Analog model for Euclidean wormholes: Bose-Einstein condensate with dirty surfaces

Gustavo O. Heymans; Isaque P. de Freitas; Nami F. Svaiter

REVIEW 4 major objections 4 minor 84 references

Analog model for Euclidean wormholes: Bose-Einstein condensate with dirty surfaces

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read In a Bose-Einstein condensate between two planar surfaces, random fields produced by the non-condensed atomic cloud generate non-local terms in the effective action that have the same mathematical structure as Euclidean wormhole…

desk verdict A physically motivated analog-gravity setup undermined by a sequence of load-bearing math errors, from the quenched free energy expansion to the final Casimir pressure. read the letter →

arxiv 2412.11204 v2 pith:VVOOM2VS submitted 2024-12-15 gr-qc cond-mat.dis-nn

Isaque P. de Freitas , Nami F. Svaiter , Gustavo O. Heymans This is my paper · ORCID

classification gr-qccond-mat.dis-nn

keywords Bose-EinsteincondensatequencheddisorderanalogmodelEuclideanwormholesCasimirpressuredistributionalzetafunctionnon-localeffectiveactionrandomsurfacefields

topics Quantum Gravity

open problems 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 establish that the thermal cloud of non-condensed atoms surrounding a Bose-Einstein condensate, modeled as quenched disorder concentrated on the condensate's planar surfaces, generates non-local terms in the condensate's effective action. Because those terms have the same bilinear, distance-independent form as the wormhole insertions of Euclidean quantum gravity, the authors propose the disordered condensate as a condensed-matter analog of Euclidean wormholes. They further compute the leading disorder-induced Casimir pressure between the two surfaces through zeta-function regularization and find that its sign alternates with the integer $l_0$ that labels the dominant moment of the partition function. If the construction holds, a tabletop ultracold-atom system would display a signature of spacetime-topology physics in a measurable force.

What carries the argument

The load-bearing object is the distributional zeta function $\Phi(s)=\int [dh][dh^*]P(h,h^*)Z(h,h^*)^{-s}$, whose derivative at $s=0$ gives the quenched free energy $E[\ln Z]$, together with the moment expansion $E[\ln Z]=\sum_{k=1}^\infty (-1)^{k+1}c^k/(k!\,k)E[Z^k]+\ln c+\gamma-R(c)$ (Eq. (21)). The method converts disorder averaging into replicated partition functions: the integer $k$ in the series becomes a number of field copies, and after diagonalizing the $k\times k$ matrix of quadratic fluctuations one copy carries the disorder kernel $F(x,z;x',z')$ while the others remain bare. Choosing the covariance $F=\delta^2(x-x')C(z-z')$ makes that kernel non-local in $z$, and the step-function choice $C(z-z')=b_1\theta(z-z')+b_2\theta(z'-z)$ is rewritten with fractional derivatives, producing the spectrum whose spectral zeta function $\zeta_O(s)$ yields the Casimir pressure.

What would settle it

Set the disorder to zero so that the partition function is deterministic, say $Z=1$, and evaluate both sides of Eq. (21). The left side is zero, while the right side forces $R(c)=E_1(c)+2\gamma+2\ln c$, which grows like $2\ln c$ for large $c$ and violates the paper's own bound $|R(c)|\le e^{-Z(0)c}/(cZ(0))$. This single check rules out the expansion as written; a numerical test on a non-trivial single-mode Gaussian disorder model would confirm whether a corrected moment-series identity can still produce the claimed non-local effective action.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that averaging the quenched free energy of a Bose-Einstein condensate with additive surface disorder produces an effective action containing the non-local term $-k\sigma^2 \int d^2x \int_0^L dz \int_0^L dz'\, \phi^*(x,z)C(z-z')\phi(x,z')$, with $C(z-z')$ encoding the disorder correlation along the confinement axis. This term has the structure of the wormhole insertion $\phi_i(x)C_{ij}(x,y)\phi_j(y)$ in the Euclidean quantum-gravity partition function, with $C$ playing the role of the wormhole kernel that does not depend on spacetime separation. Regularizing the resulting spectrum $q_x^2+q_z^2+k\sigma^2\epsilon|q_z|+m_0^2$ with the spectral zeta function, the paper obtains a leading Casimir pressure $P_c = \frac{(-1)^{l_0}}{16\, l_0\, l_0!\, L^4}\,[l_0^2(l_0+1)^2-\tfrac{1}{30}]$, whose sign is controlled by the integer $l_0$ selected by the correlation length. The paper concludes that the non-condensed cloud acts as the analog counterpart of Euclidean wormholes and can modify the usual attractive Casimir effect of the condensate.

