Radial Quantization

Goal

Radial quantization is one of the places where conformal field theory suddenly becomes simple. In ordinary quantum mechanics one chooses a time coordinate, quantizes on constant-time slices, and studies time evolution generated by the Hamiltonian. In radial quantization one instead chooses the radius from an origin as Euclidean time. Spheres around the origin are the equal-time slices, and dilatations play the role of the Hamiltonian.

The construction is especially natural in a CFT because the flat metric on punctured Euclidean space is conformally equivalent to the cylinder:

$$\mathbb R^d\setminus\{0\} \simeq S^{d-1}\times \mathbb R_\tau, \qquad \tau=\log r.$$

The central result is the state-operator correspondence:

$$\boxed{ \text{local operators at the origin} \quad\longleftrightarrow\quad \text{states on }S^{d-1} }$$

and, for a primary operator $\mathcal O$ of scaling dimension $\Delta$,

$$\boxed{ D|\mathcal O\rangle=\Delta |\mathcal O\rangle. }$$

Thus the scaling dimension of a local operator is the energy of the corresponding cylinder state. This is one of the most important conceptual bridges to AdS/CFT: in global AdS, boundary time translations act as the CFT Hamiltonian on $S^{d-1}$, and a bulk particle of energy $E$ is dual to a CFT state with dimension $\Delta=E$ in units of the AdS radius.

From flat space to the cylinder

Write a point in Euclidean space as

$$x^\mu=r n^\mu, \qquad r=|x|, \qquad n^\mu n_\mu=1,$$

where $n^\mu$ is a point on the unit sphere $S^{d-1}$. The flat metric becomes

$$ds^2_{\mathbb R^d}=dr^2+r^2 d\Omega_{d-1}^2.$$

Now introduce the radial time coordinate

$$\tau=\log r, \qquad r=e^\tau.$$

Then

$$dr=e^\tau d\tau,$$

and therefore

$$ds^2_{\mathbb R^d} = e^{2\tau}\left(d\tau^2+d\Omega_{d-1}^2\right).$$

The metric

$$ds^2_{\rm cyl}=d\tau^2+d\Omega_{d-1}^2$$

is the Euclidean cylinder metric on $S^{d-1}\times\mathbb R_\tau$. Thus flat space minus the origin is Weyl-equivalent to the cylinder:

$$ds^2_{\mathbb R^d}=e^{2\tau}ds^2_{\rm cyl}.$$

For a conformal field theory, the local physics is invariant under Weyl rescaling, up to the usual anomaly subtleties in even dimensions. So a CFT on $\mathbb R^d\setminus\{0\}$ can be viewed as a CFT on the cylinder. The origin $r=0$ becomes $\tau=-\infty$, and infinity $r=\infty$ becomes $\tau=+\infty$.

Radial quantization. Spheres $S^{d-1}_r$ around the origin in $\mathbb R^d$ become constant-$\tau$ slices of the cylinder $S^{d-1}\times\mathbb R_\tau$. A local operator inserted at the origin prepares a state on $S^{d-1}$, and radial evolution is generated by the dilatation operator $D=H_{\rm cyl}$.

This is the geometric reason that the dilatation operator is a Hamiltonian. Since

$$\tau=\log r,$$

translation in $\tau$ means scaling in $r$:

$$\tau\to \tau+a \quad\Longleftrightarrow\quad r\to e^a r.$$

The vector field generating translations in $\tau$ is

$$\frac{\partial}{\partial \tau} = r\frac{\partial}{\partial r} = x^\mu\partial_\mu.$$

This is precisely the vector field generating dilatations. Therefore the cylinder Hamiltonian is

$$H_{\rm cyl}=D.$$

Strictly speaking, this equality is a convention-dependent statement about the normalization of the cylinder radius. If the sphere has radius $R$, then

$$H_{\rm cyl}=\frac{D}{R}.$$

Throughout this page we set $R=1$.

