Heavy States, Coherent States, and Semiclassical Geometries

A boundary state does not become a geometry merely because it is heavy. A state supports one semiclassical bulk solution only relative to a specified family of observables and only when three kinds of evidence agree: its mean data fit one classical solution, its connected fluctuations are smaller than the claimed resolution, and that solution remains inside a controlled bulk effective theory. Energy of order $C_T/L$ is the natural scale for a smooth, AdS-sized deformation of the metric, but localized strong gravity can occur at lower total energy. Conversely, a heavy primary, a coherent state, and a thermal density matrix can have the same mean energy while carrying different fluctuation and state information.

Required background. Weakly Coupled Bulk Fields from Connected Correlators supplies the large-$C_T$ factorization criterion for a perturbative bulk. States, Geometries, and Radial Quantization supplies the vacuum-relative cylinder energy, state–operator map, and localization test used below.

Helpful background. Thermal States and One-Point Data supplies the thermal one-point functions used in the comparison.

Geometry is relative to a code sector and an observable resolution

Let $C_T$ be the coefficient of the stress-tensor two-point function in a declared normalization. In a two-derivative Einstein regime,

$$C_T=\kappa_d\frac{L^{d-1}}{G_{d+1}},$$

where the positive numerical factor $\kappa_d$ depends on conventions Penedones 2017, §3.3, p. 30, eqs. (118)–(121), Open PDF. The useful expansion parameter is therefore $C_T^{-1}\ll1$. The familiar scaling $C_T\propto N^2$ holds in many adjoint matrix theories, but it is not universal; vector and other large-$N$ limits can scale differently.

Before asking whether a state has a geometry, specify four pieces of data.

• Code sector. Choose a state family $\mathcal H_{\mathrm{code}}$ in which the bulk field content, boundary conditions, gauge dressing, cutoff, and reconstruction map are held fixed. This is an operational restriction here; the quantum-error-correcting formulation belongs to Code Subspaces, Logical Algebras, and Encoding Maps and Almheiri, Dong, and Harlow 2015, §§2–3, Open PDF.
• Generating family. Choose a finite or band-limited set $\mathcal G_{\mathrm{coarse}}=\{\mathcal O_A[f_A]\}$ of renormalized, smeared light single-trace observables. State their spacetime support, bandwidth, species, and charge sector. Smearing is essential because unsmeared local-field variances are generally ultraviolet divergent.
• Products and complexity. State how many products of the generators may be measured, or an equivalent complexity cutoff. The products generate a coarse algebra $\mathcal A_{\mathrm{coarse}}$; the single-trace generating family is not itself an algebra.
• Accuracy and time window. Choose tolerances $\epsilon_k$, a bulk region, and a time interval. A claim about one geometry means that one classical solution reproduces the declared observables within those tolerances—not that the exact boundary state has been reconstructed.

This distinction prevents a logical mistake that matters later. Agreement on every expectation value in an entire algebra would include agreement on its products. The nonuniqueness test on this page instead fixes the one-point functions of a declared generating family and then varies higher connected products.

An operational state-to-geometry checklist

Give each generator a state-independent classical scale $s_A$ and define

$$x_A=\frac{\mathcal O_A[f_A]}{s_A}, \qquad \delta x_A=x_A-\langle x_A\rangle_\rho.$$

For an extensive holographic observable whose classical one-point function is $O(C_T)$, one normally takes $s_A=C_T$. More generally, $s_A$ carries the macroscopic scaling of the classical solution family; the measurement or reconstruction resolution belongs separately in $\epsilon_k$. For an observable with zero mean, use that independent classical normalization rather than dividing by $\langle x_A\rangle$.

The first test is mean-field consistency. The one-point profiles must reconstruct a smooth candidate $(g_{\mathrm{cl}},\phi_{\mathrm{cl}},\ldots)$ that obeys the nonlinear bulk constraints, matter equations, boundary conditions, and conserved charges. If two inequivalent saddles fit the same declared data at the same accuracy, the data have not selected one geometry.

