Fixed-Theory, Ensemble, and Superselection Claims

A thermal average in one theory, a statistical average over different theories, and conditioning on a superselection sector are different operations. A bulk calculation must say what is integrated, what is averaged, what is held fixed, and which factorization test the result should satisfy.

Required background. Observable and Regime Matrix for Quantum Gravity fixes the observable, while Dictionary Completeness and Global Data supplies the sector and global data.

Helpful background. Statistical Ensembles and Field Configurations distinguishes state ensembles within one theory. Thermodynamic Limits, Phases, and Ensemble Equivalence explains when thermodynamic ensembles agree.

Three averaging operations

For one Hamiltonian $H$, a canonical thermal expectation is

$$\langle A\rangle_{\beta,H} =\frac{\operatorname{Tr}_{\mathcal H}(e^{-\beta H}A)} {\operatorname{Tr}_{\mathcal H}e^{-\beta H}}.$$

This averages over states in a fixed Hilbert space. By contrast, a disorder or theory average has a probability measure $P(H)$,

$$\mathbb E_H[Z_H(\beta)] =\int \mathrm dP(H)\,Z_H(\beta),$$

and changes the theory being sampled. A superselection decomposition

$$\mathcal H=\bigoplus_{\alpha}\mathcal H_\alpha$$

introduces a third choice: compute in one sector $\alpha$, form a classical mixture of sectors, or sum sector amplitudes according to a specified rule.

Factorization is the diagnostic

For two decoupled boundaries in a fixed theory, one normally expects

$$Z_H(\beta_1,\beta_2)=Z_H(\beta_1)Z_H(\beta_2).$$

After an ensemble average,

$$\mathbb E_H[Z_H(\beta_1)Z_H(\beta_2)] -\mathbb E_H[Z_H(\beta_1)]\mathbb E_H[Z_H(\beta_2)] =\operatorname{Cov}_H(Z_1,Z_2)$$

need not vanish. The connected term is then ensemble covariance, not an interaction between two copies of one fixed boundary theory.

Thermal state, disorder average, and gravity

Construction	Held fixed	Averaged	Appropriate check
Canonical ensemble	Hamiltonian and couplings	States or energies	KMS and thermodynamic identities
Disorder ensemble	Sampling measure	Hamiltonians or couplings	Mean, variance, and sample dependence
Superselection mixture	Sector algebra	Classical sector label	Sector-resolved observables
Two-boundary gravity amplitude	Boundary data and contour as declared	Geometries or topologies as declared	Compare with the proposed boundary factorization rule

In JT gravity, the genus expansion is reproduced by a matrix integral Saad, Shenker, and Stanford 2019. This is concrete evidence for an ensemble interpretation of that model, but the nonperturbative matrix completion is not unique. It cannot be promoted silently to a universal claim about every holographic theory; fixed-theory factorization is a separate question emphasized by McNamara and Vafa 2020.

Changing the prescription

Take a random variable $Z_H$. With no theory average, two copies satisfy $Z_H^2$ exactly. Under $P(H)$, the connected quantity is the variance

$$\mathbb E_H[Z_H^2]-\mathbb E_H[Z_H]^2\ge0.$$

Changing $P(H)$ changes the answer. Calling that same connected term both fixed-theory nonfactorization and ensemble covariance is inconsistent. The strongest conclusion is conditional: the bulk quantity matches the declared averaged or sector-resolved boundary object.

Chapter 18 develops JT and matrix-integral realizations; Chapter 20 treats topology sums, baby universes, and factorization. This page supplies only the distinction needed before those calculations.

Evidence cutoff. The ensemble-interpretation discussion is fixed to 25 July 2026 and is not a current verdict on live factorization disputes.

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

• McNamara, John, and Cumrun Vafa. 2020. “Baby Universes, Holography, and the Swampland,” arXiv:2004.06738.
• Saad, Phil, Stephen H. Shenker, and Douglas Stanford. 2019. “JT Gravity as a Matrix Integral,” arXiv:1903.11115.
• Witten, Edward. 2020. “Matrix Models and Deformations of JT Gravity,” Proceedings of the Royal Society A 476, 20200582.