On quantum creation of a toroidal universe

Alan H. Guth; Alexander Vilenkin

On quantum creation of a toroidal universe

T0 review · 0 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A toroidal universe cannot be quantum-created from nothing in semiclassical gravity: the instantons are singular, so earlier probability comparisons with spherical universes are unfounded.

desk verdict Guth and Vilenkin convincingly refute earlier claims that toroidal universes can nucleate from nothing, but the core conclusion is conditional on the tunneling boundary condition's regularity postulate, which the paper states clearly. read the letter →

arxiv 2508.08747 v2 pith:CG66G34S submitted 2025-08-12 gr-qc astro-ph.COhep-th

Alan H. Guth , Alexander Vilenkin This is my paper · ORCID

classification gr-qcastro-ph.COhep-th PACS 04.60.-m98.80.Qc

keywords quantumcreationoftheuniversetunnelingboundaryconditiontoroidaltopologyCasimirenergysingularinstantonsgeodesicincompletenessWheeler-DeWittequationsemiclassicalgravity

topics Quantum Gravity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to settle whether a universe shaped like a three-dimensional torus can be quantum-created from nothing, a possibility earlier work claimed was far more likely than the creation of a spherical universe. The authors argue it cannot, at least within semiclassical quantum gravity: the Euclidean instantons describing a toroidal universe are singular, the zero-size boundary cannot be a slice of a regular four-geometry, and the total Casimir energy diverges as the torus shrinks. They also show that the vacuum-energy-only toroidal spacetime is geodesically past-incomplete and classically begins at a singularity, with no unique quantum state for matter fields. The conclusion is that previously computed tunneling amplitudes for toroidal universes do not describe creation from nothing, and comparing their probability with spherical creation is not justified.

What carries the argument

The load-bearing object is the tunneling boundary condition on the Wheeler-DeWitt equation: the rule that the universe's wave function may receive probability flux only through vanishing 3-geometries that are slices of a regular four-geometry, like the three-sphere slices near the pole of a four-sphere. The paper applies this regularity test to toroidal universes and finds that a torus of zero size fails it, while the Casimir-energy instanton is singular at $a = 0$ and therefore fails to be a stationary point of the Euclidean action. This same criterion is what makes the spherical no-boundary calculation trustworthy, and what makes the toroidal one fall outside the semiclassical quantum-creation framework.

What would settle it

Construct a regular Euclidean instanton with toroidal topology—a smooth compact four-geometry whose shrinking slices are tori, with finite action and finite matter energy as the torus shrinks to zero—which would show that toroidal creation can be described semiclassically after all; alternatively, a numerical integration showing that a non-comoving past-directed timelike geodesic in the identified flat-sliced de Sitter torus has infinite proper time would overturn the incompleteness claim.

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Extended reading notes

Core claim

The paper's central claim is that the WKB tunneling amplitudes for toroidal universes computed by earlier authors are not amplitudes for creation from nothing, because the instantons they rely on are singular. In the vacuum-plus-Casimir model, the scale factor behaves as $a \propto \tau^{1/2}$ near Euclidean time $\tau = 0$, producing a curvature singularity and a total matter energy $E = -C/a$ that diverges to $-\infty$; in the vacuum-only case, the flat-sliced de Sitter covering space is only half of de Sitter space, so after toroidal identifications past-directed non-comoving geodesics still reach $t = -\infty$ in finite proper time. The wave function near $a \to 0$ does not satisfy the tunneling boundary condition, which admits probability flux only through vanishing 3-geometries that are slices of a regular four-geometry. As a result, the creation of a toroidal universe is either impossible or depends essentially on Planck-scale physics, and no semiclassical prediction follows about whether toroidal universes are more probable than spherical ones.

Load-bearing premise

Everything rests on defining 'nothing' by the tunneling boundary condition that admits only nonsingular vanishing 3-geometries; allow singular or Planck-scale origins and the toroidal creation picture could be reinstated, but then it would depend on unknown Planck-scale physics.

