On topology changes in quantum field theory and quantum gravity

TL;DR

This paper investigates the compatibility of quantum field theories with topology changes in spacetime, highlighting mathematical challenges and proposing stochastic and Regge calculus approaches for quantum gravity.

Contribution

It demonstrates the difficulties in defining quantum field theories across topology changes and suggests stochastic differential equations and Regge calculus as potential solutions.

Findings

Path integrals require finite norms, problematic for degenerate metrics.

QFTs can be defined around conical singularities from topology change.

Topology changes during universe expansion may induce frequent quantum gravity transitions.

Abstract

Two singularity theorems can be proven if one attempts to let a Lorentzian cobordism interpolate between two topologically distinct manifolds. On the other hand, Cartier and DeWitt-Morette have given a rigorous definition for quantum field theories (qfts) by means of path integrals. This article uses their results to study whether qfts can be made compatible with topology changes. We show that path integrals over metrics need a finite norm for the latter and for degenerate metrics, this problem can sometimes be resolved with tetrads. We prove that already in the neighborhood of some cuspidal singularities, difficulties can arise to define certain qfts. On the other hand, we show that simple qfts can be defined around conical singularities that result from a topology change in a simple setup. We argue that the ground state of many theories of quantum gravity will imply a small cosmological constant and, during the expansion of the universe, will cause frequent topology changes. Unfortunately, it is difficult to describe the transition amplitudes consistently due to the aforementioned problems. We argue that one needs to describe qfts by stochastic differential equations, and in the case of gravity, by Regge calculus in order to resolve this problem.