Radial Quantization and State–Operator Correspondence

Radial quantization justifies a conformal spectral decomposition by turning nested spheres into Euclidean-time slices. Local insertions prepare states; the logarithmic plane–cylinder map turns dilatations into a Hamiltonian; inversion defines the radial adjoint; reflection positivity makes the pairing nonnegative; Gram matrices reveal null states; and a complete state resolution produces conformal-family expansions. A proposed decomposition is trustworthy only when every link in that chain uses the same sector, normalization, Weyl factors, adjoint, and spectral measure. The geometric construction and its state–operator consequences are derived in Rychkov 2017, §3.1.

Helpful background. Primaries, Descendants, and Conformal Multiplets supplies conformal-family language. Reflection Positivity within Osterwalder–Schrader Reconstruction supplies the Euclidean positivity hypothesis, and Bilinear and Hermitian Forms, Adjoints, and Isometries supplies the underlying linear algebra. These are preparation routes, not conditions for reading this overview.

How radial quantization organizes CFT data

The chapter has two logically distinct layers.

The first is geometric. A sphere centered at the origin is a spatial slice, $\tau=\log(r/R)$ is Euclidean time, and a ball path integral prepares a state. Punctured flat space is Weyl-equivalent to the cylinder, so a primary of dimension $\Delta$ becomes a cylinder state with excitation energy $\Delta/R$.

The second is spectral. Inversion exchanges the inside and outside of the sphere and defines the adjoint. Reflection positivity turns this pairing into a positive inner product, permitting norm inequalities. Descendants form level-by-level Gram matrices; exact kernels produce null submodules. Finally, a resolution of the identity—discrete or continuous, and sector by sector—turns a four-point function into a sum or integral of conformal families. Simmons-Duffin 2017, §§6–9, pp. 24–48, Open PDF develops this radial-quantization route from states and positivity to the OPE and conformal blocks.

The chapter does not rederive general Euclidean reconstruction, define local operators from first principles, or treat a cylinder as automatically thermal. Those subjects belong respectively to Foundations of QFT, the local-operator chapters there, and Thermal and Nonequilibrium QFT. It also does not assume that nonunitary, logarithmic, or noncompact theories possess a positive discrete basis.

Check your preparation

Use the row that matches the claim you want to make. No total score is intended.

Can you do this?	Ready for	Quick criterion	Repair and return
Explain why $x\mapsto\lambda x$ moves one sphere to another and distinguish this from a translation	Radial time and the plane–cylinder map	You can write $\tau=\log(r/R)$ and identify $\Delta\tau=\log\lambda$	Review Conformal Geometry, Maps, and Compactification, then begin with radial time.
Interpret a Euclidean path integral with fixed boundary data as a wavefunctional	State preparation	You can say which region is integrated over, which field data are fixed, and how gluing composes amplitudes	Review Euclidean Correlators and Schwinger Functions, then return to radial time.
Distinguish a Hermitian form from a positive-definite one	Radial conjugation and norms	You can give an indefinite Hermitian matrix and explain why a basis change cannot make it positive	Review forms and adjoints, then enter conjugation.
Compute a commutator representation on a primary	Descendant Gram matrices	You can use $K\lvert\mathcal O\rangle=0$ and $[K,P]$ to evaluate $\langle\mathcal O\rvert KP\lvert\mathcal O\rangle$	Review the conformal algebra, then enter Gram matrices.
State the difference between a basis, a complete basis, and a truncated set	Completeness and OPE decompositions	You can write a resolution of the identity and identify an omitted-tail operator	Begin with Completeness and the Operator Basis and use its repair links as needed.

Choose a route

Reader goal	Route	Stop when you can…
Build the conceptual spine	Radial Time and Quantization on Spheres → The State–Operator Correspondence → From Flat Space to the Cylinder	Map an insertion to a sphere state and recover its cylinder energy and correlator Weyl factor
Establish a unitary representation bound	Conjugation and Reflection Positivity → Descendant States and Gram Matrices	State the reflected inner product, construct a complete level basis, and identify the exact null condition
Justify a conformal block expansion	Conceptual spine → Completeness and the Operator Basis	Insert the correct sum or integral of family projectors on a separating sphere and name its convergence domain
Diagnose a nonunitary or noncompact case	State–operator map → conjugation → completeness	Replace positive norms or discrete sums only where the theory requires it, then continue to nonrational two-dimensional CFT
Check a calculation symbolically	Algebra → conjugation → Gram matrices	Reproduce the scalar level-two zero, Jacobi identities, and low-level Gram spectra exactly

The arrows show efficient reading orders. Hard dependencies are stated on each leaf; a reader who already has the relevant capability may enter later.

