Spacetime foam: a review

Probably not a substantial claim: may not pass S3 (load-bearing).

Questions about spacetime foam, including whether it exists, currently have answers that depend on one's approach to quantum gravity.

“Note also that for now, questions about spacetime foam, including the most basic onewhether it exists-will have answers that depend on one's approach to quantum gravity.”

Probably not a substantial claim: may not pass S3 (load-bearing).

It is not known which choice about constraints in the path integral is correct.

“For now, we do not know which is the correct choice.”

Probably not a substantial claim: may not pass S3 (load-bearing).

There is no consensus over whether fluctuations of topology are possible.

“There is also no consensus over whether fluctuations of topology are possible.”

May combine two claims.

If classical field equations are imposed, spatial topology change requires violation of a positive energy condition, and local regions of nontrivial topology evolve to singularities.

“If one additionally imposes the classical field equations, spatial topology change requires a violation of a positive energy condition [18], and local regions of nontrivial topology evolve to singularities [19,20].”

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No approach to quantum gravity has yet given a complete theory.

“We have a wide range of research programs aimed at quantizing gravity [8,9], and although some are more popular than others, it's fair to say that none has yet given us a complete theory.”

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May combine two claims.

Equating the energy density of a graviton of wavelength L with the effective energy density of a small metric fluctuation δg yields δg ∼ lp/L, confirming that metric fluctuations become order one at the Planck scale.

“Equating these two expressions, we obtain δg ∼ ℓ p L , (1.4) confirming that metric fluctuations become order one at the Planck scale.”

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Classified as probably not a substantial claim.

Wheeler conjectured that quantum fluctuations of the metric are of order one at the Planck scale, producing large fluctuations in spacetime geometry and topology, which he called spacetime foam.

“More than 65 years ago, John Wheeler suggested that quantum uncertainties of the metric would be of order one at the Planck scale, leading to large fluctuations in spacetime geometry and topology, which he termed”

• Also stated in: § 1 Wheeler's foam · plain

  “He argued that quantum fluctuations of the metric should be of order one at the Planck scale, leading to wild and rapidly varying fluctuations in spacetime geometry and topology, which he called”

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Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

There are loopholes to the conclusion that quantum fluctuations become large at the Planck scale; for instance, if gravity is asymptotically free, interactions may become weak near the Planck scale.

“There are, of course, loopholes: if gravity is asymptotically free, for instance, interactions may become weak near the Planck scale.”

May combine two claims.

The condition of vanishing extrinsic curvature is strong but can be understood as part of the requirement for an extremal action.

“The condition of vanishing extrinsic curvature is rather strong, but can be understood as part of the requirement for an extremal action [90].”

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Probably not a substantial claim: may not pass S3 (load-bearing).

The sphere S^4 has its usual constant curvature metric, which is the Einstein metric with the largest normalized volume.

“The sphere S 4 has its usual constant curvature metric, which can be shown to be the Einstein metric with the largest normalized volume ṽ [60].”

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May combine two claims.

The Hitchin-Thorpe inequality is necessary but not sufficient; stronger limits have been found.

“This is a necessary condition, not a sufficient one; stronger limits have been found by a number of authors [56][57][58][59].”

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In four dimensions, exotic smooth structures are not uncommon: even R^4 has infinitely many, as does R × Σ^3 for any compact three-manifold Σ^3.

“In four dimensions,”

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Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

For Λ < 0, there are an infinite number of compact hyperbolic three-manifolds, whose properties have been extensively, although not completely, characterized.

“For Λ < 0, there are an infinite number of compact hyperbolic three-manifolds, whose properties have been extensively, although not completely, characterized [103,105,106].”

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For the four-dimensional case, exact results are not available, but there are good estimates and bounds; results are qualitatively similar to 2D, though more sensitive to smearing.

“For the more realistic four-dimensional case, exact results are not available, but there are now good estimates and bounds [117][118][119].The results are qualitatively very similar, although more sensitive to the choice of smearing function.”

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May combine two claims.

The precise values of the correlators depend on the choice of smearing, but some key qualitative features are largely independent of w.

“The precise values of correlators will depend on the choice of smearing, but some key qualitative features are largely independent of w.”

May combine two claims.