Load-bearing premise

The argument stands on Eq. (21) in Sec. 3, an identity expressing the averaged logarithm of the partition function as a series of moments plus a remainder; the identity is asserted rather than proved, and every subsequent effective action, spectrum, and Casimir pressure inherits its validity.

Editorial extensions

If this is right

The non-condensed atomic cloud is not just a source of noise: after disorder averaging it induces non-local, distance-independent couplings inside the condensate, with the same mathematical form as Euclidean wormhole insertions.
The disorder-induced Casimir pressure between the confining surfaces can be attractive or repulsive depending on the integer $l_0$, which is fixed by the ratio of the correlation length to the plate separation; this contrasts with the purely attractive pressure of the ideal condensate slab.
The pressure scales as $1/L^4$, so the effect is strongest at small separations and can in principle be distinguished from the standard $1/d^4$ Casimir force by its $l_0$-dependent prefactor and alternating sign.
The replica structure behind the effective action means the quenched free energy is assembled from moments $E[Z^k]$, so the analog wormhole amplitude is governed by the same moment expansion used for the Casimir energy.

Reading between the lines

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

A natural extension, not stated in the paper, is that any confined quantum fluid with surface disorder could exhibit wormhole-like non-local terms, so the analog may be testable in other slab geometries such as superfluid helium films or polariton condensates.
A direct experimental check would measure the force between the two confining plates as a function of $L$; the predicted alternating sign would appear as changes in whether the plates are pulled together or pushed apart as $L$ crosses values corresponding to successive integers $l_0$.
Because the non-local kernel connects points at the same transverse coordinate $x$ but arbitrary $z$, the analog wormhole is effectively one-dimensional along the confinement axis; density-density correlations between the surface region and the bulk at fixed $x$ would carry the signature of the kernel $C(z-z')$.

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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

Read the BEC wormhole paper. The setup is genuinely interesting: modeling the non-condensed atomic cloud as quenched surface disorder, then showing that additive disorder generates a non-local bilinear effective action (Eq. 35) that resembles the wormhole insertion in Euclidean quantum gravity. That is a fair extension of your group's earlier program, and the contrast with multiplicative disorder (which gives quartic terms) is clearly explained. If the distributional zeta-function expansion in Eq. (21) were valid, the route to Eq. (35) would be plausible.

But Eq. (21) is not established, and it looks wrong. For a deterministic partition function Z=1, the quenched free energy is zero, yet the series plus log c + gamma minus R(c) cannot vanish with the stated bound on R(c). The identity is asserted without proof, and everything downstream depends on it. So the effective action itself rests on an unstable foundation.

Even granting Eq. (21), Section 6 breaks down. The step kernel C(z-z') does not diagonalize on sine modes: acting on sin(n pi z/L) gives a constant plus cos(n pi z/L), not a multiple of the original mode. Therefore the propagator in Eq. (41), with the |q_z| term, is not the Green's function of Eq. (37), and the zeta function built on that dispersion (Eq. 43) is not the spectrum of the model. Then Eqs. (50)-(52) are internally inconsistent: differentiating E_c as defined in Eq. (50) yields an exponential factor and no 1/2, while Eq. (51) has a 1/2 and no exponential. The reported P_c is not the derivative of the stated Casimir energy.

There is also a free-parameter problem: k, l0, and c are chosen by hand, and the regularization only works for a special set of k values imported from prior papers. With those choices the final pressure sign alternates with l0, but nothing anchors those choices to experiment or to a first-principles selection rule.

What is good: the physical motivation is solid, the literature on BEC cloud coupling and analog gravity is cited sensibly, and the qualitative idea—disorder averaging can mimic non-local wormhole effects—is worth exploring. But the quantitative claim, Eq. (52), is unsupported.

I would not publish this in its current form. The authors need to supply a correct derivation of the quenched free energy expansion and redo the spectral problem in the slab geometry, or state clearly that they are treating an infinite system with a regulator. The concept is non-trivial and the errors are specific and fixable in principle, so it deserves a serious referee rather than a desk reject. I would not cite the Casimir result in my own work until the derivation is corrected.

Recommendation: send to a referee with background in both analog gravity and disordered systems, and ask them to check Eq. (21) and the spectral problem in Section 6.

