Quantization of the static space-times

A 4-dimensional Lorentzian static space-time is shown to be equivalent to 3-dimensional Euclidean gravity coupled to an imaginary massless scalar field. By quantizing canonically the coupling model in the framework of loop quantum gravity, we obtain a quantum theory which 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 angle of view, this quantized model is supposed to give certain enlightenment on a few physical problems concerning quantum gravity. E-mail: [email protected] New address: Dept of Physics, Beijing Normal University, Beijing 100875, China. 1