Radial quantization of a plane CFT

Let the Euclidean plane be the standard complex plane with its usual orientation, metric $d{s}^2=d{x}^2+d{y}^2=\mathrm{\mid }dz{\mathrm{\mid }}^2$, dimensionless coordinate $z=x+iy$, and chosen origin $0$. Write ${\mathbb{C}}^{\times }=\mathbb{C}\setminus \{0\}$. The map $(\tau ,\theta )⟼z=e^{\tau +i\theta },\tau \in \mathbb{R},\theta \in \mathbb{R}\mathrm{/}(2\pi \mathbb{Z}),$ is a diffeomorphism $\mathbb{R}\times S^1\to {\mathbb{C}}^{\times }$; its inverse is $\tau =\log \mathrm{\mid }z\mathrm{\mid }$ and $\theta =\arg z(\mathrm{m}\mathrm{o}\mathrm{d}2\pi )$. Thus $\tau$ is Euclidean radial time, and a circle $\mathrm{\mid }z\mathrm{\mid }=r$ is the time slice $\tau =\log r$. A radially ordered product is defined for insertions at distinct radii by placing the factor with larger $\mathrm{\mid }z\mathrm{\mid }$ to the left. This definition assigns no order to equal-radius factors; their equal-time ordering is separate model data.

A selected radial model supplies its state and field data rather than receiving them from the coordinate change. In the positive-Hilbert branch, these data include a complex Hilbert space $\mathcal{H}$, with the inner product linear in the first argument and complete for its induced norm, a unit vacuum vector $\mathrm{\Omega }$, a densely defined radial energy operator $K:D(K)\to \mathcal{H}$, a common dense linear field domain $\mathcal{D}\subseteq \mathcal{H}$, and the specified radial-reflection and field-adjoint operations. The model states the domains and how its radial evolution and smeared fields act on them. A weakly smeared field means a linear test-function assignment whose matrix coefficients on $\mathcal{D}$ have the declared continuity; point evaluation is not assumed to be an operator. The compact pointed-worldsheet supplier fixes the compatible holomorphic-chart convention when the plane calculation is used inside a pointed-surface model; it does not assert that the plane is compact.

For a point $z=r{e}^{i\theta }\in {\mathbb{C}}^{\times }$, radial reflection is the involution $\mathrm{\Theta }(z)=\frac{1}{\bar{z}}=r^{-1}{e}^{i\theta }\mathrm{.}$ It sends $\tau$ to $-\tau$ and fixes the unit circle. In a positive-Hilbert model, the field data include an antilinear isometric involution $J$ on the declared domain and a corresponding adjoint on smeared field labels: if $A(\phi )$ is defined on $\mathcal{D}$, its reflected adjoint $A^{\mathrm{♯}}({\mathrm{\Theta }}_{\ast }\phi )$ satisfies $⟨A(\phi )u,v⟩=⟨u,A^{\mathrm{♯}}({\mathrm{\Theta }}_{\ast }\phi )v⟩,u,v\in \mathcal{D}\mathrm{.}$ Here ${\mathrm{\Theta }}_{\ast }\phi$ denotes the test function obtained from the model's declared pullback, Jacobian, and any field-weight multiplier. Those data are specified by the model. This is not a theorem that every CFT has a positive Hilbert realization or these fields.

The geometry fixes radial time and ordering only. It does not imply a spectrum condition, a discrete energy basis, existence of radial insertion limits, field-domain invariance, radial reflection positivity, or convergence of an OPE; each such condition is supplied or proved separately in the selected model.

Pointed analytic worldsheet and family

Real and complex inner-product spaces and their induced length

Hilbert space

The diagram is schematic: the left concentric circles are level sets of $r=\mathrm{\mid }z\mathrm{\mid }$; the right horizontal circles are level sets of $\tau =\log r$. The arrow preserves $\theta$ and sends each radius $r>0$ to $\tau =\log r$. No proof uses the drawing.

A formal or analytic CFT may use the same radial coordinates without a positive Hilbert-space interpretation. Any such alternative state-space convention must be stated in its own model data.