REVIEW 3 major objections 6 minor 18 references
Signature change as phase transition in holography
T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read The paper argues that entangled holographic states stay classically connected through Euclidean spacetime regions when wormhole bridges would be unstable.
desk verdict A likeable essay with a broken central step: the claimed necessity of Euclidean regions rests on confusing thermodynamic instability with nonexistence of the Lorentzian saddle. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the bulk two-point function expressed as a path sum over continuous curves, $\langle\Psi|O(x)O(y)|\Psi\rangle = \int_{\gamma \subset M} [D\gamma]\, e^{i m l[\gamma(x,y)]}$, which turns the existence of boundary entanglement into a statement about geometric connectivity. The second ingredient is the glued spacetime $\mathcal{M} = \mathcal{M}_- \cup \mathcal{M}_L \cup \mathcal{M}_+$: a Euclidean saddle, a Lorentzian middle slice, and its time reverse, glued along common surfaces. When the Lorentzian slice shrinks away, the geometry is Euclidean AdS with period $\beta$, and this is the configuration that computes the low-temperature correlator. The sum over geodesic paths, with winding number $k$, is what exposes the Euclidean bridge: the closed form of the correlator contains terms $\cos[(t_2-t_1)+i\beta(k-\tfrac12)]-\cos(\phi_2-\phi_1)$, whose imaginary time separations are the signature of curves probing Euclidean regions.
What would settle it
A direct large-$N$ computation of the two-point function below the transition that keeps the unstable two-sided black hole in the path integral would settle the question: if that saddle alone gives the nonzero correlator, Euclidean regions are not required. Conversely, a full computation of the mutual information between the two boundary theories that finds it vanishing below the transition would contradict the proposed Euclidean connectivity.
Extended reading notes
Core claim
The central claim is that the dual of an entangled state is a classically connected geometry, and that when the entangled structure cannot be supported by a stable two-sided black hole, the connecting curves must pass through a Euclidean-signature region. Concretely, the author considers the thermal field double state $|\Psi_\beta\rangle = \sum_n e^{-\beta E_n/2} Z^{-1/2} |E_n\rangle_1 \otimes |E_n\rangle_2$ and the bulk two-point function $\langle\Psi_\beta|O(t_1,\phi_1)O(t_2,\phi_2)|\Psi_\beta\rangle$. The correlator is nonzero for boundary points on the two causally independent conformal boundaries, so by the curve-sum interpretation there must be continuous bulk paths joining them. At high temperature these paths traverse the two-sided black hole; at low temperature, where that black hole is unstable, the computation instead uses a geometry built from Euclidean saddles glued to an interpolating Lorentzian piece, $\mathcal{M} = \mathcal{M}_- \cup \mathcal{M}_L \cup \mathcal{M}_+$, and the connecting geodesics run through the Euclidean part. The conclusion is that the dual of the low-temperature thermal field double state contains an Euclidean region whose role is to preserve exactly the connectivity that entanglement requires.
Load-bearing premise
The argument assumes that below the thermal phase-transition temperature the two-sided black hole stops being the correct bulk description of the entangled thermal state; if that unstable black hole still contributes to the actual quantum state, the Lorentzian wormhole connection might survive without any Euclidean region.
Editorial extensions
If this is right
- Below the AdS thermal phase-transition temperature, the thermal field double state still has a classically connected bulk dual; the connecting curves pass through a Euclidean region instead of a two-sided black hole.
- The correspondence between entanglement and bulk connectivity acquires two geometric phases: a Lorentzian wormhole phase at high temperature and a Euclidean-bridge phase at low temperature.
- Thermal correlators in the low-temperature entangled state are explicitly computable by cutting the Euclidean circle, inserting real-time intervals, and summing over winding geodesics; the shortest Euclidean path dominates.
- If the Lorentzian segment shrinks completely, the construction reduces to the standard Euclidean thermal AdS, reproducing previously known thermal correlators.
Reading between the lines
- The paper does not say so explicitly, but the same argument would predict that mutual information between the two boundaries stays nonzero at arbitrarily low temperature, since classical connectivity via Euclidean regions should maintain some amount of boundary correlation; a direct large-$N$ mutual-information calculation below the transition would test this.
