Exact Dirac Quantization of All 2-D Dilaton Gravity Theories

TL;DR

The paper presents an exact Dirac quantization of the most general two-dimensional dilaton gravity theory. It conducts a Hamiltonian analysis with the metric parametrization $ds^2 = e^{2\rho}[-\sigma^2 dt^2 + (dx+M dt)^2]$, identifies the first-class constraints ${\cal F}$ and ${\cal G}$, and shows the reduced phase space is two-dimensional, characterized by a global observable $C$ and its conjugate $P$. Through quantization in the functional Schrödinger representation, the quantum constraints are solved exactly, yielding a physical wave functional $\Psi_{phys}[C;\phi,\rho] = \chi[C] \exp\left(\frac{i}{\hbar} \int dx \left[ Q + \phi' \ln\left(\frac{2\phi' - Q}{2\phi' + Q}\right) \\right] \right)$ that is invariant under spatial diffeomorphisms and annihilated by the Hamiltonian constraint. Because the reduced Hamiltonian vanishes, $\chi[C]$ remains arbitrary, providing a consistent diffeomorphism-invariant quantum description for all theories in this class and offering insights into time-slicing and quantum aspects of black hole physics in 2D dilaton gravity.

Abstract

The most general dilaton gravity theory in 2 spacetime dimensions is considered. A Hamiltonian analysis is performed and the reduced phase space, which is two dimensional, is explicitly constructed in a suitable parametrization of the fields. The theory is then quantized via the Dirac method in a functional Schrodinger representation. The quantum constraints are solved exactly to yield the (spatial) diffeomorphism invariant quantum wave functional for all theories considered. This wave function depends explicitly on the (single) configuration space coordinate as well as on the imbedding of space into spacetime (i.e. on the choice of time).