Radial state/operator data in a selected CFT model

Fix the positive-Hilbert radial model of Radial quantization of a plane CFT. It consists of a complex Hilbert space $\mathcal{H}$, a unit vector $\mathrm{\Omega }\in \mathcal{H}$ fixed by radial evolution, and a densely defined self-adjoint operator $K:D(K)\subseteq \mathcal{H}\to \mathcal{H}$ such that $⟨Ku,u⟩\ge 0$ for $u\in D(K)$, $\mathrm{\Omega }\in D(K)$, and $K\mathrm{\Omega }=0$. Self-adjointness uses the domain convention of the supplied operator definition. The model supplies a strongly continuous radial contraction semigroup $U(t)$ for $t\ge 0$, meaning bounded operators with $U(0)=I$, $U(s+t)=U(s)U(t)$, $\mathrm{\parallel }U(t)u-u\mathrm{\parallel }\to 0$ as $t\downarrow 0$ for every $u\in \mathcal{H}$, and generator $K$ in the convention $D(K)=\{u:{\lim }_{t\downarrow 0}\frac{U(t)u-u}{t}\text{ exists in }\mathcal{H}\},Ku=-{\lim }_{t\downarrow 0}\frac{U(t)u-u}{t}\mathrm{.}$ The equality of this generator with the specified nonnegative self-adjoint $K$ is model data; no semigroup representation theorem is invoked here. The contraction condition is $\mathrm{\parallel }U(t)u\mathrm{\parallel }\le \mathrm{\parallel }u\mathrm{\parallel }$ for every $t\ge 0$ and $u\in \mathcal{H}$.

A common dense linear field domain $\mathcal{D}$ is contained in $D(K)$ and is invariant under $K$ and every declared smeared local field. A local insertion label is a pair $a=(A,\eta )$ consisting of a local field $A$ and its declared smooth angular test function $\eta \in C^{\mathrm{\infty }}(S^1)$, together with the model's specified radial normalization. The label space $F_{\mathrm{l}\mathrm{o}\mathrm{c}}$ is the vector space of finite linear combinations of these labels. For each $a\in F_{\mathrm{l}\mathrm{o}\mathrm{c}}$, the model specifies a radius $r_a>0$ and a vector-valued insertion map $v_a:(0,r_a)\to \mathcal{H}$, linear in $a$. The vector $v_a(r)$ is the model's normalized, angularly smeared insertion into $\mathrm{\Omega }$ at radius $r$. Its smearing, field domain, and normalization are part of the adopted model.

Define $F_{\mathrm{r}\mathrm{a}\mathrm{d}}:=\{a\in F_{\mathrm{l}\mathrm{o}\mathrm{c}}:{\lim }_{r\downarrow 0}{v}_a(r)\text{ exists in Hilbert norm}\},\sigma (a):={\lim }_{r\downarrow 0}{v}_a(r),{\mathcal{H}}_{\mathrm{l}\mathrm{o}\mathrm{c}}:=\sigma (F_{\mathrm{r}\mathrm{a}\mathrm{d}})\mathrm{.}$ Linearity of each $v_a(r)$ and continuity of vector addition and scalar multiplication imply that $F_{\mathrm{r}\mathrm{a}\mathrm{d}}$ is a vector subspace and $\sigma :F_{\mathrm{r}\mathrm{a}\mathrm{d}}\to {\mathcal{H}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ is linear. The model state/operator identification is the induced map $\bar{\sigma }:F_{\mathrm{r}\mathrm{a}\mathrm{d}}\mathrm{/}\ker \sigma ⟶{\mathcal{H}}_{\mathrm{l}\mathrm{o}\mathrm{c}},[a]⟼\sigma (a)\mathrm{.}$ It is well defined because $\sigma (a-b)=0$ exactly when $a-b\in \ker \sigma$, injective by that same equivalence, and surjective by the definition of ${\mathcal{H}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$. Thus it is an isomorphism onto its declared image by definition.

This definition does not assert that every field has a radial limit, that ${\mathcal{H}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ is dense in $\mathcal{H}$, or that the image is all of $\mathcal{H}$. A later theorem that uses a target sector ${\mathcal{H}}_{\ast }\subseteq \mathcal{H}$, cyclicity, density, surjectivity, or a complete energy eigenbasis must state that condition separately. In particular, no pure-point-spectrum or Osterwalder–Schrader reconstruction statement is included.

Radial quantization of a plane CFT

Primary field in a holomorphic coordinate

Real and complex inner-product spaces and their induced length

Hilbert space

Symmetric, self-adjoint and essentially self-adjoint operators

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)