Quantization and the Issue of Time for Various Two-Dimensional Models of Gravity

TL;DR

This paper demonstrates the embedding of various 2D gravity models into the Katanaev-Volovich framework and explores multiple quantization methods, revealing consistent finite-dimensional quantum systems.

Contribution

It introduces a unified embedding of several 2D gravity models into a non-Einsteinian framework and compares different quantization approaches, showing their equivalence.

Findings

Exact solutions to Dirac constraints in momentum space

Path integral can be fully integrated

Different quantization methods yield the same quantum system

Abstract

It is shown that the models of 2D Liouville Gravity, 2D Black Hole- and $R^2$-Gravity are {\em embedded} in the Katanaev-Volovich model of 2D NonEinsteinian Gravity. Different approaches to the formulation of a quantum theory for the above systems are then presented: The Dirac constraints can be solved exactly in the momentum representation, the path integral can be integrated out, and the constraint algebra can be {\em explicitely} canonically abelianized, thus allowing also for a (superficial) reduced phase space quantization. Non--trivial dynamics are obtained by means of time dependent gauges. All of these approaches lead to the {\em same} finite dimensional quantum mechanical system.