Remarks on 2D quantum cosmology

TL;DR

This paper explores two-dimensional quantum cosmology with a positive cosmological constant, analyzing classical and quantum properties, solving the Wheeler-DeWitt equation exactly, and connecting wavefunctionals to path integrals in Liouville theory.

Contribution

It provides a complete classical phase space analysis and an exact quantum solution, linking wavefunctionals to path integrals and uncovering a quantum bouncing spacetime in 2D quantum gravity.

Findings

Classical phase space includes bouncing and big bang/crunch cosmologies.

Exact solutions to the Wheeler-DeWitt equation are obtained.

Quantum bouncing spacetime is identified and related to classical cosmologies.

Abstract

We consider two-dimensional quantum gravity endowed with a positive cosmological constant and coupled to a conformal field theory of large and positive central charge. We study cosmological properties at the classical and quantum level. We provide a complete ADM analysis of the classical phase space, revealing a family of either bouncing or big bang/crunch type cosmologies. At the quantum level, we solve the Wheeler-DeWitt equation exactly. In the semiclassical limit, we link the Wheeler-DeWitt state space to the classical phase space. Wavefunctionals of the Hartle-Hawking and Vilenkin type are identified, and we uncover a quantum version of the bouncing spacetime. We retrieve the Hartle-Hawking wavefunction from the disk path integral of timelike Liouville theory. To do so, we must select a particular contour in the space of complexified fields. The quantum information content of the big bang cosmology is discussed, and contrasted with the de Sitter horizon entropy as computed by a gravitational path integral over the two-sphere.