The second test is joint sharpness. Introduce the symmetrized covariance matrix

$$\Sigma_{AB} =\frac12\left\langle \delta x_A\delta x_B+\delta x_B\delta x_A \right\rangle_\rho$$

and the connected joint cumulants. Because the generators need not commute, their ordering must be part of the observable protocol: Wightman, time ordered, contour ordered, or symmetrized. For definiteness, the equal-time diagnostic below uses fully symmetrized cumulants,

$$\kappa_{A_1\cdots A_k} =\left\langle \operatorname{Sym}\!\left( \delta x_{A_1}\cdots\delta x_{A_k} \right) \right\rangle_{\rho,c}.$$

In the declared normalized basis, a simple operational requirement is

$$\lambda_{\max}(\Sigma)\le\epsilon_2^2, \qquad \left|\kappa_{A_1\cdots A_k}\right|\le\epsilon_k \quad (k\ge3).$$

A standard factorizing semiclassical family has, at fixed order $k$, fixed species, fixed smearings, and fixed kinematics,

$$\kappa_{A_1\cdots A_k} =O\!\left(C_T^{\,1-k}\right), \qquad k\ge2.$$

This is a scaling target, not a theorem about arbitrary heavy states. It need not remain uniform when the number of insertions, the frequency, the observation time, or the number of species grows with $C_T$. The large-$N$ bulk criterion itself is conditional rather than proved in full generality Harlow 2018, §2.4, pp. 9–11, Open PDF.

The third test is effective-theory control. Bulk loops, higher-derivative terms, string or Kaluza–Klein excitations, local curvature, occupation-enhanced nonlinearities, and secular growth must all remain below the requested accuracy.

On a narrow screen, the table reflows into labeled cards; at intermediate widths, scroll horizontally without shrinking the text.

Operational state-to-geometry tests

Test	Input that must be reported	What passing licenses	What failure means
Mean-field consistency	smeared one-point profiles, charges, boundary conditions, and constraint residuals	a candidate classical solution	no solution, or the wrong solution, fits the declared means
Sharpness	covariance eigenvalues and relevant higher connected cumulants	one phase-space packet at the stated resolution	a broad state, mixture, or macroscopic superposition remains possible
Uniqueness within scope	comparison with all competing saddles in the same code sector	one solution for the chosen region and observable family	the inverse problem is unresolved even if each saddle is smooth
Bulk control	loop, derivative, curvature, localization, species, and time-window estimates	a semiclassical effective-theory description	the classical-looking mean lies outside the controlled expansion

These tests are complementary. Factorization without the equations of motion does not construct a spacetime. A mean configuration that solves the classical equations without small fluctuations can equally well be the average of distinct branches.

Coherent occupation separates a classical field from a backreacted metric

For canonically normalized, weakly interacting bulk normal modes, define

$$D(\{\alpha_n\}) =\exp\!\left[ \sum_n\left(\alpha_na_n^\dagger-\alpha_n^*a_n\right) \right], \qquad |\{\alpha_n\}\rangle=D(\{\alpha_n\})|0\rangle.$$

The state obeys $a_n|\{\alpha\}\rangle=\alpha_n|\{\alpha\}\rangle$. For one mode,

$$q=\frac{a+a^\dagger}{\sqrt{2\omega}}, \qquad p=-i\sqrt{\frac{\omega}{2}}(a-a^\dagger),$$

and the dimensionless canonical quadratures

$$Q=\sqrt{\omega}\,q=\frac{a+a^\dagger}{\sqrt2}, \qquad P=\frac{p}{\sqrt{\omega}}=\frac{a-a^\dagger}{i\sqrt2}, \qquad [Q,P]=i.$$

Then

$$\langle Q\rangle=\sqrt2\operatorname{Re}\alpha, \qquad \langle P\rangle=\sqrt2\operatorname{Im}\alpha,$$

while $\operatorname{Var}(Q)=\operatorname{Var}(P)=1/2$. Increasing $|\alpha|$ moves the center of the packet without widening it. This is the familiar coherent-state route from quantum occupation to a classical phase-space trajectory, and it is a concrete instance of the generalized-coherent-state large-$N$ limit developed by Yaffe 1982, pp. 407–435.

