Elliptic de Sitter Space: dS/Z_2

Abstract

We propose that for every event in de Sitter space, there is a CPT-conjugate event at its antipode. Such an ``elliptic'' $Z_2$-identification of de Sitter space provides a concrete realization of observer complementarity: every observer has complete information. It is possible to define the analog of an S-matrix for quantum gravity in elliptic de Sitter space that is measurable by all observers. In a holographic description, S-matrix elements may be represented by correlation functions of a dual (conformal field) theory that lives on the single boundary sphere. S-matrix elements are de Sitter-invariant, but have different interpretations for different observers. We argue that Hilbert states do not necessarily form representations of the full de Sitter group, but just of the subgroup of rotations. As a result, the Hilbert space can be finite-dimensional and still have positive norm. We also discuss the elliptic interpretation of de Sitter space in the context of type IIB* string theory.

Figures (6)

• Figure 1: The antipodal map reverses the local arrow of time.
• Figure 2: These Penrose diagrams of de Sitter space have been opened up to make all antipodal points distinct. The left and right edges of a diagram are identified, and every point in the interior (except on the central vertical line) now signifies an $\mathbb{R}P^{D-2}$, instead of a $S^{D-2}$. The antipode of a given point is reached by reflecting about the dashed horizontal line, and moving horizontally by half the width of the diagram. Two antipodes, marked $p$ and $\bar{p}$, are shown. In (a) an observer traveling from $i^-$ to $i^+$ has $p$ but not $\bar{p}$ in his causal past (shaded), while in (b) an observer with a different worldline can see $\bar{p}$ but not $p$. The antipodal image of a shaded region is the unshaded region, giving every observer complete information after the $\mathbb{Z}_2$ identification.
• Figure 3: Penrose diagram of de Sitter space. Region $I~(II)$ corresponds to the static patch of an observer on the south (north) pole. The solid lines indicate equal time slices in the static time, they are Cauchy surfaces for region $I$. The dotted lines are their antipodal images, and constitute Cauchy surfaces for region $II$. When a solid line is continued through the horizon, onto its antipodal image, it constitutes a Cauchy surface for the whole space.
• Figure 4: In the far past, an observer at the south pole might describe the state of the world by an initial state $|i \rangle$ on $\mathcal{I}^-$. This evolves in time until it becomes a final state $\langle f|$ on $\mathcal{I}^+$. The antipodal map relates this again to a state on $\mathcal{I}^-$. In- and out-states are therefore associated with a single surface, as in a conventional CFT.
• Figure 5: Radial quantization on an $S^{D-1}$. In-states and out-states are at antipodal points. The Hamiltonian is the dilation operator. Each surface corresponding to constant time for the observer in the bulk is an $S^{D-2}$.