Radial ordering

In ordinary Euclidean QFT, correlation functions do not come with an obvious time ordering unless we choose a time direction. Radial quantization chooses $\tau=\log r$ as Euclidean time. Thus the ordering operation is radial ordering.

For two operators,

$$\mathcal R\left[\mathcal O_1(x_1)\mathcal O_2(x_2)\right] = \begin{cases} \mathcal O_1(x_1)\mathcal O_2(x_2), & |x_1|>|x_2|,\\ \mathcal O_2(x_2)\mathcal O_1(x_1), & |x_2|>|x_1|. \end{cases}$$

For many operators, radial ordering places larger-radius insertions to the left. Equivalently, on the cylinder it places later Euclidean time insertions to the left.

This simple definition is the hidden mechanism behind much of CFT. It turns correlation functions on flat space into matrix elements in a Hilbert space:

$$\langle \mathcal O_1(x_1)\cdots \mathcal O_n(x_n)\rangle_{\mathbb R^d} \quad\longleftrightarrow\quad \langle 0|\mathcal R\{\mathcal O_{{\rm cyl},1}(\tau_1,n_1)\cdots \mathcal O_{{\rm cyl},n}(\tau_n,n_n)\}|0\rangle.$$

The role of the Hilbert space is played by the space of states on a sphere $S^{d-1}$. The role of Euclidean time evolution is played by

$$e^{-\tau D}.$$

This is why radial quantization makes the OPE feel like ordinary spectral decomposition.

Weyl transformation of operators

Let $\mathcal O$ be a scalar primary of scaling dimension $\Delta$. The flat-space and cylinder operators are related by the Weyl factor. Since

$$ds^2_{\mathbb R^d}=e^{2\tau}ds^2_{\rm cyl},$$

the corresponding local operators obey

$$\mathcal O_{\rm cyl}(\tau,n) = e^{\Delta\tau}\mathcal O_{\mathbb R^d}(x=e^\tau n).$$

This factor is crucial. Without it, the limit $r\to0$ would usually be singular. With it, a primary insertion at the origin becomes a finite cylinder state:

$$|\mathcal O\rangle = \lim_{\tau\to-\infty} \mathcal O_{\rm cyl}(\tau,n)|0\rangle = \lim_{r\to0}r^\Delta\mathcal O_{\mathbb R^d}(rn)|0\rangle.$$

For a scalar primary, the state is independent of the direction $n$. For spinning primaries, the state carries the corresponding representation of $SO(d)$, and one should keep track of polarization data on the sphere.

The inverse statement is equally important. A state prepared on the sphere by a local insertion at the origin can be evolved outward by radial time evolution. A local operator is not just an observable; in radial quantization it is also a state-creating device.

Path-integral definition of the state

The state $|\mathcal O\rangle$ can be defined without assuming a Lagrangian, but the path-integral picture is very useful when a Lagrangian exists. Choose a sphere $S^{d-1}_R$ of radius $R$ around the origin. The path integral over the ball $B_R$ with an insertion $\mathcal O(0)$ prepares a wavefunctional of boundary data on $S^{d-1}_R$:

$$\Psi_{\mathcal O}[\varphi] = \int_{\phi|_{S_R}=\varphi}\mathcal D\phi\, \mathcal O(0)e^{-S[\phi]}.$$

This wavefunctional is the state $|\mathcal O\rangle$ on the sphere. Changing $R$ corresponds to evolving the state in radial time. In a CFT this evolution is controlled by $D$.

This is the operator-state correspondence in its most physical form:

$$\mathcal O(0) \quad\longmapsto\quad |\mathcal O\rangle_{S^{d-1}}.$$

Conversely, given a state on the sphere, shrink the sphere to the origin. If the theory has a good local operator expansion, the state can be represented by a local operator inserted at the origin. This is why CFT Hilbert spaces and local operator spectra are so tightly linked.

