Factorization, Ensembles, and Gravitational Path Integrals

Disconnected boundaries pose a sharp question for gravitational path integrals. In one fixed boundary theory, independent components give a product of partition functions. Yet the bulk saddle sum can contain Euclidean wormholes joining them and hence a connected contribution. This chapter develops the tension, shows why solvable JT gravity naturally computes matrix-ensemble averages, and examines baby-universe sectors, spectral evidence, and proposals for an exact single-theory completion without treating any one interpretation as universal.

Helpful background. Semiclassical Gravity supplies Euclidean actions and topology expansions; Quantum Information supplies tensor products and conditional states; Quantum Fields in Curved Spacetime supplies partition functions and correlation functions; Black-Hole Information supplies replica wormholes; Quantum Gravity supplies the nonperturbative-definition problem and gravitational observables.

The factorization equation

Let $B_1$ and $B_2$ be disconnected boundaries and let $Z_H(B)$ be the partition function of one fixed microscopic theory $H$. Independence gives

$$Z_H(B_1\sqcup B_2) =Z_H(B_1)Z_H(B_2).$$

There is no connected correlator of two numbers in the same fixed theory. A semiclassical bulk sum, however, can have the form

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

If the complete $Z_{\mathrm{grav}}$ is claimed to equal $Z_H$, any connected term that survives the full sum must be reconciled with exact factorization. Individual semiclassical saddles need not factorize term by term: additional contributions can cancel them. The tension therefore depends on the observable’s normalization, the boundary conditions, and the claim that the full path integral defines one fixed theory.

For an ensemble with probability measure $d\mu(H)$,

$$\langle Z_H(B_1)Z_H(B_2)\rangle = \langle Z_H(B_1)\rangle \langle Z_H(B_2)\rangle +\operatorname{Cov}\!\left(Z_1,Z_2\right).$$

The connected term can equal the covariance. This identity makes wormholes natural in an ensemble-valued theory but does not prove that every gravitational path integral is an ensemble average.

Routes through the chapter

The chapter sequence is The Factorization Problem, Euclidean Wormholes and Connected Correlators, Ensemble Interpretation in Solvable Models, Baby Universes and Alpha Sectors, Spectral Form Factors and Ramp–Plateau Physics, and Single-Theory Completion and Open Questions.

The path-integral route follows that order, tracking a connected saddle through topology weights, ensemble covariance, and conditional alpha sectors. It is the shortest route to understanding which proposed resolutions alter the object computed rather than the saddle itself.

The spectral route reads the factorization problem, the ensemble page, and the spectral-form-factor analysis before returning to wormholes and completion. It emphasizes what dip, ramp, and plateau data do and do not diagnose. A ramp can reveal correlations between energy levels after smoothing; it does not uniquely identify a Euclidean saddle or ensemble ontology.

Both routes keep three levels separate: an exact statement about one Hamiltonian, a controlled result for an ensemble or low-dimensional gravity model, and a conjectured interpretation of a higher-dimensional single theory.

JT gravity, ensembles, and alpha sectors

JT gravity provides an unusually explicit bridge. Its genus expansion is reproduced order by order by a double-scaled random matrix integral. Boundaries correspond to traces such as $Z(\beta)=\operatorname{Tr}e^{-\beta H}$, and connected bulk topologies reproduce connected ensemble correlators. The perturbative equivalence, spectral curve, and topology expansion are developed in Saad, Shenker, and Stanford 2019, §§2–5.

This exact perturbative organization has two qualifications. First, a matrix integral is an ensemble definition, whereas a specified Hamiltonian is one sample. Second, the genus series does not by itself select a unique nonperturbative completion. The page on solvable models makes those boundaries explicit rather than treating “JT equals random matrix” as an unqualified statement.

Baby-universe language rewrites connected amplitudes as correlations mediated by an auxiliary Hilbert space. Commuting operators that act like fluctuating couplings can be diagonalized in alpha states. Conditional on a sharp alpha sector, amplitudes may factorize even though an unconditioned state averages over sectors. The construction and its operational meaning are analyzed in Marolf and Maxfield 2020, §§2–5. Conditional factorization is a possible organization of the answer, not automatically a microscopic identification of the sector selected by a boundary theory.

Results and open questions as of 2026-08-19

The cutoff for mutable evidence in this chapter is 2026-08-19.