Until recently, most work on passive fluctuations focused on small fluctuations treatable at low order in perturbation theory; such fluctuations are not really spacetime foam.

“Until recently, though, most of this work focused on small fluctuations that could be treated at low order in perturbation theory.Such fluctuations can have interesting physical effects-see, for instance, [115]-but they are not really spacetime foam.”

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For any integer k one can find a four-manifold with k smooth structures that admit different Einstein metrics and infinitely many other smooth structures that admit no Einstein metric.

“In fact, for any integer k one can find a four-manifold with k smooth structures that admit (different) Einstein metrics and infinitely many other smooth structures that admit no Einstein metric [39].”

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Simply connected manifolds are under better control, providing a practical impetus to focus on such topologies.

“Simply connected manifolds are under better control [35], providing a practical impetus to focus on such topologies [12].”

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CP^2 admits an Einstein metric, as do connected sums CP^2 # k CP^2 with 3 ≤ k ≤ 8.

“The complex projective plane CP 2 admit an Einstein metric, as do connected sums CP 2 # kCP 2 with 3 ≤ k ≤ 8 [61].”

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Einstein manifolds in three dimensions necessarily have constant curvature and can always be represented as quotients M = M~/Γ, where M~ is S^3, R^3, or H^3 and Γ is a discrete group of isometries.

“Einstein manifolds in three dimensions are further restricted: they necessarily have constant curvature (that is, they satisfy (2.9)), and can always be represented as quotients”

Probably not a substantial claim: may not pass S3 (load-bearing).

We will sometimes find sums for which no obvious regularization exists.

“But we will sometimes find sums for which no obvious regularization exists.”

May combine two claims.

In the 4D case, there is still a strict lower bound and an infinite positive tail, which decays slower than exponentially, typically as e^{-x^{1/3}}.

“In particular, there is still a strict lower bound and an infinite positive tail, which now decays slower than exponentially, typically as e -x 1/3 .”

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May combine two claims.

For a conformal field theory in two spacetime dimensions, the probability distribution for quantum fluctuations of smeared energy density around its vacuum value can be calculated exactly and is largely independent of the smearing function.

“For a conformal field theory in two spacetime dimensions, the answer can be calculated exactly [116], and is largely independent of the smearing function [122].”

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May combine two claims.

If stable causality is required, local fluctuations in topology are forbidden, although a fixed spatial topology may be very complex.

“In particular, if one requires stable causality-a slightly stronger version of the condition that there be no closed timelike curves-then local fluctuations in topology are forbidden [93], although a fixed spatial topology may be very complex [94].”

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Probably not a substantial claim: may not pass S3 (load-bearing).

Different approaches to quantum gravity suggest different possibilities for allowed topologies.

“Not surprisingly, different approaches to quantum gravity suggest different possibilities for allowed topologies.”

May combine two claims.

Three-manifolds are classifiable, although this does not necessarily imply that product manifolds R × Σ^3 are.

“Note that three-manifolds are classifiable [36], although this does not necessarily imply that product manifolds R × Σ 3 are.”

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For Λ = 0, there are a total of ten flat closed three-manifolds.

“For Λ = 0, there are a total of ten flat closed three-manifolds; the three-torus is the simplest, but the sides of a cube can also be identified with various twists [101,103,104].”

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It is not known how much of the space of initial data is covered by the Chrusciel-Isenberg-Pollack construction.

“It is not known how much of the space of initial data is covered by this Chrusciel-Isenberg-Pollack construction.”

May combine two claims.

Although Birkhoff's theorem implies a spherically symmetric vacuum spacetime must be locally isometric to some region of Schwarzschild, local patches can be sewn together to form spacetimes that look drastically different.

“But while Birkhoff's theorem implies that a spherically symmetric vacuum spacetime must be locally isometric to some region of Schwarzschild, local patches can be sewn together to form spacetimes that look drastically different [84].”

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For Λ < 0, some known Einstein manifolds look more like Wheeler's foam; Anderson's Dehn surgery construction yields manifolds with nontrivial geometry in discrete bubbles near the tips of would-be cusps, but these regions are far apart rather than at Planck density.

“For the case Λ < 0, some of the known Einstein manifolds look more like Wheeler's foam.Anderson's Dehn surgery construction [53,66], for instance, yields manifolds in which the nontrivial geometry manifests itself in a discrete set of”

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Probably not a substantial claim: may not pass S3 (load-bearing).

