Fixed-Theory, Ensemble, and Superselection Claims

A thermal state of one theory, a probability distribution over theories, a mixture of superselection sectors, and a gravitational sum over geometries can produce similar-looking brackets. They are nevertheless different mathematical objects. The reliable diagnostic is to state the sample space, the fixed data, the normalization, whether two insertions share one draw or sector label, and the factorization rule before interpreting any connected term.

Required background. Observable and Regime Matrix for Quantum Gravity fixes the observable and approximation, while Dictionary Completeness and Global Data fixes the global form, sectors, and boundary prescription.

Helpful background. Statistical Ensembles and Field Configurations develops Gibbs states within one theory. Thermodynamic Limits, Phases, and Ensemble Equivalence explains when thermodynamic ensembles become equivalent and when they do not.

Reading path. Begin with the six operations, then compare independent copies with the shared-draw example. The gravitational interpretation and JT laboratory lead to the reproducible contract and exercises.

Thermal, disorder, and sector operations

For this page, a fixed theory means that its Hamiltonian or action, couplings, Hilbert space or observable algebra, global form, boundary conditions, and nonperturbative prescription have all been fixed. Choosing a state inside that theory does not create a theory ensemble. Conversely, writing an integral sign does not make a gravitational prescription a probability distribution.

Operations that must be distinguished before interpreting a holographic average

Construction	Held fixed	Varied, conditioned, or summed	Defining check
Gibbs state	One theory, Hamiltonian, and observable algebra	Trace weights inside one Hilbert space	Normalization, KMS, and thermodynamic identities
Disorder or theory ensemble	Sample space, measure, and observable definition	Couplings, Hamiltonian, or another theory label	Sample normalization, moments, and draw dependence
Sharp superselection sector	Observable algebra and one sector representation	Nothing outside the selected sector	Every allowed observable preserves the sector
Classical sector mixture	Inequivalent sector states represented block-diagonally	A classical sector label with stated weights	Conditioning recovers sector-resolved answers
Gauging or sector sum	Action, backgrounds, phases, and normalization rule	Bundles, fluxes, or twisted sectors	The result defines the intended new fixed theory
Gravitational topology sum	Boundary data, action, contour, and approximation	Metrics, fields, and admitted topologies	Compare all boundary amplitudes with the proposed exact object

A Gibbs state is a positive normalized quantum state. Disorder averages and classical sector mixtures instead use classical probability measures, while conditioning on a sharp sector introduces no mixture. Gauge-theory and gravitational topology sums need not be probabilistic: their phases, Euclidean weights, contour choices, or asymptotic saddle series are not automatically probability measures.

A Gibbs state keeps the theory fixed

For a Hermitian Hamiltonian $H$ at real inverse temperature $\beta>0$, assume that $e^{-\beta H}$ is trace class and that $0<Z_H(\beta)<\infty$. Then

$$\rho_{\beta,H}=\frac{e^{-\beta H}}{Z_H(\beta)}, \qquad Z_H(\beta)=\operatorname{Tr}_{\mathcal H}e^{-\beta H},$$

and

$$\langle A\rangle_{\beta,H} =\operatorname{Tr}_{\mathcal H}(\rho_{\beta,H}A).$$

The trace combines contributions from states in the same Hilbert space using Gibbs weights. The Hamiltonian, couplings, and operator algebra have not been sampled. This is why the unnormalized partition function $Z_H$ and the normalized expectation $\langle A\rangle_{\beta,H}$ must not be interchanged.

In an infinite-volume continuum QFT, a global Gibbs density matrix generally does not exist. Thermal equilibrium is then formulated algebraically through the KMS condition; this changes the representation of the state, not the distinction between varying a state and varying the theory.

The laboratory analogy is one apparatus prepared in thermal equilibrium. A disorder average instead rebuilds the apparatus with different frozen couplings. Superselection sectors are more like sealed rooms that no allowed observable can connect. The analogy stops at gravitational topology sums, whose weights may be complex or only asymptotic.

