Aspects of Quantum Gravity in de Sitter Spaces

TL;DR

This work surveys multiple approaches to quantum gravity in de Sitter spaces, focusing on holographic ideas (dS/CFT) and the tension with Banks–Fischler’s finite Hilbert space idea. It analyzes the geometry and conserved charges of de Sitter space, reviews no-go theorems for string embeddings and viable circumventions, and discusses spacelike branes as time-dependent pathways to acceleration. The discussion extends to the dS/CFT correspondence, including 3D gravity and Liouville theory as a boundary description, and to the debate over whether the dS Hilbert space is finite, presenting several reconciliatory proposals. Finally, it covers the thermodynamics of de Sitter space, including horizon entropies, temperatures, the Nariai limit, and the intriguing possibility of negative temperatures, highlighting fundamental tensions and potential resolutions in holography and quantum gravity.

Abstract

In these lectures we give a review of recent attempts to understand quantum gravity on de Sitter spaces. In particular, we discuss the holographic correspondence between de Sitter gravity and conformal field theories proposed by Hull and by Strominger, and how this may be reconciled with the finite-dimensional Hilbert space proposal by Banks and Fischler. Furthermore we review the no-go theorems that forbid an embedding of de Sitter spaces in string theory, and discuss how they can be circumvented. Finally, some curious issues concerning the thermal nature of de Sitter space are elucidated.

Figures (5)

• Figure 1: Relative intensity of light (corresponding to relative distance) of type Ia supernovae measured as a function of the redshift (corresponding to relative velocity) Riess:1998cb. The measurements indicate that type Ia supernovae are about 25% dimmer than forecast, which means that the cosmic expansion is accelerating.
• Figure 2: De Sitter space dS$_D$ as a hypersurface in $(D+1)$-dimensional Minkowski spacetime.
• Figure 3: Carter-Penrose diagram for de Sitter space HawkEllis. The future and past horizon of a static observer sitting at the south pole $r=0$ (with $r$ being the radial coordinate in (\ref{['metrstatic']})) of the spatial $(D-1)$-sphere are shown. The static patch is shaded. Two spatial slices $\Sigma$ of constant time in inflationary coordinates are indicated.
• Figure 4: Carter-Penrose diagram for Schwarz-schild-de Sitter space with mass parameter $m>m_N$. The solution starts from an asymptotically de Sitter geometry at past infinity ${\cal I}^-$ ($r=\infty$), and finishes in a big crunch at $r=0$ (curvature singularity). There is also a time inverted solution with a curvature singularity in the past.
• Figure 5: A typical system that exhibits negative temperatures: $N$ atoms with spin $1/2$ and magnetic moment $\mu$ on a one-dimensional wire, in an external magnetic field $B$ pointing down, with spin-flip the only degree of freedom.