A compact orientable two-manifold has topology determined by its genus g.

“A compact, orientable two-manifold-that is, a closed surface-has a topology determined by its genus g, the”

Probably not a substantial claim: may not pass S3 (load-bearing).

Morrow-Jones, Witt, and Schleich have carried out detailed studies of locally spherically symmetric initial data.

“Morrow-Jones, Witt, and Schleich have carried out detailed studies of locally spherically symmetric initial data [82][83][84].”

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May combine two claims.

The expectation value ⟨Tμν⟩ of the stress-energy diverges and must be renormalized.

“The expectation value ⟨T µν ⟩ of the stress-energy diverges, and must be renormalized.There are a number of ways of doing this; see, for example, [120].”

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May combine two claims.

Vacuum fluctuations frequently carry negative energy, so the positive energy condition obstruction to topology change may not be a problem for the quantum theory.

“As we will see in §3, though, vacuum fluctuations frequently carry negative energy, so this may not be a problem for the quantum theory.”

May combine two claims.

If all topologies are allowed, four-manifolds are not classifiable: there is no algorithm to determine whether two arbitrary four-manifolds are homeomorphic.

“If we choose to allow all topologies, an added complication appears: four-manifolds are not classifiable, in the sense that there is no algorithm that can determine whether two arbitrary four-manifolds are homeomorphic [32,33].”

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Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

At the topological level, all three-manifolds have a decomposition as connected sums, unique for orientable manifolds and almost unique for nonorientable ones.

“At the topological level, all three-manifolds have a decomposition as connected sums, unique for orientable manifolds and almost unique for nonorientable ones [87,88].”

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Passive fluctuations have been studied for decades, notably through stochastic gravity.

“Passive fluctuations have been studied for decades [113], notably through a model known as stochastic gravity [114].”

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May combine two claims.

Quantum fluctuations of the metric come in two forms: active fluctuations from the quantum nature of the metric itself, and passive fluctuations, the metric's response to quantum fluctuations of the stress-energy tensor.

“Broadly speaking, quantum fluctuations of the metric come in two forms [112]:”

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Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

On occasion, sums over geometries or topologies diverge, and one must make sense of this behavior.

“On occasion, we will find that certain sums over geometries or topologies diverge, and we will have to make sense of this behavior.”

Probably not a substantial claim: may not pass S3 (load-bearing).

There are phenomena such as the Casimir effect in which an appropriately regulated sum of vacuum fluctuations gives an experimentally verifiable prediction.

“Indeed, there are phenomena such as the Casimir effect in which an appropriately regulated sum of vacuum fluctuations gives an experimentally verifiable prediction [111].”

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Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing). May combine two claims. The quote only partly states this claim.

It is important not to overinterpret divergences; a divergent sum might indicate summing too many terms, but it could also be that adding more geometries or topologies would give a physically sensible answer.

“But it could be that we were summing too few terms, and that the addition of more geometries or topologies might give us a physically sensible answer.”

Probably not a substantial claim: may not pass S1 (asserted), S3 (load-bearing).

For a boundary with a single component, the density of topologies ρ(ṽ,h) has sharp peaks at certain symmetric values of the boundary metric.

“The”

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Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

For Einstein manifolds with boundary, only a few examples are known for Λ ≥ 0, but large classes are known for Λ < 0.

“As in the compact case, only a few examples of Einstein manifolds with boundary are known for Λ ≥ 0, but large classes are known for Λ < 0.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

There is some evidence that perturbative quantum gravity is asymptotically safe and that corrections to the action beyond quadratic order become unimportant at very short distances, but this is far from proven.

“There is some evidence that perturbative quantum gravity is”

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Myers' theorem implies that a complete compact Einstein manifold with Λ > 0 necessarily has a finite fundamental group, so there are no Euclidean wormholes with positive cosmological constant.

“Notably, Myers' theorem implies that a complete compact Einstein manifold with Λ > 0 necessarily has a finite fundamental group: there are no”

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For Λ < 0, there are infinitely many hyperbolic four-manifolds with constant negative curvature; any such manifold is a quotient H^4/π1 and the constant negative curvature metric is its unique Einstein metric.

