Diving into booklet wormholes

Abstract: ArXiv:2508.17898 proposed the booklet wormhole as the holographic dual of the GHZ state. This paper extends the investigation into this geometry, particularly focusing on the junction conditions for matter fields. We show that the symmetry of the GHZ state requires the bulk to admit special Killing vector fields that standard manifolds cannot realize. Moreover, these bulk symmetries require unprecedented quantum non-local junction conditions at the multi-way interface: Observers entering from different horizons will perceive different states inside the wormhole, where the junction conditions appear as constraints on the observables of different sets of observers. We finally discuss how to render booklet wormholes traversable via boundary deformations. A localized wave packet injected from one page generally evolves into a non-local mixed state on each remaining page, with the information encoded in the entanglement between different pages.

• The paper develops a quantum description of booklet-wormhole junctions, showing that thermal GHZ symmetries impose the non-local constraint \(\hat P_A+\hat P_B+\hat P_C=0\) on infalling observers.
• The analysis treats the multi-way interface as topological and gauge-dependent in position, while identifying first-class constraints that require physical matter operators to be jointly dressed across pages.
• The paper finds that ordinary double-trace couplings cannot make booklet wormholes traversable, whereas tripartite boundary interactions can transmit information non-locally through entanglement between the output pages.

Overview

This paper extends the program initiated in "The holographic dual of the GHZ state" (Jiang et al., 25 Aug 2025), which proposed the booklet wormhole—a multi-way junction gluing more than two asymptotically AdS spacetimes—as the bulk dual of GHZ-type entanglement. The present work addresses two questions left open there: what junction conditions matter fields must satisfy at a multi-way interface, and whether such wormholes can be made traversable. The central results are that (i) the multi-way junction is topological and its position is a gauge degree of freedom; (ii) the symmetries of thermal GHZ states force a genuinely quantum, non-local junction condition $\hat P_A+\hat P_B+\hat P_C=0$ on conserved quantities measured by observers falling in from different pages; and (iii) traversability requires multipartite boundary couplings, with teleported information encoded non-locally across output pages.

Preference states and the non-uniqueness of holographic TFDs

The paper first argues that the canonical TFD state $\sum_i e^{-\beta E_i/2}\ket{i}_L(\Theta\ket{i}_L)$ is not the unique CFT state dual to a smooth two-sided black hole. Applying a unitary $U$ to the right Hilbert space while simultaneously transforming $H_R\to UH_RU^{-1}$ yields a "generic TFD" whose reduced density matrix remains thermal and whose bulk geometry is unchanged except for a transparent topological interface inserted between the horizons. The firewall diagnostic—whether double-trace teleportation transports high-energy interior excitations—gives an identical result after the basis change, so no firewall is induced. This observation matters because it removes the obstruction to generalizing to multipartite states: since no canonical singlet-like form exists for GHZ states, the authors construct thermal GHZ states of the form $$\sum_i e^{-\beta E_i/2}\ket{i}_1(\mathcal{R}_{12}\ket{i}_1)(\mathcal{R}_{13}\ket{i}_1)$$ and argue these admit regular booklet-wormhole duals.

A subtlety is that some states are nevertheless distinguished by complexity. The paper introduces the notion of a preference state: within each orbit of maximally entangled states generated by real-time evolution, the member of minimal complexity that prepares the shortest wormhole. The holographic dual is then claimed to depend not only on the state and Hamiltonian but also on the chosen complexity measure—an assertion the authors themselves note is not fully controlled, since it remains unclear how inserting a non-preference maximally entangled state deforms the Euclidean bulk geometry away from a regular black hole.

Multi-way topological interfaces

The GHZ junction inherits the defining property of a topological interface: the partition function is invariant under deformations of its position. Concretely, the thermal GHZ state is invariant under imaginary-time evolutions satisfying $\sum_\alpha t_\alpha=0$, so Euclidean leg lengths can be redistributed among pages without physical effect. Two consequences follow. First, the interface must have exactly zero reflection; otherwise scattering could localize it, contradicting topologicality. Second, single-page correlation functions are identical to those of a two-sided black hole, reflecting the equality of reduced density matrices between thermal GHZ and TFD states.

