Analysis of the Wheeler-DeWitt Equation beyond Planck Scale and Dimensional Reduction

We solve the Wheeler-DeWitt equation for four-dimensional Einstein gravity as an expansion in powers of the Planck mass by means of a heat kernel regularization. Our results suggest that in the universe with a very small radius or with a very large curvature beyond a Planck scale expectation values of operators are reduced to calculations in a path integral representation of three-dimensional Einstein gravity. 1 Work supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture (No.30183817). 2 E-mail address: [email protected] 3 E-mail address: [email protected] 4 E-mail address: [email protected] As the evolution of the universe is followed backwards in time, we would expect quantum gravitational effects to become important. Nonperturbative effects would drastically change the notion of space-time beyond the Planck scale and invalidate perturbative calculations around a fixed background metric. Quantum geometrodynamics based on the Wheeler-DeWitt equation [1] is a natural framework which may answer the question of quantum space-time structure at the early stage of the universe. A lot of work has been done along this line and various interesting results have been obtained [1]-[5]. However, in contrast with its grand motivation, most of studies have been devoted on simple minisuperspace models in the semiclassical approximation. Short distance physics beyond the Planck scale is not still uncovered. In this paper, we propose a new approximation scheme to solve the Wheeler-DeWitt equation and discuss physical meanings of the wave function of the universe to leading order. Our results may suggest that beyond the Planck scale dimensional reduction of four-dimensional Einstein gravity occurs and a new phase appears. The (unregulated) Wheeler-DeWitt equation without matter is [ − mp Gijkl(x) δ δhkl(x) δ δhij(x) + mp 16π √