“Any such manifold is completely determined by its fundamental group π 1 ; it is, in fact, a quotient H 4 /π 1 of hyperbolic space, and the constant negative curvature metric is its unique Einstein metric [63].”

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The coefficients α and β of the quadratic curvature terms are expected to be of order lp^2, and the corrections are likely negligible at ordinary scales but may become important near the Planck scale.

“The coefficients α and β are expected to be of order ℓ p 2 , and the corrections are likely to be negligible at ordinary scales, but they may become important near the Planck scale.”

• AssumptionRests on an assumption: The coefficients α and β of the quadratic curvature corrections are of order ℓp^2.

Probably not a substantial claim: may not pass S3 (load-bearing).

The quadratic action also has other extrema whose role is not well understood.

“But (2.4) also has other extrema, whose role is not well understood.”

May combine two claims.

Not every four-manifold admits an Einstein metric with any value of Λ; T^4 # T^4 and S^1 × S^3 have no Einstein metrics.

“Note first that not every four-manifold admits an Einstein metric at all, with any value of Λ.Indeed, even such simple manifolds as T 4 #T 4 , the connected sum of two tori, * and S 1 ×S 3 have no Einstein metrics [52].”

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There is a strict lower bound to the energy of a vacuum fluctuation, determined by quantum energy inequalities, with numerical value determined by the smearing function.

“2. There is, however, a strict lower bound to the energy of a vacuum fluctuation, determined by quantum energy inequalities [123], with a numerical value determined by the smearing function.”

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The topology of three-manifolds is completely classified, thanks to Perelman's proof of Thurston's geometrization conjecture.

“The topology of three-manifolds is now completely classified, thanks to Perelman's proof of Thurston's geometrization conjecture [36].”

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Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

A constant curvature metric on a closed surface necessarily has positive curvature on a sphere (g=0), vanishing curvature on a torus (g=1), and negative curvature on a surface of genus g>1.

“A constant curvature metric necessarily has positive curvature on a sphere (g = 0), vanishing curvature on a torus (g = 1), and negative curvature on a surface of genus g > 1.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Many routine divergences can be eliminated with zeta function regularization or Borel summation.

“Of course, many routine divergences can be eliminated with, for example, zeta function regularization [109] or Borel summation [110].”

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Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The question of what manifolds to allow in a description of spacetime foam is complicated, and there is no consensus or even understanding of how a decision might be reached.

“The question of what manifolds to allow in a description of spacetime foam is thus complicated, and we seem rather far from a consensus, or even an understanding of how a decision might be reached.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

At the geometrical level, the structure of the necks connecting components in the Chrusciel-Isenberg-Pollack construction is rather special.

“At the geometrical level, though, the structure of the”

May combine two claims.

For Λ ≥ 0, known Einstein metrics have complicated topologies that may contribute to the path integral comparably to the simplest S^4 topology, but viewing them as Planck-scale fluctuations is not obvious; by Myers' theorem there are no wormhole-like contributions.

“For Λ ≥ 0, the known examples of Einstein metrics fit one aspect of Wheeler's definition of spacetime foam: they have complicated topologies that may contribute to the path integral in an amount comparable to the simplest S 4 topology.But viewing these as Planck scale fluctuations is not so obvious.The topological structure seems more global; in particular, by Myers' theorem, there are no wormhole”

May combine two claims.

Specifying the trace K of extrinsic curvature and conformal class h_ij at the boundary may tame the divergence because the number of Einstein manifolds with fixed (K,h) need not grow superexponentially even though compact Einstein manifolds do.

“This may tame the divergence-while the number of compact Einstein manifolds grows superexponentially with ṽ, this need not imply that the number of Einstein manifolds with fixed (K, h) does.”

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Quantum gravity has no fixed background structure that can be used to measure the passage of time, and finding a general enough relational time is not easy.

“Quantum gravity has no fixed background structure that can be used to measure the passage of time, and finding a general enough “relational” time is not easy.”

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Scattering off quantum foam is likely to be energy dependent, plausibly modifying dispersion relations for light.

“For instance, scattering off quantum foam is likely to be energy dependent, plausibly modifying dispersion relations for light.”

May combine two claims.

In three dimensions, the Einstein-Hilbert action can be rewritten as a Chern-Simons action, allowing the path integral often to be computed exactly.

