Nonperturbative solutions for canonical quantum gravity: an overview

TL;DR

This paper surveys known solutions to the Wheeler-DeWitt equation in canonical quantum gravity, focusing on Wilson-loop and topological solutions, and discusses their interpretations and mathematical structures.

Contribution

It provides an overview of existing solutions in Ashtekar's formulation, highlighting the current status and potential developments in Wilson-loop and topological approaches.

Findings

Topological solutions can be interpreted in terms of spacetime geometry.

Explicit formulas for measures on spin network states are provided.

Wilson-loop solutions are still in early development stages.

Abstract

In this paper we will make a survey of solutions to the Wheeler-Dewitt equation which have been found up to now in Ashtekar's formulation for canonical quantum gravity. Roughly speaking they are classified into two categories, namely, Wilson-loop solutions and topological solutions. While the program of finding solutions which are composed of Wilson loops is still in its infancy, it is expected to be developed in the near future. Topological solutions are the only solutions at present which we can give their interpretation in terms of spacetime geometry. While the analysis made here is formal in the sense that we do not deal with rigorously regularized constraint equations, these topological solutions are expected to exist even in the fully regularized theory and they are considered to yield vacuum states of quantum gravity. We also make an attempt to review the spin network states as intuitively as possible. In particular, the explicit formulae for two kinds of measures on the space of spin network states are given.