States, Geometries, and Radial Quantization

Radial quantization fixes the excitation spectrum on the boundary cylinder: a primary of dimension $\Delta$ lies $\Delta/L$ above the cylinder vacuum when the sphere has radius $L$. Holography identifies the same generator with global-AdS time evolution. This exact kinematic map organizes bulk particles and their normal modes, but it does not turn every boundary state—or even every high-energy state—into one classical spacetime. A classical field needs a suitably occupied and sharp state; a classical metric also needs controlled backreaction, gravitational constraints, and small fluctuations for a declared set of observables.

Required background. Anti-de Sitter geometry supplies global time and the radius-$L$ boundary cylinder, while the state–operator correspondence supplies radial states. Helpful background. The flat-space-to-cylinder map derives the cylinder Hamiltonian and its vacuum-energy qualification; coherent states and the classical limit supplies the oscillator construction used below.

Cylinder excitation energy and global-AdS energy

Let $d$ be the boundary spacetime dimension, so the bulk is $(d+1)$-dimensional, and let $\Omega\in S^{d-1}$ label a point on the unit sphere. Choose the Euclidean cylinder representative with sphere radius $L$. In flat-space polar coordinates, set

$$r=L e^{\tau_E/L}.$$

Then

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

The coordinate change is followed by a Weyl transformation. For a scalar primary at separated points,

$$\mathcal O_{\mathrm{cyl}}(\tau_E,\Omega) =e^{\Delta\tau_E/L}\mathcal O_{\mathbb R^d}(r\Omega).$$

The dimensionless dilatation generator $D$ translates $\tau_E/L$. The origin is the past end $\tau_E\to-\infty$, and the renormalized insertion defines

$$\lvert\mathcal O\rangle \equiv\mathcal O_{\mathbb R^d}(0)\lvert0\rangle, \qquad D\lvert\mathcal O\rangle=\Delta\lvert\mathcal O\rangle.$$

Analytic continuation of $\tau_E$ gives Lorentzian cylinder time. Let $E_{\mathrm{vac}}$ be the absolute cylinder vacuum energy in the chosen anomaly and counterterm convention. The physical Lorentzian Hamiltonian, with units of inverse length, obeys

$$H_{\mathrm{cyl}}-E_{\mathrm{vac}}\mathbf1=\frac{D}{L}, \qquad \delta E_{\mathcal O} \equiv E_{\mathcal O}-E_{\mathrm{vac}} =\frac{\Delta}{L}.$$

The subtraction matters: the Weyl map can produce a cylinder Casimir term Penedones 2017, §2.4, pp. 9–10, footnote i, Open PDF. In four dimensions its finite value is scheme dependent because an allowed $R^2$ counterterm shifts it Assel et al. 2015, §1, pp. 2–3, eqs. (1.2)–(1.6), Open PDF. The state–operator map fixes the difference $\delta E$, not a universal value of $E_{\mathrm{vac}}$.

In a proposed AdS/CFT duality, the boundary cylinder generator is matched to the renormalized global-AdS charge. With physical global time $t=L t_{\mathrm{global}}$,

$$\left(E_{\mathrm{AdS},\mathcal O}-E_{\mathrm{AdS},0}\right)L =\Delta.$$

This is an exact representation-theoretic statement conditional on the Hilbert-space dictionary; it is not an independent proof of that dictionary Witten 1998, “Anti-de Sitter Space and Holography,” §3.3, preprint pp. 34–35, Open PDF.

A generic descendant at level $k$ has excitation energy

$$\delta E_k L=\Delta+k.$$

For a free bulk scalar, or a linearized leading-large-$N$ bulk field, of mass $m^2L^2=\Delta(\Delta-d)$ in standard quantization, separation into a radial node number $n$ and sphere angular momentum $\ell$ gives

$$\omega_{n\ell}L=\Delta+2n+\ell, \qquad n,\ell=0,1,2,\ldots,$$

so this scalar sector realizes $k=2n+\ell$. The last formula is a linearized scalar normal-mode result, not a universal spectrum for arbitrary spin or a shortened conformal multiplet Hamilton et al. 2006, §2.1, p. 4, eq. (4), Open PDF.

