Radial Quantization and State–Operator Correspondence

Radial quantization turns nested spheres into Euclidean time slices. Local operators prepare states on a sphere, and scale transformations evolve them. This chapter explains when that geometric picture supplies the positive inner product and complete state expansion needed to decompose a correlator into conformal families.

Enter this chapter

Begin with Radial Time and Quantization on Spheres to understand state preparation by a ball path integral.

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.

Topics

1. Radial Time and Quantization on Spheres — Interprets radius as time and an annulus as evolution between sphere states.
2. The State–Operator Correspondence — Establishes the operator–state map in the local vacuum sector and identifies excluded sectors.
3. From Flat Space to the Cylinder — Relates scaling dimensions to excitation energies on a cylinder.
4. Conjugation and Reflection Positivity — Constructs the radial adjoint and states the positivity condition on its inner product.
5. Descendant States and Gram Matrices — Calculates descendant norms and removes exact null submodules.
6. Completeness and the Operator Basis — Distinguishes a complete family expansion from a finite approximation, including continuous spectra.

How the ideas fit together

The sphere provides the bridge between geometry and a spectral expansion. Operators inside it prepare a ket; reflected operators outside it prepare a bra. Reflection positivity makes their pairing nonnegative, while the state–operator correspondence identifies the families that can appear between them. Resolving the identity then groups the correlator into contributions from those families. Simmons-Duffin 2017, §§ 6–9.

Geometry, positivity, and completeness are distinct inputs. On a cylinder of radius $R$, a state of scaling dimension $\Delta$ has excitation energy $\Delta/R$ above the vacuum; this relation does not determine an additive vacuum energy. Nor does choosing finitely many families make that set complete.

Review the chapter

Does a positive Gram matrix for every descendant retained in a calculation justify an exact conformal-block decomposition?

Solution

No. It checks the inner product on the retained subspace. An exact decomposition also needs the complete state resolution, the correct sectors and null quotients, and convergence in the chosen domain. A truncated expansion needs separate control of the omitted states.

Where to go next

Continue to Correlators, OPE, and Conformal Blocks to use the state expansion in crossing equations.

References

• 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.