Fredenhagen–Jörß fields and their energy bounds

Let $(\mathcal{A},U,\mathcal{H},\mathrm{\Omega })$ be a conformal net with the interval, Hilbert-space, commutant, and projective covariance conventions fixed earlier on this page. Write $L_0\ge 0$ for the rotation generator, $H^{\mathrm{\infty }}={\bigcap }_{k\ge 0}D((1+L_0{)}^k)$ for its smooth finite-energy domain, and $H_{\mathrm{f}\mathrm{i}\mathrm{n}}$ for the algebraic span of all eigenspaces of $L_0$. A vector $a\in \mathcal{H}$ is quasi-primary of weight $d$ when $L_{0}a=da$, $d\in {\mathbb{Z}}_{\ge 0}$, and $L_{1}a=0$. The field data include an antiunitary involution $\theta$ with ${\theta }^2=1$ and $\theta \mathrm{\Omega }=\mathrm{\Omega }$; it is the stated anti-linear involution used in the theorem.

For an interval $I$, the vacuum is required to be cyclic and separating for $\mathcal{A}(I)$: cyclic means $\overline{\mathcal{A}(I)\mathrm{\Omega }}=\mathcal{H}$, and separating means $X\mathrm{\Omega }=0$ for $X\in \mathcal{A}(I)$ implies $X=0$. The Tomita operator $S_I$ is the closure of the antilinear map $\mathcal{A}(I)\mathrm{\Omega }\to \mathcal{H}$, $X\mathrm{\Omega }\mapsto X^{\ast }\mathrm{\Omega }$, and $\mathcal{A}(I{)}^{\mathrm{\prime }}\mathrm{\Omega }=\{X\mathrm{\Omega }:X\in \mathcal{A}(I{)}^{\mathrm{\prime }}\}$ is the dense commutant-vacuum domain. For a quasi-primary $a$ of weight $d$ and $f\in C_c^{\mathrm{\infty }}(I)$, with ${\widehat{f}}_m={\int }_{-\pi }^{\pi }f(e^{i\vartheta })e^{-im\vartheta }d\vartheta \mathrm{/}(2\pi )$, define the Fredenhagen–Jörß vector $a(f)={\sum }_{n\ge 0}{\widehat{f}}_{-n-d}\frac{L_{-1}^{n}a}{n!}\mathrm{.}$ When $a(f)\in D(S_I)$ and $S_{I}a(f)=(-1{)}^d(\theta a)(\bar{f})$, the densely defined operator $T_{I,a,f}^0$ on $\mathcal{A}(I{)}^{\mathrm{\prime }}\mathrm{\Omega }$ given by $T_{I,a,f}^0(X\mathrm{\Omega })=Xa(f)$ is closable; its closure is the Fredenhagen–Jörß smeared field $Y_I(a,f)$. The local gluing lemma below uses only the linearity and interval-compatibility properties explicitly recorded at the end.

For a closed densely defined operator $T$, say $T$ is affiliated with a von Neumann algebra $M$ when every unitary $u\in M^{\mathrm{\prime }}$ maps $D(T)$ onto itself and $uT\xi =Tu\xi$ for every $\xi \in D(T)$. The supplied polar decomposition gives $T=V_T\mathrm{\mid }T\mathrm{\mid }$, where $V_T$ is a bounded partial isometry and $\mathrm{\mid }T\mathrm{\mid }=(T^{\ast }T{)}^{1\mathrm{/}2}$ is positive self-adjoint with $D(\mathrm{\mid }T\mathrm{\mid })=D(T)$. Let $E_{\mathrm{\mid }T\mathrm{\mid }}(B)$ be the spectral projection of $\mathrm{\mid }T\mathrm{\mid }$ for a Borel set $B\subseteq [0,\mathrm{\infty })$. For a family $\mathcal{T}$ of such operators, define $W^{\ast }(\mathcal{T})=W^{\ast }(\{V_T:T\in \mathcal{T}\}\cup \{E_{\mathrm{\mid }T\mathrm{\mid }}(B):T\in \mathcal{T},B\subseteq [0,\mathrm{\infty })\text{ Borel}\})$, where the outer $W^{\ast }$ is the weak-operator closure of the bounded unital $\ast$-algebra generated by this set. Every $T\in \mathcal{T}$ is affiliated with this algebra: a unitary in its commutant commutes with $V_T$ and all spectral projections of $\mathrm{\mid }T\mathrm{\mid }$, hence preserves $D(\mathrm{\mid }T\mathrm{\mid })$ and commutes with $\mathrm{\mid }T\mathrm{\mid }$ by the spectral calculus, and therefore preserves $D(T)$ and commutes with $T=V_T\mathrm{\mid }T\mathrm{\mid }$. This is the convention for generating local algebras from closed, possibly unbounded, fields. A family $F\subset H_{\mathrm{f}\mathrm{i}\mathrm{n}}$ is a generating family of quasi-primary vectors when every $a\in F$ is quasi-primary and, for every $I\in \mathcal{I}$,

$$\mathcal{A}(I)=W^{\ast }\text{\hspace{0.17em}⁣}(\{Y_{I_1}(a,f):I_1\in \mathcal{I},I_1\subset I,a\in F,f\in C_c^{\mathrm{\infty }}(I_1)\})\mathrm{.}$$

The family is PCT-invariant when $\theta F=F$. Its Fredenhagen–Jörß fields satisfy polynomial energy bounds when for each $a\in F$ there are $M_a>0$ and integers $s_a,k_a\ge 0$ such that $\mathrm{\parallel }{Y}_I(a,f)b\mathrm{\parallel }\le M_a\mathrm{\parallel }f{\mathrm{\parallel }}_{s_a}\mathrm{\parallel }(1+L_0{)}^{k_a}b\mathrm{\parallel },\mathrm{\parallel }f{\mathrm{\parallel }}_s={\sum }_{n\in \mathbb{Z}}(1+\mathrm{\mid }n\mathrm{\mid }{)}^s\mathrm{\mid }{\widehat{f}}_n\mathrm{\mid },$ for every interval $I$, $f\in C_c^{\mathrm{\infty }}(I)$, and $b\in \mathcal{A}(I{)}^{\mathrm{\prime }}\mathrm{\Omega }\cap H^{\mathrm{\infty }}$. The finite-dimensional-energy condition means every eigenspace $\ker (L_0-\lambda )$ is finite-dimensional for each eigenvalue $\lambda \ge 0$. All these conditions are data or hypotheses; no statement that every conformal net admits such fields is made.

The fields are linear in test functions on the common invariant core $H^{\mathrm{\infty }}$ and satisfy interval compatibility: if $I_1\subset I_2$ and $\mathrm{supp}f\subset I_1$, then $Y_{I_2}(a,f)b=Y_{I_1}(a,f)b$ for every $b\in H^{\mathrm{\infty }}$. These are the exact properties used by the local gluing lemma below.

Möbius and conformal nets on the circle

Quasi-primary translation series converges in the Hilbert space

Hilbert space

Von Neumann algebras and commutants

Strong and weak operator topologies

The Axiom of Choice

Polar decomposition of a closed densely defined operator

Unbounded Borel functional calculus: domains, products, spectral mapping