Editorial extensions

If this is right

The WKB tunneling probabilities for toroidal universes in the earlier literature do not represent creation from nothing, so they should not be used in probability comparisons with spherical universes.
A toroidal vacuum-energy universe classically starts at a singularity and is geodesically past-incomplete, so it has no unique set of initial conditions.
At the perturbative level, the Wheeler-DeWitt equation plus the regularity condition leaves the quantum state of matter fields on a toroidal universe largely undetermined, in contrast with the spherical case.
Including Casimir energy creates a classically forbidden region and a finite WKB tunneling action, but the resulting instanton is singular and has infinite total energy at $a=0$, so it cannot be treated as a valid stationary point.
If the conclusion is right, the question of whether the universe began as a torus cannot be answered by semiclassical quantum cosmology and is currently beyond the reach of testable theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: the same regularity test would apply to other compact spatial topologies whose zero-size slices are not slices of a regular four-geometry, so their semiclassical creation probabilities would be unquantified by the same argument.
Beyond the paper: if toroidal topology were ever inferred observationally, this analysis implies the initial quantum state of matter fields would have to be fixed by physics outside the Wheeler-DeWitt equation, since semiclassical boundary conditions do not determine it.
Beyond the paper: a consistent Planck-scale completion that predicts toroidal creation would have to produce a nonsingular instanton with finite energy at zero size, and candidates that do not would be excluded by this argument.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does something useful: it takes the old Zeldovich-Starobinsky, Coule-Martin, and Linde claims about toroidal universe creation seriously and shows they do not survive contact with the tunneling boundary condition as usually defined. The geodesic incompleteness argument is elementary but correct - flat-sliced de Sitter covers only half the manifold, and periodic identifications do not change the affine parameter time for non-comoving geodesics. The WDW analysis adds a second, independent leg: the wave function (65) with alpha=0 is a legitimate solution with outgoing flux, but it enters superspace through a singular 3-geometry, which the tunneling condition excludes. And the perturbative superspace argument that the scalar field state is left almost entirely unconstrained in the vacuum-energy case is a real point, not a rhetorical one. The Casimir case is handled honestly: the instanton is singular, the energy diverges as a goes to 0, and the singularity means the Euclidean action is not stationary. The paper also flags its own simplifications, like dropping the trace anomaly, and the numerical coefficient C is the one free parameter, taken from the earlier literature. No fitted parameters, no circularity: the BGV theorem is invoked but also re-derived directly for this spacetime, and Ref. [42] stands on its own as a falsifiable counterexample to singular instantons generally. The soft spot is the load-bearing premise, and the stress-test note is right about it. The argument that a toroidal universe cannot be created from nothing rests on the requirement that probability flux enter superspace only through a slice of a regular four-geometry. If you relax that requirement, the vacuum-energy wave function is a perfectly good semiclassical solution with flux emerging from a singular boundary. The authors' counter - that such a description is non-predictive because the quantum state is unconstrained - is a powerful practical objection, but it is not a mathematical no-go. So the abstract's phrasing 'cannot be described in the context of semiclassical quantum gravity' is one step stronger than what is demonstrated. The demonstration is 'cannot be described under the tunneling boundary condition as defined, and without that condition the description loses predictivity.' This is a qualifier, not a fatal flaw - the paper is explicit about the boundary condition in Sec. 2 and returns to it throughout - but it should be carried into the abstract. This is a careful, well-written contribution by people who know the framework from the inside. The refutation of the earlier probability comparisons is solid under the stated assumptions, and the paper deserves a serious referee. My only recommendation is to soften the abstract slightly and let the conditionality show. I would cite it and bring it to reading group.