Conceptual map

The chapter can be summarized as a typed chain:

$$\begin{gathered} \text{ball path integral} \xrightarrow{\text{prepares}} \text{state on }S^{d-1} \xleftrightarrow{\text{local sector}} \text{operator at the origin},\\ \text{radial scale} \xrightarrow{\tau=\log(r/R)} \text{cylinder time}, \qquad D\xrightarrow{\text{generator map}}RH_{\mathrm{cyl}},\\ \text{inversion + conjugation} \xrightarrow{\text{defines}} \text{radial adjoint} \xrightarrow{\text{reflection positivity}} \text{positive inner product},\\ \text{conformal commutators} \xrightarrow{\text{compute}} \text{Gram matrices} \xrightarrow{\text{kernel and quotient}} \text{irreducible families},\\ \text{complete family projectors} \xrightarrow{\text{insert on a sphere}} \text{OPE and block expansion}. \end{gathered}$$

Each arrow has a different logical force. The plane–cylinder relation is geometric; the adjoint is an antilinear convention tied to reflection; positivity is an extra property of the Euclidean theory; a Gram determinant is a representation calculation; and completeness is a state-space hypothesis or theorem. Collapsing those statements into “radial quantization works” hides exactly where an argument can fail.

The six pages

The order below is the chapter and sidebar order.

1. Radial Time and Quantization on Spheres asks how radius becomes Euclidean time. It constructs inside and outside path-integral states and fixes the sign of $e^{-\Delta\Delta\tau}$. Continue when you can distinguish radial ordering from an arbitrary Lorentzian ordering.
2. The State–Operator Correspondence states both directions of the local-sector map, including null, sector, topology, gauge, and extended-operator qualifications. Continue when you can say which sphere states a point insertion does not create.
3. From Flat Space to the Cylinder derives the logarithmic Weyl transformation, primary factors, $H_{\mathrm{cyl}}=D/R$, and the transformed two-point function. Its stopping boundary is the absolute vacuum energy and general curved-background response.
4. Conjugation and Reflection Positivity turns inversion into the radial adjoint and identifies the precise condition behind positive norms and nonnegative identical-scalar block weights. It explicitly withdraws those conclusions in nonunitary sectors.
5. Descendant States and Gram Matrices gives an implementable PBW, projection, commutator, kernel, and quotient procedure. The scalar level-two example exposes the bound $\Delta\geq(d-2)/2$ and the null state at saturation.
6. Completeness and the Operator Basis writes discrete and continuous resolutions of the identity, inserts family projectors into a four-point function, and separates completeness, convergence, and truncation error. It prepares the OPE and conformal-block chapter.

Conventions that recur

The chapter inherits the site’s Euclidean positive metric and general mathematical notation. Its local conformal convention is

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

Radial time and the cylinder radius are

$$\tau=R\log\frac rR, \qquad H_{\mathrm{cyl}}=\frac DR,$$

where some pages instead use the dimensionless time $\tau/R=\log(r/R)$. Every displayed exponential makes the choice clear. A scalar primary transforms as

$$\mathcal O_{\mathrm{cyl}}(\tau,n) =e^{\Delta\tau/R}\mathcal O_{\mathrm{flat}}(R e^{\tau/R}n),$$

and its radial adjoint on a unit sphere is

$$\mathcal O(x)^\dagger =(x^2)^{-\Delta} \mathcal O(x/x^2)$$

for a Hermitian scalar. Spin matrices, internal conjugation, statistics, null quotients, and spectral measures remain local data.

Two checks settle most convention disputes:

• the plane two-point function must round-trip to the cylinder kernel and back; and
• the algebraic identity $P_\mu^\dagger=K_\mu$ must reproduce both the level-one norm and the scalar level-two null point.

Synthesis: what licenses a block decomposition

Suppose a four-point function is written as a sum of conformal blocks with nonnegative coefficients. The chapter separates the obligations behind that line.