Normalizable modes are state data

The state/source distinction can be seen directly near the AdS boundary. Let $z\to0$ be a Fefferman–Graham defining coordinate and let $x$ denote boundary coordinates. For a scalar in standard quantization, away from logarithmic cases, write schematically

$$\phi(z,x) =z^{d-\Delta}J(x)+z^\Delta A(x)+\cdots, \qquad \Delta>\frac d2.$$

The boundary condition holds the leading coefficient $J$ fixed; $J$ is the source for $\mathcal O$. A normalizable excitation changes the subleading coefficient $A$ while leaving the source fixed, often at $J=0$. At a classical saddle, the renormalized one-point function is proportional to $A$, with a proportionality and local terms fixed by the bulk action and counterterm convention. This is the Lorentzian state/source split developed explicitly in Balasubramanian et al. 1999, §2.2, pp. 5–7, especially eqs. (12)–(17), Open PDF.

Three qualifications prevent a common overinterpretation.

• $A(x)$ must satisfy the bulk equations, constraints, regularity conditions, and the chosen boundary condition; it is not arbitrary boundary data.
• One-point data do not specify an entire quantum state. A state vector or density operator also fixes connected correlators and, when relevant, superselection or global-sector data.
• In alternate or mixed quantization the source/response assignment changes. Boundary Conditions, Alternate Quantization, and Deformations owns that change of dictionary.

The next page constructs states whose normalizable coefficients are selected by Euclidean caps. Here the only needed conclusion is that fixing the theory and its sources still leaves genuine state data.

One quantum and a coherent bulk field

A state–operator map alone does not make a primary a perturbative bulk particle. The additional input is a unit-normalized, light single-trace primary in a generalized-free large-$N$ sector—one in which connected correlators factorize at leading order—dual to a weakly coupled bulk field. Large-$N$ factorization then gives a Fock-like sector in which single-trace modes create one-particle states and multi-trace modes create multiparticle states Penedones 2017, §2.8, especially eqs. (44)–(45), Open PDF.

Consider one normalizable scalar mode of frequency $\omega$ with

$$[a,a^\dagger]=1, \qquad N_a=a^\dagger a.$$

The state $a^\dagger\lvert0\rangle$ carries one quantum and vacuum-subtracted energy $\omega$. It is not a classical wave: $\langle a\rangle=0$, and its field quadratures have quantum-scale fluctuations.

The coherent state

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

For this linearized harmonic mode, the following identities are exact:

$$\begin{aligned} \langle a\rangle_\alpha&=\alpha,\\ \bar n\equiv\langle N_a\rangle_\alpha&=\lvert\alpha\rvert^2,\\ \operatorname{Var}_\alpha(N_a)&=\bar n,\\ \sigma_{N_a}\equiv\sqrt{\operatorname{Var}_\alpha(N_a)}&=\sqrt{\bar n},\\ \frac{\sigma_{N_a}}{\bar n}&=\frac1{\sqrt{\bar n}}, \qquad \bar n>0,\\ \delta E_\alpha L&=\bar n\,(\omega L). \end{aligned}$$

For the quadrature $Q=(a+a^\dagger)/\sqrt2$, $\operatorname{Var}(Q)=1/2$ while its mean is $O(\!\sqrt{\bar n})$ at a phase where that mean is nonzero. The conjugate quadrature also retains vacuum-scale variance. Thus the quadrature standard deviation divided by the phase-space displacement is $O(\!\bar n^{-1/2})$; equivalently, covariance divided by displacement squared is $O(\!\bar n^{-1})$. This—not small relative number variance by itself—is the field-classicality test: a high-occupation number eigenstate has no such phase-space displacement. Finite bulk interactions introduce mode mixing and coupling-, $1/N$-, occupation-, and time-dependent corrections to this free-mode description. In the free or linearized scalar construction, suitable Euclidean source profiles prepare generally multimode coherent states at leading large $N$ Botta-Cantcheff, Martínez, and Silva 2016, §4.6, eqs. (4.33)–(4.35), but the preparation contour belongs to the next page.

Backreaction depends on energy and localization

Let $C_T$ denote the coefficient of the stress-tensor two-point function in a declared normalization. At leading classical order in a two-derivative Einstein bulk,

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

where the positive numerical factor $\kappa_d$ depends on the $C_T$ and gravitational-action conventions Penedones 2017, §3.3, p. 30, eqs. (118)–(121), Open PDF. For a smooth lowest global mode whose size is $O(L)$, the order-one backreaction parameter scales as

$$\varepsilon_{\mathrm{global}} \sim\frac{G_{d+1}\,\delta E}{L^{d-2}} \sim\frac{\delta E L}{C_T}.$$

Consequently a coherent global mode has a useful fixed-background window

$$1\ll\bar n, \qquad \bar n\,(\omega L)\ll C_T:$$

the matter field is classical while its metric backreaction is parametrically small.

