$(2+1)$-Dimensional gravity coupled to a dust shell: Quantization in terms of global phase space variables

We perform a canonical analysis of a model in which gravity is coupled to a spherically symmetric dust shell in $2{+}1$ space–time dimensions. The result is a reduced action depending on a finite number of degrees of freedom. We emphasize finding canonical variables supporting a global parameterization for the entire phase space of the model. It turns out that different regions of the momentum space corresponding to different branches of the solution of the Einstein equation form a single manifold in the ADS$_2$ geometry. The Euler angles support a global parameterization of that manifold. Quantization in these variables leads to noncommutativity and also to discreteness in the coordinate space, which allows resolving the central singularity. We also find the map between the ADS$_2$ momentum space obtained here and the momentum space in Kuchař variables, which could be helpful in extending the obtained results to $3{+}1$ dimensions.