Factorization and Microscopic Definition

A semiclassical gravitational saddle is not yet a microscopic definition. To decide what a recovery proposal describes, one must identify whether its path integral computes an observable in one theory, an average over theories, a conditioned baby-universe sector, a code-subspace matrix element, or only an asymptotic expansion. The factorization requirement exposes the difference: partition functions of independent boundary systems factorize in a fixed quantum theory, whereas a connected bulk wormhole can contribute to the same semiclassical calculation.

Required background. Quantum Information for Black-Hole Physics supplies tensor products, algebras, channels, and conditioned states.

Helpful background. Black-Hole Thermodynamics and Semiclassical Gravity supplies Euclidean saddles; Quantum Fields in Curved Spacetime supplies path-integral and state qualifications; Black-Hole Information supplies replica wormholes and reconstruction; Quantum Gravity supplies nonperturbative microscopic questions.

The fixed-theory factorization equation

Consider two boundary systems with independent sources $J_1$ and $J_2$. If their microscopic Hilbert space is $\mathcal H_1\otimes\mathcal H_2$, their Hamiltonian is $H_1\otimes I_2+I_1\otimes H_2$, and their state is uncorrelated, the generating functional obeys

$$Z_{\mathrm{fixed}}[J_1,J_2] =Z_{\mathrm{fixed}}[J_1]Z_{\mathrm{fixed}}[J_2].$$

Equivalently, the connected correlator between operators belonging to the two systems vanishes. This is not an optional aesthetic preference. It follows from the definition of two decoupled quantum systems. A proposed microscopic completion must therefore identify any shared constraint, state, sector, or interaction that would invalidate the premise before accepting a connected answer.

A gravitational path integral may instead sum over connected bulk topologies,

$$Z_{\mathrm{grav}}[J_1,J_2] =Z_1Z_2+Z_{\mathrm{wh}}[J_1,J_2]+\cdots.$$

The apparent contradiction is sharp only when both sides are asserted to compute the same fixed-theory object. Euclidean wormholes and their possible interpretations are analyzed in Marolf and Maxfield 2020, §§2–5.

Five objects that must not be conflated

Object being computed	Defining data	Factorization question
Fixed theory	One Hilbert space, Hamiltonian, observables, state, and boundary conditions	Independent boundary factors must factorize exactly.
Ensemble average	Measure $d\mu(H)$ and an averaged product such as $\langle Z_1Z_2\rangle_\mu$	The connected covariance can be nonzero even though every member factorizes.
Alpha sector	Baby-universe state fixing eigenvalues of commuting coupling operators	Conditional amplitudes may factorize while the unconditioned state averages sectors.
Code subspace	Encoder, logical algebra, physical region, state domain, and error	A bulk tensor split, direct-sum algebra, or approximate factorization is a declared code property, not a definition of the full theory.
Semiclassical saddle expansion	Contour, boundary data, topology weights, loop order, and asymptotic parameter	A connected term needs a microscopic interpretation and nonperturbative completion.

The matrix-integral description of JT gravity gives a particularly explicit example: its genus expansion matches the double-scaled ensemble expansion order by order, while a chosen matrix integral supplies additional nonperturbative data. The matching and spectral curve are developed in Saad, Shenker, and Stanford 2019, §§2–5. The perturbative gravity series alone does not select a unique nonperturbative completion, and the ensemble dictionary does not imply that every holographic boundary theory is averaged.

What a microscopic definition must contain

A claimed single theory should provide at least

$$\mathfrak M=(\mathcal H,H,\mathcal A,\rho, \mathcal Z,\mathcal D),$$

where $\mathcal H$ is the state space, $H$ the dynamics, $\mathcal A$ the observable algebra, $\rho$ the state family, $\mathcal Z$ the nonperturbative definition of amplitudes, and $\mathcal D$ the dictionary to semiclassical bulk quantities. Gauge constraints, null states, superselection sectors, branes, and baby universes must appear in these data rather than in an after-the-fact interpretation.

Three checks then follow. First, reproduce the semiclassical connected saddle with its action, zero modes, contour, and topology weight. Second, show how the exact microscopic observable factorizes or why the boundary systems were not independent. Third, specify the nonperturbative ambiguity invisible in the genus expansion and how the chosen completion fixes it. A formal all-orders series can miss terms of order $e^{-S}$ that decide exact factorization.

Conditioning is not erasing the question

Suppose baby-universe coupling operators $\widehat\alpha_i$ commute and a state $\lvert\alpha\rangle$ fixes their eigenvalues. Conditional amplitudes can take the form

$$Z_\alpha[J_1,J_2] =Z_\alpha[J_1]Z_\alpha[J_2],$$

while averaging over $\alpha$ restores a connected covariance. This resolves an algebraic factorization problem only if the proposal explains why the physical state is a fixed alpha sector, how it is prepared or selected, and which observables can detect sector fluctuations. Conditioning without that state specification merely moves the missing microscopic datum.

