No boundary density matrix in elliptic de Sitter dS/$\mathbb{Z}_2$

TL;DR

This paper explores the properties of elliptic de Sitter spacetime, proposing a no-boundary density matrix instead of a wavefunction, and computes entanglement entropies in a specific fermionic conformal field theory.

Contribution

It introduces the concept of a no-boundary density matrix for elliptic de Sitter and analytically calculates entanglement measures in a 2D free Dirac fermion CFT.

Findings

Computed von Neumann and Rényi entropies for the density matrix.

Reduced the calculation to correlation functions on non-orientable surfaces.

Analyzed the time evolution of entanglement entropy after a crosscap quench.

Abstract

Elliptic de Sitter (dS) spacetime dS$/\mathbb{Z}_2$ is a non-time-orientable spacetime obtained by imposing an antipodal identification to global dS. Unlike QFT on global dS, whose vacuum state can be prepared by a no-boundary Euclidean path integral, the Euclidean elliptic dS does not define a wavefunction in the usual sense. We propose instead that the path integral on the Euclidean elliptic dS defines a no-boundary density matrix. As an explicit example, we study the free Dirac fermion CFT in two-dimensional elliptic dS and analytically compute the von Neumann and the R\'enyi entropies of this density matrix. The calculation reduces to correlation functions of vertex operators on non-orientable surfaces. As a by-product, we compute the time evolution of entanglement entropy following a crosscap quench in free Dirac fermion CFT. We also comment on a striking feature of free QFT in elliptic dS: its global Hilbert space is one-dimensional, wheres the Hilbert space associated to each observer is a nontrivial Fock space.