Disorder averaging changes the sample

Let $H$ be drawn from a normalized measure $\mathrm d\mu(H)$. The notation

$$\mathbb E_\mu[X_H]=\int \mathrm d\mu(H)\,X_H$$

is incomplete until $X_H$ is specified. In particular, averaging a normalized expectation sample by sample gives

$$\mathbb E_\mu[\langle A\rangle_{\beta,H}],$$

whereas averaging numerator and denominator separately gives

$$\frac{\mathbb E_\mu[\operatorname{Tr}(e^{-\beta H}A)]} {\mathbb E_\mu[Z_H(\beta)]} =\frac{\mathbb E_\mu[Z_H(\beta)\langle A\rangle_{\beta,H}]} {\mathbb E_\mu[Z_H(\beta)]}.$$

The second expression reweights samples by $Z_H$ and generally differs from the first. The same distinction appears in the annealed and quenched free energies,

$$F_{\mathrm{ann}}=-\frac{1}{\beta}\log\mathbb E_\mu[Z_H], \qquad F_{\mathrm{que}}=-\frac{1}{\beta}\mathbb E_\mu[\log Z_H].$$

For positive $Z_H$ and $\beta>0$, Jensen’s inequality gives

$$F_{\mathrm{que}}-F_{\mathrm{ann}} =\frac{1}{\beta} \left(\log\mathbb E_\mu[Z_H]-\mathbb E_\mu[\log Z_H]\right) \ge 0,$$

with equality precisely when $Z_H$ is constant almost surely. A finite set of integer moments, or a formal or asymptotic expansion for them, does not automatically determine $\mathbb E[\log Z_H]$. That inference needs a moment-determinate completion and log-integrability, or an independently controlled continuation to $n\to0$; integer replica data alone do not supply it. The definitions and their low-temperature limitations are reviewed in Baldwin and Swingle 2020, § II, eqs. (8)–(11).

Superselection is relative to an observable algebra

In the general algebraic formulation, superselection sectors are mutually inequivalent representations $\pi_\alpha$ of one allowed observable algebra $\mathcal A$. If they are displayed together on

$$\mathcal H=\bigoplus_\alpha\mathcal H_\alpha, \qquad \pi(A)=\bigoplus_\alpha\pi_\alpha(A),$$

then the projections $P_\alpha$ onto the summands obey

$$P_\alpha P_\gamma=\delta_{\alpha\gamma}P_\alpha, \qquad \sum_\alpha P_\alpha=1, \qquad [\pi(A),P_\alpha]=0\quad(A\in\mathcal A).$$

These projectors commute with the represented observables but need not be elements of the abstract algebra $\mathcal A$. No allowed observable connects two inequivalent sectors. A classical mixture of sector states is therefore described without assuming density matrices:

$$\omega(A)=\sum_\alpha p_\alpha\,\omega_\alpha(A), \qquad p_\alpha\ge0, \qquad \sum_\alpha p_\alpha=1,$$

where each $\omega_\alpha$ is a positive normalized state in sector $\alpha$. Conditioning on sector $\alpha$ retains one state $\omega_\alpha$; retaining the weights $p_\alpha$ gives a classical mixture. Relative phases between vectors in different sectors are invisible to $\mathcal A$, although a larger field algebra containing sector-changing intertwiners would change that conclusion. This algebra-relative framework and its field-algebra reconstruction are developed in Doplicher and Roberts 1990, §§ 2–5, pp. 55–94.

For intuition, a finite type-I toy model permits density matrices and block observables, so that $\omega(A)=\sum_\alpha p_\alpha\operatorname{Tr}(\rho_\alpha A_\alpha)$. Only when the chosen fixed-theory prescription traces over every block does $Z=\sum_\alpha Z_\alpha$ and hence $p_\alpha=Z_\alpha/Z$; general thermal or KMS sector weights are not fixed by this formula.

Gauging is different again: it changes the observable algebra and prescribes a sum over backgrounds or bundles with definite normalization and topological phases. The preceding page develops that distinction under background fields and sector sums.