The paper carefully distinguishes two superficially similar symmetries: moving the topological interface (a gauge transformation, tied to vanishing interface energy) versus genuine global symmetry of the state under generators such as $(a+b)H_1-aH_2-bH_3$ (a physical isometry). This distinction becomes essential for the junction conditions later.

For correlation functions, tripartite correlators reduce via the topological property to TFD two-point functions of an effective operator built from the Hadamard product: $O_{\text{eff}}=O_2\odot O_3$, with matrix elements $(O_2)_{ij}(O_3)_{ij}$ in the preference-state basis. Operators crossing the junction transform non-trivially—transposition for EPR states, and for GHZ states a map involving the projector $P_{23}=\sum_i\ket{ii}\bra{ii}$, so a local operation on one page appears as a joint operation on the others. A notable special case: if any inserted operator commutes with the Hamiltonian, only diagonal matrix elements contribute, so the correlator probes purely classical correlations reproducible by a local hidden-variable model. The Hadamard product of primaries is generally not primary, making generic tripartite correlators theory-dependent—the authors concede this limits quantitative predictions.

Symmetries of the Lorentzian bulk

Extending the CFT symmetries to the Lorentzian booklet wormhole, the authors posit a multi-boost symmetry $\sum_i e^{-\beta E_i/2}\ket{i}_L(\Theta\ket{i}_L)$0. Crucially, although each observer defines a conserved quantity associated with $\sum_i e^{-\beta E_i/2}\ket{i}_L(\Theta\ket{i}_L)$1 on their page, only two independent Killing vector fields exist globally. The paper argues this provides independent evidence against any manifold dual: distinct Killing fields ($\sum_i e^{-\beta E_i/2}\ket{i}_L(\Theta\ket{i}_L)$2 and $\sum_i e^{-\beta E_i/2}\ket{i}_L(\Theta\ket{i}_L)$3) would coincide on Page 1's entanglement wedge, which is impossible on a connected manifold. An immediate consequence is that static exterior observers perceive the thermal GHZ state as the unique multi-boost-invariant state, just as TFD observers perceive the boost-invariant TFD.

What infalling observers see

Two thought experiments establish that local junction conditions fail. In the first, three photons meeting at the junction is not multi-boost invariant: a large boost sends photons from Pages 1 and 2 into the singularity before reaching the junction. In the second, a boundary excitation's causal wedge on Page 2 can be removed by boosting Pages 1 and 3 while leaving Page 2 untouched. Both imply causality between pages cannot be taken for granted and demand a non-local junction condition.

The main result follows from counting: three infalling observers (Alice, Bob, Charlie) each define a conserved momentum $\sum_i e^{-\beta E_i/2}\ket{i}_L(\Theta\ket{i}_L)$4 inside the horizons, but the global symmetry algebra has rank two. The $\sum_i e^{-\beta E_i/2}\ket{i}_L(\Theta\ket{i}_L)$5 symmetry of the GHZ state plus vanishing vacuum expectation value forces

$\sum_i e^{-\beta E_i/2}\ket{i}_L(\Theta\ket{i}_L)$6

with analogous constraints $\sum_i e^{-\beta E_i/2}\ket{i}_L(\Theta\ket{i}_L)$7 for every conserved charge. Consequently, an excitation entering from Page 1 with momentum $\sum_i e^{-\beta E_i/2}\ket{i}_L(\Theta\ket{i}_L)$8 is perceived by Bob and Charlie as carrying $\sum_i e^{-\beta E_i/2}\ket{i}_L(\Theta\ket{i}_L)$9 each, and their observations are correlated as $U$0—an expression the authors note is strictly ill-defined because $U$1 and $U$2 are not spacelike-separated subsystems. When Alice observes a localized wave packet, Bob or Charlie sees a completely decohered mixed state spread over the entire page.

