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.
Highlights
Version of Vasiliev theory in four dimensions [4, 5], but that model is far from traditional Einstein gravity
We focus on the quantum mechanics of the model at past and future infinity
We would like to answer the following questions: what is the Hilbert space of asymptotic states? What is the inner product on the space? What is the bulk interpretation of operators which act on this space? Is infinite-time evolution unitary? Is the no-boundary state normalizable? We begin by re-examining the simplest amplitudes, from 1-boundary states to 1-boundary states, and recast this into the language of single-particle quantum mechanics
Summary
Let us briefly review Jackiw-Teitelboim (JT) gravity with positive cosmological constant (see [21, 22]). One may exactly evaluate the JT partition function Zg,n,m on a genus g surface with n future boundaries and m past boundaries. While the metric (2.2) of global nearly dS2 space is symmetric under T, the dilaton profile is antisymmetric It is for this reason that we introduced the ± into the dilaton part of the nearly dS2 boundary condition, so T maps a past circle with some to a future circle with the same. Beyond the disk and annulus, there is significant evidence [22] that Zg,n,m is an analytic continuation of the partition function Zg,m+n of JT gravity in Euclidean AdS on a genus g surface with m + n boundaries, recently obtained in [24]
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