Let $\Omega_n=\omega_nL$ and, in the free-mode approximation, define

$$\bar n=\sum_n|\alpha_n|^2, \qquad \mathcal E_\alpha =L\langle H-E_{\mathrm{vac}}\rangle_\alpha =\sum_n\Omega_n|\alpha_n|^2.$$

For a fixed finite set of smooth modes with $\Omega_n=O(1)$, three regimes must be separated.

• If $1\ll\bar n\ll C_T$, the field is relatively sharp, but $\mathcal E_\alpha/C_T\ll1$. It is a classical matter wave on nearly fixed AdS, not an order-one new metric.
• If $\bar n=O(C_T)$, the packet is still relatively sharp and order-one global backreaction becomes possible. One must now solve the coupled nonlinear Einstein–matter equations.
• If the occupation, frequency, localization, or evolution time grows still faster, large $C_T$ alone gives no control. Collapse, high curvature, string-scale gradients, or secular effects must be tested directly.

The Euclidean Preparation and Lorentzian State Dictionaries page constructs the corresponding coherent eigenvalue relation at linearized saddle level; the explicit mode-space statement appears in Botta-Cantcheff, Martínez, and Silva 2016, §4.6, eqs. (4.33)–(4.35), Open PDF. It does not follow that an $O(C_T)$ excitation is merely a linear solution with small $1/N$ corrections. Schematically, a canonically normalized $m$-point tree coupling and a macroscopic mode amplitude scale as

$$g_m\sim C_T^{\,1-m/2}, \qquad \Phi_{\mathrm{cl}}\sim\sqrt{C_T}, \qquad g_m\Phi_{\mathrm{cl}}^{\,m-2}=O(1).$$

Large $C_T$ suppresses loops; it does not suppress the classical nonlinear saddle created by the macroscopic occupation. Moreover, sharply localized initial data can require sources that become too large for the perturbative construction Marolf et al. 2018, §2, eqs. (4)–(10), §3 before eq. (12), and §4.1, pp. 20–22, Open PDF.

Heavy energy is evidence, not yet a geometry

Use the vacuum-relative, dimensionless cylinder energy

$$\mathcal E_\rho =L\,\operatorname{Tr}\!\left[\rho(H-E_{\mathrm{vac}})\right].$$

For a primary state $|H\rangle$ created by an operator of dimension $\Delta_H$,

$$\mathcal E_H=\Delta_H.$$

On this page, backreaction-heavy means $\mathcal E_H=O(C_T)$. For a smooth profile spread over an AdS radius, the associated orthonormal-frame stress tensor has the dimensional scaling

$$L^d\langle T_{\hat\mu\hat\nu}\rangle_H=O(C_T),$$

and an order-one global metric deformation is possible. This energy scale is not necessary for localized strong gravity. For example, a small Schwarzschild–AdS black hole in $d>2$ obeys parametrically

$$\frac{\mathcal E}{C_T} \sim\left(\frac{r_h}{L}\right)^{d-2} \ll1$$

while its near-horizon gravity is order one, provided the classical window $\ell_{\mathrm{UV}}\ll r_h\ll L$ remains open Witten 1998, §2.3, preprint pp. 7–8, eqs. (2.6)–(2.9), Open PDF. Equivalently, a localized lump requires a separate compactness check $\varepsilon_{\mathrm{loc}}\sim G_{d+1}\delta E/R_{\mathrm{loc}}^{d-2}$; $d=2$ has the distinct BTZ threshold. The detailed derivation belongs to States, Geometries, and Radial Quantization.

An energy eigenstate has $\operatorname{Var}(\mathcal E)=0$, but this does not make the stress tensor or other light observables jointly sharp. Heavy-state connected correlators, spatial profiles, and competing saddle solutions must therefore be tested rather than inferred from $\Delta_H$.

