Generalized Sums over Histories for Quantum Gravity I. Smooth Conifolds

TL;DR

This paper introduces conifolds as a broader class of topological spaces for Euclidean quantum gravity path integrals, enabling inclusion of nonmanifold stationary points and convergence of Einstein sequences, thus enriching the study of topological effects.

Contribution

It generalizes the set of histories in quantum gravity path integrals to include conifolds, allowing for a more comprehensive semiclassical analysis.

Findings

Sequences of Einstein manifolds and conifolds converge to Einstein conifolds.

Conifold histories can produce semiclassical amplitudes approaching stationary points.

Sum over conifold histories offers a new approach to topological effects in quantum gravity.

Abstract

This paper proposes to generalize the histories included in Euclidean functional integrals from manifolds to a more general set of compact topological spaces. This new set of spaces, called conifolds, includes nonmanifold stationary points that arise naturally in a semiclasssical evaluation of such integrals; additionally, it can be proven that sequences of approximately Einstein manifolds and sequences of approximately Einstein conifolds both converge to Einstein conifolds. Consequently, generalized Euclidean functional integrals based on these conifold histories yield semiclassical amplitudes for sequences of both manifold and conifold histories that approach a stationary point of the Einstein action. Therefore sums over conifold histories provide a useful and self-consistent starting point for further study of topological effects in quantum gravity. Postscript figures available via anonymous ftp at black-hole.physics.ubc.ca (137.82.43.40) in file gen1.ps.