Dimensional reduction and renormalizability of the Wheeler-De Witt equation : Next-leading-order contribution

The author solves the Wheeler-DeWitt equation for the wave function as an expansion in powers of the Planck mass by using heat kernel regularization. To solve the next-leading-order equation in this expansion, additional terms which are proportional to three-dimensional scalar curvature and Ricci tensor in heat equation are introduced. The result is an approximate wave function up to next-leading-order in this expansion. Expectation values computed with the leading-order approximation are reduced to the expectation value in three- dimensional Euclidean Einstein gravity theory in the region which is much smaller than the Planck scale. This means that the new phase (the dynamical system described by the three-dimensional quantum Einstein gravity) exists in the region beyond the Planck scale. The renormalization group equation for the wave function of the Wheeler-DeWitt equation is also discussed