Factorization, Ensembles, and the Gravitational Path Integral

For two genuinely independent copies of one fixed boundary theory, the partition function factorizes exactly. A gravitational prescription that instead gives a connected two-boundary term must therefore identify a different object—such as an ensemble moment or an unconditioned baby-universe state—or supply additional nonperturbative contributions that restore fixed-theory factorization.

Required background. Euclidean Gravitational Path Integrals, Topology Sums, and Boundary Conditions defines the multi-boundary amplitude. Fixed-Theory, Ensemble, and Superselection Claims fixes the claim domains.

Helpful background. Why Continuum QFT Does Not Factorize Naively separates spatial-algebra subtleties from independent-theory factorization. Non-Unique JT Matrix-Integral Completion: Fixed-Theory, Disorder-Average, and Ensemble Distinctions provides the central example.

The fixed-theory factorization equation

Let theory $T$ have Hilbert space $\mathcal H_T$ and Hamiltonian $H_T$. Two noninteracting copies have

$$\mathcal H_{T\otimes T}=\mathcal H_T\otimes\mathcal H_T,\qquad H_{12}=H_T\otimes1+1\otimes H_T.$$

Therefore

$$Z_{T\otimes T}(\beta_1,\beta_2) =\operatorname{Tr}_{\mathcal H_T\otimes\mathcal H_T} e^{-\beta_1H_T}\otimes e^{-\beta_2H_T} =Z_T(\beta_1)Z_T(\beta_2).$$

This is not the disputed tensor factorization of adjacent spatial regions in a gauge theory. It follows from specifying two independent systems. Sources on the two copies likewise factorize unless the state preparation or action couples them.

Application: ensemble covariance

For an ensemble of Hamiltonians with probability measure $d\mu(T)$,

$$\overline{Z(\beta_1)Z(\beta_2)} =\int d\mu(T)\,Z_T(\beta_1)Z_T(\beta_2).$$

In general,

$$\overline{Z(\beta_1)Z(\beta_2)} -\overline{Z(\beta_1)}\,\overline{Z(\beta_2)} =\operatorname{Cov}_\mu(Z_1,Z_2)\ne0.$$

Thus a connected gravitational amplitude is naturally compatible with an ensemble moment. In JT gravity the genus expansion and its double-trumpet contribution are reproduced by a double-scaled random-matrix integral Saad, Shenker, and Stanford 2019. The matrix integral independently identifies the brackets as an ensemble average; the Euclidean saddle alone does not.

An unconditioned baby-universe state produces the same algebraic pattern:

$$\langle Z_1Z_2\rangle_\Psi =\int d\alpha\,p_\Psi(\alpha)Z_\alpha(B_1)Z_\alpha(B_2).$$

Conditioning on a sharp $\alpha$ gives $Z_\alpha(B_1)Z_\alpha(B_2)$ and removes this covariance. Calling this a fixed theory is justified only if the alpha-conditioned sector and its observable algebra are independently defined.

Three logically distinct resolutions

There are three common responses to $Z_{\mathrm{conn}}\ne0$.

1. Ensemble object. The gravitational path integral computes averaged observables. Factorization was never expected before conditioning on a member.
2. Superselection object. The path integral averages over alpha sectors, while an actual experiment remains in one sector. One must define sector preparation and show asymptotic observables preserve it.
3. Incomplete semiclassics. A fixed boundary theory is intended, and omitted corrections cancel or reorganize the connected result. One must exhibit those corrections at the required order.

These are alternatives, not interchangeable descriptions.

Adversarial conditioning test

Hold the genus expansion fixed and demand a single theory $T$. The ensemble covariance must then vanish in the exact two-copy observable. If it does not, the proposed dictionary is not computing $Z_T(\beta_1)Z_T(\beta_2)$. If conditioning on alpha removes it, verify that other connected terms do not survive and that alpha is not changed by allowed operations. If exponentially small corrections are invoked, calculate their scaling and phase: merely noting that they could exist does not demonstrate cancellation.

Also vary the ultraviolet regulator for spatial subregions while retaining two independent copies. Edge-mode or center choices can alter subregion entropy but cannot create a covariance between two uncoupled theories. This distinguishes the gravitational factorization problem from local-QFT factorization failure.

Limits and handoff

Factorization is one necessary test, not a complete definition of quantum gravity. Positivity, a Hilbert space, finite observables, spectral data, and controlled Lorentzian continuation are also required. The combined test is developed on Fixed-Theory Factorization and Nonperturbative Completion Tests.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Marolf, D., and H. Maxfield. “Transcending the Ensemble: Baby Universes, Spacetime Wormholes, and the Order and Disorder of Black Hole Information.” Journal of High Energy Physics 2020, 8 (2020): 044. DOI.
• Saad, P., S. H. Shenker, and D. Stanford. “JT Gravity as a Matrix Integral.” arXiv:1903.11115.