Heavy States, Coherent States, and Semiclassical Geometries

A boundary state represents a single semiclassical bulk geometry only when the observables used to reconstruct that geometry are sharply peaked and mutually consistent. Large energy is necessary for order-one gravitational backreaction but is not sufficient: a heavy primary, a coherent state, and a thermal mixture can have the same mean energy while encoding very different fluctuations, entropy, and interior information. Here the semiclassical parameter is $C_T\sim L^{d-1}/G_N\sim N^2$, and all comparisons refer to a specified low-energy, single-trace algebra.

Required background. Weakly Coupled Bulk Fields from Connected Correlators supplies the large-$N$ factorization criterion for a perturbative bulk. States, Geometries, and Radial Quantization supplies the cylinder-energy and state–operator maps.

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

What it means for a state to define a geometry

Let $\mathcal A_{\rm coarse}$ be the boundary algebra dual to bulk fields with wavelengths above a chosen cutoff in a causal region. A candidate state $\rho$ defines a semiclassical geometry for this algebra when three conditions hold.

First, its stress-tensor expectation value has the backreaction scaling

$$\langle T_{\mu\nu}\rangle_\rho=O(C_T),$$

when the geometry differs from vacuum AdS at order one. Second, the connected fluctuations of the normalized collective observables are small. If $X$ is scaled so that $\langle X\rangle=O(1)$, then a standard coherent semiclassical family has

$$\langle X^k\rangle_{c}=O\!\left(C_T^{1-k}\right), \qquad k\ge2.$$

Third, the one-point data solve the bulk constraint equations with boundary conditions and charges appropriate to the state. Here $X$ is normalized by its extensive $O(C_T)$ scale, so its variance is $O(C_T^{-1})$ and its relative standard deviation is $O(C_T^{-1/2})$. Factorization without the constraints does not create a spacetime, and the constraints without small fluctuations describe only an ensemble mean.

These conditions are algebra-relative. Two microstates can be indistinguishable to $\mathcal A_{\rm coarse}$ yet differ on very high-point, nonlocal, or exponentially precise probes. This distinction underlies the difference between typicality and exact thermality emphasized by Balasubramanian et al. 2008, §§2–4.

Coherent states and the large-N scaling test

For weakly interacting normal modes $a_n$, a bulk coherent state is

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

It obeys $a_n|\alpha\rangle=\alpha_n|\alpha\rangle$. If $|\alpha_n|=O(\sqrt{C_T})$, then the classical field and its energy are order one in gravitational units while quadrature fluctuations remain $O(1)$; their relative size is $O(C_T^{-1/2})$. Euclidean cap sources construct precisely such states at linear order Skenderis and van Rees 2008, §§2–3, with interactions producing controlled $1/N$ corrections. The state is semiclassical only while curvatures, occupation densities, and string-scale gradients remain within the bulk effective theory.

A primary state $|H\rangle$ with $\Delta_H=O(C_T)$ is also heavy because

$$E_H=\frac{\Delta_H}{L}=O\!\left(\frac{C_T}{L}\right).$$

But the scaling of $E_H$ says nothing by itself about connected correlators of other single-trace operators. Symmetry may fix a stress-tensor one-point function while leaving higher moments broad. Heavy-state correlators must therefore be tested, not assumed, before assigning a smooth geometry.

First application: three states at the same mean energy

Fix a mean cylinder energy $E=O(C_T/L)$ and compare:

State	Sharp data	Fluctuation or entropy diagnostic	Geometric conclusion
Heavy primary $\lvert H\rangle$	Exact energy and global charges	Other connected correlators are state dependent	Energy alone does not select a geometry
Coherent multi-trace state $\lvert\alpha\rangle$	Bulk field quadratures and energy are relatively sharp	Relative fluctuations scale as $C_T^{-1/2}$	A classical exterior is justified within the effective-field-theory cutoff
Canonical thermal state $\rho_\beta$	Stationary one-point functions fixed by $\beta$	$S(\rho_\beta)>0$ and $\operatorname{Var}(E)=T^2 C_V$	A thermal saddle describes ensemble observables, not a specified pure microstate

The energies can be matched by choosing $\Delta_H=EL$, $\sum_n\omega_n|\alpha_n|^2=E$ to leading order, and $\beta$ so that $-\partial_\beta\log Z=E$. The three states can nevertheless disagree in a scalar one-point function, in connected four-point functions, and in entropy. A common mean stress tensor is evidence for a common coarse exterior only after its fluctuations and the relevant constraint data have also been matched.

Adversarial check: identical means, different higher moments

Consider one macroscopically occupied mode with $\alpha=O(\sqrt{C_T})$ and real squeezing parameter $r$. The two states $D(\alpha)S(r)|0\rangle$ and $D(\alpha)S(-r)|0\rangle$ have the same field mean $\langle a\rangle=\alpha$ and the same oscillator energy

$$\langle H\rangle=\omega\left(|\alpha|^2+\sinh^2r+\frac12\right).$$

Their position and momentum variances are interchanged:

$$\operatorname{Var}(q)=\frac{e^{-2r}}{2\omega}, \qquad \operatorname{Var}(p)=\frac{\omega e^{2r}}{2}.$$

Thus the declared one-point data and total energy agree exactly while connected two-point data differ. If $e^{2|r|}$ scales with $C_T$, one state has a macroscopically broad quadrature and fails the sharpness condition. This adversarial construction rules out any criterion based only on mean fields and shows why a small set of exterior one-point functions cannot select a unique interior.

Controlled limits and handoff

The analysis establishes a semiclassical geometry only for the declared coarse algebra and only to an accuracy set by $1/N$, the derivative expansion, and the state’s fluctuation bounds. It does not prove that a heavy pure state has a horizon, that an ensemble saddle captures exponentially fine observables, or that equal exterior data imply equal interiors. The boundary distinction between pure heavy states and thermal ensembles was already visible in early AdS dynamics Banks et al. 1998, §§3–5. Later chapters will add black-hole saddles, entanglement wedges, and interior reconstruction; this page supplies the state-quality tests those constructions require.

Exercises

For a harmonic oscillator coherent state, show that $\operatorname{Var}(q)/\langle q\rangle^2=O(C_T^{-1})$ at a time when $\langle q\rangle=O(\sqrt{C_T})$.

Solution

With $q=(a+a^\dagger)/\sqrt{2\omega}$, a coherent state has $\operatorname{Var}(q)=1/(2\omega)$, independent of $\alpha$. At a phase where $\langle q\rangle=\sqrt{2/\omega}\,\operatorname{Re}\alpha=O(\sqrt{C_T})$, the ratio is $O(C_T^{-1})$.

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

• 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/s10714-008-0617-0. arXiv:hep-th/0701122.
• Banks, Tom, Michael R. Douglas, Gary T. Horowitz, and Emil Martinec. “AdS Dynamics from Conformal Field Theory.” 1998. arXiv:hep-th/9808016.
• Skenderis, Kostas, and Balt C. van Rees. “Real-Time Gauge/Gravity Duality.” Physical Review Letters 101 (2008): 081601. doi:10.1103/PhysRevLett.101.081601. arXiv:0805.0150.