Quantum Liouville Cosmology

TL;DR

This paper analyzes the disk path integral of timelike Liouville theory as a quantum cosmology model, deriving wavefunctions and proposing an inner product structure for Euclidean histories.

Contribution

It provides a detailed computation of wavefunctions in timelike Liouville quantum cosmology and suggests a framework for an inner product on Euclidean histories.

Findings

Computed one-loop wavefunctions in fixed K-representation.

Proposed a K-independent quantity for the inner product.

Analyzed ensembles including fixed area and static patch perspectives.

Abstract

We provide a detailed analysis of the disk path integral of timelike Liouville theory, conceived as a tractable and precise toy-model quantum cosmology in two dimensions. Disk path integrals with the insertion of matter field operators, taken along a judiciously chosen complex contour, yield states akin to the Hartle-Hawking wavefunction. Working in the fixed $K$-representation, where $K$ is the trace of the extrinsic curvature, we compute the one-loop wavefunctions and put forward a conjecture for the all-loop expressions. A suitable pairing of Liouville disk path integrals yields a $K$-independent quantity that may form the basis for a well-defined inner product on the space of Euclidean histories. We also consider other ensembles, including one with fixed area, and provide a static patch perspective with a timelike feature.