Radially ordered operator product expansion

Fix a selected mathematical radial model on an open coordinate patch $U\subseteq {\mathbb{C}}^{\times }$, with dimensionless coordinates and $d^{2}z=dxdy$. Write $\mathcal{D}(U)=C_c^{\mathrm{\infty }}(U;\mathbb{C})$ with the compact-support test-function topology, and let $\mathcal{D}$ be the common complex field domain, with algebraic dual ${\mathcal{D}}^{\ast }$. A weak field is a linear map $A:\mathcal{D}(U)\to {\mathrm{Hom}}_{\mathbb{C}}(\mathcal{D},{\mathcal{D}}^{\ast })$ whose coefficients $(A(\phi )u)(v)$ are continuous in $\phi$ for every $u,v\in \mathcal{D}$; this is the supplied primary-field convention, without requiring that $A$ be primary. Such fields have neither point evaluations nor composable operator products by definition.

Set $Q_{>}=\{(z,w)\in U^2:\mathrm{\mid }z\mathrm{\mid }>\mathrm{\mid }w\mathrm{\mid }\}$ and $Q_{<}=\{(z,w)\in U^2:\mathrm{\mid }w\mathrm{\mid }>\mathrm{\mid }z\mathrm{\mid }\}$. Work in a specified open simply connected region $Q\subseteq Q_{>}$ on which the chosen coefficient branches exist. In a weak realization the model additionally supplies a two-insertion product $P_{AB}:\mathcal{D}(Q)⟶{\mathrm{Hom}}_{\mathbb{C}}(\mathcal{D},{\mathcal{D}}^{\ast }),$ linear with continuous scalar coefficients, and separately the reversed product on a specified region in $Q_{<}$. The notation $A(z)B(w)$ on $Q$ denotes this product distribution, not composition of weak fields. Numerical weak coefficients may be used only where the model supplies smooth functions $p_{u,v}$ representing it, meaning $(P_{AB}(\chi )u)(v)={\int }_Q\chi (z,w)p_{u,v}(z,w)d^{2}z{d}^{2}w$. Alternatively a purely formal realization supplies fields as coefficient arrays with endomorphisms of a common complex vector space $V$, and a specified iterated generalized-power series space in which the products and re-expansions used are well defined coefficientwise. Formal products mean coefficientwise composition on $V$, not evaluation at points; every coefficient sum must be finite or have a separately specified convergence meaning. A formal realization alone asserts no weak realization. If both are used, their scalar coefficient compatibility must be supplied. Larger-radius factors stand on the left; the ungraded reversed order is $B(w)A(z)$, while a graded model must specify its interchange sign. Equal radii have no order unless the model declares one.

Choose a label space $\mathcal{F}={⨁}_{\lambda \in \mathrm{\Lambda }}{\mathcal{F}}_{\lambda }$, a specified indexed spanning family $(a_{\lambda i}{)}_{i\in I_{\lambda }}$ in each summand, and an associated field $A_{\lambda i}$ for each label. The full index set is $J=\{(\lambda ,i):\lambda \in \mathrm{\Lambda },i\in I_{\lambda }\}$, including descendant and multiplicity labels; it is not the set of all vectors in $\mathcal{F}$. Coefficients $C_{AB,\lambda i}$, normalizations and branches are supplied data, without an assertion of unique coefficients for a redundant spanning family. In a Hilbert realization, also specify a closed channel sector $H_{\ast }$ and vectors $s_{\lambda i}\in H_{\ast }$ assigned to the labels. The family is complete for this channel when $\stackrel{}{\mathrm{span}}\{s_{\lambda i}:(\lambda ,i)\in J\}=H_{\ast }$. Neither existence of such a family nor a state assignment is inferred from weak fields alone.

The schematic OPE is $A(z)B(w)\sim {\sum }_{(\lambda ,i)\in J}{C}_{AB,\lambda i}(z,w)A_{\lambda i}(w)\mathrm{.}$ In a weak realization require $C_{AB,\lambda i}\in C^{\mathrm{\infty }}(Q)$ on the chosen branch. Each individual term is the well-defined distribution $T_{\lambda i}(\chi ):=A_{\lambda i}(g_{\lambda i,\chi }),g_{\lambda i,\chi }(w):={\int }_U{C}_{AB,\lambda i}(z,w)\chi (z,w)d^{2}z\mathrm{.}$ Here the smooth product $C_{AB,\lambda i}\chi$, for $\chi \in \mathcal{D}(Q)$, is extended by zero to $U^2$ before integration; thus $g_{\lambda i,\chi }\in \mathcal{D}(U)$ and the map $\chi \mapsto g_{\lambda i,\chi }$ is continuous. No multiplication of two distributions is involved. Let $S_F={\sum }_{j\in F}{T}_j$ for finite $F\subseteq J$ in this realization, or the corresponding formal partial sum. Finite subsets are directed by inclusion. The notation $\sim$ alone selects no meaning for the sum; the model declares the following meaning or meanings separately.

