Exact solutions of the Wheeler-DeWitt equation and the Yamabe construction

TL;DR

This paper presents exact solutions to the Wheeler-DeWitt equation in four-dimensional Lorentzian gravity, characterized by constant scalar curvature 3-metrics, and discusses their properties and invariance features.

Contribution

It provides the first explicit exact solutions of the Wheeler-DeWitt equation with support on constant scalar curvature 3-metrics, linking them to the Yamabe construction.

Findings

Solutions are Gaussian with minimum uncertainty.

Solutions recover 3D diffeomorphism and gauge invariance when the regulator is removed.

Support on 3-metrics of constant scalar curvature.

Abstract

Exact solutions of the Wheeler-DeWitt equation of the full theory of four dimensional gravity of Lorentzian signature are obtained. They are characterized by Schr\"odinger wavefunctionals having support on 3-metrics of constant spatial scalar curvature, and thus contain two full physical field degrees of freedom in accordance with the Yamabe construction. These solutions are moreover Gaussians of minimum uncertainty and they are naturally associated with a rigged Hilbert space. In addition, in the limit the regulator is removed, exact 3-dimensional diffeomorphism and local gauge invariance of the solutions are recovered.