Referee Report

4 major / 4 minor

Summary. The manuscript studies a Bose-Einstein condensate interacting with a non-condensed thermal cloud, modeling the interaction as quenched disorder. It proposes a distributional zeta-function representation of the quenched free energy and derives effective actions for multiplicative and additive disorder. With the condensate confined between two planar surfaces, the paper claims that random surface fields generate non-local bilinear terms in the effective action and that these terms constitute an analog model of Euclidean wormholes. The final section computes a disorder-dependent Casimir pressure whose sign alternates with an integer parameter l0. The central claim is that the nonlocal effects in the condensed-matter system define a Euclidean-wormhole analog and that the leading Casimir pressure due to these effects is obtained.

Significance. If the derivation were sound, the paper would offer a creative condensed-matter analog in which quenched surface disorder produces a nonlocal effective action structurally similar to the wormhole bilinear in Euclidean quantum gravity, together with a sign-controllable Casimir pressure. The conceptual link between replica-style disorder averages and Euclidean wormholes is interesting and merits attention. However, the significance is conditional: the paper does not derive the nonlocal kernel from the microscopic BEC/cloud interaction, and the mathematical steps leading to the pressure contain load-bearing errors. The manuscript does not provide reproducible code, machine-checked proofs, or an experimental calibration, so its value rests entirely on the validity of the analytic derivation, which is currently not established.

major comments (4)

[Sec. 3, Eq. (21)] Equation (21) is asserted without proof and fails a simple consistency test. For a deterministic partition function Z=1, E[ln Z]=0 and E[Z^k]=1 for all k. The right-hand side of Eq. (21) is then S(c)+ln c+gamma-R(c), where S(c)=sum_{k>=1} (-1)^{k+1} c^k/(k! k). Since S(c)=gamma+ln c-Ei(-c), the equality forces R(c) to be approximately 2gamma+2ln c for large c, which grows logarithmically. This directly contradicts the stated bound |R(c)| <= e^{-Z(0)c}/(c Z(0)), which decays exponentially. Thus Eq. (21) cannot be a valid representation of the quenched free energy as stated. Because Eqs. (24), (25), (27), and (35) all use this expansion, the derivation of the nonlocal effective action is not grounded.
[Sec. 6, Eq. (41)] The Fourier-space propagator in Eq. (41) does not follow from the equation of motion (37) with the step kernel (38). On the finite interval [0,L] the kernel is not translation invariant, and Fourier transformation in q_z is not legitimate under Dirichlet boundary conditions. Acting on f(z)=sin(n pi z / L), the integral term gives [b1 - b2 (-1)^n]/(n pi/L) + (b2 - b1)/(n pi/L) cos(n pi z / L), which is not proportional to f(z). The sine modes therefore do not diagonalize the operator, and the dispersion relation q_x^2 + q_z^2 + k sigma^2 epsilon |q_z| + m_0^2 in Eq. (41) is not the spectrum of the model. Since Eqs. (43)-(46) build the spectral zeta function on this dispersion relation, the subsequent Casimir energy and pressure inherit the error. In addition, the sign of the nonlocal term in Eq. (40) appears opposite to that in Eq. (37) unless an unexplained sign convention is being used.
[Sec. 6, Eqs. (50)-(52)] Equation (51) is not the derivative of the Casimir energy defined in Eq. (50). Differentiating Ec = (-1)^{l0}/(l0 l0!) exp[l0 ln c - zeta_O(-1/2)] with respect to L produces a factor exp[l0 ln c - zeta_O(-1/2)] and no factor 1/2. Equation (51) contains a factor 1/2 and no exponential. If c is chosen to maximize the exponential, as stated in the text after Eq. (50), then c becomes L-dependent and its derivative must also be included. Consequently Eq. (52) is not the pressure obtained from the stated Casimir energy, and the sign-alternating prediction in Fig. 3 is unsupported.
[Sec. 5, Eqs. (27)-(35); Sec. 6, Eq. (38)] The nonlocal kernel C(z-z') is chosen by hand rather than derived from the physical model. The correlation function obtained from the surface-disorder assumption is F=delta^2(x-x')[delta(z)+delta(z-L)], which gives boundary-local terms. The wormhole-like bilinear action only appears after postulating F=delta^2(x-x') C(z-z') with an unspecified C, and the specific step kernel in Eq. (38) is introduced as the simplest form. The paper therefore does not demonstrate that random surface fields generate the nonlocal wormhole structure from the BEC/cloud dynamics; it inserts that structure as a modeling assumption. The subsequent regularization choices k = floor(2 m_0/(sigma^2 epsilon)) l and l0 = floor(sqrt(m_0^2 L^2 / pi)) are imported from Ref. [66] without derivation, so the final pressure depends on special parameter choices rather than on a systematic calculation.