Independent copies factorize exactly

Take two noninteracting copies of one fixed theory, with no shared projection, constraint, dynamical field, coupling, or random label. Then

$$\mathcal H_{12}=\mathcal H\otimes\mathcal H, \qquad H_{12}=H\otimes1+1\otimes H,$$

and the thermal partition function factorizes:

$$\begin{aligned} Z_H^{(2)}(\beta_1,\beta_2) &=\operatorname{Tr}_{\mathcal H\otimes\mathcal H} \exp[-\beta_1(H\otimes1)-\beta_2(1\otimes H)]\\ &=Z_H(\beta_1)Z_H(\beta_2). \end{aligned}$$

This statement concerns independent copies or disconnected boundary theories. It is not the disputed tensor-factorization of adjacent spatial regions in gauge theory or gravity; Why Continuum QFT Does Not Factorize Naively covers that separate issue. Nor does it say that every correlator in an entangled state vanishes: a thermofield-double state can have connected cross-boundary correlators while the partition function of two uncoupled copies still factorizes.

Two-copy prescriptions and the meaning of a connected term

Prescription	What the two factors share	Connected contribution
One fixed theory, independent copies	Only identical fixed defining data	Zero for the disconnected partition function
One common disorder draw	The same sampled Hamiltonian or couplings	Ensemble covariance can be nonzero
Two independent disorder draws	The measure, but not the sampled label	Zero after subtracting the product of means
One sharp superselection sector	The fixed sector representation	Zero if no other coupling remains
One shared sector mixture	A single classically sampled sector label	Sector covariance can be nonzero
Entangled state in a fixed theory	State preparation	Correlators may connect; this is not partition-function failure
Connected gravitational topology	A bulk component joining the boundaries	A bulk connected term whose boundary interpretation still must be shown

Worked example: one draw or two?

Let a common random label choose $H_a$ with probability $p$ and $H_b$ with probability $1-p$. Write $z_{ai}=Z_{H_a}(B_i)$ and $z_{bi}=Z_{H_b}(B_i)$ for two boundary conditions. A shared draw gives

$$\mathbb E[Z_1Z_2] =p\,z_{a1}z_{a2}+(1-p)z_{b1}z_{b2},$$

and direct subtraction yields

$$\operatorname{Cov}(Z_1,Z_2) =p(1-p)(z_{a1}-z_{b1})(z_{a2}-z_{b2}).$$

For real partition functions at equal boundary data, this becomes the nonnegative variance

$$\operatorname{Var}(Z)=p(1-p)(z_a-z_b)^2.$$

For $p=1/2$ and $(z_a,z_b)=(1,3)$, one finds $\mathbb E[Z]=2$, $\mathbb E[Z^2]=5$, and a connected part equal to $1$. This number measures sample-to-sample fluctuation; it is not an interaction between two fixed copies.

If the copies draw $H_1$ and $H_2$ independently, then

$$\mathbb E_{H_1,H_2}[Z_{H_1}(B_1)Z_{H_2}(B_2)] =\mathbb E_H[Z_H(B_1)]\,\mathbb E_H[Z_H(B_2)],$$

so the connected part vanishes. It also vanishes for $p=0$, $p=1$, or identical samples $H_a=H_b$. For complex sources or complex temperature, the positive equal-data quantity is

$$\mathbb E|Z-\mathbb E Z|^2 =p(1-p)|z_a-z_b|^2;$$

a cross-covariance at different complex arguments has no fixed sign.

What a connected gravitational saddle licenses

Suppose a specified semiclassical prescription gives

$$Z_{\mathrm{grav}}[B_1\sqcup B_2] =Z_{\mathrm{disc}}[B_1,B_2]+Z_{\mathrm{conn}}[B_1,B_2].$$

A nonzero connected saddle establishes a connected contribution in that bulk approximation. It does not by itself decide whether the exact boundary object is an ensemble moment, an expectation in an unconditioned baby-universe state, a coupled system, or an incomplete fixed-theory saddle sum whose omitted contributions restore factorization.