Worked comparison at fixed excitation energy

For this worked comparison, use a neutral scalar heavy primary and neutral coherent modes, with no additional conserved charge active in the declared observable family. No charge projection is performed. Fix

$$\overline{\mathcal E}=O(C_T).$$

For the canonical state, use the vacuum-shifted partition function

$$Z_{\mathrm{exc}}(\beta) =\operatorname{Tr}\exp[-\beta(H-E_{\mathrm{vac}})].$$

The three mean energies can then be matched consistently by choosing

$$\Delta_H =\sum_n\Omega_n|\alpha_n|^2 =-L\,\partial_\beta\log Z_{\mathrm{exc}} =\overline{\mathcal E}.$$

This equality is deliberately weak: it does not match the spatial stress profile or any higher connected correlator. In a charged problem one must additionally specify chemical potentials or a fixed-charge ensemble and compare charge fluctuations. Projecting a coherent state onto a fixed charge generally destroys the independent Poisson-mode formulas used below.

On a narrow screen, the comparison reflows into one labeled card per state; at intermediate widths, scroll horizontally without shrinking the text.

Same mean excitation energy, different geometric evidence

State	State kind	Excitation-energy data	Other diagnostic data	Strongest licensed bulk claim
Heavy primary	pure; zero global von Neumann entropy	mean excitation energy equals the primary dimension; energy variance is zero	neutral charge is fixed in this example, but light connected correlators and spatial profiles are state dependent	a backreaction-heavy candidate, not a uniquely selected geometry
Coherent multimode state	pure; zero global von Neumann entropy	the mean is the mode-frequency-weighted occupation; the variance is its squared-frequency-weighted analogue for free modes	phase-space means and covariances are explicit; nonlinear constraints and local EFT controls remain to be checked	a classical field packet; a backreacted geometry only after the full checklist passes
Canonical thermal state	mixed; positive entropy away from a pure zero-temperature limit	the mean is the temperature derivative of the shifted partition function; the physical energy variance is temperature squared times the heat capacity	thermal connected correlators and possible coexistence of several saddles	an ensemble saddle for coarse observables only where one stable phase dominates

For independent coherent modes, each occupation number is Poisson distributed, which gives

$$\operatorname{Var}_\alpha(\mathcal E) =\sum_n\Omega_n^2|\alpha_n|^2.$$

If a fixed $O(1)$ frequency band carries $O(C_T)$ quanta, then $\sigma_{\mathcal E}/\overline{\mathcal E}=O(C_T^{-1/2})$. By contrast, a one-mode coherent state with mean occupation $\bar n=1$ and $\Omega=O(C_T)$ has the same heavy mean-energy scaling but order-one relative energy width and no parametrically sharp semiclassical field, so the frequency and occupation assumptions are indispensable.

For a well-defined canonical ensemble with $k_B=1$,

$$\operatorname{Var}_\beta(E)=T^2C_V.$$

When $C_V=O(C_T)$, $TL=O(1)$, and one stable phase dominates, the relative energy width is also $O(C_T^{-1/2})$. Near phase coexistence or criticality, the energy distribution can instead be broad or bimodal; between-saddle fluctuations can be $O(C_T^2)$. A thermal density matrix can therefore support one classical exterior saddle for coarse observables without identifying any particular pure microstate or its interior. Global entropy helps classify the states in the table, but it need not itself belong to the low-complexity observable family.

Typical microstates may reproduce selected thermal observables with entropically small variation, yet exact thermality does not follow. The distinction and its observable dependence are analyzed in Balasubramanian et al. 2008, §§2.1–2.2 and §3.1, Open PDF.