“Moreover, the Einstein-Hilbert action can be rewritten as a Chern-Simons action [168][169][170], a form in which the path integral (4.2) can often be computed exactly.”

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In a D-brane foam model, recoil off solitons induces an energy-dependent vacuum refractive index that might be astrophysically observable.

“They find that recoil off such solitons induces an energy dependent vacuum refractive index, which might be astrophysically observable.”

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Classified as probably not a substantial claim.

It remains possible that the cumulative effect of many Planck scale fluctuations might be observable.

“It remains possible, though, that the cumulative effect of many Planck scale fluctuations might be observable.”

Probably not a substantial claim: may not pass S1 (asserted), S3 (load-bearing).

Quantum gravitational fluctuations at the Planck scale might smear the light cone, potentially eliminating the source of divergences.

“It was noted long ago [236][237][238] that quantum gravitational fluctuations at the Planck scale might “smear” the light cone, potentially eliminating the source of divergences.”

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May combine two claims.

This decoherence can pick out a preferred direction of time, leading to CPT violation.

“This decoherence can pick out a preferred direction of time, leading to CPT violation.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Friedman et al. concluded that fermions would acquire Planck-scale masses unless either the particles had a finite size larger than typical quantum fluctuations or the path integral included a sum over spin structures and an arbitrary restriction to only CP-reversing handles.

“Friedman et al. analyzed fermion propagation on manifolds with nonorientable handles [205], and concluded that fermions would acquire Planck-scale masses unless either the particles had a finite size (larger than that of typical quantum fluctuations) or the path integral included a sum over spin structures and an arbitrary restriction to only CP-reversing handles.”

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May combine two claims.

In 2+1 dimensions, the partition function for an arbitrary number of wormholes was computed, and the sum over wormholes was shown to diverge and not even be Borel summable.

“In [174], the partition function for an arbitrary number of wormholes was computed, in the saddle point approximation for Λ > 0 and with one loop corrections included for Λ < 0. The sum over wormholes was shown to diverge, and to not even be Borel summable.”

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The stress-energy fluctuations can be treated as a stochastic source term for the Einstein equations, a procedure shown to be a particular next order extension of semiclassical gravity.

“We can treat these as a stochastic source term for the Einstein equations, a procedure that has been shown to be a particular next order extension of semiclassical gravity [114].”

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  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

The two-dimensional results should probably not be taken too literally because they neglect correlations between successive fluctuations and do not account for the quantum nature of the gravitational field.

“These results should probably not be taken too literally: they neglect the correlations between successive fluctuations, and do not account for the quantum nature of the gravitational field itself.”

May combine two claims.

In causal dynamical triangulations, a time slicing fixes spatial topology, prohibits topology change, and provides an unambiguous direction of time.

“In causal dynamical triangulations [13], a time slicing is chosen in a way that fixes the spatial topology, prohibiting topology change and at the same time providing an unambiguous direction of time that makes the spacetime more “Lorentzian.””

• cites [13]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The distribution of fluctuations obtained from (3.2) depends on a smearing length; this might most naturally be taken to be the Planck length, but one can explore the effects of changing the scale.

“Of course, the distribution of fluctuations obtained from (3.2) depends on a smearing length; this might most naturally taken to be the Planck length, but one can explore the effects of changing the scale.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Virtual handles and similar nontrivial fluctuations of topology can lead to effective nonlocal interactions, which can perhaps act as a thermal bath.

“Moreover, virtual handles and similar nontrivial fluctuations of topology can lead to effective nonlocal interactions, which can perhaps act as a thermal bath [201,222].”

• cites [201]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [222]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Classified as probably not a substantial claim.

The real motivation for quantum foam comes from quantum gravity.

“While vacuum fluctuations of matter certainly affect the small scale structure of spacetime, the real motivation for quantum foam comes from quantum gravity.”

May combine two claims.

Such a loss of coherence would lead to the disappearance of diffraction patterns-Airy rings-from extragalactic sources, as well as angular broadening and blurring of distant point sources.

“Such a loss of coherence would lead to the disappearance of diffraction patterns-Airy rings-from extragalactic sources, as well as angular broadening and blurring of distant point sources.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

A similar dimensional reduction occurs in a number of other approaches to quantum gravity; it is not known whether the phenomenon is generically connected to spacetime foam.