To establish an ordinary ensemble interpretation, one needs a single normalized positive measure $\mu$ whose moments reproduce the entire compatible family of multi-boundary amplitudes,

$$\frac{Z_{\mathrm{grav}}[B_1,\ldots,B_n]}{Z_{\mathrm{grav}}[\varnothing]} =\int\mathrm d\mu(\lambda)\, \prod_{i=1}^{n}Z_\lambda[B_i], \qquad \mathrm d\mu(\lambda)\ge0,$$

together with reflection positivity and consistent normalization. Matching one two-boundary term is insufficient. If the contour or weights are complex, “ensemble” may be only a formal analogy rather than a probability interpretation.

In a reflection-positive baby-universe construction, the boundary-insertion operators may commute and admit simultaneous alpha eigenstates. For a discrete spectrum of normalized alpha states, Marolf and Maxfield’s decomposition can be written

$$\frac{\langle\prod_i\widehat Z[J_i]\rangle}{\langle1\rangle} =\sum_\alpha p_\alpha\prod_i Z_\alpha[J_i],$$

while a sharp alpha state factorizes,

$$\langle\alpha|\prod_i\widehat Z[J_i]|\alpha\rangle =\prod_i Z_\alpha[J_i].$$

For a continuous alpha spectrum, the sum becomes a spectral integral and the alpha eigenstates may be delta-normalized; sharp-alpha factorization then refers to the corresponding generalized spectral value, not an ordinary normalized vector expectation. These conclusions require the exact positive Hilbert-space construction and commuting boundary algebra in Marolf and Maxfield 2020, §§ 2.1–2.3, eqs. (1)–(2) and (16)–(23). A single wormhole saddle does not supply those hypotheses.

JT gravity is a controlled ensemble laboratory

For JT gravity, the connected multi-boundary genus expansion obeys the model-specific correspondence

$$\begin{aligned} Z_{\mathrm{conn}}^{\mathrm{JT}}(\beta_1,\ldots,\beta_n) &\sim\sum_{g\ge0}e^{S_0(2-2g-n)} Z_{g,n}(\beta_1,\ldots,\beta_n),\\ &\longleftrightarrow \left\langle\prod_{i=1}^{n}\operatorname{Tr}e^{-\beta_iH} \right\rangle_{\mathrm{MM},c} \quad\text{order by order}. \end{aligned}$$

Here the disk and cylinder are the lowest-topology cases $(g,n)=(0,1)$ and $(0,2)$. Saad, Shenker, and Stanford show the correspondence to all orders in the formal genus expansion and explicitly state that the matrix integral gives a non-unique nonperturbative completion 2019, § 1, eqs. (1)–(7), and §§ 3.4–3.5, 5. Witten’s deformed-JT construction likewise identifies a specific class of models with random matrix ensembles 2020, §§ 1 and 8–9. These are annealed moments and cumulants; they do not automatically determine $\mathbb E\log Z$ or one exact Hamiltonian.

Published proposals reach different, assumption-dependent conclusions about how exact factorization might be recovered and how much closed-universe information boundary observers can access: Saad et al. 2024, §§ 2–4, McNamara and Vafa 2020, §§ 1, 3.1–3.3, and 4.1–4.2, Usatyuk and Zhao 2025, § 2, §§ 3.1, 3.3, and Appendix A, and Antonini et al. 2025, § 2.5, §§ 4.1–4.3, and § 5. Because the models, state preparations, observable algebras, and notions of access differ, this page does not combine them into one verdict; the linked Chapter 18, Chapter 20, and Research pages below explain the mechanisms and scope.