Evidence or result	Controlled scope	Limitation or open inference
JT topology expansion equals matrix-integral expansion, Saad, Shenker, and Stanford 2019, §§2–5	Perturbative JT gravity with a specified matrix-integral completion	Ensemble valued; perturbation theory alone does not choose one nonperturbative fixed theory
Baby-universe alpha sectors, Marolf and Maxfield 2020, §§2–5	Gravitational amplitudes organized by a baby-universe Hilbert space and commuting observables	Requires a state or conditioning prescription and does not settle every UV completion
Spectral ramp and plateau, Cotler et al. 2017, §§2–4	Random-matrix ensembles and selected chaotic spectra after averaging or smoothing	A ramp is not unique to one bulk saddle and finite samples fluctuate
Reflection-positive reconstruction, McNamara and Wang 2026, §1.1, Theorem 1.1; §§4–5, 9, and 11	Given the paper’s source-category setup and its five axioms—finiteness, reality, multiplicativity, continuity, and reflection positivity—the closed-manifold partition function reconstructs a unique unitary compact functorial QFT	Part (c) identifies manifold-preparable states with the invariant sector of a maximal compact symmetry; charged-spectrum completion belongs to the later factorization interpretation, not to the reconstruction hypotheses, and the theorem does not construct UV-complete gravity

The 2026 reconstruction result separates a theorem from its gravitational interpretation. Parts (a) and (b) reconstruct a unique unitary compact functorial QFT from the source-category setup and five axioms—finiteness, reality, multiplicativity, continuity, and reflection positivity. Part (c) separately identifies the subspace preparable by the chosen manifolds and sources with the invariant sector of a maximal compact Hausdorff symmetry. In the paper’s factorization interpretation, enlarging the source or charged-state spectrum may restore states hidden by that invariant-sector restriction; such completeness is not an additional hypothesis of parts (a) and (b). Applying either step to gravity still requires verifying the stated axioms and identifying the physically correct sources.

The ensemble position regards nonfactorizing gravitational amplitudes as computing averages from the outset. The alpha-sector position restores conditional factorization after choosing a superselection-like state. A single-theory position expects branes, null states, microscopic degrees of freedom, or another nonperturbative mechanism to cancel or reinterpret connected terms. These competing positions can agree on the same perturbative wormhole calculation while disagreeing on what it means. The question remains unresolved beyond controlled models.

An adverse check is to choose one finite Hamiltonian and calculate $Z_H(\beta_1)Z_H(\beta_2)$: it factorizes exactly even if a time or energy-window average of its spectrum displays random-matrix correlations. Conversely, averaging over Hamiltonians produces a covariance without demonstrating a spacetime wormhole. Either observation blocks a one-way inference.

The conclusions could change after a microscopic fixed-theory construction matches all connected gravitational amplitudes, after a nonperturbative JT completion changes the topology-to-ensemble dictionary, or after a theorem determines which baby-universe sector and factorization rule a given boundary condition selects.

Review the chapter

Diagnostic. An ensemble contains two Hamiltonians with equal probability and partition functions $Z_a=1$ and $Z_b=3$ at a chosen $\beta$. Compute $\langle Z^2\rangle$, $\langle Z\rangle^2$, and the connected covariance. Does either sample fail to factorize?

Solution

$\langle Z^2\rangle=(1^2+3^2)/2=5$, while $\langle Z\rangle^2=[(1+3)/2]^2=4$. The connected covariance is $1$. Each fixed sample still obeys $Z_aZ_a=1$ or $Z_bZ_b=9$ exactly; the nonzero connected quantity arises only after ensemble averaging. Thus ensemble nonfactorization cannot be assigned to either Hamiltonian without changing the observable.

The next chapter translates entropic wedges into operator-algebra recovery and asks what, if anything, can be reconstructed in one code subspace.

References

• Cotler et al. 2017, Jordan S. Cotler et al., “Black Holes and Random Matrices,” Journal of High Energy Physics 05 (2017) 118.
• 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.
• McNamara and Wang 2026, Jacob McNamara and Zhencheng Wang, “Wormholes as Red Herrings: Reflection Positivity and the Reconstruction of Unitary Quantum Field Theories,” arXiv:2607.01322 (2026).
• Saad, Shenker, and Stanford 2019, Phil Saad, Stephen H. Shenker, and Douglas Stanford, “JT Gravity as a Matrix Integral,” arXiv:1903.11115 (2019).

Dated Research assessment. For open questions and recent results, follow the link below. Each assessment states the date through which it considers results and the limits of its conclusions.

• Factorization, Ensembles, and Gravitational Path Integrals