“A similar “dimensional reduction” occurs in a number of other approaches to quantum gravity [150]; it is not known whether the phenomenon is generically connected to spacetime foam”

• cites [150]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S4 (based).

Components of tensors depend on a choice of basis, which can vary from point to point.

“-Components of tensors depend on a choice of basis, which can vary from point to point.”

Probably not a substantial claim: may not pass S3 (load-bearing).

These averaging problems are clearly relevant to understanding macroscopic effects of spacetime foam.

“But they are clearly relevant to the understanding of macroscopic effects of spacetime foam-both for averaging over geometries and, for a fixed inhomogeneous geometry, for comparing different locations and times.”

May combine two claims.

Classically, the initial data of §2.2 leads to singularities: the necks of connected sums typically contain marginally outer trapped surfaces, essentially black hole horizons, and standard singularity theorems apply.

“Classically, the initial data of §2.2 leads to singularities: the “necks” of the connected sums typically contain marginally outer trapped surfaces, essentially black hole horizons, and the standard singularity theorems apply [161].”

• cites [161]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Spacetime foam could provide a preferred reference frame, allowing for a violation of Lorentz invariance.

“As discussed in §5.1, spacetime foam could provide a preferred reference frame, allowing for a violation of Lorentz invariance.”

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

It is likely that something like the Wheeler-DeWitt equation will hold in any eventual quantum theory of gravity.

“something like the Wheeler-DeWitt equation [129] is likely to hold”

• cites [129]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

Thus most fluctuations are negative, but there is a fat tail of very high positive energy fluctuations.

“Thus most fluctuations are negative, but there is a”

May combine two claims.

In a matrix model inspired by causal dynamical triangulations, transition amplitudes between spatial slices can be obtained; at weak coupling slices have fixed S^2 topology, while at strong coupling they disintegrate into a foam of multiple components connected by wormholes.

“In a somewhat different approach, a matrix model inspired by causal dynamical triangulations ( §4.2) can be used to obtain transition amplitudes between spatial slices [176].While the slices have a fixed S 2 topology at weak coupling, in the strong coupling regime they “disintegrate” into a foam of multiple components connected by wormholes.”

• cites [176]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Most physicists currently working on quantum gravity expect black hole formation and evaporation to be ultimately unitary, but the question is still hotly debated.

“I think it is fair to say that most physicists currently working on quantum gravity expect that the process is ultimately unitary, but the question is still hotly debated [218,219].”

• cites [218]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [219]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

Fields in a foamy spacetime may behave as open systems, with Planck scale fluctuations acting as an environment, leading to loss of coherence if the fields are observed and the spacetime is not.

“Fields in a foamy spacetime may behave as open systems, with the Planck scale fluctuations of the spacetime acting as an environment; if the fields are observed and the spacetime is not, the result can be a loss of coherence [221].”

• cites [221]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

In causal dynamical triangulations, while topology fluctuations are excluded, large fluctuations of geometry can occur.

“But while fluctuations of topology are excluded, large fluctuations of geometry can certainly occur.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

A number of relatively ad hoc smearings have been investigated and do indeed cut off divergences; Ohanian argued this behavior is independent of the details of the smearing.

“A number of relatively ad hoc smearings have investigated [239][240][241][242], and do indeed cut off divergences; Ohanian has argued that this behavior is independent of the details of the smearing [243].”

• cites [239]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [240]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [241]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [242]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [243]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

Causal dynamical triangulations eliminate quantum fluctuations in topology, although the initial spatial slice may have arbitrarily complicated topology.

“This eliminates one version of spacetime foam, the conjectured quantum fluctuations in topology, although the initial spatial slice may have an arbitrarily complicated topology.”

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing), S4 (based).

It seems likely that some echo of the Wheeler-DeWitt equation will remain in whatever the eventual final theory of quantum gravity is.

“But it seems likely that whatever the eventual final theory of quantum gravity, some echo of this equation will remain.”

Probably not a substantial claim: may not pass S4 (based).

If the standard picture of spacetime foam is right, quantum fluctuations of spacetime only become large near the Planck scale.

“If the standard picture of spacetime foam is right, quantum fluctuations of spacetime only become large near the Planck scale.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Wheeler suggested that it is essential to allow for fluctuations in the metric and gravitational interactions in any proper treatment of the compensation problem.