The dilatation charge from the stress tensor

The current associated with an infinitesimal conformal transformation generated by a conformal Killing vector $\xi^\mu$ is

$$j^\mu_\xi=T^{\mu\nu}\xi_\nu.$$

For dilatations,

$$\xi^\mu=x^\mu.$$

Thus the dilatation current is

$$j_D^\mu=T^{\mu\nu}x_\nu.$$

Its divergence is

$$\partial_\mu j_D^\mu = \partial_\mu(T^{\mu\nu}x_\nu) = (\partial_\mu T^{\mu\nu})x_\nu+T^\mu{}_{\mu}.$$

In a CFT on flat space,

$$\partial_\mu T^{\mu\nu}=0, \qquad T^\mu{}_{\mu}=0,$$

up to contact terms and possible anomaly effects on curved backgrounds. Therefore

$$\partial_\mu j_D^\mu=0.$$

The charge is obtained by integrating this current over a sphere surrounding the origin:

$$D = \int_{S^{d-1}_r}d\Sigma_\mu\,x_\nu T^{\mu\nu}.$$

Because the current is conserved, the value of the integral is independent of the radius $r$, as long as the sphere is not moved across an operator insertion. This is the radial-quantization analogue of conservation of ordinary energy.

This is also the geometric origin of Ward identities. If a sphere surrounds a collection of local insertions, the charge integral acts on those insertions by the corresponding conformal transformation.

Primary states

A scalar primary operator obeys

$$[D,\mathcal O(x)] = (x^\mu\partial_\mu+\Delta)\mathcal O(x),$$

and

$$[K_\mu,\mathcal O(0)]=0.$$

At the origin, the first relation gives

$$[D,\mathcal O(0)]=\Delta\mathcal O(0).$$

Acting on the vacuum, which is conformally invariant,

$$D|0\rangle=0, \qquad K_\mu|0\rangle=0, \qquad P_\mu|0\rangle=0,$$

we find

$$D|\mathcal O\rangle = D\mathcal O(0)|0\rangle = \Delta\mathcal O(0)|0\rangle = \Delta |\mathcal O\rangle.$$

Thus the state created by a primary operator has cylinder energy $\Delta$.

The second relation gives

$$K_\mu|\mathcal O\rangle=0.$$

So a primary state is a lowest-energy state in a conformal multiplet. The special conformal generators $K_\mu$ lower the cylinder energy, and the translation generators $P_\mu$ raise it.

Indeed, the conformal algebra contains

$$[D,P_\mu]=P_\mu, \qquad [D,K_\mu]=-K_\mu.$$

Therefore, if

$$D|\psi\rangle=E|\psi\rangle,$$

then

$$D(P_\mu|\psi\rangle)=(E+1)P_\mu|\psi\rangle,$$

and

$$D(K_\mu|\psi\rangle)=(E-1)K_\mu|\psi\rangle,$$

provided the states are nonzero.

This is the higher-dimensional analogue of the familiar two-dimensional statement that a primary state is annihilated by the positive Virasoro modes and descendants are obtained by acting with lowering modes. The terminology differs slightly across dimensions, but the representation-theoretic idea is the same: primaries generate conformal families.

Descendants

Starting with a primary state $|\mathcal O\rangle$, the descendants are obtained by acting with translations:

$$P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle.$$

Their cylinder energy is

$$D\,P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle = (\Delta+n)P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle.$$

On the operator side, descendants are derivatives of the primary:

$$P_\mu|\mathcal O\rangle \quad\longleftrightarrow\quad \partial_\mu\mathcal O(0),$$

and more generally

$$P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle \quad\longleftrightarrow\quad \partial_{\mu_1}\cdots\partial_{\mu_n}\mathcal O(0).$$

Thus one conformal family has the schematic form

$$\mathcal V_{\mathcal O} = \operatorname{span}\left\{ |\mathcal O\rangle, P_\mu|\mathcal O\rangle, P_\mu P_\nu|\mathcal O\rangle, \ldots \right\},$$

modulo possible null states or equations of motion.