Other approaches use brane sectors, half-wormholes, correlated disorder, or null-state relations to reconcile semiclassical connected pieces with a fixed description. Their value is tested by the same three checks, not by whether they share a geometric name. General holographic dictionary construction is outside this page’s scope; here the issue is what object the information- recovery calculation actually defines.

Results and open questions through 2026-08-19

Model or proposal	Assumptions	Controlled result	Limitation
JT matrix integral, Saad, Shenker, and Stanford 2019, §§2–5	Perturbative JT topology expansion and a specified double-scaled matrix-integral completion	Matches the genus expansion and ensemble spectral observables order by order	Low-dimensional ensemble result; the perturbative gravity series alone does not choose a unique completion or fixed Hamiltonian
Euclidean AdS wormholes, Maldacena and Maoz 2004, §§2–4	Named boundary conditions, semiclassical saddle set, contour, and stability analysis	Exhibits connected saddles with multiple asymptotic boundaries	Contribution, contour, stability, and fixed-theory interpretation remain model-dependent
Baby-universe alpha sectors, Marolf and Maxfield 2020, §§2–5	Baby-universe Hilbert space, commuting boundary operators, and conditioning on a sharp alpha state	Gives algebraic conditional factorization while the unconditioned state can average sectors	State selection, observable meaning, and universal UV realization remain unresolved
Half-wormholes at fixed couplings, Saad et al. 2021, §§2–5	Fixed-coupling SYK-like disorder models and their collective-field saddles	Reorganizes averaged wormhole effects into realization-dependent wormhole and half-wormhole contributions that restore factorization in the models	Extension beyond selected disorder models and relation to gravity without disorder data remain open
Holographic code subspaces, Almheiri, Dong, and Harlow 2015, §§2–4	Finite code, declared logical algebra, physical region, and exact or approximate error criterion	Gives algebraic recovery and multiple representations without requiring a global bulk tensor factor	Does not define the entire microscopic theory or its nonperturbative path integral
Reflection-positive QFT reconstruction, McNamara and Wang 2026, §1.1, Theorem 1.1; §§5.2–5.3, 9, and 11	Finite, multiplicative, reflection-positive closed-manifold partition function in the theorem’s compact source class, with the full charged spectrum	Reconstructs a unitary compact functorial QFT; restoring omitted charged states can remove apparent Hilbert-space nonfactorization	Does not establish that a proposed gravitational path integral satisfies the hypotheses or construct its ultraviolet completion

The last row supplies a genuinely different competing position at the cutoff: apparent Hilbert-space nonfactorization can result from reconstructing with an incomplete charged spectrum rather than from an ensemble or a failure of unitarity. Applying the theorem to gravity requires the exact finite partition-function object, its full allowed sources, multiplicativity, and reflection positivity. A semiclassical wormhole saddle by itself verifies none of those hypotheses.

Competing positions agree on the semiclassical connected calculations but differ on their microscopic object: ensemble moment, conditioned sector, full-spectrum formal QFT, state-dependent observable, or incomplete saddle expansion. The conclusions could change if a fixed UV theory derives the connected contribution and exact factorization simultaneously, if a contour theorem removes a proposed saddle, or if a sector construction gives preparation and measurement rules rather than only formal conditional amplitudes.

Check your understanding

Why can $\langle Z_1Z_2\rangle_\mu-\langle Z_1\rangle_\mu\langle Z_2\rangle_\mu$ be nonzero without violating factorization of each theory in the ensemble?

Solution

For every fixed Hamiltonian $H$, the two decoupled systems can satisfy $Z_H[J_1,J_2]=Z_H[J_1]Z_H[J_2]$. Averaging the product over $H$ does not equal the product of separate averages when the two factors depend on the same random variable. The difference is their ensemble covariance. A violation would arise only if that averaged quantity were incorrectly identified with the product observable of one fixed theory without a sector, shared state, or other microscopic explanation.

References

• Almheiri, Ahmed, Xi Dong, and Daniel Harlow, “Bulk Locality and Quantum Error Correction in AdS/CFT,” Journal of High Energy Physics 04 (2015) 163. DOI. Open manuscript
• Maldacena, Juan and Liat Maoz, “Wormholes in AdS,” Journal of High Energy Physics 02 (2004) 053. arXiv
• Marolf, Donald and Henry Maxfield, “Transcending the Ensemble: Baby Universes, Spacetime Wormholes, and the Order and Disorder of Black Hole Information,” Journal of High Energy Physics 08 (2020) 044. arXiv
• McNamara, Jacob and Zhencheng Wang, “Wormholes as Red Herrings: Reflection Positivity and the Reconstruction of Unitary Quantum Field Theories,” arXiv:2607.01322 (2026). Open manuscript
• Saad, Phil, Stephen H. Shenker, and Douglas Stanford, “JT Gravity as a Matrix Integral,” arXiv preprint (2019). arXiv
• Saad, Phil, Stephen H. Shenker, Douglas Stanford, and Shunyu Yao, “Wormholes without Averaging,” arXiv preprint (2021). arXiv