Quantum Theory of Dilaton Gravity in 1+1 Dimensions

TL;DR

This paper develops a quantum theory for 1+1 dimensional dilaton gravity, analyzing the measure, gauge fixing, and state conditions, with implications for black hole singularities and evaporation.

Contribution

It provides an explicit evaluation of the functional measures and fixes diffeomorphism invariance in 1+1D dilaton gravity using conformal gauge techniques.

Findings

The physical state conditions reduce to Wheeler-DeWitt equations at large dilaton values.

A singularity appears at a specific dilaton value related to matter fields.

The final black hole evaporation stage involves Liouville dynamics from the metric measure.

Abstract

We discuss the quantum theory of 1+1 dimensional dilaton gravity, which is an interesting model with analogous features to the spherically symmetric gravitational systems in 3+1 dimensions. The functional measures over the metrics and the dilaton field are explicitly evaluated and the diffeomorphism invariance is completely fixed in conformal gauge by using the technique developed in the two dimensional quantum gravity. We argue the relations to the ADM formalism. The physical state conditions reduce to the usual Wheeler-DeWitt equations when the dilaton $\df^2 ~ (=\e^{-2\phi}) $ is large enough compared with $\kappa =(N-51/2)/12$, where $N $ is the number of matter fields. This corresponds to the large mass limit in the black hole geometry. A singularity appears at $\df^2 =\kappa (>0) $. The final stage of the black hole evaporation corresponds to the region $\df^2 \sim \kappa $, where the Liouville term becomes important, which just comes from the measure of the metrics. If $\kappa < 0 $, the singularity disappears.