The full Hilbert space on the sphere decomposes into conformal families:

$$\mathcal H_{S^{d-1}} = \bigoplus_{\mathcal O\in \text{primaries}} \mathcal V_{\mathcal O}.$$

This decomposition is one of the fundamental reasons CFT is solvable in principle. Instead of diagonalizing an arbitrary Hamiltonian, one classifies representations of the conformal algebra.

Cylinder two-point function

The flat-space scalar two-point function is

$$\langle \mathcal O(x_1)\mathcal O(x_2)\rangle_{\mathbb R^d} = \frac{C_{\mathcal O}}{|x_{12}|^{2\Delta}}.$$

Use

$$x_i=e^{\tau_i}n_i, \qquad n_i^2=1.$$

Then

$$|x_{12}|^2 = |e^{\tau_1}n_1-e^{\tau_2}n_2|^2 = e^{\tau_1+\tau_2}\left(2\cosh\tau_{12}-2 n_1\cdot n_2\right),$$

where

$$\tau_{12}=\tau_1-\tau_2.$$

Multiplying by the Weyl factors for the two cylinder operators gives

$$\langle \mathcal O_{\rm cyl}(\tau_1,n_1) \mathcal O_{\rm cyl}(\tau_2,n_2) \rangle = \frac{C_{\mathcal O}}{ \left[2\left(\cosh\tau_{12}-n_1\cdot n_2\right)\right]^\Delta }.$$

This formula is worth remembering. It shows explicitly how a power law on flat space becomes exponential decay on the cylinder at large Euclidean time separation.

For large positive $\tau_{12}$,

$$\cosh\tau_{12}\sim \frac12 e^{\tau_{12}},$$

so

$$\langle \mathcal O_{\rm cyl}(\tau_1,n_1) \mathcal O_{\rm cyl}(\tau_2,n_2) \rangle \sim C_{\mathcal O}e^{-\Delta\tau_{12}}.$$

This is exactly the behavior of a state of energy $\Delta$ propagating for Euclidean time $\tau_{12}$:

$$\langle \mathcal O|e^{-\tau_{12}H_{\rm cyl}}|\mathcal O\rangle \sim e^{-\Delta\tau_{12}}.$$

So the formula again confirms

$$E_{\rm cyl}=\Delta.$$

The bra state and insertion at infinity

The ket $|\mathcal O\rangle$ is created by an insertion at the origin. The corresponding bra is created by an insertion at infinity. For a scalar primary, one defines

$$\langle \mathcal O| = \lim_{|x|\to\infty} |x|^{2\Delta}\langle 0|\mathcal O(x).$$

Then the flat-space two-point function gives

$$\langle \mathcal O|\mathcal O\rangle = \lim_{|x|\to\infty}|x|^{2\Delta} \langle \mathcal O(x)\mathcal O(0)\rangle = C_{\mathcal O}.$$

If we choose the standard normalization

$$C_{\mathcal O}=1,$$

then

$$\langle \mathcal O|\mathcal O\rangle=1.$$

The precise Hermitian conjugation operation in radial quantization is not ordinary complex conjugation at fixed Euclidean time. It involves inversion,

$$x^\mu\mapsto \frac{x^\mu}{x^2},$$

which exchanges the origin and infinity. The next page develops this carefully because it is the starting point for reflection positivity and unitarity bounds.

OPE as spectral decomposition

Radial quantization also explains why the OPE is so powerful.