“In his first proposal of spacetime foam, Wheeler suggested that . . . it is essential to allow for fluctuations in the metric and gravitational interactions in any proper treatment of the compensation problem-the problem of compensation of “infinite” energies that is so central to the physics of fields and particles [3].”

• cites [3]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

For light with wavelength λ, one might expect random phase fluctuations on the order δφ ∼ 2πδl/λ ∼ 2π λ l^{1-α} l_p^α.

“Similarly, for light with wavelength λ one might expect random phase fluctuations on the order [210,211] δϕ ∼ 2πδℓ/λ ∼ 2π λ ℓ 1-α ℓ α p .(5.3)”

• cites [210]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [211]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

Different approaches to quantum gravity may give very different answers to questions about spacetime foam.

“For now, though, different approaches to quantum gravity may give very different answers to questions about spacetime foam.”

May combine two claims.

In the midisuperspace model, the Wheeler-DeWitt equation can be solved in a WKB approximation, and formally time-independent stationary states exist and frequently give high probabilities for foamy geometries.

“The Wheeler-DeWitt equation can be solved in a WKB approximation, and one can look for formally time-independent stationary states.Such states exist, and they frequently give high probabilities for “foamy” geometries.”

• cites [79]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

Both the definition of a time slice and the geometry of geodesic balls depend on the metric, making it difficult to compare results among different manifolds or locations.

“But both the definition of the slice (say as a slice of constant mean curvature) and the geometry of the geodesic balls will depend on the metric, making it difficult to compare results among different manifolds, or even different locations in the same manifold.”

May combine two claims.

Results so far indicate that foamy structures with rapidly varying geometries occur with fairly high probability in the midisuperspace model, though questions about the proper probability measure remain.

“The results so far indicate that foamy structures with rapidly varying geometries occur with fairly high probability, though there remain some unsettled questions about the proper probability measure.”

• cites [79]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [80]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

In that same early attempt, the foamy structure induced Planck-size λφ^4 interactions and possibly Planck-scale tachyonic masses for scalar fields.

“but induced Planck-size λφ 4 interactions, and possibly Planck-scale tachyonic masses, for scalar fields.”

• cites [200]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [204]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Hawking argued that virtual black holes could form coherently and then evaporate thermally, leading to a failure of unitarity in quantum field theory.

“The idea that spacetime foam could lead to a failure of unitarity in quantum field theory dates back to work by Hawking [216,217], who argued that virtual black holes could form coherently and then evaporate thermally.”

• cites [216]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [217]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

With the paper's choice of renormalization, the vacuum expectation value of the stress-energy tensor is zero (up to possible higher order curvature corrections), but the fluctuations can be very large.

“With our choice of renormalization, the vacuum expectation value of the stress-energy tensor is zero (up to the possible higher order curvature corrections discussed briefly in §2), but the fluctuations can be very large.”

Classified as probably not a substantial claim.

It is possible that a foamy structure could influence the propagation of other fields.

“But spacetime is also the arena in which other fields propagate, and it is possible that a foamy structure could influence that propagation.”

Probably not a substantial claim: may not pass S3 (load-bearing).

It is not at all clear that canonical quantization allows topology change.

“Finally, though, even if canonical quantization allows complicated spatial geometry and topology, it is not at all clear that it allows topology change.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Our most energetic probes have wavelengths far larger than the Planck length and thus cannot resolve individual Planck scale events.

“Even our most energetic probes have wavelengths far larger than ℓ p , and thus cannot resolve individual Planck scale events.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

In an inhomogeneous spacetime, the foam provides a background structure.

“This is not necessarily a problem, since in an inhomogeneous spacetime the “foam” provides a background structure.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

For Λ>0, the set of Einstein metrics is even more poorly understood, and much less can be said.

“For Λ > 0, the set of Einstein metrics is even more poorly understood, and much less can be said.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The construction of an inner product on the space of solutions of the Wheeler-DeWitt equation is a deep and largely unsolved problem.

“the construction of an inner product on the space of solutions is a deep and largely unsolved problem [154].”

• cites [154]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S4 (based).

Any observations we can make will probe a region much larger than Planck size.

“In practice, any observations we can make will probe a region much larger than Planck size.”