For parametric control, read this as a double-scaling family: $C_T\to\infty$ while $\bar n\to\infty$ more slowly, so $\bar n(\omega L)/C_T\to0$, commonly with $\omega L=O(1)$. The approximation also requires gradients and occupancies within the bulk effective theory and times short enough that occupation-enhanced interactions, secular growth, or dephasing have not become important.

Total energy is not a universal local-gravity test. For a roughly spherical bulk lump of proper size $R_{\mathrm{loc}}\ll L$ in $(d+1)$ dimensions with $d>2$,

$$\varepsilon_{\mathrm{loc}} \sim\frac{G_{d+1}\,\delta E}{R_{\mathrm{loc}}^{d-2}} \sim\frac{\delta E L}{C_T} \left(\frac{L}{R_{\mathrm{loc}}}\right)^{d-2}.$$

A controlled small AdS-Schwarzschild black hole is the decisive counterexample. If $\ell_{\mathrm{UV}}$ is the largest microscopic bulk length, choose $\ell_{\mathrm{UV}}\ll r_h\ll L$; then

$$\frac{\delta E L}{C_T} \sim\left(\frac{r_h}{L}\right)^{d-2} \left[1+O\!\left(\frac{r_h^2}{L^2}\right)\right] \ll1,$$

while the near-horizon backreaction is order one Witten 1998, “Thermal Phase Transition and Confinement,” §2.3, preprint pp. 7–8, eqs. (2.6)–(2.9), Open PDF. Energy of order $C_T/L$ therefore makes order-one global backreaction possible; lower energy does not exclude strong gravity after localization.

The $d=2$ case has the distinct BTZ threshold structure and is not described by this small-lump power law.

The following comparison keeps the supported conclusion separate from the tempting overclaim. Here $O(1)$ and $O(C_T)$ refer to scaling as $C_T\to\infty$, not to a numerical threshold in one fixed theory.

Bulk-interpretation evidence

State and large-CT scaling	Observable evidence	Licensed claim and limit
One light single-trace quantum, δE L = ωL = O(1)	One-particle occupation; no macroscopic field mean	A perturbative bulk quantum, not a classical field or metric
Smooth coherent mode, 1 ≪ n̄ and n̄(ωL) ≪ CT	An O(√n̄) phase-space displacement with vacuum-scale covariance in both canonical quadratures, within the controlled time window	A classical matter wave on nearly fixed AdS, not an order-one new geometry
State with δE L = O(CT), or with εloc = O(1)	Backreaction can be order one	A geometry candidate only; energy does not establish smoothness, uniqueness, or semiclassical sharpness

When a state can support one geometry

Choose a coarse observable algebra $\mathcal A_{\mathrm{coarse}}$: a declared set of gauge-invariant, low-energy boundary observables used to infer a particular bulk region at a stated resolution. Examples include suitably smeared one-point functions and connected correlators of the stress tensor and other light single-trace operators.

A state can support one semiclassical geometry for that algebra only if:

• the one-point data are compatible with one bulk solution, including its constraints, charges, boundary conditions, and matter equations;
• covariances and higher connected fluctuations lie below the claimed resolution;
• bulk loops, higher-derivative terms, string-scale gradients, and finite-$N$ effects are controlled; and
• the proposed geometry is unique only up to diffeomorphism and only in the declared region and observable algebra.

For a quantum state $\rho$—a density operator, with a pure state represented by $\rho=\lvert\psi\rangle\langle\psi\rvert$—and a Hermitian observable $X$ with a nonzero classical scale $x_{\mathrm{cl}}$, one elementary sharpness test is

$$\frac{\operatorname{Var}_\rho(X)}{x_{\mathrm{cl}}^2}\ll1.$$

One should not divide by $\langle X\rangle^2$ when the classical mean vanishes; in that case use an independently declared classical scale or a connected-cumulant bound. The later Heavy States, Coherent States, and Semiclassical Geometries page develops the full algebra-relative hierarchy. Both a pure vector and a mixed density operator are quantum states; an ensemble-averaged or coarse saddle is a description of selected observables and need not identify one underlying branch. A thermal density matrix can share the same mean energy and even the same simple one-point functions as a pure state while differing in entropy and connected correlators, so energy alone cannot identify a particular pure state or establish one classical geometry.

