The Factorization Problem

The factorization problem is the apparent conflict between two statements: a single fixed boundary theory assigns a product partition function to disconnected boundaries, while a gravitational path integral may admit a connected Euclidean wormhole saddle between them. The conflict is precise only after the boundary theory, observable, normalization, topology sum, and ensemble meaning are specified. A connected saddle is evidence to interpret, not by itself a contradiction.

Required background. Quantum Information supplies tensor products, conditional states, and the distinction between a fixed system and a probabilistic mixture.

Helpful background. Semiclassical Gravity supplies Euclidean saddle expansions; Quantum Fields in Curved Spacetime supplies generating functionals; Black-Hole Information supplies replica wormholes; Quantum Gravity supplies microscopic and nonperturbative completion questions.

Factorization on disconnected boundaries

Let $H$ denote one fixed microscopic boundary theory and let $B_1$, $B_2$ be disconnected components with independent sources $J_1$, $J_2$. With no shared dynamical degrees of freedom, the Hilbert space and evolution factorize, giving

$$Z_H[B_1\sqcup B_2;J_1,J_2] = Z_H[B_1;J_1]Z_H[B_2;J_2].$$

Equivalently, the connected cross-boundary functional derivative of $W_H=\log Z_H$ vanishes. This is an exact algebraic statement for the specified fixed theory; it is not a large-$N$ approximation.

Normalization matters. If each boundary prepares a normalized state, an overall constant in $Z_H$ must be removed consistently before comparing connected quantities. Shared constraints, gauging a diagonal symmetry, summing a common coupling, or preparing an entangled two-boundary state changes the premises and can create correlations without violating the factorization equation for independent preparations.

The connected bulk term

A semiclassical gravitational computation instead has a saddle expansion

$$Z_{\mathrm{grav}}[B_1\sqcup B_2] \simeq Z_{\mathrm{disc}}[B_1]Z_{\mathrm{disc}}[B_2] +Z_{\mathrm{wh}}[B_1,B_2]+\cdots .$$

$Z_{\mathrm{wh}}$ integrates over a connected bulk geometry with the same two asymptotic boundaries. The on-shell action, boundary terms, matter determinant, zero-mode measure, topology weight, phase, and contour coefficient all enter its value. Examples of such asymptotically disconnected Euclidean geometries and their interpretive tension appear in Maldacena and Maoz 2004, §§2–4.

If one asserts $Z_{\mathrm{grav}}=Z_H$ for one fixed theory and $Z_{\mathrm{wh}}\ne0$ survives the exact sum, the equations conflict. Possible resolutions include cancellation by other saddles, a nonperturbative completion that changes the sum, hidden shared sectors or constraints, or a reinterpretation of $Z_{\mathrm{grav}}$ as an averaged quantity.

The problem cannot be solved by saying that the wormhole is exponentially small. Exact factorization requires the connected term to vanish exactly, although its smallness determines when perturbative calculations can ignore it.

This test concerns disconnected boundary path-integral data with independent preparations. It does not forbid entangled states of two noninteracting systems. For example, a thermofield-double state can have nonzero left–right correlators even though its two Hamiltonians are fixed and uncoupled. Entangled preparation and averaging over a common Hamiltonian are two different ways of correlating outcomes; neither is the independent product preparation assumed above.

Nor is the normalization issue cosmetic. Dividing a two-boundary amplitude by independent one-boundary partition functions tests a connected normalized observable; subtracting the product tests a connected cumulant. Mixing these definitions can leave a spurious constant that looks like a factorization defect. The same definition must be used on the boundary and bulk sides.