Evidence status of fixed-theory and ensemble claims

Claim	Status	Licensed conclusion
Independent fixed copies factorize	Exact under the stated tensor-product assumptions	A nonzero exact connected partition function requires a changed object or assumption
A shared random label produces covariance	Probability identity	Nonfactorization can reflect sample fluctuation rather than interaction
JT genus amplitudes match matrix cumulants	Established order by order in the formal genus expansion	JT has a concrete ensemble realization, with nonunique completion
Sharp alpha conditioning factorizes the commuting boundary-insertion algebra	Conditional on an exact positive commuting-operator framework	The unconditioned covariance of those insertions may be sector uncertainty
Every UV-complete AdS gravity averages theories	Not established	No generic ensemble conclusion follows from topology sums alone

A reproducible ensemble-status contract

Before comparing a bulk result with boundary data, record:

1. Object. Is the quantity a partition function, normalized expectation, correlator, entropy, spectral statistic, or amplitude?
2. Fixed theory data. Name the action or Hamiltonian, couplings, global form, observable algebra, regulator, boundary conditions, and nonperturbative prescription.
3. State. Specify the density matrix or state-preparation contour independently of any theory average.
4. Variable and weights. State exactly what is sampled, conditioned, gauged, or summed, with its measure, phases, and normalization.
5. Sector rule. Distinguish a sharp sector, a classical mixture, a direct-sum trace, and a gauged theory.
6. Replica rule. Say whether different boundaries or replicas share one draw or receive independent draws.
7. Normalization. Distinguish an average of ratios from a ratio of averages and annealed from quenched quantities.
8. Factorization test. Compute the disconnected product and connected remainder, including the zero-variance and independent-draw controls.
9. Claim ceiling. State whether the result is exact, order-by-order, model-specific, conditional, or open, and name the first omitted effect that could change it.

The strongest conclusion is the one that survives this contract. Merely declaring a bulk expression to be an ensemble or a fixed-theory observable does not establish the dictionary.

Common pitfalls

Calling a Gibbs state a theory average. The Gibbs weights vary contributions inside one fixed Hilbert space. They do not randomize the Hamiltonian.

Suppressing the draw label. The same-draw moment $\mathbb E[Z_H(B_1)Z_H(B_2)]$ and the independent-draw moment have different connected parts. Always label which prescription is used.

Treating a sector mixture as gauging. Conditioning, mixing states from inequivalent sectors in a block-diagonal representation, tracing over a fixed direct sum, and summing gauge bundles define different observables or theories. Similar summation signs do not make them interchangeable.

Letting a saddle interpret itself. A connected geometry is a bulk contribution. An ensemble measure, alpha-state expectation, or fixed-theory cancellation mechanism requires additional structure.

Confusing independent-copy and subregion factorization. Gauge constraints can obstruct tensor factorization of neighboring spatial regions without correlating two otherwise independent boundary theories.

Exercises

Annealed and quenched free energies. Use Jensen’s inequality to prove $F_{\mathrm{que}}\ge F_{\mathrm{ann}}$ for $Z_H>0$. Evaluate both at $\beta=1$ when $Z_H$ equals $1$ or $3$ with equal probability.

Solution

Concavity of $\log$ gives $\mathbb E[\log Z]\le\log\mathbb E[Z]$. Multiplication by $-1/\beta$ reverses the inequality, so $F_{\mathrm{que}}\ge F_{\mathrm{ann}}$. For the two samples,

$$F_{\mathrm{ann}}=-\log2, \qquad F_{\mathrm{que}}=-\frac12\log3.$$

Since $\log2>\tfrac12\log3$, the claimed ordering follows. The inequality is strict because the partition function fluctuates.

Shared and independent draws. Derive the covariance formula for the two-Hamiltonian example at general $p$. Then repeat the calculation when the two factors draw independently.

Solution

For a shared draw,

$$\begin{aligned} \mathbb E[Z_1Z_2] &=p z_{a1}z_{a2}+(1-p)z_{b1}z_{b2},\\ \mathbb E[Z_1]\mathbb E[Z_2] &=[p z_{a1}+(1-p)z_{b1}] [p z_{a2}+(1-p)z_{b2}]. \end{aligned}$$

Subtracting gives $p(1-p)(z_{a1}-z_{b1})(z_{a2}-z_{b2})$. With independent labels, the joint probability factors, so the first line is already $\mathbb E[Z_1]\mathbb E[Z_2]$ and the connected part is zero.