Adversarial check: a macroscopic superposition

Let $\lvert g_1\rangle$ and $\lvert g_2\rangle$ be individually sharp semiclassical states associated with macroscopically distinct branches, and write $s=\langle g_1\vert g_2\rangle$. The correctly normalized superposition is

$$\lvert\Psi\rangle =\frac{\lvert g_1\rangle+\lvert g_2\rangle} {\sqrt{2+2\operatorname{Re}s}}.$$

Choose a Hermitian, gauge-invariant boundary observable $X\in\mathcal A_{\mathrm{coarse}}$ that distinguishes the branches. Let

$$x_i=\langle g_i\vert X\vert g_i\rangle, \qquad \sigma_i^2=\operatorname{Var}_{g_i}(X).$$

If $s$ and the off-diagonal matrix elements of $X$ and $X^2$ are negligible at the claimed resolution, then

$$\begin{aligned} \langle X\rangle_\Psi &\simeq\frac{x_1+x_2}{2},\\ \operatorname{Var}_\Psi(X) &\simeq \frac{\sigma_1^2+\sigma_2^2}{2} +\frac{(x_1-x_2)^2}{4}. \end{aligned}$$

The last term is the between-branch variance. It is macroscopic when the branches predict macroscopically different $x_i$, even if each $\sigma_i$ is small. Thus the mean can resemble a smooth branch average while the state is not peaked on that mean. Moreover, averaging two metrics does not generally solve the nonlinear Einstein equation.

The failed hypothesis is now precise: the state is not sharply peaked on one branch for the declared observable algebra. The strongest surviving claim is a branch-averaged coarse profile, similar to an ensemble description for observables that cannot see interference. Neglecting those interference matrix elements for the coarse algebra neither decoheres the pure superposition nor turns it into a mixed density operator; a larger algebra may still detect the interference. The profile is not one unique semiclassical geometry.

What is exact, what is semiclassical, and where to continue

The vacuum-relative relation $\delta E L=\Delta$ follows exactly from radial quantization and the matched global-time generator, conditional on the duality. The scalar normal-mode formula is a linearized bulk calculation. The one-particle interpretation uses leading large-$N$ factorization. The coherent-field and geometry statements are semiclassical inferences controlled parametrically by occupation, localization, $1/C_T$, the bulk derivative expansion, and the chosen observable resolution. No model-independent numerical error bar exists until a particular CFT, bulk action, state, and observable set are specified.

Next. Euclidean Preparation and Lorentzian State Dictionaries constructs normalizable states and their continuation contours. Heavy States, Coherent States, and Semiclassical Geometries develops the fluctuation hierarchy used to license geometry claims.

Later applications. Thermal and real-time holography owns general Lorentzian response and state contours, while bulk reconstruction owns the map to localized, gravitationally dressed bulk operators. State interpretations in low-dimensional holography, black-hole microstates, wormholes and ensembles, and black-hole information add model-specific geometry, ensemble, and interior qualifications.

Common pitfalls

Using an absolute energy where only an excitation energy is fixed. Radial quantization determines $(E-E_{\mathrm{vac}})L=\Delta$. The cylinder vacuum offset requires an anomaly and subtraction convention.

Calling every single-trace state one particle. The Fock-space interpretation needs a light operator in a generalized-free large-$N$ sector with a weakly coupled bulk field. It is not supplied by conformal representation theory alone.

Equating large occupation with a classical metric. Large occupation can make a matter-field quadrature sharp. A state-dependent classical metric interpretation additionally needs controlled gravitational response and sharp source and constraint data; only an order-one departure from AdS requires $\varepsilon=O(1)$.

Using total energy without a localization scale. The global ratio $\delta E L/C_T$ controls a smooth AdS-scale disturbance. Localized energy can generate strong gravity at a much smaller total energy.

Mistaking a mean for a sharp state. A macroscopic superposition can have a placid-looking mean and a macroscopic between-branch variance.

Confusing source data with state data. In standard quantization, changing $J$ changes the source or boundary problem; changing a normalizable coefficient at fixed $J$ changes the state data.

Exercises

Descendant levels and scalar modes

A primary has dimension $\Delta$ on a cylinder of radius $L$. Find the vacuum-relative energy of a level-$k$ descendant. For a leading-order scalar mode with radial number $n$ and angular momentum $\ell$, identify $k$ and its frequency. Does either result determine $E_{\mathrm{vac}}$?

