Generalized Sums over Histories for Quantum Gravity II. Simplicial Conifolds

TL;DR

This paper explores the implementation of sums over conifolds in four-dimensional Euclidean quantum gravity, demonstrating that 4-conifolds are algorithmically decidable and providing algorithms for summing over them within Regge calculus.

Contribution

It introduces a concrete method for summing over 4-conifolds in quantum gravity, addressing the algorithmic decidability issue in four dimensions.

Findings

4-conifolds are algorithmically decidable in four dimensions

Presented explicit algorithms for summing over 4-conifolds

Provides a foundation for concrete Euclidean sums over histories in 4D quantum gravity

Abstract

This paper examines the issues involved with concretely implementing a sum over conifolds in the formulation of Euclidean sums over histories for gravity. The first step in precisely formulating any sum over topological spaces is that one must have an algorithmically implementable method of generating a list of all spaces in the set to be summed over. This requirement causes well known problems in the formulation of sums over manifolds in four or more dimensions; there is no algorithmic method of determining whether or not a topological space is an n-manifold in five or more dimensions and the issue of whether or not such an algorithm exists is open in four. However, as this paper shows, conifolds are algorithmically decidable in four dimensions. Thus the set of 4-conifolds provides a starting point for a concrete implementation of Euclidean sums over histories in four dimensions. Explicit algorithms for summing over various sets of 4-conifolds are presented in the context of Regge calculus. Postscript figures available via anonymous ftp at black-hole.physics.ubc.ca (137.82.43.40) in file gen2.ps.