant way, and show that the effective entropy satisfies the quantum extremal surface formula. When the quantum fields carry a significant amount of entanglement, the quantum extremal surface can have a topology transition, after which an entanglement island region appears. Our result generalizes the Hubeny-Rangamani-Takayanagi formula of holographic entropy (with quantum corrections) to general geometries without asymptotic AdS boundary, and provides a more solid framework for addressing problems such as the Page curve of evaporating black holes in asymptotic flat spacetime. We apply the formula to two example systems, a closed two-dimensional universe and a four-dimensional maximally extended Schwarzchild black hole. We discuss the analog of the effective entropy in random tensor network models, which provides more concrete understanding of quantum information properties in general dynamical geometries. By introducing ancilla systems, we show how quantum information in the entanglement island can be reconstructed in a state-dependent and observer-dependent map. We study the closed universe (without spatial boundary) case and discuss how it is related to open universe.

Effective Entropy of Quantum Fields Coupled with Gravity

The paper "Effective Entropy of Quantum Fields Coupled with Gravity" by Xi Dong and colleagues tackles the intricate problem of defining entanglement entropy in quantum field theories (QFT) when these fields are embedded in a curved space-time influenced by dynamical gravity. Traditionally, entanglement entropy, or von Neumann entropy, quantifies the uncertainty of quantum states on fixed, non-dynamic backgrounds. The authors propose a novel generalization where this entropy is computed with gravity as a dynamic participant in the system.

Key Contributions and Methods

The central contribution of the paper is the introduction of the concept of effective entropy, which extends the notion of QFT entanglement entropy to systems where gravity cannot be considered merely as a static backdrop, but is a dynamic entity that interacts with matter fields. To formulate this, the paper employs the replica trick traditionally used in computing the Renyi entropy. The computation involves a gravitational path integral over replicated geometries, supplemented by a co-dimension-2 brane introduced to manage the associated conical singularities.

The authors assert that this effective entropy obeys the quantum extremal surface (QES) formula, a significant result that supports the extension of the Hubeny-Rangamani-Takayanagi (HRT) formula to more general non-AdS geometries. In situations with substantial field entanglement, the geometry can undergo a topology transition, resulting in entanglement islands—regions that significantly alter expected entropy behavior, a concept related to the evaluation of the Page curve for black hole evaporation scenarios.

Examples and Applications

Two specific case studies demonstrate the utility of the effective entropy framework:

1. Closed Two-Dimensional Universe: The entropy of a spatial interval within a closed universe is examined, employing Jackiw-Teitelboim gravity models. This example showcases the implementation and calculative framework for closed universes where no explicit spatial boundary exists.
2. Four-Dimensional Schwarzschild Black Hole: The entropy considerations include entanglement islands in asymptotically flat spacetime, relevant for evaluating Hawking radiation emitted since the black hole's formation, connecting to the enduring enigma of the black hole information paradox.

Observations and Implications

The paper also explores the analogy between effective entropy calculations and random tensor network (RTN) models. RTN models act as a simplified backdrop for understanding gravitational dynamics and entropic calculations, underpinning how information can be reconstructed in black hole spacetimes. The framework indicates the importance of observer-dependent factors in determining quantum information distribution across different entangled regions—a notion echoing quantum error correction mechanisms.

Concluding Remarks

The research fundamentally broadens the application of entanglement entropy in quantum gravity scenarios, laying groundwork for future exploration in systems where gravity is not static. The implication that the quantum extremal surface can dictate entropy behavior in dynamically changing geometries provides a robust platform for the theoretical exploration of phenomena like the black hole information paradox and entropy calculation in non-trivial spacetimes.

In conclusion, this work opens pathways for extensive research into cosmological models where dynamical effects of gravity are non-negligible, offering a rewardingly complex entanglement structure crucial for understanding quantum information flow in our universe. The presentation of formidable computational and conceptual tools foreshadows significant advancements in how entropy is calculated and understood in the context of quantum fields influenced by gravity.