minor comments (4)

[Sec. 6, Fig. 3] The figure labels curves by fixed values of l0 while L is varied, but l0 is defined as the integer part of sqrt(m_0^2 L^2 / pi), so l0 depends on L. Treating l0 as an independent curve label while plotting against L is inconsistent.
[Sec. 2, Eqs. (9)-(14)] The passage from Eq. (10) to Eq. (11) uses several unexplained numerical factors (for example 9/10 and 1/10) and redefinitions without showing the algebra; a reader cannot reproduce the effective Hamiltonian from the stated starting point.
[Sec. 4, Eq. (24)] The notation phi_j^{*2}(x,z) phi_i^2(x',z') is ambiguous: it is not clear whether these are complex squares, moduli squared, or replica indices, and the distinction matters for the meaning of the effective action.
[References] Several reference entries contain typographical artifacts, such as 'Word Scientific' in Ref. [83] and the spacing in 'Funda¸ c˜ ao'; these should be corrected in a final version.

Circularity Check

2 steps flagged · score 6.0 of 10

The wormhole analog and the Casimir-pressure prediction are partly circular: the nonlocal kernel is inserted by hand, and the regularized spectrum and k-set used for the pressure are imported from the authors' own Ref. [66] rather than derived from the slab equation of motion.

renaming known result [Sec. 5, Eq. (35) and surrounding paragraph]
"However, for an analog model of Euclidean wormholes, the contribution of F(x,z;x',z') is interesting when it is not δ-correlated along the z axis. Therefore, consider F(x,z;x',z') = δ²(x−x')C(z−z'), in which C(z−z') is expected to encode the non-locality. ... The above result shows similarity to the action for Euclidean quantum gravity (see Eq. (1)). In this comparison, C(z−z') would play the same role as C_ij(x,y), thus configuring the Bose-Einstein condensate interacting with the non-condensed atomic gas cloud as an analogous model for Euclidean wormholes."

The abstract's claim that 'random surface fields generate non-local terms' is not derived: the non-locality is put into the disorder covariance F by hand. The effective action S_σ then reproduces the wormhole bilinear by construction, because C(z−z') is chosen precisely to make the action 'similar' to Eq. (1). The later Casimir calculation inherits this hand-picked C, so the analog-model conclusion is equivalent to the ansatz rather than a first-principles result.
uniqueness imported from authors [Sec. 6, Eqs. (40)-(46)]
"Proceeding in an analogous way to Ref. [66], we can obtain the Fourier representation of the two-point correlation function of the k-th equation of motion, Eq. (37), as: G^(k)_0(q_x,q_z)=1/(q_x^2+q_z^2+kσ²ε|q_z|+m0²) ... As explicitly obtained in Ref. [66], only a set of k's can be regularized; such a set is given by k=⌊2m0/(σ²ε)⌋ l, l∈N ... therefore, the main contribution to the Casimir energy will be related to ζ_O(s)=... (46)."

The propagator (41), the special allowed k-values, and the l0 choice that turn Eq. (45) into the Hurwitz-zeta expression (46) are imported from Ref. [66], a prior paper by the same authors, rather than derived from the equation of motion (37) for this slab. Because the zeta-regularized Casimir energy (47)-(49) and the final pressure (52) are built entirely on that imported spectrum and k-set, the central 'prediction' is controlled by a self-cited construction. No independent derivation or external check is supplied for the imported regularization.

full rationale

The derivation chain is not self-contained. The nonlocal wormhole-like action is obtained by selecting a nonlocal covariance F=δ²C, so the advertised 'generation' of nonlocal terms is an input ansatz; the final Casimir pressure then depends on that arbitrary C and on the k/l0 restrictions borrowed from the authors' prior Ref. [66]. This makes the central claim partially circular. I do not count the unproved Eq. (21) expansion or the apparent derivative inconsistency in Eq. (51) as circularity: Eq. (21) is a questionable mathematical identity, and Eq. (51) not following from Eq. (50) is an internal-error/correctness issue, not a reduction of output to input. If the only concerns were errors, the score would be low; the circularity score is raised by the hand-inserted kernel and the self-citation load-bearing regularization.

