Quantum Gravity and Black Hole Dynamics in 1+1 Dimensions

TL;DR

This paper investigates the quantum properties of 1+1 dimensional dilaton gravity as a simplified model for black hole dynamics, revealing new features of the quantum constraints and potential resolutions to the information loss paradox.

Contribution

It explicitly evaluates the functional measures, derives the physical state conditions, and uncovers new features not seen in ADM formalism, including the role of singularities and the quantum region.

Findings

Constraints form the Virasoro algebra without central charge

Singularity at a specific value of the dilaton field squared

Quantum region extends behind the classical singularity

Abstract

We study the quantum theory of 1+1 dimensional dilaton gravity, which is an interesting toy model of the black hole dynamics. The functional measures are explicitly evaluated and the physical state conditions corresponding to the Hamiltonian and the momentum constraints are derived. It is pointed out that the constraints form the Virasoro algebra without central charge. In ADM formalism the measures are very ambiguous, but in our formalism they are explicitly defined. Then the new features which are not seen in ADM formalism come out. A singularity appears at $\df^2 =\kappa (>0) $, where $\kappa =(N-51/2)/12 $ and $ N$ is the number of matter fields. Behind the singularity the quantum mechanical region $\kappa > \df^2 >0 $ extends, where the sign of the kinetic term in the Hamiltonian constraint changes. If $\kappa <0 $, the singularity disappears. We discuss the quantum dynamics of black hole and then give a suggestion for the resolution of the information loss paradox. We also argue the quantization of the spherically symmetric gravitational system in 3+1 dimensions. In appendix the differences between the other quantum dilaton gravities and ours are clarified and our status is stressed.