States, Geometries, and Radial Quantization

Boundary radial quantization maps a primary of dimension $\Delta$ to a state of cylinder energy $\Delta$, which the AdS dictionary identifies with global energy $E=\Delta/L$. A low-occupation state is a quantum bulk excitation, not a classical geometry. A semiclassical field or metric requires appropriate large occupation or heavy scaling together with suppressed connected fluctuations on a declared observable algebra.

Required background. Anti-de Sitter geometry supplies global time, and the state–operator correspondence supplies radial states. Helpful background. The flat-space-to-cylinder map derives the cylinder Hamiltonian.

From dilatation weight to global-AdS energy

In Euclidean boundary space write $r=e^\tau$. A Weyl transformation maps

$$\mathrm ds_{\mathbb R^d}^2=\mathrm dr^2+r^2\mathrm d\Omega_{d-1}^2 =e^{2\tau}\left(\mathrm d\tau^2+\mathrm d\Omega_{d-1}^2\right)$$

to the cylinder. Translation in $\tau$ is generated by the dilatation operator $D$. A primary inserted at the origin produces

$$\lvert\mathcal O\rangle =\lim_{r\to0}\mathcal O(r,\Omega)\lvert0\rangle, \qquad D\lvert\mathcal O\rangle=\Delta\lvert\mathcal O\rangle.$$

After continuation to Lorentzian cylinder time and matching its dimensionless time to global AdS time, the Hamiltonians obey

$$H_{\mathrm{AdS}}=\frac{D}{L}, \qquad E_{\mathcal O}=\frac{\Delta}{L}.$$

Descendants add integer energies, matching global normal modes $\omega L=\Delta+2n+\ell$. This spectral agreement and its state interpretation appear in Witten 1998, §§2–3 and Banks et al. 1998, §§2–4. It is a kinematic dictionary check; it does not say that every energy eigenstate has a classical bulk description.

One particle, a coherent field, and a geometry

A unit-normalized single-trace primary acting once creates a one-particle state at leading large $N$. Its field expectation value may vanish while its two-point function records one quantum. By contrast, a coherent state of a weakly coupled mode,

$$\lvert\alpha\rangle =e^{-\lvert\alpha\rvert^2/2} e^{\alpha a^\dagger}\lvert0\rangle,$$

has

$$\langle a\rangle=\alpha, \qquad \frac{\Delta N_a}{\langle N_a\rangle} =\frac{1}{\lvert\alpha\rvert}.$$

For $\lvert\alpha\rvert\gg1$, the relative fluctuation is small and the bulk field follows a classical solution to the accuracy set by interactions and $1/N$. If its dimensionless energy $EL$ is parametrically below $C_T\sim L^{d-1}/G_{d+1}$, it propagates on approximately fixed AdS. When energy is of order $C_T/L$, backreaction can be order one and a semiclassical metric may be needed.

A classical geometry requires more than large energy:

• a family of one-point functions consistent with classical constraints;
• connected fluctuations small compared with products on the chosen coarse algebra;
• controlled higher-derivative and bulk-loop corrections;
• a state, not merely an ensemble average, unless the claim is explicitly coarse-grained;
• sufficient observables to distinguish competing saddles at the claimed accuracy.

The distinction between normalizable state data and nonnormalizable sources was already central to early state reconstructions Balasubramanian et al. 1999, §§2–3.

First application: a primary and a coherent excitation

For a scalar primary of dimension $\Delta=O(1)$ as $N\to\infty$, $\mathcal O\lvert0\rangle$ has global energy $\Delta/L$ and is a perturbative quantum. A coherent superposition with occupation $n\gg1$ but $n\ll C_T$ has a classical scalar profile with negligible metric backreaction. A heavy primary with $\Delta\sim C_T$ can source order-one geometry, but its one-point data need not select a unique smooth saddle. The comparison separates energy scaling from the fluctuation evidence required for a geometry.

The same energy expectation can occur in a canonical thermal density matrix. The mixture has thermal entropy and KMS correlators, unlike a pure coherent state. Energy alone therefore cannot distinguish a black-hole ensemble, a pure heavy state, and a coherent wavepacket.

Adversarial check: macroscopic superpositions

Let $\lvert g_1\rangle$ and $\lvert g_2\rangle$ be semiclassical states peaked on macroscopically distinct metrics with negligible overlap, and form

$$\lvert\Psi\rangle=\frac{\lvert g_1\rangle+\lvert g_2\rangle}{\sqrt2}.$$

For an observable whose interference terms are negligible, its one-point function is the average of the two branches. That average need not solve the nonlinear Einstein equation and does not define a geometry on which all higher correlators factorize. Large variance or branch-sensitive observables reveal the failure.

The strongest statement available from mean fields alone is a coarse expectation-value profile. A unique semiclassical geometry requires small relative fluctuations and branch stability for the declared observables. State dependence and interior reconstruction require additional arguments beyond this chapter.

Controlled limits and handoff

The energy map $E=\Delta/L$ is exact once the cylinder representative is fixed, but interpreting a state as a geometry is a large-$N$, coarse-observable statement. Euclidean Preparation and Lorentzian State Dictionaries constructs explicit normalizable states, and Heavy States, Coherent States, and Semiclassical Geometries turns fluctuation suppression into a quantitative geometry criterion.

Exercise

At what occupation does a coherent scalar begin to backreact strongly if each quantum has energy $O(L^{-1})$ and $C_T\sim L^{d-1}/G_{d+1}$?

Solution

The AdS curvature scale carries energy of order $C_T/L$. A coherent mode with $n$ quanta has energy $n/L$, so order-one gravitational backreaction begins parametrically at $n\sim C_T$. The estimate does not decide whether the resulting geometry is smooth or a black hole; that depends on localization, charges, and dynamics.

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, Per Kraus, Albion Lawrence, and Sandip P. Trivedi. “Holographic Probes of Anti-de Sitter Spacetimes.” Physical Review D 59 (1999): 104021. arXiv. DOI.
• Banks, Tom, Michael R. Douglas, Gary T. Horowitz, and Emil Martinec. “AdS Dynamics from Conformal Field Theory.” (1998). arXiv.
• Witten, Edward. “Anti-de Sitter Space and Holography.” Advances in Theoretical and Mathematical Physics 2 (1998): 253–291. arXiv. DOI.