De Sitter Holography with a Finite Number of States

TL;DR

De Sitter holography with a finite number of states addresses how quantum gravity in de Sitter space can be encoded in a finite Hilbert space while preserving spacetime symmetries. The authors propose observer complementarity and let the de Sitter group act on the S-matrix rather than on single states, yielding a finite-dimensional, unitary, $O(1,d)$-invariant S-matrix. They realize this holographically through two constructions on the boundary: (i) a Dirac-spinor toy model and (ii) finite conformal spinors on the boundary sphere, both exploiting a Fermi exclusion principle to truncate the state space. An antipodal identification maps the antipodal horizon sector to final states, so that de Sitter-invariant tensor products become S-matrix elements and unitarity is maintained in explicit examples; in the large-representation limit the framework reproduces area-law entropy and reduces to Minkowski space, supporting a finite-state holographic description of de Sitter space.

Abstract

We investigate the possibility that, in a combined theory of quantum mechanics and gravity, de Sitter space is described by finitely many states. The notion of observer complementarity, which states that each observer has complete but complementary information, implies that, for a single observer, the complete Hilbert space describes one side of the horizon. Observer complementarity is implemented by identifying antipodal states with outgoing states. The de Sitter group acts on S-matrix elements. Despite the fact that the de Sitter group has no nontrivial finite-dimensional unitary representations, we show that it is possible to construct an S-matrix that is finite-dimensional, unitary, and de Sitter-invariant. We present a class of examples that realize this idea holographically in terms of spinor fields on the boundary sphere. The finite dimensionality is due to Fermi statistics and an `exclusion principle' that truncates the orthonormal basis in which the spinor fields can be expanded.

Figures (3)

• Figure 1: 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. The global state is defined on a Cauchy surface for the whole space and therefore lives in the tensor product of the Fock spaces of regions $I$ and $II$.
• Figure 2: A graphical depiction of the $\Delta=5/2$ representation in $d=4$ with 40 states. The states are organized with respect to their $L_0$ eigenvalue, their $J_3$ eigenvalue, and their total spin $J$. Both $L_0$ and $J_3$ take half integer values between $-\Delta$ and $\Delta$, in this case $-5/2$ and $5/2$. The total spin $J$ goes from $1/2$ till $\Delta=5/2$. We plot $\Delta-J-|L_0|+1$ instead of $J$ since this gives a clearer picture. From the diagram it is obvious how to extend it to representations with other values of $\Delta$
• Figure 3: The antipodal map reverses the local arrow of time.