Suppose two operators are inserted inside a sphere, and all other operators are inserted outside the sphere. From the viewpoint of the sphere, the inner insertions prepare a state. Since the Hilbert space on the sphere has a basis of energy eigenstates, this state can be expanded in conformal families:

$$\mathcal O_i(x)\mathcal O_j(0)|0\rangle = \sum_{k}\sum_{\alpha\in\mathcal V_k} C_{ij}^{\;\;k,\alpha}(x)|k,\alpha\rangle.$$

Translated back into local operators, this becomes the operator product expansion:

$$\mathcal O_i(x)\mathcal O_j(0) \sim \sum_k C_{ijk}\,|x|^{\Delta_k-\Delta_i-\Delta_j} \left(\mathcal O_k(0)+\text{descendants}\right).$$

The OPE is not merely a formal short-distance expansion. In a unitary CFT, radial quantization gives it a Hilbert-space interpretation: it is a convergent expansion of a state, as long as the sphere separating the inner and outer insertions exists.

This is the conceptual seed of conformal blocks. A four-point function can be decomposed by inserting a complete set of cylinder energy eigenstates between pairs of operators. The sum over descendants inside one conformal family is a conformal block.

Subtleties on the cylinder

There are a few subtleties worth keeping visible.

First, radial time is Euclidean time, not physical Lorentzian time. After Wick rotation on the cylinder, $S^{d-1}\times\mathbb R$ becomes the Lorentzian cylinder. In AdS/CFT this Lorentzian cylinder is the natural boundary of global AdS.

Second, the vacuum energy on the cylinder can differ from zero because of Weyl anomalies and Casimir effects. In two-dimensional CFT one famously finds a shift by $-c/12$ in the cylinder Hamiltonian when expressed in terms of plane Virasoro generators:

$$H_{\rm cyl}^{(d=2)}=L_0+\bar L_0-\frac{c}{12},$$

for the usual cylinder with angular circle of circumference $2\pi$. The plane dilatation operator itself is

$$D=L_0+\bar L_0.$$

The distinction matters for torus partition functions and finite-size energies. It does not change the basic local statement that a primary operator of dimension $\Delta$ creates a state whose energy above the appropriate vacuum is governed by $\Delta$.

Third, noncompact CFTs can have continuous spectra. Then the direct sum over primaries should be replaced by a direct integral over representations. The local logic of radial quantization still works, but the Hilbert-space decomposition is less discrete.

AdS/CFT checkpoint

Radial quantization is one of the cleanest entry points into the AdS/CFT dictionary.

The boundary of global Euclidean AdS is conformal to $S^{d-1}\times\mathbb R$. The CFT Hamiltonian on this cylinder is the dilatation operator $D$. Therefore:

$$\boxed{ \text{CFT scaling dimension }\Delta = \text{global AdS energy }E }$$

in units where the AdS radius is one.

This explains several standard pieces of holographic language:

• A single-trace primary corresponds to a one-particle bulk state.
• Descendants correspond to the same bulk particle with boundary momentum or angular excitations.
• Multi-trace operators correspond, at large $N$, to multi-particle states.
• The CFT vacuum on $S^{d-1}$ corresponds to global AdS.
• The stress tensor state corresponds to the graviton sector.

The celebrated scalar mass-dimension relation

$$\Delta(\Delta-d)=m^2R_{\rm AdS}^2$$

should be understood in this representation-theoretic context. The bulk field furnishes a representation of the AdS isometry group $SO(d,2)$, and the boundary primary furnishes the corresponding conformal representation. Radial quantization is the CFT-side construction that makes this statement concrete.

Summary

Radial quantization replaces flat-space time by logarithmic radius:

$$\tau=\log r.$$

Because

$$\mathbb R^d\setminus\{0\} \sim S^{d-1}\times\mathbb R_\tau,$$

CFT correlation functions can be interpreted as cylinder transition amplitudes. The dilatation operator becomes the cylinder Hamiltonian,

$$H_{\rm cyl}=D,$$

and every primary operator creates a cylinder energy eigenstate:

$$\mathcal O(0) \longleftrightarrow |\mathcal O\rangle, \qquad D|\mathcal O\rangle=\Delta|\mathcal O\rangle.$$

Descendants are obtained by acting with $P_\mu$, and their energies are $\Delta+n$. The OPE becomes spectral decomposition on the sphere. This is why radial quantization is not just a technical trick; it is the Hilbert-space foundation of CFT and the representation-theoretic foundation of the AdS/CFT dictionary.

