Emergent unitarity in de Sitter from matrix integrals

TL;DR

The paper investigates de Sitter holography using a tractable 1+1D model, JT gravity with a positive cosmological constant, to understand the quantum mechanics at past and future infinity. It shows that an asymptotic Hilbert space and an infinite-time evolution operator emerge from the bulk path integral, but the evolution is non-unitary overall; on a positive-momentum subspace it behaves unitarily at leading genus order, with non-perturbative topology-changing effects (baby universes) providing corrections. A dual double-scaled matrix integral is shown to reproduce these features, with unitarity and the Hilbert space arising from universal eigenvalue-repulsion phenomena in random matrix theory. The results point to a robust mechanism by which bulk time, a Hilbert space, and approximate unitarity emerge from a 0+0 dimensional matrix model, with potential generalizations to other dS holographic setups and symmetry classes.

Abstract

We study Jackiw-Teitelboim gravity with positive cosmological constant as a model for de Sitter quantum gravity. We focus on the quantum mechanics of the model at past and future infinity. There is a Hilbert space of asymptotic states and an infinite-time evolution operator between the far past and far future. This evolution is not unitary, although we find that it acts unitarily on a subspace up to non-perturbative corrections. These corrections come from processes which involve changes in the spatial topology, including the nucleation of baby universes. There is significant evidence that this 1+1 dimensional model is dual to a 0+0 dimensional matrix integral in the double-scaled limit. So the bulk quantum mechanics, including the Hilbert space and approximately unitary evolution, emerge from a classical integral. We find that this emergence is a robust consequence of the level repulsion of eigenvalues along with the double scaling limit, and so is rather universal in random matrix theory.

Figures (4)

• Figure 1: The transition amplitude $\bra{\ell_1,\ell_2}\widehat{\mathcal{U}}\ket{\ell_1',\ell_2'}$ is a sum over geometries which connect two circles at past infinity with two circles at future infinity. At leading order in the genus expansion, the amplitude is a sum of two terms, analogous to Wick contractions, as given in Eq. \ref{['E:2to2']}. The dots indicate subleading terms in the genus expansion. Each boundary circle is "wiggly," on account of the Schwarzian reparameterization mode living on it.
• Figure 2: The inner product $\braket{\ell|\ell'}$ is a sum over patches of spacetime in the asymptotic past. This patch runs between a circle with renormalized signed length $\ell'$, on which we impose the past version of the nearly dS$_2$ boundary conditions \ref{['E:BC']}, and another circle with $\ell$ on which we impose the future version of the nearly dS$_2$ boundary conditions.
• Figure 3: The basic observable of de Sitter JT gravity is the annulus partition function, $Z_{0,1,1}(\ell,\ell')$, the path integral over global nearly dS$_2$ space and metric/dilaton configurations connected to it. Properly normalized it gives the leading expression for the infinite-time evolution operator in \ref{['E:leadingU']}.
• Figure 4: An entry in the dictionary between the matrix integral and de Sitter JT gravity. The JT partition function on a genus $g$ surface with two future boundaries and one past boundary is equal to the indicated three-point function of the matrix ensemble. See \ref{['E:dictionary']}.