Towards Canonical Quantum Gravity for Geometries Admitting Maximally Symmetric Two-dimensional Surfaces

TL;DR

This paper develops a canonical quantization framework for geometries with maximally symmetric two-dimensional surfaces, providing an explicit solution to the Wheeler-deWitt equation based on a re-normalized manifold.

Contribution

It introduces a specific re-normalization assumption and requirement enabling the canonical quantization of such geometries, with an analytical solution to the Wheeler-deWitt equation.

Findings

Analytical solutions to the Wheeler-deWitt equation are obtained.

A new re-normalization approach simplifies the quantization process.

The solution space is characterized by three scalar functionals.

Abstract

The 3+1 (canonical) decomposition of all geometries admitting two-dimensional space-like surfaces is exhibited. A proposal consisting of a specific re-normalization {\bf Assumption} and an accompanying {\bf Requirement} is put forward, which enables the canonical quantization of these geometries. The resulting Wheeler-deWitt equation is based on a re-normalized manifold parameterized by three smooth scalar functionals. The entire space of solutions to this equation is analytically given, exploiting the freedom left by the imposition of the {\bf Requirement} and contained in the third functional.