1. Formal OPE: equality coefficient by coefficient in the specified iterated generalized-power series space on the declared branch or line bundle (ordinary Laurent distributions are the integer-power special case). The sum over $J$ must be defined coefficientwise in that space; no numerical convergence is implied.

2. Asymptotic OPE: the model supplies a directed family of finite partial sums, a normed or seminormed target, and representatives $R_F(z,w)$ of the product minus the partial sum in that target, together with real orders $q_F\to +\mathrm{\infty }$. For each fixed $F$, require $p(R_F(z,w))=o(\mathrm{\mid }z-w{\mathrm{\mid }}^{q_F})$ as $z\to w$ in a declared approach region within $Q$. This pointwise version requires the stated representatives. A distributional version instead specifies test functions ${\chi }_{\epsilon }\in \mathcal{D}(Q)$ approaching the diagonal, their normalization and support scaling, and requires $p((P_{AB}-S_F)({\chi }_{\epsilon }))=o({\epsilon }^{q_F})$ as $\epsilon \downarrow 0$ for each $F$; the seminorm $p$ is on the declared space of smeared outputs.

3. Hilbert-convergent OPE: the model supplies a fused-pair vector ${\mathrm{\Phi }}_{AB}(z,w)\in H_{\ast }$ and coefficient-weighted term vectors $t_j(z,w)\in H_{\ast }$ at every allowed pair, or after an explicitly specified smearing. For every such fixed configuration the net ${\sum }_{j\in F}{t}_j$ converges in Hilbert norm to ${\mathrm{\Phi }}_{AB}$. These vector assignments, their relation to the field labels and coefficients, and their insertion domains are additional data, not point evaluation of weak fields.

4. Weakly convergent correlator OPE: in a weak realization require $(S_F(\chi )u)(v)\to (P_{AB}(\chi )u)(v)$ for each declared $\chi ,u,v$. In a Hilbert correlator realization, specify a set of finite-norm complementary states $\mathrm{\Psi }$, an overlap order, and scalar compatibility: the correlator equals ${\mathrm{\ell }}_{\mathrm{\Psi }}({\mathrm{\Phi }}_{AB})$ and its $j$th OPE term equals ${\mathrm{\ell }}_{\mathrm{\Psi }}(t_j)$. Require ${\sum }_{j\in F}{\mathrm{\ell }}_{\mathrm{\Psi }}(t_j)\to {\mathrm{\ell }}_{\mathrm{\Psi }}({\mathrm{\Phi }}_{AB})$ for each such state and fixed allowed configuration. With the supplied first-argument-linear inner product, either ${\mathrm{\ell }}_{\mathrm{\Psi }}(v)=⟨v,\mathrm{\Psi }⟩$ or ${\mathrm{\ell }}_{\mathrm{\Psi }}(v)=⟨\mathrm{\Psi },v⟩$ may be declared; the latter conjugates scalar coefficients in $t_j$. Unsmeared scalar assertions require the separately supplied scalar or vector representatives just described.

When coefficient functions contain complex powers, their branch or line-bundle data are part of the selected model and are fixed on the simply connected radial configuration region being used. For a two-dimensional global-conformal decomposition, a block is the subseries obtained by grouping all descendants assigned to one intermediate global-conformal primary label $\lambda$; these blocks are the global-conformal (quasi-primary-family) blocks and are not full Virasoro blocks unless that stronger grouping is separately supplied. This definition distinguishes formal, asymptotic, Hilbert, and scalar-correlator convergence; it asserts none of them for a particular theory.

Primary field in a holomorphic coordinate

Radial quantization of a plane CFT

A source may provide a formal OPE or a source-reported convergence result without proving its hypotheses in the selected model. A local theorem that uses the Hilbert or weak-correlator meaning must state the finite-norm, completeness, coefficient/state, and insertion-separation hypotheses explicitly.