Dirac constraint analysis: all such constraints are first-class, generating gauge transformations. Single-particle creation operators are gauge-dependent and excluded from the physical Hilbert space; physical operators must be dressed, e.g., $U$3.

Quantum reference frames: adopting Alice's frame via $U$4 renders her subsystem unconstrained while the reference absorbs the constraint. In this relational picture, a local unitary by Alice appears non-local and non-unitary to Bob and Charlie, and the redundancy among observer descriptions is interpreted as gauge redundancy arising from multiple observers probing one shared system—not fake degrees of freedom.

Several caveats are stated plainly. The framework assumes integrability: a QFT has infinitely many degrees of freedom but finitely many conserved charges, so the complete junction condition cannot be fixed in non-integrable (chaotic) systems, and no exact example is provided. Charge quantization poses an apparent contradiction (how does one perceive $U$5?), resolved only approximately: exciting a charge breaks the exact GHZ symmetry, so backreaction disrupts the bulk isometries and the constraint holds only approximately once matter is added. Finally, the Euclidean interpretation of the constraint—relating it to $U$6 per observer—is argued qualitatively but lacks a satisfactory excited-bulk construction due to the complicated bulk dual of the Hadamard product.

Traversability and holographic teleportation

Double-trace deformations provably fail: the relevant translated correlator is time-independent in the thermal GHZ state, so the shockwave mechanism of Maldacena–Stanford–Yang cannot open the wormhole. Instead, a tripartite interaction $U$7 is required. Working in the $U$8-symmetric subspace with the modified momentum decomposition ($U$9), the key correlator is $H_R\to UH_RU^{-1}$0. If $H_R\to UH_RU^{-1}$1 is itself primary with matching dimension, this reduces exactly to the two-sided result and traversability holds identically. If $H_R\to UH_RU^{-1}$2 are individually primary, descendant contributions shift $H_R\to UH_RU^{-1}$3, requiring stronger coupling but preserving the conclusion.

The qualitative claim about the teleported state is significant: a pure wave packet injected by Alice emerges as a mixed, highly non-local state on each output page individually, with the information encoded in entanglement between Pages 2 and 3—which become genuinely entangled only once the state resides in their spacelike-separated exteriors. The authors acknowledge this picture is heuristic: the negative shockwave destroys the Killing fields used to derive the junction condition, and they defer quantitative tests to many-body teleportation studies on the CFT side.

Limitations and open questions

The paper is explicit about several open problems. Whether topological multi-way junctions exist in generic dynamical spacetimes without Killing vector fields is unresolved; the generalized Israel junction conditions were derived only classically, and the authors conjecture a non-local gravitational junction condition should replace them in full quantum gravity. Stability of tensionless junctions is addressed only by appeal to entanglement protection, not by a dynamical calculation. The complete matter-field junction condition for perturbed or non-integrable systems is unknown, though the authors guess it constrains certain global (non-conserved) observables. Quantum reference frames in QFT remain an active problem, blocking a rigorous account of how Alice's operations manifest for Bob and Charlie. Finally, the bulk dual of the Hadamard product—and hence an excited Euclidean construction testing the observer-dependent state preparation—is absent.

Conclusion

This work establishes that the booklet wormhole's multi-way junction demands junction conditions of a fundamentally new type: constraints relating the conserved observables of different infalling observers, with no classical limit, interpretable as first-class constraints in a gauge theory of observer perspectives. Combined with the earlier entropy-saddle multiplicity, this identifies the junction as intrinsically quantum even at small $H_R\to UH_RU^{-1}$4. The extension of holographic teleportation to multipartite couplings suggests a concrete experimental handle via many-body protocols, and the observer-dependence of locality and purity sketched here offers a structural lesson—though not a solution—for the firewall and information paradoxes.