Exercises

Exercise 1 — Flat space as a Weyl-rescaled cylinder

Show that the flat Euclidean metric on $\mathbb R^d\setminus\{0\}$ is Weyl-equivalent to the cylinder metric on $S^{d-1}\times\mathbb R$.

Solution

Write

$$x^\mu=r n^\mu, \qquad n^\mu n_\mu=1.$$

Then

$$ds^2=dr^2+r^2d\Omega_{d-1}^2.$$

Set

$$r=e^\tau.$$

Then

$$dr=e^\tau d\tau,$$

so

$$dr^2=e^{2\tau}d\tau^2, \qquad r^2d\Omega_{d-1}^2=e^{2\tau}d\Omega_{d-1}^2.$$

Therefore

$$ds^2=e^{2\tau}\left(d\tau^2+d\Omega_{d-1}^2\right).$$

Thus

$$ds^2_{\mathbb R^d}=e^{2\tau}ds^2_{\rm cyl},$$

with

$$ds^2_{\rm cyl}=d\tau^2+d\Omega_{d-1}^2.$$

Exercise 2 — Primary energy on the cylinder

Let $\mathcal O$ be a scalar primary of scaling dimension $\Delta$, and define

$$|\mathcal O\rangle=\mathcal O(0)|0\rangle.$$

Using

$$[D,\mathcal O(x)]=(x^\mu\partial_\mu+\Delta)\mathcal O(x),$$

show that

$$D|\mathcal O\rangle=\Delta|\mathcal O\rangle.$$

Solution

At the origin, the derivative term vanishes:

$$[D,\mathcal O(0)]=\Delta\mathcal O(0).$$

Assume the vacuum is conformally invariant, so

$$D|0\rangle=0.$$

Then

$$D|\mathcal O\rangle =D\mathcal O(0)|0\rangle.$$

Insert the commutator:

$$D\mathcal O(0)|0\rangle = [D,\mathcal O(0)]|0\rangle+\mathcal O(0)D|0\rangle.$$

The second term vanishes, and the first gives

$$D|\mathcal O\rangle = \Delta\mathcal O(0)|0\rangle = \Delta|\mathcal O\rangle.$$

So the cylinder energy of the primary state is $\Delta$.

Exercise 3 — Descendant energies

Use

$$[D,P_\mu]=P_\mu$$

to show that

$$D P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle = (\Delta+n)P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle.$$

Solution

First consider one $P_\mu$:

$$D P_\mu|\mathcal O\rangle = [D,P_\mu]|\mathcal O\rangle+P_\mu D|\mathcal O\rangle.$$

Using

$$[D,P_\mu]=P_\mu, \qquad D|\mathcal O\rangle=\Delta|\mathcal O\rangle,$$

we get

$$D P_\mu|\mathcal O\rangle =P_\mu|\mathcal O\rangle+ \Delta P_\mu|\mathcal O\rangle =(\Delta+1)P_\mu|\mathcal O\rangle.$$

For $n$ momenta, commute $D$ through each $P_{\mu_i}$. Each commutator contributes one extra unit of energy. Therefore

$$D P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle = (\Delta+n)P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle.$$

Exercise 4 — Cylinder two-point function

Starting from

$$\langle \mathcal O(x_1)\mathcal O(x_2)\rangle_{\mathbb R^d} = \frac{C_{\mathcal O}}{|x_{12}|^{2\Delta}},$$

with

$$x_i=e^{\tau_i}n_i, \qquad n_i^2=1,$$

show that

$$\langle \mathcal O_{\rm cyl}(\tau_1,n_1) \mathcal O_{\rm cyl}(\tau_2,n_2) \rangle = \frac{C_{\mathcal O}}{ \left[2\left(\cosh\tau_{12}-n_1\cdot n_2\right)\right]^\Delta }.$$