Adversarial checks: fixed means, different connected data

A one-mode squeezing example shows the first failure. Let $H_{\mathrm{osc}}=\omega(a^\dagger a+\tfrac12)$. For real $r>0$, define

$$S(r)=\exp\!\left[\frac r2\left(a^2-a^{\dagger2}\right)\right]$$

and take real $\alpha$. The states

$$|\psi_+\rangle=D(\alpha)S(r)|0\rangle, \qquad |\psi_-\rangle=D(\alpha)S(-r)|0\rangle$$

have the same declared linear means and the same oscillator energy,

$$\langle Q\rangle=\sqrt2\alpha, \qquad \langle P\rangle=0, \qquad \langle H_{\mathrm{osc}}\rangle =\omega\left(\alpha^2+\sinh^2r+\frac12\right).$$

Their covariance matrices in the dimensionless ordered basis $(Q,P)$ are

$$\Sigma_+ =\frac12\begin{pmatrix} e^{-2r} & 0\\ 0 & e^{2r} \end{pmatrix}, \qquad \Sigma_- =\frac12\begin{pmatrix} e^{2r} & 0\\ 0 & e^{-2r} \end{pmatrix}.$$

Thus the fixed data are the means of the declared list $\{Q,P,H_{\mathrm{osc}}\}$, not the expectation value of every operator: $Q^2$ and $P^2$ already distinguish the states. If $r=O(1)$ and $\alpha^2=O(C_T)$, both covariance matrices are small compared with the squared displacement $Q_{\mathrm{cl}}^2+P_{\mathrm{cl}}^2=2\alpha^2=O(C_T)$. The states can support the same leading classical field while differing in subleading quantum data. If $e^{2r}=O(C_T)$, each state has one macroscopically broad dimensionless quadrature—the broad direction is exchanged—and both fail a joint $(Q,P)$ sharpness test.

A stronger one-mode counterexample holds all linear and quadratic moments fixed. Let $N=a^\dagger a$, choose integers $n\ge m\ge2$, and define

$$|\chi_1\rangle=|n\rangle, \qquad |\chi_2\rangle =\frac{|n-m\rangle+|n+m\rangle}{\sqrt2}.$$

Orthogonality and the selection rules of $a$ give

$$\langle a\rangle_1=\langle a\rangle_2=0, \qquad \langle a^2\rangle_1=\langle a^2\rangle_2=0, \qquad \langle N\rangle_1=\langle N\rangle_2=n.$$

Consequently the two states have identical one-point functions for every linear and quadratic expression in this mode, including its free quadratic stress-tensor contribution and mean energy. Their number fluctuations nevertheless differ:

$$\operatorname{Var}_{\chi_1}(N)=0, \qquad \operatorname{Var}_{\chi_2}(N)=m^2.$$

Applying the same displacement $D(\alpha)$ to both states preserves equality of their linear and quadratic moments while giving the same nonzero field mean; higher connected moments remain different. Taking $n,m=O(C_T)$ makes the difference macroscopic. This is an exact moment-problem counterexample in the free-mode model. A macroscopic holographic realization would require a nonlinear completion, and the example is not, by itself, evidence for distinct black-hole interiors.

What the criterion licenses—and what it cannot

After specifying $\mathcal H_{\mathrm{code}}$, $\mathcal G_{\mathrm{coarse}}$, its product order, a region, a time window, and tolerances, the checklist supports the following conditional statement:

Within the assumed holographic dictionary and bulk effective theory, the state is represented by one semiclassical saddle for the declared observables to the stated accuracy.

The uncertainty is not captured by one universal power of $1/N$. Depending on the observable, it includes state fluctuations often of order $C_T^{-1/2}$, loop corrections often of order $C_T^{-1}$, occupation-enhanced classical nonlinearities that must be solved rather than expanded away, derivative or string corrections, and errors that can grow with time.

The criterion does not establish a unique microscopic boundary state, a horizon, exact thermality, or an interior. Euclidean Saddles and Hawking–Page adds the competition and stability of thermal saddles. Conical Defects, Orbifolds, and Heavy States treats the special AdS$_3$/CFT$_2$ heavy-state thresholds. Interior Reconstruction, Recovery, and Scrambling addresses the much stronger interior claim.

Common pitfalls

Equating heavy with geometric. Heavy energy can make AdS-scale backreaction possible, but it says nothing by itself about smoothness, sharpness, or uniqueness. Localized strong gravity can also occur below the global heavy scale.

