[Paper Review] Further remarks on de Sitter space, extremal surfaces and time entanglement

K. Narayan|arXiv (Cornell University)|Sep 30, 2023

Black Holes and Theoretical PhysicsPhysics and Astronomy3 citations

TL;DR

This paper extends recent work on de Sitter space by showing that no-boundary extremal surface areas—key to understanding time entanglement and holography—can be derived via analytic continuation from AdS extremal surfaces, equivalent to space-time rotations. The resulting complex-valued areas suggest a heuristic Lewkowycz-Maldacena replica formulation on the de Sitter wavefunction, interpreting the areas as pseudo-entropy, with novel entropy inequalities reflecting non-unitary structure.

ABSTRACT

We develop further the investigations in arXiv:2210.12963 [hep-th] on de Sitter space, extremal surfaces and time entanglement. We discuss the no-boundary de Sitter extremal surface areas as certain analytic continuations from $AdS$ while also amounting to space-time rotations. The structure of the extremal surfaces suggests a geometric picture of the time-entanglement or pseudo-entanglement wedge. We also study some entropy relations for multiple subregions. The analytic continuation suggests a heuristic Lewkowycz-Maldacena formulation of the extremal surface areas. In the bulk, this is now a replica formulation on the Wavefunction which suggests interpretation as pseudo-entropy. Finally we also discuss aspects of future-past entangled states and time evolution.

Motivation & Objective

• To understand the geometric and holographic structure of extremal surfaces in de Sitter space, particularly those stretching between future and past null boundaries.
• To establish a connection between no-boundary de Sitter extremal surfaces and analytic continuations of AdS RT/HRT surfaces, suggesting a space-time rotation interpretation.
• To develop a heuristic replica formulation for extremal surface areas in de Sitter space, analogous to the Lewkowycz-Maldacena method in AdS, now applied to the wavefunction rather than a density matrix.
• To explore the implications of complex-valued surface areas for entropy relations in multiple subregions, including mutual and tripartite information.
• To investigate the role of future-past entangled states and time evolution in quantum mechanics, linking the time evolution operator to the existence of such entangled states.

Proposed method

• Using analytic continuation from AdS extremal surfaces to de Sitter, the paper maps RT/HRT surfaces in AdS to timelike and spacelike segments in de Sitter, with the no-boundary condition joining a Euclidean hemisphere to a timelike surface.
• The method involves a geometric reinterpretation of the extremal surface as a 'pseudo-entanglement wedge' bounded by a subregion at future null infinity and the extremal surface, analogous to subregion duality in AdS.
• A heuristic replica approach is applied to the de Sitter wavefunction, generalizing the Lewkowycz-Maldacena method to derive surface areas as pseudo-entropy, with the codimension-2 brane becoming time-evolving and part Euclidean, part timelike.
• Complex-valued area expressions are derived, with the real part from the Euclidean hemisphere (half de Sitter entropy) and the imaginary part from the timelike segment.
• Entropy inequalities for multiple subregions are analyzed in dS₃, yielding novel complex constraints: Re(Iₜ[A,B]) ≥ 0, Im(Iₜ[A,B]) ≤ 0, Im(I₃ᵗ[A,B,C]) ≥ 0, which encode positivity from AdS via analytic continuation.
• The paper connects antipodal observers at I⁺ to codimension-2 surfaces via geometric duality, suggesting a non-trivial bulk-boundary encoding via |Ψ_dS|² and the area relation.

Experimental results

Research questions

• RQ1 How can extremal surface areas in de Sitter space be understood as analytic continuations from AdS RT/HRT surfaces?
• RQ2 What is the geometric and physical interpretation of the no-boundary extremal surface in de Sitter space, particularly in terms of space-time rotations?
• RQ3 Can the Lewkowycz-Maldacena replica method be generalized to de Sitter space, and what does it imply for the wavefunction and pseudo-entropy?
• RQ4 How do entropy relations such as mutual and tripartite information generalize in de Sitter space with complex-valued surface areas?
• RQ5 What is the connection between future-past entangled states, time evolution, and the structure of the de Sitter wavefunction?

Key findings

• The no-boundary extremal surface in de Sitter space arises as an analytic continuation from AdS, with the surface structure suggesting a space-time rotation from the AdS entanglement wedge.
• The real part of the extremal surface area corresponds to half the de Sitter entropy and originates from the Euclidean hemisphere, while the imaginary part comes from the timelike segment.
• A heuristic replica formulation on the de Sitter wavefunction leads to a natural interpretation of the area as pseudo-entropy, with the codimension-2 brane becoming time-evolving and part Euclidean, part timelike.
• Complex-valued entropy inequalities emerge: Re(Iₜ[A,B]) ≥ 0, Im(Iₜ[A,B]) ≤ 0, and Im(I₃ᵗ[A,B,C]) ≥ 0, which are novel and encode known positivity in AdS via analytic continuation.
• The geometric construction of the time-entanglement or pseudo-entanglement wedge is shown to be consistent with subregion duality in de Sitter, based on domain of dependence bounded by boundary subregions and extremal surfaces.
• The existence of future-past entangled states is linked to the time evolution operator, suggesting a deep connection between time evolution and non-unitary holography in de Sitter space.