Solution: excitation energies

The conformal algebra gives

$$\delta E_k=\frac{\Delta+k}{L}.$$

For a linearized scalar, $k=2n+\ell$, hence

$$\omega_{n\ell}=\frac{\Delta+2n+\ell}{L}.$$

Both statements concern excitation energies. Neither fixes the additive cylinder or AdS vacuum energy.

Source data or state data?

Work in standard quantization with fixed bulk theory and boundary metric. Classify each operation: (a) vary $J(x)$; (b) keep $J=0$ and add a regular normalizable solution that changes $A(x)$; (c) prescribe an arbitrary $A(x)$ that violates a bulk constraint. Which operation changes a source, which can change a state, and which is inadmissible?

Solution: hold the boundary problem fixed

Operation (a) changes the source and therefore the boundary-value problem. Operation (b) changes normalizable data at fixed source and can prepare a different state. Operation (c) is not valid state data: the response coefficient must arise from a solution satisfying the bulk equations, constraints, regularity conditions, and boundary conditions.

Coherent occupation and backreaction

Take one smooth global scalar mode with frequency $\omega$ and coherent amplitude $\alpha$. Compute its mean occupation, number variance, vacuum-subtracted energy, and relative number fluctuation. State the windows for a classical field on nearly fixed AdS and for possible order-one global backreaction.

Solution: two parametric thresholds

In the linearized harmonic-mode approximation, the coherent state gives

$$\begin{aligned} \bar n&=\lvert\alpha\rvert^2, &\operatorname{Var}(N_a)&=\bar n,\\ \frac{\sigma_{N_a}}{\bar n}&=\bar n^{-1/2}, &\delta E&=\bar n\omega. \end{aligned}$$

The last equality receives interaction corrections beyond the linearized mode approximation.

For a wavelength and localization scale of order $L$, the classical-field, weak-backreaction window is the double-scaling limit

$$\bar n\to\infty, \qquad \frac{\bar n(\omega L)}{C_T}\to0.$$

Order-one global backreaction becomes possible when $\bar n(\omega L)=O(C_T)$. This is not a universal collapse threshold and does not establish a smooth geometry. If the excitation is localized on $R_{\mathrm{loc}}\ll L$, the stronger local parameter $G_{d+1}\delta E/R_{\mathrm{loc}}^{d-2}$ must be checked.

The variance of a two-branch state

Assume two orthogonal branch states have negligible off-diagonal matrix elements of $X$ and $X^2$. Derive the variance of $X$ in $(\lvert g_1\rangle+\lvert g_2\rangle)/\sqrt2$. When $\lvert x_1-x_2\rvert$ is macroscopic, state the strongest geometric claim that survives.

Solution: within-branch and between-branch variance

The first two moments are

$$\langle X\rangle_\Psi=\frac{x_1+x_2}{2}, \qquad \langle X^2\rangle_\Psi =\frac{\sigma_1^2+x_1^2+\sigma_2^2+x_2^2}{2}.$$

Subtracting the squared mean gives

$$\operatorname{Var}_\Psi(X) =\frac{\sigma_1^2+\sigma_2^2}{2} +\frac{(x_1-x_2)^2}{4}.$$

When the branch separation is macroscopic, the second term is macroscopic even though both branches are individually sharp. The mean remains a valid branch average for the stated coarse observables; it does not define one sharp geometry.

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

• Assel, Benjamin, Davide Cassani, Lorenzo Di Pietro, Zohar Komargodski, Jakob Lorenzen, and Dario Martelli. “The Casimir Energy in Curved Space and Its Supersymmetric Counterpart.” Journal of High Energy Physics 07 (2015): 043. doi:10.1007/JHEP07(2015)043. Open PDF.
• Balasubramanian, Vijay, Per Kraus, Albion Lawrence, and Sandip P. Trivedi. “Holographic Probes of Anti-de Sitter Spacetimes.” Physical Review D 59 (1999): 104021. doi:10.1103/PhysRevD.59.104021. 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. Open PDF.
• Hamilton, Alex, Daniel Kabat, Gilad Lifschytz, and David A. Lowe. “Holographic Representation of Local Bulk Operators.” Physical Review D 74 (2006): 066009. doi:10.1103/PhysRevD.74.066009. 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. Open PDF.
• Witten, Edward. “Anti-de Sitter Space and Holography.” Advances in Theoretical and Mathematical Physics 2 (1998): 253–291. doi:10.4310/ATMP.1998.v2.n2.a2. 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. Open PDF.