May combine two claims.

The three-dimensional Boulatov group field theory can be made Borel summable, but it is not currently known whether this extends to higher dimensions.

“The three-dimensional version due to Boulatov can be made Borel summable [179], but it is not currently known whether this extends to higher dimensions.”

• cites [179]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

It is plausible that some form of holographic bulk-boundary duality will survive in any eventual quantum theory of gravity.

“it is plausible that some form of the “holographic” bulk-boundary duality [130] will survive.”

• cites [130]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

For three-dimensional lens spaces, the exact path integral was computed and the sum fails to converge even with zeta function regularization, though it might change if more topologies are incorporated.

“Building on [171], the exact path integral was computed in [173] for one infinite class of such manifolds, the lens spaces.The sum fails to converge, even with zeta function regularization, although again this might change if more topologies are incorporated.”

• cites [173]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing). The quote only partly states this claim.

The answer to whether a typical physically sensible wave function gives significant probabilities for foamy three-geometries at the Planck scale is plausibly yes.

“We do not have an answer to any of these questions, but the answer to the first is plausibly “yes.””

Probably not a substantial claim: may not pass S3 (load-bearing).

Crane and Smolin argued that a gas of virtual black holes alters the spectrum of perturbations in a way that imposes a high energy cutoff on the density of states.

“In a more explicitly “foamy” construction, Crane and Smolin have considered the effect of a gas of virtual black holes on field propagation [198,247], arguing that their presence alters the spectrum of perturbations in a way that imposes a high energy cutoff on the density of states.”

• cites [198]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [247]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

Coleman argued that spacetime foam can change the status of Λ as a fixed parameter.

“Coleman has argued that spacetime foam can change this [67,225].”

• cites [67]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [225]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

The observed absence of such effects has been used to rule out certain phenomenological models of spacetime foam, although other models suggest the effects may still be unobservably small.

“The observed absence of such effects has been used to rule out certain phenomenological models of spacetime foam [210,[265][266][267][268], although other models suggest that the effects may still be unobservably small [269].”

• cites [210,[265][266][267][268]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [269]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

If axions exist, their shift symmetry can be broken by spacetime fluctuations, in particular by wormholes.

“If axions exist, their shift symmetry can be broken by spacetime fluctuations, in particular by wormholes [280].”

• cites [280]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

If photons can lose energy through interactions with spacetime foam, one effect should be a broadening of spectral lines.

“If photons can lose energy through interactions with spacetime foam, one effect should be a broadening of spectral lines.”

Classified as probably not a substantial claim. May combine two claims.

All forms of energy should include the energy of vacuum fluctuations, which should manifest as a cosmological constant Λ.

“The problem is that “all forms of energy” should include the energy of vacuum fluctuations, which should manifest as a cosmological constant Λ.”

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

Spacetime foam could take many different forms.

“As we have seen, spacetime foam could take many different forms.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In a later paper, Unruh and Wang suggest that a large negative bare cosmological constant could combine with a positive contribution from spacetime foam to produce a microcyclic structure to the Universe, with Planck sized regions independently undergoing oscillations but with a net overall expansion.

“In a later paper [298], they suggest that a large negative bare cosmological constant could combine with a positive contribution from spacetime foam to produce a “microcyclic” structure to the Universe, with Planck sized regions independently undergoing oscillations but with a net overall expansion.”

• cites [298]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Vacuum fluctuations of bosonic and fermionic fields contribute to the cosmological constant with opposite signs, and quantum field theoretical considerations do not tell us whether Λ should be positive or negative.

“Vacuum fluctuations of bosonic and fermionic fields contribute to the cosmological constant with opposite signs, and quantum field theoretical considerations do not tell us whether Λ should be positive or negative.”

Probably not a substantial claim: may not pass S1 (asserted), S3 (load-bearing).

It has been suggested that this mechanism could explain the absence of a large negative cosmological constant.

“It has been suggested that this mechanism could explain the absence of a large negative cosmological constant [51].”

• cites [51]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

That assumption is consistent with present knowledge of quantum field theory, but it requires extrapolations to scales far beyond what can be probed by current observations.

“While this is consistent with our present knowledge of quantum field theory, it requires extrapolations to scales far beyond what can be probed by current observations.”

May combine two claims.