Radial Time and Quantization on Spheres

Radial quantization treats the logarithm of distance from a chosen origin as Euclidean time. A path integral over the ball inside a sphere prepares a state on that sphere, while the annulus between two spheres evolves the state by a scale transformation. In a conformal theory this identifies the radial Hamiltonian with the dilatation generator and turns scaling dimensions into energies on the cylinder. The construction is geometric and does not by itself prove positivity or completeness; those require the state-space assumptions developed later in this chapter.

Required background. Conformal Geometry, Maps, and Compactification supplies the action of dilatations and the distinction between a local conformal map and a global symmetry. Euclidean Correlators and Schwinger Functions supplies the Euclidean path-integral and correlation-function setting. Helpful background. Reflection Positivity within Osterwalder–Schrader Reconstruction explains the additional condition needed for a positive Hilbert-space inner product.

Radius as Euclidean time

Work on Euclidean $\mathbb R^d$ with a chosen origin and write

$$x^\mu=r n^\mu, \qquad r>0, \qquad n^\mu n_\mu=1.$$

A dilatation $x\mapsto \lambda x$ moves one sphere to another without changing the angular point $n$. Fix a reference radius $R$ and define the dimensionless radial time

$$\tau=\log\frac rR.$$

Then a finite change $r_1\to r_2$ is the time translation

$$\Delta\tau=\log\frac{r_2}{r_1}.$$

The flat metric makes the cylinder hidden in this coordinate choice:

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

Thus punctured flat space is Weyl-equivalent to $\mathbb R_\tau\times S^{d-1}$. The present page needs only the foliation by spheres; the operator Weyl factors, possible vacuum-energy shift, and precise cylinder Hamiltonian are derived in From Flat Space to the Cylinder. Rychkov 2017, §3.1.1, pp. 35–37, Open PDF develops the same construction directly from quantization on nested spheres.

Radial ordering is ordinary Euclidean-time ordering in $\tau$. For local operators at distinct radii,

$$\mathcal R\{\mathcal O_1(x_1)\mathcal O_2(x_2)\} = \begin{cases} \mathcal O_1(x_1)\mathcal O_2(x_2),&r_1>r_2,\\ \mathcal O_2(x_2)\mathcal O_1(x_1),&r_2>r_1, \end{cases}$$

with the usual graded sign when two fermionic operators are exchanged. Coincident radii require an angular ordering or a regulated limiting prescription; the symbol $\mathcal R$ does not resolve coincident singularities by itself.

States from path integrals on balls

Choose a sphere $S_r^{d-1}$ and denote its boundary field configuration schematically by $\varphi(\Omega)$. Insert local operators strictly inside the sphere and integrate over the interior. The resulting wavefunctional

$$\Psi_r[\varphi] = \int_{\substack{\Phi|_{S_r}=\varphi\\ |x|<r}} \mathcal D\Phi\; e^{-S_E[\Phi]} \prod_i\mathcal O_i(x_i)$$

is the state prepared on $S_r^{d-1}$. This is the radial analogue of preparing a state by a Euclidean path integral over a half-space. It is useful to keep three pieces of data explicit:

• the sphere orientation and which side of it is integrated over;
• the boundary conditions and any background or topological sector held fixed; and
• the normalization of the vacuum path integral and inserted operators.

The annular path integral between $r_1$ and $r_2$ composes with the two boundary wavefunctionals. In a CFT, scale invariance gives the radial transfer operator

$$U(r_2,r_1) =\exp\!\left[-\log\!\left(\frac{r_2}{r_1}\right)D\right], \qquad r_2>r_1,$$

in the Euclidean convention where a state of dimension $\Delta$ decays as $e^{-\Delta\Delta\tau}$. Composition is immediate:

$$U(r_3,r_2)U(r_2,r_1)=U(r_3,r_1).$$

This statement identifies the generator of radial evolution; it does not yet assert that $D$ is self-adjoint or that its spectrum is nonnegative. Those properties follow only after the conjugation and positivity structure is supplied. The quantization logic and surface-state interpretation are explained in Simmons-Duffin 2017, §§2.2 and 6, pp. 8–9 and 24–30, Open PDF.

A scalar insertion at the origin

Let $\mathcal O$ be a scalar primary with dimension $\Delta$. An insertion at $x=0$ lies inside every sphere centered at the origin and prepares the state

$$|\mathcal O\rangle \equiv \lim_{x\to0}\mathcal O(x)|0\rangle.$$

The commutator with dilatations at the origin gives

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

so propagation through an annulus yields

$$U(r_2,r_1)|\mathcal O\rangle =\left(\frac{r_1}{r_2}\right)^\Delta |\mathcal O\rangle.$$

This is the simplest independent check of the sign in $U$. If $r_2>r_1$ and $\Delta>0$, a normalized Euclidean amplitude must decay toward the larger sphere. The same dependence follows from the scalar two-point function: after placing one insertion at the origin and the conjugate insertion on the exterior side, the radial separation contributes $e^{-\Delta(\tau_2-\tau_1)}$. The exact normalization and the exterior insertion are handled by Conjugation and Reflection Positivity.

Translations create descendants. Since

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

the state $P_\mu|\mathcal O\rangle$ has radial energy $\Delta+1$. Repeated action of $P_\mu$ therefore organizes local Taylor data into levels of increasing cylinder energy. This is the bridge from spherical state preparation to conformal multiplets.

What the construction does and does not establish

Radial quantization is local to the chosen center and the vacuum sector unless further data are supplied. Several qualifications matter.

Positivity is additional. A Euclidean functional integral may define radially ordered correlators without defining a positive inner product. Reflection positivity is the condition that turns the inside/outside pairing into a Hilbert-space norm.

Local insertions need not span every sector. States carrying flux through the sphere, states created by extended operators, twisted sectors, and sectors selected by boundary conditions can require insertions or boundary data not represented by ordinary local operators at the origin.

The spectrum need not be a discrete sum. On a compact spatial sphere, a well-behaved unitary CFT often has a discrete spectrum with finite degeneracies, but noncompact or nonunitary theories can require continuous measures, generalized eigenstates, or indecomposable modules. Completeness and the Operator Basis states the appropriate resolution of the identity rather than assuming one universal form.

The cylinder here is not a thermal ensemble. Euclidean cylinder time is the logarithm of radius. Thermal physics additionally identifies Euclidean time periodically and specifies an ensemble; that subject belongs to Modular and Thermal Bootstrap.

The reliable conclusion is therefore precise: nested spheres define radial evolution, and conformal symmetry identifies its generator with $D$. A positive, complete Hilbert space and a particular spectral resolution require separate hypotheses.

Check your understanding

Take a scalar primary state with dimension $\Delta$ prepared on $S_{r_1}^{d-1}$. If $r_2=e^a r_1$, what annular factor multiplies the state, and what changes for the descendant $P_\mu|\mathcal O\rangle$?

Answer

The primary acquires $e^{-a\Delta}$. Because $[D,P_\mu]=P_\mu$, the descendant has dimension $\Delta+1$ and acquires $e^{-a(\Delta+1)}$. The relative extra factor $e^{-a}$ checks both the sign of radial evolution and the descendant level.

References

• 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.