$dS$ extremal surfaces, replicas, boundary Renyi entropies in $dS/CFT$ and time entanglement

TL;DR

This work establishes a concrete replica-based framework for de Sitter extremal surfaces and boundary Renyi entropies within dS/CFT, grounded in smooth bulk replica geometries and the no-boundary prescription. It uncovers that boundary Renyi entropies are generically complex, reflecting the ghost-like nature of the dS/CFT duals and the timelike components of dS extremal surfaces, while the n->1 limit recovers entanglement-like quantities tied to IR slices. In parallel, it develops a QM-centric theory of time entanglement, introducing the reduced time-evolution operator, weak values, and the novel dsT transition matrix to study autocorrelations and future-past entanglement, and then synthesizes these ideas into a cosmological transition-matrix picture. The results illuminate how dS entropy and time-entanglement notions can be related via analytic continuation from AdS, and they raise intriguing questions about the emergence of bulk time and the role of cosmic branes in holographic cosmology. Collectively, the paper advances our understanding of holographic entanglement in de Sitter space and of time-related entanglement structures in quantum theory, with implications for quantum cosmology and ghost-like CFT duals.

Abstract

We develop further previous work on de Sitter extremal surfaces and time entanglement structures in quantum mechanics. In the first part, we first discuss explicit quotient geometries. Then we construct smooth bulk geometries with replica boundary conditions at the future boundary and evaluate boundary Renyi entropies in $dS/CFT$. The bulk calculation pertains to the semiclassical de Sitter Wavefunction and thus evaluates pseudo-Renyi entropies. In 3-dimensions, the geometry in quotient variables is Schwarzschild de Sitter. The 4-dim $dS$ geometry involves hyperbolic foliations and is a complex geometry satisfying a regularity criterion that amounts to requiring a smooth Euclidean continuation. Overall this puts on a firmer footing previous Lewkowycz-Maldacena replica arguments based on analytic continuation for the extremal surface areas via appropriate cosmic branes. In the second part (independent of de Sitter), we study various aspects of time entanglement in quantum mechanics, in particular the reduced time evolution operator, weak values of operators localized to subregions, a transition matrix operator with two copies of the time evolution operator, autocorrelation functions for operators localized to subregions, and finally future-past entangled states and factorization. Based on these, we then give some comments on a cosmological transition matrix using the de Sitter Wavefunction.

Figures (10)

• Figure 1: No-boundary de Sitter space, with the top Lorentzian region continuing smoothly into the Euclidean hemisphere region ending at the no-boundary point. Also shown are IR no-boundary extremal surfaces (blue) anchored at the future boundary $I^+$ dipping into the time direction, timelike in the Lorentzian region and going around the hemisphere.
• Figure 2: The boundary $S^2$ with the IR subregion $\theta_1=[0,\pi]$ on the boundary Euclidean time slice given by the equatorial plane $\theta_2=0$. The quotient space arises from the replica-like space with $n$ copies, and contains conical singularities at the North/South Pole endpoints. The box shows the bulk quotient space and the cosmic brane (red).
• Figure 3: Depicted (top) is the boundary $S^2$ with a conformal transformation to the cylinder. Also shown is the maximal (IR) subregion as the semicircle $\theta=[0,\pi]$ on the boundary Euclidean time slice given by the equatorial plane $\phi=0$, which maps to a line along the entire cylinder length. The bottom shows the $n=2$ replica space obtained by gluing two copies of the cylinder cyclically along the cuts as $1^+\rightarrow 2^-\rightarrow 2^+\rightarrow 1^-$ and back to $1^+$. Since the gluing is along the entire cylinder length, this gives a fatter cylinder with $\phi$-periodicity $4\pi$.
• Figure 4: The reduced transition matrix for states $|\psi\rangle$ and $|\phi\rangle$.
• Figure 5: The weak value of $A$ localized to subregion-1 with state $|\psi\rangle$ and its time-evolution.