- An extension the author leaves implicit is that any entangled pure state with a well-defined Euclidean saddle construction, not just the thermal field double, would develop Euclidean regions whenever no stable Lorentzian horizon exists, making instanton-like geometries a generic symptom of entanglement rather than a thermal artifact.
- One could probe the Euclidean bridge in the dual picture by scattering bulk probes between the two boundaries: geodesic lengths would acquire imaginary components characteristic of Euclidean traversal, showing up as phases in two-point functions at low temperature.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper argues that in holography, entangled states whose Lorentzian Einstein-Rosen bridges are thermodynamically unstable must instead be dual to geometries containing Euclidean-signature regions. The argument uses the thermofield double state (4.1), the geodesic approximation to boundary correlators (2.1)-(2.2), and a Hartle-Hawking wavefunction decomposition M = M- ∪ ML ∪ M+ to claim that below the Hawking-Page temperature nonzero two-sided correlators force curves connecting the two boundaries to traverse Euclidean regions. The paper presents this as an extension of ER-EPR to regimes where wormholes are unstable.
Significance. If the central claim were correct, it would generalize ER-EPR beyond stable Lorentzian wormholes and give a phase-transition interpretation of signature change in holography. The paper draws on well-established real-time AdS/CFT tools, and the framing using geodesic probes of emergent geometry is a useful way to think about connectivity. However, the argument fails at a load-bearing point: it ignores the existence of small, thermodynamically unstable AdS black holes that are nevertheless valid two-sided Lorentzian solutions with Einstein-Rosen bridges. The quoted correlator formula is standard and does not select Euclidean mediation. The proposed scenario is interesting as a conjecture, but the paper does not establish necessity of Euclidean regions.
major comments (3)
- [Section 3] The dichotomy stated in Section 3, 'there are only the following two possibilities' (Lorentzian ER bridge or Euclidean region), is incomplete. Below the Hawking-Page temperature, small AdS black holes still exist as Lorentzian two-sided geometries with ER bridges; the Hawking-Page transition concerns which Euclidean saddle dominates the canonical partition function Z(β)=Tr e^{-βH}, not whether a two-sided Lorentzian solution exists. For example, BTZ black holes exist for every inverse temperature β, and higher-dimensional Schwarzschild-AdS solutions exist above a minimum temperature. These unstable black holes admit the usual maximally extended Lorentzian geometry with geodesics connecting the two boundaries. Thus the correlator (2.1) can be mediated by an unstable but existing Lorentzian bridge, and the inference that Euclidean regions are required does not follow.
- [Section 4] The key step in Section 4 is the assertion that when 'the black holes are unstable, the argument above cannot work.' This conflates canonical-ensemble dominance with the existence of a bulk saddle for the pure TFD state (4.1). The TFD state is a pure state in H1⊗H2, and its Hartle-Hawking wavefunction receives contributions from all saddles, including the two-sided Lorentzian black hole, regardless of whether that black hole dominates the thermal partition function of a single CFT. Thermodynamic instability does not delete the Lorentzian saddle, nor the geodesics that connect the two boundaries. Therefore the conclusion 'This proves our claim' is unsupported.
- [Section 4, displayed correlator formula] The displayed formula for ⟨Ψβ|O(t1,φ1)O(t2,φ2)|Ψβ⟩ is quoted without derivation and does not discriminate between Lorentzian and Euclidean mediation. This formula is the standard two-sided thermal correlator for the eternal black hole; in the BTZ case its geodesic interpretation includes geodesics that pass through the Lorentzian horizon and connect the two asymptotic regions. The sum over k can be understood as a sum over image geodesics around the black hole, not necessarily as paths that must probe Euclidean regions. The claim that 'the main contribution comes from the geodesic paths that probe the Euclidean regions' is therefore not established by the formula.
minor comments (6)
- [Abstract] In the abstract, 'regimes whether wormholes' should read 'regimes where wormholes', and 'entangled structure of the dual state persists' repeats 'state' twice in the same sentence.
- [Section 4] The notation for the TFD state contains a rendering error: 'flflΨβfi' should be displayed as |Ψβ⟩, and the bra-ket notation in ⟨Ψβ|Ψβ⟩ uses the wrong angle-bracket glyphs.