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The central result depends on a set of hand-chosen parameters (sigma, epsilon, c, l0, k, m0) and on modeling assumptions that are either borrowed from the authors' prior work or introduced ad hoc. The nonlocal kernel that produces the wormhole-like action is an invented mathematical structure with no independent experimental support. The final Casimir pressure is therefore a function of arbitrary choices rather than a parameter-free prediction.

free parameters (6)

sigma (disorder strength) = not specified
Enters the effective action in Eq. (27) and the spectrum in Eq. (41); no microscopic value is derived from the BEC or gas-cloud parameters.
epsilon = |b2 - b1| = not specified
Difference between the constants in the step-function kernel C(z-z'), Eq. (38); controls the |qz| term in Eq. (41).
c = not specified
Parameter in the quenched free energy expansion, Eq. (21); taken 'large enough' or chosen to maximize the exponential in Eq. (50), but never fixed.
l0 = floor(sqrt(m0^2 L^2 / pi))
Integer selected in Sec. 6 as the dominating moment; it fixes the sign and magnitude of the Casimir pressure, Eq. (52).
k = floor(2 m0 / (sigma^2 epsilon)) * l, l in N
Replica index and k-th field; only this set is regularizable according to Ref. [66], and it is not derived from the BEC parameters.
m0 = positive constant
Constant effective mass m0^2 = V_trap - mu, assumed in Sec. 6 to reduce the equation of motion to the Fourier form in Eq. (41).

assumptions (5)

ad hoc to paper The distributional zeta expansion Eq. (21) is a valid identity for E[ln Z].
Used in Secs. 4, 5, and 6; no proof is given, and the identity appears inconsistent for a deterministic partition function.
domain assumption The disorder correlation has the form delta^2(x - x') C(z - z').
Eq. (23) and Sec. 5; the non-condensate cloud is concentrated on the surfaces and delta-correlated in the planar directions.
domain assumption The Hartree-Fock-Bogoliubov-Popov approximation discards m(r) and h(r).
Sec. 4; used to reduce the condensate action to the multiplicative-disorder form in Eq. (22).
ad hoc to paper The step-function kernel C(z-z') is equivalent to fractional derivatives D^1_0 and D^1_L.
Eqs. (38)-(40); the replacement is formal, and the resulting Fourier symbol |qz| is not derived for the finite slab with Dirichlet boundaries.
ad hoc to paper Only the k set and l = l0 choices are regularizable in the spectral zeta function.
Sec. 6, text after Eq. (45); this restriction is borrowed from Ref. [66] and not derived in the present system.

invented entities (1)

Non-local surface disorder kernel C(z-z') = b1 theta(z-z') + b2 theta(z'-z)
purpose: Inserts the non-local bilinear term into the effective action and is the piece interpreted as the Euclidean wormhole analog.
Introduced by hand in Eq. (38); its step-function form is not derived from the microscopic gas-condensate interaction, and it carries no falsifiable prediction outside the paper.

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Cite this review

Pith. "Pith review of Analog model for Euclidean wormholes: Bose-Einstein condensate with dirty surfaces." pith.science (2026). https://pith.science/paper/VVOOM2VS

@misc{pith2026241211204,
  author       = {Pith},
  title        = {Pith review of: Analog model for Euclidean wormholes: Bose-Einstein condensate with dirty surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VVOOM2VS}},
  note         = {Machine review of arXiv:2412.11204}
}

read the original abstract

We study a Bose-Einstein condensate under the effects of the non-condensate atomic cloud. We model the resulting linear interaction of the condensate with the atomic gas as a quenched disorder. Using the distributional zeta function method, we obtain a representation for the quenched free energy as a series of integral moments of the partition function. Assuming that the Bose-Einstein condensate is confined between two planar surfaces, we show that random surface fields generate non-local terms in the effective action. The non-local effects in this condensed matter system define an analog model of a Euclidean wormhole. The leading contribution of the non-local interactions to the Casimir pressure is obtained.

Figures

Figures reproduced from arXiv: 2412.11204 by the authors.

**Figure 2.** Illustrative diagram of non-local connections in the disordered Bose-Einstein condensate. [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Plot of the Casimir pressure Pc, Eq. (52), for l0 = 1, 2, 3, 4. 7. Conclusions Many approaches to quantum gravity discuss modifications of standard physics at short distances. However, it has become increasingly clear that short-distance modifications alone cannot address many fundamental problems in quantum gravity. One must understand long-distance physics and even topology change. Instead of discussing analog mo… view at source ↗

Discussion (0). Continue with ORCID to comment.

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