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

A bstract 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.