Summary
We solve 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, we introduce additional terms which are proportional to three-dimensional scalar curvature and Ricci tensor in the heat equation. We have 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. We also discuss the renormalization group equation for the wave function of the Wheeler-DeWitt equation.
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Horiguchi, T. Dimensional reduction and renormalizability of the Wheeler-DeWitt equation: next-leading-order contribution. Nuov Cim B 111, 165–191 (1996). https://doi.org/10.1007/BF02724644
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DOI: https://doi.org/10.1007/BF02724644