Single theory versus ensemble average

For an ensemble of theories or Hamiltonians with probability measure $d\mu(H)$,

$$\left\langle Z_H(B_1)Z_H(B_2)\right\rangle - \left\langle Z_H(B_1)\right\rangle \left\langle Z_H(B_2)\right\rangle = \operatorname{Cov}_\mu(Z_1,Z_2),$$

which need not vanish. JT gravity realizes this structure perturbatively: connected bulk topologies match connected correlators in a matrix integral, as shown in Saad, Shenker, and Stanford 2019, §§2–5. In that controlled model the ensemble interpretation is part of the dictionary, not an embarrassment.

The extrapolation to a higher-dimensional holographic dual believed to be one fixed CFT is the disputed step. An ensemble match in JT does not establish ensemble averaging in every theory. Conversely, exact factorization of one CFT does not explain why a semiclassical wormhole gives accurate averaged spectral correlations.

Baby-universe formulations provide a conditional alternative. Amplitudes may be averaged in a state that superposes alpha sectors, yet factorize after conditioning on a simultaneous eigenstate of commuting coupling operators. The construction is analyzed in Marolf and Maxfield 2020, §§2–5. It moves the question to which state and sector the boundary conditions select.

Results and open questions as of 2026-08-19

This discussion reflects results available through 2026-08-19.

Evidence or result	Model or formulation	Limitation or unresolved implication
Euclidean AdS wormhole solutions, Maldacena and Maoz 2004, §§2–4	Specified supergravity examples with multiple asymptotic boundaries	Existence does not prove dominance, contour inclusion, or a nonzero exact fixed-theory correlator
JT/matrix-integral correspondence, Saad, Shenker, and Stanford 2019, §§2–5	Perturbative JT topology expansion with matrix-ensemble interpretation	Does not supply a unique nonperturbative single Hamiltonian
Alpha-sector factorization, Marolf and Maxfield 2020, §§2–5	Baby-universe Hilbert space with commuting observables and conditional states	Requires a sector-selection rule and does not settle every UV completion

The ensemble position identifies $Z_{\mathrm{wh}}$ with covariance and accepts nonfactorization before conditioning. The cancellation position expects an exact fixed theory to include additional saddles or microscopic effects that remove the connected term. The alpha-sector position regards factorization as conditional on a baby-universe state. These competing positions agree on some semiclassical calculations but assign different observables and Hilbert spaces; the general single-theory resolution remains unresolved.

An adverse check fixes one Hamiltonian $H$. Its numerical partition functions are ordinary numbers, so $Z_H(B_1)Z_H(B_2)$ factorizes identically even if a smoothed spectral statistic resembles an ensemble average. A second adverse check averages a two-valued random coupling: its nonzero covariance appears without any geometric wormhole. Neither direction permits a unique inference from connectedness alone.

The conclusions could change after an exact microscopic calculation cancels or reproduces a bulk connected term in a fixed theory, after a nonperturbative contour decides the relevant wormhole coefficient, or after boundary conditions are shown to select a definite alpha sector with all multi-boundary amplitudes factorizing.

Exercises

Let an ensemble variable $z$ equal $2$ with probability $1/4$ and $4$ with probability $3/4$. Compute $\langle z\rangle^2$, $\langle z^2\rangle$, and the connected covariance. Then state the factorized two-boundary value within each fixed sample.

Solution

$\langle z\rangle=3.5$, so $\langle z\rangle^2=12.25$. Also $\langle z^2\rangle=(1/4)4+(3/4)16=13$, giving covariance $0.75$. Within the $z=2$ sample the product is exactly $4$; within the $z=4$ sample it is exactly $16$. The connected $0.75$ belongs to the averaging operation and does not signal a failure of multiplication inside either fixed theory.

References

• Maldacena and Maoz 2004, Juan Maldacena and Liat Maoz, “Wormholes in AdS,” Journal of High Energy Physics 02 (2004) 053.
• Marolf and Maxfield 2020, Donald Marolf 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.
• Saad, Shenker, and Stanford 2019, Phil Saad, Stephen H. Shenker, and Douglas Stanford, “JT Gravity as a Matrix Integral,” arXiv:1903.11115 (2019).