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On topology changes in quantum field theory and quantum gravity
General Relativity and Quantum Cosmology · Submitted 13 Nov 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Two singularity theorems can be proven if one attempts to let a Lorentzian cobordism interpolate between two topologically distinct manifolds. | background 0.74 | 0.52 | 0.08 | asserted 0.92 |
| s1 | On the other hand, Cartier and DeWitt-Morette have given a rigorous definition for quantum field theories (qfts) by means of path integrals. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s2 | This article uses their results to study whether qfts can be made compatible with topology changes. | contribution 0.80 | 0.56 | 0.05 | asserted 0.80 |
| s3 | 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. | result 0.58 | 0.89 | 0.09 | asserted 0.62 |
| s4 | We prove that already in the neighborhood of some cuspidal singularities, difficulties can arise to define certain qfts. | result 0.97 | 0.92 | 0.08 | asserted 1.00 |
| s5 | 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. | result 0.99 | 0.92 | 0.12 | asserted 1.00 |
| s6 | 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. | interpretation 0.89 | 0.51 | 0.18 | hedged possibility 0.44 |
| s7 | Unfortunately, it is difficult to describe the transition amplitudes consistently due to the aforementioned problems. | limitation 0.80 | 0.29 | 0.08 | asserted 0.92 |
| s8 | 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. | future work 0.63 | 0.52 | 0.59 | asserted 0.66 |
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The paper
Original 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.
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How the literature uses it
1 arXiv paper cite this in 1 sentence
What it cites · 23 sentences without a confident label
Contests 1 paper
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It is puzzling that the authors of [ref] do not come to this logical conclusion.
Twisted Poisson Structures and Non-commutative/non-associative Closed String Geometry · How the description of topology changes does not work · contest 0.84
Cites as background 3 papers
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Physicists also have tried to apply repeated applications of T duality transformations in various directions on backgrounds of the conformal field theory (CFT) that results from string theory [ref] .
Twisted Poisson Structures and Non-commutative/non-associative Closed String Geometry · How the description of topology changes does not work · 0.82
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More recent articles on that topic are the ones from Giulini [ref] and Anderson [ref] .
Spaces of Spaces · The path integrals that we investigate in this article · 0.99
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We mention these results and proofs here, even though they are known in the mathematics community, because cobordisms have recently attracted attention of the string theory community in form of a so-called cobordism hypothesis [ref] .
Cobordism Classes and the Swampland · Space-time topology · 0.92
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