A semiclassical interpretation of the topological solutions for canonical quantum gravity

TL;DR

This paper explores the topological solutions in Ashtekar's formulation of canonical quantum gravity, linking points in the moduli space of flat connections to Lorentzian spacetime structures, highlighting both pathological and regular cases.

Contribution

It provides a semiclassical interpretation of topological solutions by relating moduli space points to Lorentzian geometries in canonical quantum gravity.

Findings

Most moduli space points lead to singular or causality-violating spacetimes.

A subspace of the moduli space corresponds to regular, well-behaved spacetimes.

Detailed analysis is performed for spacetimes homeomorphic to R×T^3.

Abstract

Ashtekar's formulation for canonical quantum gravity is known to possess the topological solutions which have their supports only on the moduli space $\CN$ of flat $SL(2,C)$ connections. We show that each point on the moduli space $\CN$ corresponds to a geometric structure, or more precisely the Lorentz group part of a family of Lorentzian structures, on the flat (3+1)-dimensional spacetime. A detailed analysis is given in the case where the spacetime is homeomorphic to $R\times T^{3}$. Most of the points on the moduli space $\CN$ yield pathological spacetimes which suffers from singularities on each spatial hypersurface or which violates the strong causality condition. There is, however, a subspace of $\CN$ on which each point corresponds to a family of regular spacetimes.