Referee Report

0 major / 4 minor

Summary. This paper re-examines the proposal of Zeldovich, Starobinsky, Coule, Martin, and Linde that a toroidal (T^3) universe can be quantum-created from nothing, and argues the contrary. The authors first review the spherical de Sitter tunneling calculation in Sec. 2, establishing the tunneling boundary condition and its regularity postulate. They then show that a vacuum-energy-dominated toroidal spacetime is past-incomplete (Sec. 3.1), that the corresponding minisuperspace wavefunction (Eq. 65) does not satisfy the tunneling boundary condition because the a=0 boundary is not a nonsingular slice of a regular 4-geometry (Sec. 3.2), that in the perturbative superspace the scalar-field quantum state is non-unique (Sec. 3.3), and that the Casimir-energy instanton is singular with divergent total energy (Sec. 4). The paper concludes that quantum creation of a toroidal universe from nothing cannot be described in semiclassical quantum gravity, and that previous probability comparisons with spherical universes are unfounded.

Significance. If the conclusions are correct, this paper removes a widely cited counterexample to the view that only spherical, positively curved universes nucleate semiclassically, and it reframes the ZS/Coule-Martin/Linde amplitudes as attempts to describe a classical singularity rather than quantum creation. The paper's strengths include a direct covering-space geodesic-incompleteness argument that is independent of the authors' own BGV theorem, explicit closed-form WDW solutions, and a clear demonstration of the loss of predictivity when singular origins are admitted. There are no fitted parameters and no circularity: the direct geodesic argument reproduces the BGV result for this case, and the paper's conclusions are falsifiable in the sense that a regular toroidal instanton would invalidate the argument. The main caveat is that the no-go is conditional on the regularity postulate of the tunneling boundary condition; the body of the paper states this conditionality explicitly, so it does not undermine the core argument, but the abstract should make the qualifier visible.

minor comments (4)

[Abstract and Sec. 5] The sentence in the abstract and in Sec. 5 that quantum creation of a toroidal universe “cannot be described in the context of semiclassical quantum gravity” is one step stronger than the demonstration, which is conditional on the regularity postulate of the tunneling boundary condition adopted in Sec. 2. As the stress-test discussion notes, without that postulate the pure-vacuum solution (65) with α=0 is a legitimate WDW solution, finite at a=0 and outgoing at infinity; what fails is the requirement that probability flux enter superspace only through nonsingular slices. The body already says this, but the abstract should carry the qualifier, for example by stating that the impossibility holds within the tunneling boundary condition framework adopted here and that alternative descriptions would depend on Planck-scale physics.
[Sec. 3.2] The claim that a vanishing 3-torus “cannot be obtained as a slice of a regular 4-geometry” is load-bearing but is asserted rather than proved. A brief justification would help, e.g., by noting that in the flat slicing of de Sitter the limit t→−∞ sends all points to the same boundary point, so the T^3 identification degenerates and does not extend through a regular de Sitter boundary. This would make the contrast with the smooth pole slicing of the four-sphere in footnote 3 fully explicit.
[Sec. 4.2 and Sec. 3.3] The discontinuity between the C=0 case (non-unique B_n) and the C→0+ limit (which selects B_n=0) is emphasized as a symptom of treating a singular instanton as legitimate. The argument is suggestive; since the formalisms differ because the classically forbidden region disappears exactly at C=0, I suggest adding one sentence noting that the discontinuity is in the boundary-condition structure and not by itself a derivation of invalidity, to preempt a potential criticism.
[Eq. (74) and surrounding text] The definitions of N_b and N_f are compressed; spelling out that N_1 counts vector fields with two transverse polarizations and that N_{1/2} counts chiral spinor fields would make the Casimir coefficient formula easier to follow.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivations are self-contained, and the author self-citations are corroborated by direct arguments in the paper.

