research35 min studyLesson 732 in Physics

Radial quantization and the state-operator map

Turn scale transformations into cylinder evolution with the correct adjoint and energy reference.

Before you begin

On this page

The question for today

What you’ll understand

Map flat radii to cylinder time.
Compute a scalar radial norm by inversion.
Relate scaling gaps to cylinder energy differences with explicit radius units.

Ordinary Euclidean quantization cuts space along constant time. A conformal theory also permits cuts along concentric spheres. The path integral inside a sphere prepares a state on that sphere, and a local operator at the center selects a corresponding state. Radial quantization turns an operator expansion into a Hilbert-space expansion. Reflection positivity and completeness are essential assumptions for its norm and convergence arguments; a formal coordinate change alone does not prove that a candidate theory has a positive Hilbert space.

Rewrite the metric and identify evolution

Write a nonzero point as its radius times a unit vector and introduce logarithmic radius relative to a reference length. The flat metric is a position-dependent scale factor times a cylinder metric. Weyl covariance relates the two descriptions, including field rescalings and possible anomaly contributions. Increasing logarithmic radius is cylinder Euclidean-time evolution. The generator is dilatation, so a local scaling dimension becomes an excitation eigenvalue on the cylinder rather than a flat-space particle mass.

For a cylinder with sphere radius R, energy differences acquire a factor reduced Planck constant times a reference propagation speed divided by R. The absolute vacuum energy can contain a Casimir or Weyl-anomaly offset depending on dimension and convention. It is safer to state the primary's energy relative to the vacuum. Operator-state correspondence does not imply a finite number of energy levels, and a CFT on flat space need not have ordinary massive asymptotic particle states.

r=L e^\tau,\qquad ds_{\rm flat}^2=r^2(d\tau^2+d\Omega_{d-1}^2),\qquad E_{\mathcal O}-E_{\rm vac}=\frac{\hbar v}{R}\Delta

L and R are lengths, τ is dimensionless radial time and v converts length to physical time. The energy formula refers to the radius-R cylinder and subtracts its vacuum offset.↔ Scroll wide equations horizontally.

Construct the bra through inversion

An operator approaching the origin defines a ket after the appropriate vacuum path integral. Its bra is not obtained by placing an identical insertion at the same point and taking a naive square. Radial reflection reverses cylinder time and therefore inverts the radius. For a real scalar primary, inversion comes with a power fixed by the scaling dimension. Tensor primaries also require index rotations from the inversion matrix, which cannot be discarded when computing their norms.

With a scalar two-point function normalized to a positive constant, multiplying the large-radius correlator by the matching power gives the state norm. This shows why a two-point coefficient can be chosen as one for a nonzero positive-norm operator. Zero-norm states must be quotiented appropriately. It also explains the radial adjoint relation between translations and special conformal generators: inversion exchanges operations moving an insertion outward and inward relative to the quantization sphere.

|\mathcal O\rangle=\mathcal O(0)|0\rangle,\qquad \mathcal O(x)^{\dagger_{\rm rad}}=|x|^{-2\Delta}\mathcal O(x/|x|^2),\qquad \langle\mathcal O|\mathcal O\rangle=\lim_{r\to\infty}r^{2\Delta}\langle\mathcal O(r\hat n)\mathcal O(0)\rangle

Coordinates in the inversion formula are measured in units of the chosen inversion-sphere radius. The simple scalar formula assumes a real primary; complex operators need conjugation and spinning operators need inversion tensors.↔ Scroll wide equations horizontally.

Use cylinder evolution without importing a decay process

A state of fixed scaling dimension propagating for a positive cylinder-time interval contributes an exponential suppression. Descendants have larger dimensions by integer levels and are suppressed accordingly. This is imaginary-time spectral filtering, not irreversible dissipation into a bath. A superposition evolves by different factors before any optional normalization; interpreting those factors as real-time probabilities would confuse Euclidean projection with unitary Lorentzian dynamics.

The path-integral argument assumes states on the sphere can be expanded in a suitable complete energy basis. In noncompact or continuous-spectrum settings one may need integrals and additional care with normalizability. The state-operator map therefore needs a specified class of operators and states. The pure-qubit lab can illustrate normalized overlap weights only; its phase precession is real coherent two-state geometry and does not implement the nonunitary Euclidean projection appearing in a radial expansion.

e^{-\tau D}|\mathcal O\rangle=e^{-\tau\Delta}|\mathcal O\rangle,\qquad e^{-\tau D}|n\rangle=e^{-\tau(\Delta+n)}|n\rangle

These factors describe dimensionless Euclidean cylinder evolution for a primary and its nonnull level-n descendant. They are amplitudes in an expansion, not directly normalized outcome probabilities.↔ Scroll wide equations horizontally.

Work it through

A radial norm and a cylinder gap

A real scalar primary has $\Delta=2$ and $\langle\mathcal O(x)\mathcal O(0)\rangle=3/|x|^4$ in reference-length units. Find its radial norm and normalized operator. On a cylinder with $\hbar v/R=0.20\,\mathrm{eV}$ , find its energy above vacuum.

The norm is the limit $r^4(3/r^4)=3$ . The inversion weight removes the coordinate dependence; taking the raw correlator at infinity would incorrectly give zero.
Divide the operator by $\sqrt3$ . Its two-point coefficient and state norm both become one because each of the two insertions contributes the same normalization factor.
Multiply dimension by the cylinder energy scale: $E-E_{\rm vac}=2(0.20)=0.40\,\mathrm{eV}$ . This is a finite-radius excitation gap, not a flat-space mass or an assumption that the vacuum energy vanishes.

The result

The state norm is 3, the normalized operator is O/√3, and the cylinder excitation is 0.40 eV above vacuum.

The living notebook

Explore the model

Interactive lab

Interactive lab · quantum

Normalized overlaps only

Reset and set BOTH Mixing angle θ = 60° and Relative phase φ = 0°; predict probability 3/4 for basis zero. Reset and set θ = 60° and φ = 90°; predict the same probability with a different Bloch direction. This illustrates pure-state normalization, not radial evolution, inversion conjugation or a CFT Hilbert space.

Bloch vector / P(0)P(1)Phase latitude

Probability |0⟩: 71.1%
Probability |1⟩: 28.9%
Bloch (x, y, z): 0.78, 0.45, 0.42

Mixing angle θ65°Relative phase φ30°

t = 0 s

A normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forNormalized overlaps and Euclidean projection are different mathematical operations.

Pause & reason

Check your understanding

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Read the checkpoint explanation

The radius and propagation-speed conversion set the energy units, and a possible vacuum offset must be treated separately from the excitation dimension.

Your turn

Try it yourself

For a family with primary dimension $\Delta=1.50$ , compare the Euclidean evolution amplitude of a level-two descendant with that of the primary over $\tau=\ln2$ . Find the ratio and its square. Are these automatically physical probabilities?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

The common primary factor cancels. Keep amplitude suppression distinct from an independently normalized measurement probability.

Compare with the solution

The descendant has dimension $3.50$ , so the amplitude ratio is $\exp[-(\ln2)(3.50-1.50)]=2^{-2}=1/4$ . Its squared magnitude is $1/16$ . These are relative Euclidean filtering factors for specified initial coefficients, not automatically probabilities in a normalized measurement experiment. A superposition would need its full norm and measurement specification, and Lorentzian evolution instead gives phases for energy eigenstates in a closed unitary system.

Keep the idea

Take it with you

Radial time is logarithmic radius and is Euclidean.
Inversion and field weights define radial adjoints.
Cylinder excitation energies require radius units and a vacuum reference.

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