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

TL;DR

This paper solves the Wheeler-DeWitt equation for four-dimensional Einstein gravity using heat kernel regularization, revealing a dimensional reduction to three-dimensional gravity at scales beyond the Planck length.

Contribution

It introduces a novel expansion method for the Wheeler-DeWitt equation and demonstrates dimensional reduction at sub-Planckian scales.

Findings

Operators reduce to three-dimensional gravity calculations at small scales.

The approach uses heat kernel regularization for solving the Wheeler-DeWitt equation.

Results suggest a natural dimensional reduction beyond the Planck scale.

Abstract

We solve the Wheeler-DeWitt equation for {\it 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 {\it three}-dimensional Einstein gravity.