- [Figure 1 caption] The caption says 'spacial slice' and 'entaglement'; these should be 'spatial slice' and 'entanglement'.
- [Section 2] The text refers to 'l.h.s.(1)' and 'r.h.s of this equation' where the equation is numbered (2.1); the referencing should be consistent.
- [Discussion] There is a typo 'Eclidean' in the sentence 'geometrically connected to through an Eclidean region'; it should be 'Euclidean', and the phrase 'connected to through' should be 'connected through'.
- [References] Reference [12] begins with a stray bracket '] J. M. Maldacena'; this should be cleaned up.
Circularity Check
Euclidean regions are introduced by the assumed Hartle-Hawking wavefunctional and then read back as a necessary prediction; the alternative Lorentzian saddle is dismissed by a non-circular but unsupported instability premise.
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self definitional
[Section 4, paragraph after 'This proves our claim' (Eq. (4.1) and Fig. 1/2 discussion)]
"Therefore, the geometry M of (2.1) must be promoted to M ≡ M− ∪ ML ∪ M+, which consists of three smoothly glued parts through the common surfaces Σ±, ML is Lorentzian, and M−/+ stands for the (Euclidean) saddle geometry and its respective time reflected (more details on this type of construction can be found in refs.[15, 16, 17, 18])"
The paper's claim that low-temperature entangled states require Euclidean regions is supported by 'promoting' the geometry to a union that explicitly contains Euclidean saddles (M±). The conclusion is therefore an unpacking of the assumed Hartle-Hawking representation, not an independent output. The only independent premise, low-temperature black-hole instability, concerns canonical-ensemble dominance and does not eliminate the two-sided Lorentzian saddle for the pure TFD state; the nonzero correlator is equally consistent with that Lorentzian saddle. Thus the Euclidean regions are effectively an input by construction, and 'must be promoted' is an ansatz rather than a derivation.
full rationale
The derivation chain is not wholly circular: the ER=EPR-connectivity premise and the nonzero TFD correlator provide independent motivation, and the Skenderis-van Rees real-time formalism (refs. [15,16]) is standard, with self-citations [17,18] playing only a supporting role. However, the central step that 'proves' the presence of Euclidean regions does so by assuming a Hartle-Hawking wavefunctional whose geometry already includes Euclidean sectors. The exclusion of the competing Lorentzian Einstein-Rosen bridge is based on a thermodynamic-instability argument that is a correctness gap, not a circular reduction. Because one 'prediction' (existence of Euclidean regions) reduces by construction while the interpretive connectivity claim has residual independent content, the circularity is partial, scoring 4.
Assumptions & free parameters
assumptions (4)
- domain assumption AdS/CFT correspondence and the large-N semi-classical bulk limit are valid.
- domain assumption Equation (2.1) is a valid prescription for correlators as sums over continuous curves in the bulk geometry, including complex or Euclidean pieces.
- ad hoc to paper Below the Hawking-Page temperature black holes are unstable, so the TFD state lacks a Lorentzian ER bridge.
- ad hoc to paper The Hartle-Hawking wavefunctional decomposition M = M- union ML union M+ correctly describes the TFD state.
Cite this review
Pith. "Pith review of Signature change as phase transition in holography." pith.science (2026). https://pith.science/paper/DOLZ45NP
@misc{pith2026250513349,
author = {Pith},
title = {Pith review of: Signature change as phase transition in holography},
year = {2026},
howpublished = {\url{https://pith.science/paper/DOLZ45NP}},
note = {Machine review of arXiv:2505.13349}
}
read the original abstract
In holographic quantum gravity, Euclidean pieces of the spacetime appear in the large N limit as representing semi-classical states of the theory. In this essay, we argue that the duals of entangled states are spacetime geometries that contain Euclidean regions in order to preserve classical connectivity. Thereby, the proposal is to extend the ER-EPR conjecture to regimes whether wormholes (Einstein-Rosen bridges) become unstable but the entangled structure of the dual state persists.
Figures
Reference graph
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Reviewed August 15, 2026 · model on record in the stance chip above.
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