1. A sphere must separate the chosen operator pair from the other insertions.
2. The inner and outer path integrals must prepare a ket and its appropriate bra in the same sector.
3. The state–operator map must identify the relevant primary and descendant families, modulo null states.
4. Reflection positivity must apply if coefficient nonnegativity is claimed.
5. The family projectors must contain the inverse Gram form or the correct continuous measure.
6. The sum or integral must converge in the stated Euclidean domain; analytic continuation beyond it needs additional branch and ordering data. Under the hypotheses of a unitary CFT, this convergence and the exponentially small high-dimension tail are established by Pappadopulo, Rychkov, Espin, and Rattazzi 2012, §§2–5, pp. 3–16, Open PDF.
7. A finite truncation needs an error estimate and cannot be relabeled as completeness.

The first, third, and fifth items can survive in modified form in a nonunitary theory; the fourth cannot. A continuous spectrum changes the fifth and sixth, not necessarily the geometric state preparation. A logarithmic theory changes the action of $D$ and the spectral decomposition independently of continuity. These distinctions are why the chapter treats positivity, Gram matrices, and completeness on separate pages.

Review the chapter

Retrieval and explanation. State the operator-to-state map, the inverse map’s sector qualification, and the relation between $D$ and $H_{\mathrm{cyl}}$.

Verification criteria

A complete answer says that a ball insertion prepares a state on its boundary, shrinking a local-sector state defines an insertion through correlation functions, and $R(H-E_{\mathrm{vac}})=D$ fixes excitation energies. It does not claim that point operators create every topological or extended sector.

Derivation. Starting from $\mathrm ds^2=\mathrm dr^2+r^2\mathrm d\Omega^2$, derive the cylinder metric and transform a scalar two-point function. Then extract the large-$|\tau|$ decay.

Verification criteria

The derivation uses $r=R e^{\tau/R}$, includes one factor $e^{\Delta\tau_i/R}$ per primary, and obtains decay $e^{-\Delta|\tau|/R}$. Missing Weyl factors or an increasing Euclidean exponential indicate an error.

Positivity and failure diagnosis. Compute the level-one scalar norm and explain why it does not yield the sharp scalar bound. Then compute or quote the factorized level-two trace norm.

Verification criteria

Level one gives $2\Delta\delta_{\mu\nu}$. The trace at level two gives $8d\Delta[\Delta-(d-2)/2]$, so the nonidentity scalar bound appears only after the second level. In a nonunitary theory the negative direction diagnoses failure of positivity rather than inconsistency of the conformal algebra.

Transfer. A theory has a continuous spectrum of dimensions in one sector. Rewrite the identity insertion and say which familiar statement must be removed.

Verification criteria

Replace the discrete primary sum by $\sum_R\int\mathrm d\mu_R(\Delta)\,\Pi_{\Delta,R}$. Delta-normalized states and a spectral density replace ordinary unit vectors and squared discrete OPE coefficients. Do not infer nonunitarity solely from continuity.

Continue from here

• Correlators, OPE, and Conformal Blocks uses the state resolution to define normalized conformal data, blocks, convergence domains, crossing, and positivity.
• Conformal Symmetry and Representations supplies spin-dependent unitarity bounds, shortening, conserved multiplets, and characters if the representation step needs repair.
• Nonunitary, Logarithmic, Noncompact, and Nonrational Two-Dimensional CFT develops the replacement structures when positivity, diagonalizability, or discreteness fails.
• Modular and Thermal Bootstrap adds genuine torus traces, thermal ensembles, KMS conditions, and thermal inversion; these are not part of radial quantization alone.
• For symbolic checks: reproduce the Jacobi identities and low-level Gram spectra with exact arithmetic.

References

• Pappadopulo, Duccio, Slava Rychkov, Johnny Espin, and Riccardo Rattazzi. “OPE Convergence in Conformal Field Theory.” Physical Review D 86 (2012): 105043. doi:10.1103/PhysRevD.86.105043. Open PDF.
• Rychkov, Slava. EPFL Lectures on Conformal Field Theory in $D\geq3$ Dimensions. SpringerBriefs in Physics. Cham: Springer, 2017. doi:10.1007/978-3-319-43626-5. Open PDF.
• Simmons-Duffin, David. “The Conformal Bootstrap.” In New Frontiers in Fields and Strings, 1–74. Singapore: World Scientific, 2017. doi:10.1142/9789813149441_0001. Open PDF.