Quantization of empty space

Abstract

We suggest to use "minimal" choice of quantum gravity theory, that is the quantum field theory, in which space-time is seen as Riemannian space and metric (or vierbein field) is the dynamical variable. We then suggest to use the simplest acceptable action, that is the squared curvature action. The correspondent model is renormalizable, has the correct classical limit without matter and can be explored using Euclidian path integral formalism. In order to get nonperturbative results one has to put this model on the lattice. While doing so serious problems with measure over dynamical variables are encountered, which were not solved until present. We suggest to solve them using the representation of Riemannian space as a limiting case of Riemann - Cartan space, where the Poincare group connection plays the role of dynamical variable. We construct manifestly gauge invariant discretization of Riemann - Cartan space. Lattice realization of Poincare gauge transformation naturally acts on the dynamical variables of the constructed discretization. There exists local measure invariant under this gauge transformation, which could be used as a basic element of lattice path integral methods. The correspondent lattice model appears to be useful for numerical simulations.