full rationale

The paper's central claims are derived from inputs that are not equivalent to the conclusions. Geodesic incompleteness is established both by a direct covering-space argument (the flat slicing covers only half of de Sitter space, so past-directed non-comoving geodesics reach t=-infinity in finite proper time, per Hawking & Ellis) and by the BGV theorem; the self-cited BGV result is therefore not load-bearing because the paper reproduces the argument independently. The conclusion that the toroidal wave function (65) does not satisfy the tunneling boundary condition is an application of the boundary condition explicitly postulated in Sec. 2, not a definitional renaming: the regularity requirement is stated independently as 'incoming waves are allowed only on the part of the boundary corresponding to a vanishing 3-geometry which is nonsingular, in the sense that it can be described as a slice through a nonsingular four-geometry,' and then applied to the a=0 torus. The non-uniqueness of the quantum state in Secs. 3.3 and 4.2 is proven from the first-order ODE (36) with arbitrary initial data at a=a1, so it does not rest on the authors' prior uniqueness theorems for the spherical case; those theorems are used only for contrast. The singular-instanton argument cites Ref. [42], but the paper also gives the direct reason: the curvature singularity at a=0 means the equations of motion are not satisfied there. No parameters are fitted, and no 'prediction' is a renamed input. The abstract's wording is stronger than the demonstration in that the conclusion depends on the adopted tunneling boundary condition, but the paper states this condition explicitly as a postulate; conditional dependence on an explicit assumption is not circularity.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper's central claim rests chiefly on the definition of the tunneling boundary condition and on the semiclassical instanton framework. The only numerical input from outside is the Casimir coefficient C, which is a microphysical parameter taken from prior calculations. No new entities are introduced.

free parameters (1)

Casimir coefficient C = C = (Nb - Nf) C0 with C0 = 0.8375 for a real scalar; C1 = 2 C0, C1/2 = -C0 for chiral fermion; analytic formula gives…
Input from prior Casimir calculations, not fitted here. It controls the Casimir tunneling action in Sec. 4; the central claim requires only C > 0 or specific signs per field content.

assumptions (6)

domain assumption The tunneling boundary condition of Sec. 2 is the correct formalization of quantum creation from nothing.
Enters at Sec. 3.2 where the a=0 torus is rejected as a singular boundary; the paper's conclusion depends on this definition.
domain assumption The renormalized stress tensor of quantum fields in a toroidal FRW spacetime is given by the Casimir term (73); trace anomaly and other derivative terms are neglected.
Stated explicitly in Sec. 4: 'we will follow ZS and include only the Casimir contribution.' Affects the Casimir instanton analysis.
standard math The WKB approximation is valid for the Wheeler-DeWitt equation in the regimes considered.
Used throughout; justified for H_v << 1.
domain assumption Singular instantons are not valid stationary points of the Euclidean action (Ref. [42]).
Cited rather than re-derived; used to reject the Casimir tunneling amplitude in Sec. 4.1.
domain assumption The regularity condition Im S_n > 0 correctly represents the boundary condition that the wave function does not grow at large field values.
Used in the perturbative superspace analysis to show non-uniqueness of the scalar quantum state.
standard math Borde-Guth-Vilenkin theorem: spacetimes with past-averaged positive expansion are past-incomplete.
Used as a check in Sec. 3.1; the direct covering-space argument is independent.

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Pith. "Pith review of On quantum creation of a toroidal universe." pith.science (2026). https://pith.science/paper/CG66G34S

@misc{pith2026250808747,
  author       = {Pith},
  title        = {Pith review of: On quantum creation of a toroidal universe},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CG66G34S}},
  note         = {Machine review of arXiv:2508.08747}
}

read the original abstract

We consider the quantum creation of a universe with flat spatial sections and the topology of a 3-torus, taking into account the effect of Casimir energy. We show that the corresponding instantons are singular. Since these instantons describe universes originating in a state of infinite energy, we argue that they cannot be interpreted as quantum creation from `nothing'. If quantum corrections to the energy-momentum tensor are neglected, the spacetime of the toroidal universe reduces to de Sitter space with appropriate periodic identifications. Contrary to previous claims in the literature, this spacetime is geodesically incomplete. We argue that this spacetime describes a classical universe originating at a singularity, and not a quantum origin. We conclude that the quantum creation of a toroidal universe from nothing cannot be described in the context of semiclassical quantum gravity -- it is either impossible, or it depends essentially on Planck-scale physics. We therefore see no reasonable way to compare the probability of creation of a toroidal universe, if it is possible at all, with that of a spherical universe.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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