Treating large $C_T$ as a ban on nonlinearities. Large $C_T$ suppresses quantum loops in a controlled bulk sector. A macroscopic occupation can make classical tree nonlinearities order one, so the nonlinear saddle must still be solved.

Matching energy instead of matching geometry data. Equal excitation energy does not imply equal stress profiles, charges, boundary conditions, or connected correlators. Energy is one constraint among many.

Dividing a fluctuation by a vanishing mean. A zero one-point function can be perfectly classical—for example, a classical trajectory at a turning point. Normalize by an independently declared classical scale, and compare the resulting dimensionless fluctuation with the measurement tolerance.

Calling generators an algebra. Light single-trace operators are a useful generating family, but products introduce multi-trace observables. Fixing generator one-points does not fix their connected products; fixing every element of the full algebra would.

Identifying a thermal saddle with a pure microstate. A stable saddle can reproduce coarse ensemble observables. That conclusion neither selects an individual pure state nor determines its interior.

Exercises

Let a coherent packet occupy a fixed set of modes with $\Omega_n=O(1)$ and total occupation $\bar n=C_T^\gamma$. Classify its sharpness relative to the occupied phase-space displacement norm and its global backreaction for $0<\gamma<1$, $\gamma=1$, and $\gamma>1$.

Solution

The displacement norm scales as $\sqrt{\bar n}$ while the coherent covariance stays at vacuum size. Noise relative to that norm therefore scales as $\bar n^{-1/2}=C_T^{-\gamma/2}$ for every $\gamma>0$. A particular quadrature can still have zero mean because of its phase, which is why the displacement norm—not division by that quadrature mean—is the correct comparison. Meanwhile

$$\frac{\mathcal E_\alpha}{C_T}=O\!\left(C_T^{\gamma-1}\right).$$

For $0<\gamma<1$, this is a classical matter field with parametrically small global backreaction. For $\gamma=1$, global backreaction can be order one and the coupled nonlinear saddle is required. For $\gamma>1$, the excitation exceeds the AdS-scale gravitational normalization; large $C_T$ alone gives no controlled interpretation. In every regime, unusually high frequencies or tight localization require separate checks.

For independent coherent modes, derive $\operatorname{Var}(\mathcal E)=\sum_n\Omega_n^2|\alpha_n|^2$. Compare it with a stable canonical state for which $\overline{\mathcal E}=O(C_T)$, $C_V=O(C_T)$, and $TL=O(1)$. Why does neither energy-width result establish a geometry?

Solution

Each coherent occupation $N_n$ is Poisson distributed, so $\operatorname{Var}(N_n)=|\alpha_n|^2$. Independent modes have no cross covariance, and therefore

$$\operatorname{Var}(\mathcal E) =\operatorname{Var}\!\left(\sum_n\Omega_nN_n\right) =\sum_n\Omega_n^2|\alpha_n|^2.$$

For a fixed $O(1)$ frequency band with $O(C_T)$ total occupation, this variance is $O(C_T)$ and the relative width is $O(C_T^{-1/2})$. The canonical identity gives

$$\operatorname{Var}(\mathcal E)=L^2T^2C_V=O(C_T),$$

so its relative width has the same scaling in a single stable phase. These statements test only the energy distribution. The two states still differ in purity, phase-space data, connected correlators, and possibly their mean stress profiles; all remaining checklist tests are still required.

Using the stated squeezing convention, derive the means, energy, and covariance matrices of $D(\alpha)S(\pm r)|0\rangle$ in the dimensionless $(Q,P)$ basis. Explain separately what follows when $r=O(1)$ and when $e^{2r}=O(C_T)$ with $\alpha^2=O(C_T)$.