A superselection phase. Let $|\psi\rangle=c_a|a\rangle+c_b|b\rangle$, with $|c_a|^2+|c_b|^2=1$, where every allowed observable is block diagonal between sectors $a$ and $b$. Show that the relative phase of $c_a$ and $c_b$ is invisible. State what extra datum would be needed before summing over $a$ and $b$ could be called gauging.

Solution

For $A=A_a\oplus A_b$,

$$\langle\psi|A|\psi\rangle =|c_a|^2\langle a|A_a|a\rangle +|c_b|^2\langle b|A_b|b\rangle,$$

because the off-diagonal matrix elements vanish. The restriction of the pure vector state to the observable algebra is therefore identical to a classical mixture with probabilities $|c_a|^2$ and $|c_b|^2$. Gauging would additionally require the symmetry being gauged, the allowed bundles or twisted sectors, their weights and topological phases, the normalization, and the resulting observable algebra.

Claim downgrade. A semiclassical calculation finds one connected two-boundary saddle. What is established, and what would be needed to call it an ensemble covariance or a violation of fixed-theory factorization?

Solution

The calculation establishes a connected contribution within its contour, topology set, and approximation. An ensemble claim requires a normalized positive measure reproducing a consistent family of multi-boundary moments. A fixed-theory violation requires an exact identification with two independent copies and control of all omitted saddles and nonperturbative terms. Until one of those structures is shown, the boundary interpretation remains conditional.

Scope and continuation

This page develops the distinction and its diagnostics, not the detailed models. Continue to Non-Unique JT Matrix-Integral Completion for the low-dimensional laboratory; Baby Universes, Alpha Parameters, and Proposed Superselection Sectors for sector constructions; and Factorization, Ensembles, and the Gravitational Path Integral for topology sums. Fixed-Theory Factorization and Nonperturbative Completion Tests and the next page, Nonperturbative Definition and Completion Criteria, assemble the stronger acceptance test.

Status boundary. The Holography and Quantum Gravity research guide carries its own dated literature assessment. This page states stable definitions and model-specific published results; it does not issue a current verdict on the generic factorization or baby-universe debates.

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

• Antonini, Stefano, Pratik Rath, Martin Sasieta, Brian Swingle, and Alejandro Vilar López. “The Baby Universe Is Fine and the CFT Knows It: On Holography for Closed Universes.” Journal of High Energy Physics 12 (2025): 159. DOI. Open PDF.
• Baldwin, C. L., and Brian Swingle. “Quenched versus Annealed: Glassiness from SK to SYK.” Physical Review X 10 (2020): 031026. DOI. Open PDF.
• Doplicher, Sergio, and John E. Roberts. “Why There Is a Field Algebra with a Compact Gauge Group Describing the Superselection Structure in Particle Physics.” Communications in Mathematical Physics 131 (1990): 51–107. DOI.
• 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. DOI. Open PDF.
• McNamara, Jacob, and Cumrun Vafa. “Baby Universes, Holography, and the Swampland.” arXiv:2004.06738 [hep-th] (2020). arXiv.
• Saad, Phil, Stephen H. Shenker, and Douglas Stanford. “JT Gravity as a Matrix Integral.” arXiv:1903.11115 [hep-th] (2019). arXiv.
• Saad, Phil, Stephen H. Shenker, Douglas Stanford, and Shunyu Yao. “Wormholes without Averaging.” Journal of High Energy Physics 09 (2024): 133. DOI. Open PDF.
• Usatyuk, Mykhaylo, and Ying Zhao. “Closed Universes, Factorization, and Ensemble Averaging.” Journal of High Energy Physics 02 (2025): 052. DOI. Open PDF.
• Witten, Edward. “Matrix Models and Deformations of JT Gravity.” Proceedings of the Royal Society A 476 (2020): 20200582. DOI. Open PDF.