Quantization of static space-times

A four-dimensional Lorentzian static space-time is equivalent to three-dimensional Euclidean gravity coupled to a massless Klein-Gordon field. By canonically quantizing the coupling model in the framework of loop quantum gravity, we obtain a quantum theory that actually describes quantized static space-times. The kinematical Hilbert space is the product of the Hilbert space of gravity with that of imaginary scalar fields. It turns out that the Hamiltonian constraint of the 2+1 model corresponds to a densely defined operator in the underlying Hilbert space, and hence it is finite without renormalization. As a new point of view, this quantized model might shed some light on a few physical problems concerning quantum gravity