Solution

First compute

$$|x_{12}|^2 = |e^{\tau_1}n_1-e^{\tau_2}n_2|^2.$$

Expanding,

$$|x_{12}|^2 = e^{2\tau_1}+e^{2\tau_2}-2e^{\tau_1+\tau_2}n_1\cdot n_2.$$

Factor out $e^{\tau_1+\tau_2}$:

$$|x_{12}|^2 = e^{\tau_1+\tau_2} \left(e^{\tau_1-\tau_2}+e^{\tau_2-\tau_1}-2n_1\cdot n_2\right).$$

Using

$$e^{\tau_{12}}+e^{-\tau_{12}}=2\cosh\tau_{12},$$

we get

$$|x_{12}|^2 = e^{\tau_1+\tau_2} 2\left(\cosh\tau_{12}-n_1\cdot n_2\right).$$

Therefore

$$|x_{12}|^{2\Delta} = e^{\Delta(\tau_1+\tau_2)} \left[2\left(\cosh\tau_{12}-n_1\cdot n_2\right)\right]^\Delta.$$

The cylinder operators are

$$\mathcal O_{\rm cyl}(\tau_i,n_i) = e^{\Delta\tau_i}\mathcal O_{\mathbb R^d}(x_i).$$

Multiplying the flat two-point function by $e^{\Delta\tau_1}e^{\Delta\tau_2}$ cancels the factor $e^{\Delta(\tau_1+\tau_2)}$ in the denominator. Hence

$$\langle \mathcal O_{\rm cyl}(\tau_1,n_1) \mathcal O_{\rm cyl}(\tau_2,n_2) \rangle = \frac{C_{\mathcal O}}{ \left[2\left(\cosh\tau_{12}-n_1\cdot n_2\right)\right]^\Delta }.$$

Exercise 5 — Why the OPE converges radially

Consider a four-point function with $|x_1|,|x_2|<R$ and $|x_3|,|x_4|>R$. Explain why the product $\mathcal O_1(x_1)\mathcal O_2(x_2)|0\rangle$ can be expanded in a complete basis of states on $S^{d-1}_R$, and why this gives the OPE channel for $12\to34$.

Solution

The two inner operators lie inside the sphere $S^{d-1}_R$. In radial quantization, the path integral over the ball bounded by this sphere prepares a state on $S^{d-1}_R$:

$$|\Psi_{12}\rangle = \mathcal O_1(x_1)\mathcal O_2(x_2)|0\rangle.$$

The Hilbert space on the sphere has a basis of dilatation eigenstates, organized into conformal families:

$$\mathcal H_{S^{d-1}} = \bigoplus_k \mathcal V_k.$$

Therefore

$$|\Psi_{12}\rangle = \sum_{k,\alpha}c_{k,\alpha}(x_1,x_2)|k,\alpha\rangle,$$

where $\alpha$ labels descendants inside the conformal family of the primary $\mathcal O_k$.

The two outer operators define a bra state outside the same sphere. The four-point function is the overlap of the outer bra with the inner ket:

$$\langle \Psi_{34}|\Psi_{12}\rangle.$$

Inserting the complete basis gives a sum over exchanged conformal families. Written in local-operator language, this is precisely the OPE expansion of $\mathcal O_1\mathcal O_2$, followed by evaluation of the resulting three-point functions with $\mathcal O_3$ and $\mathcal O_4$.

The expansion is radial because the separating sphere exists only when the inner insertions are strictly inside the outer ones. This geometric separation is what controls convergence.

Further reading

For two-dimensional radial quantization and the operator formalism, see Di Francesco, Mathieu, and Sénéchal, Chapter 6. For the higher-dimensional version used in the conformal bootstrap, see Rychkov’s CFT lectures and Simmons-Duffin’s TASI lectures. For AdS/CFT preparation, keep the cylinder interpretation especially close: it is the boundary Hilbert space of global AdS.