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

TL;DR

This paper investigates elliptic de Sitter spacetime as a $\,\mathbb{Z}_2$ quotient where a Euclidean path integral over $\,\mathbb{RP}^{d+1}$ cannot define a no-boundary wavefunction. It proposes a no-boundary density-matrix interpretation for Euclidean QFT on $\,\mathbb{RP}^2$ and analyzes a concrete test case: a free Dirac fermion CFT in two dimensions, computing von Neumann and Rényi entropies via replica techniques on non-orientable surfaces. The authors derive explicit entanglement formulas on RP$^2$ and study real-time entanglement evolution for subsystems, highlighting divergence as the whole spatial slice is approached and horizon-induced phase-like behavior for fixed proper length versus co-expanding intervals. They also discuss the peculiar one-dimensional global Hilbert space and observer-dependent nontrivial Fock spaces, connecting these to broader questions about dS holography and potential gravitational extensions of the no-boundary construction. Overall, the work provides a concrete, field-theoretic framework for no-boundary density matrices in elliptic de Sitter and elucidates how entanglement encodes the causal and observational structure of this spacetime.

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 Rényi 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.

Figures (14)

• Figure 1: A sketch of global dS$_2$ and elliptic dS$_2$.
• Figure 2: A sketch of $\mathbb{S}^2$ and $\mathbb{RP}^2$. They are the Euclidean counterparts of dS$_2$ and dS$_2/\mathbb{Z}_2$.
• Figure 3: A sketch of global dS$_2$ with metric $ds^2 = L_{\rm dS}^2 \left( -dt^2 + {\mathrm{cosh}}(t)^2d\phi^2\right )$ where $t\in (-\infty,\infty)$ and $\phi \in (-\pi,\pi]$ with $\phi \sim \phi + 2\pi$ is shown. The right panel shows its Penrose diagram. The static patch associated to the static observer sitting at $\phi =0$ is shaded. The static patch can be expressed as $|\phi| + \arcsin({\mathrm{tanh}} |t|) < \pi/2$. Its boundary is called the cosmological horizon.
• Figure 4: The left panel shows how a 2D elliptic de Sitter is obtained from a global dS$_2$ via the $\mathbb{Z}_2$ quotient $(t,\phi) \sim (-t, \phi+\pi)$. The right panel shows how this is reflected in the Penrose diagram.
• Figure 5: The upper left panel shows that a spacetime point living on the intersection of the light cones of $p$ and $q$ can neither send a signal to $O$ or receive a signal from them. The lower left panel shows that a spacetime point living on the light cone of $p$ ($q$) but not that of $q$ ($p$) can send a signal to (receive a signal from) $O$ but not receive a signal from (send a signal to) them. All other points can communicate with $O$. The case of points in the static patch is shown in the upper right panel, and the case of the other points is shown in the lower right panel. Note that the Penrose diagrams are shown in the unfolded way.