Solution

The transformations

$$D^\dagger(\alpha)aD(\alpha)=a+\alpha, \qquad S^\dagger(r)aS(r)=a\cosh r-a^\dagger\sinh r$$

give the common means $\langle Q\rangle=\sqrt2\,\alpha$, $\langle P\rangle=0$, and

$$\langle H_{\mathrm{osc}}\rangle =\omega\left(\alpha^2+\sinh^2r+\frac12\right).$$

They also give $\operatorname{Var}(Q)=e^{-2r}/2$ and $\operatorname{Var}(P)=e^{2r}/2$ for $+r$, with the two values exchanged for $-r$. For fixed $r$, both states are narrow compared with the squared displacement $2\alpha^2=O(C_T)$ and may share a leading classical field. When $e^{2r}=O(C_T)$, each covariance matrix has an $O(C_T)$ eigenvalue, so neither state is sharp for the full $(Q,P)$ family.

Verify the equal moments and unequal number variances of $|\chi_1\rangle$ and $|\chi_2\rangle$. Which claim survives after $Q^2$ and $P^2$ are added to the measured family, and which claim fails when $N^2$ is added?

Solution

The number eigenstate immediately gives $\langle N\rangle_1=n$ and $\langle N^2\rangle_1=n^2$. In $|\chi_2\rangle$, the two number values occur with equal probability, so

$$\langle N\rangle_2 =\frac{(n-m)+(n+m)}2=n,$$ $$\langle N^2\rangle_2 =\frac{(n-m)^2+(n+m)^2}{2} =n^2+m^2.$$

For $m\ge2$, neither $a$ nor $a^2$ connects the two components of $|\chi_2\rangle$, so their expectations vanish just as in $|n\rangle$. Hence all linear and quadratic moments—including $Q^2$ and $P^2$—agree, and the states remain equivalent for that refined mean-data family. Adding $N^2$ exposes the difference $\operatorname{Var}_2(N)-\operatorname{Var}_1(N)=m^2$. The states can share a leading geometry only if this newly resolved connected datum lies below the declared tolerance.

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

• Almheiri, Ahmed, Xi Dong, and Daniel Harlow. “Bulk Locality and Quantum Error Correction in AdS/CFT.” Journal of High Energy Physics 04 (2015): 163. doi:10.1007/JHEP04(2015)163. arXiv:1411.7041 (Open PDF).
• Balasubramanian, Vijay, Bartłomiej Czech, Veronika E. Hubeny, Klaus Larjo, Mukund Rangamani, and Joan Simón. “Typicality versus Thermality: An Analytic Distinction.” General Relativity and Gravitation 40 (2008): 1863–1890. doi:10.1007/s10714-008-0606-8. arXiv:hep-th/0701122 (Open PDF).
• Botta-Cantcheff, Marcelo, Pedro J. Martínez, and Guillermo A. Silva. “On Excited States in Real-Time AdS/CFT.” Journal of High Energy Physics 02 (2016): 171. doi:10.1007/JHEP02(2016)171. arXiv:1512.07850 (Open PDF).
• Harlow, Daniel. “TASI Lectures on the Emergence of Bulk Physics in AdS/CFT.” Proceedings of Science TASI2017 (2018): 002. doi:10.22323/1.305.0002. arXiv:1802.01040 (Open PDF).
• Marolf, Donald, Onkar Parrikar, Charles Rabideau, Ali Izadi Rad, and Mark Van Raamsdonk. “From Euclidean Sources to Lorentzian Spacetimes in Holographic Conformal Field Theories.” Journal of High Energy Physics 06 (2018): 077. doi:10.1007/JHEP06(2018)077. arXiv:1709.10101 (Open PDF).
• Penedones, João. “TASI Lectures on AdS/CFT.” In New Frontiers in Fields and Strings, 75–136. Singapore: World Scientific, 2017. doi:10.1142/9789813149441_0002. arXiv:1608.04948 (Open PDF).
• Witten, Edward. “Anti-de Sitter Space, Thermal Phase Transition, and Confinement in Gauge Theories.” Advances in Theoretical and Mathematical Physics 2 (1998): 505–532. doi:10.4310/ATMP.1998.v2.n3.a3. arXiv:hep-th/9803131 (Open PDF).
• Yaffe, Laurence G. “Large N Limits as Classical Mechanics.” Reviews of Modern Physics 54 (1982): 407–435. doi:10.1103/RevModPhys.54.407.