Covariance of Buchholz--Fredenhagen Categories under QCA, Reflection, and Time Reversal

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1. Definition 1.1: [Def.~1.1
   PF:2026.000008/v1/DEF-1.1

   {Ogata2022}}] A pure state $\psi$ satisfies \emph{approximate Haag duality} (AHD) if the following holds. For all $\varphi\in(0,2\pi)$ and $\varepsilon>0$ with $\varphi+4\varepsilon<2\pi$, there are a constant $R_{\varphi,\varepsilon}\ge0$ and, for each $\delta>0$ with $\varphi+2\varepsilon+2\delta<2\pi$, a decreasing function $f_{\varphi,\varepsilon,\delta}:[0,\infty)\to[0,\infty)$ with $f_{\varphi,\varepsilon,\delta}(t)\to0$ as $t\to\infty$, such that every cone $\Lambda$ with opening angle $\varphi$ has the two properties below. \item There is a unitary $U_{\Lambda,\varepsilon}$ on $\Hil_\psi$ with \R_\psi(\Lambda^c)' \subset \Ad(U_{\Lambda,\varepsilon})\, \R_\psi\bigl((\Lambda-R_{\varphi,\varepsilon}e_\L…

2. Theorem 2.1
   PF:2026.000008/v1/THM-2.1

   Let $\psi$ be a pure state that satisfies AHD as in Definition 1.1. \item \textbf{QCA.} Let $\alpha\in\operatorname{Aut}(\A)$ and $R\ge0$ with \alpha^{\pm1}(\A_X)\subset\A_{X^{+R}} \qquad\text{for all finite }X\subset\Gamma. Then $\psi_\alpha:=\psi\circ\alpha$ is a pure state that satisfies AHD, and \[ F_\alpha(\rho):=\rho\circ\alpha, \qquad F_\alpha(x):=x \] is a unitary braided tensor equivalence $F_\alpha:\BF(\psi)\to\BF(\psi_\alpha)$. Here $\BF(\psi_\alpha)$ is built from the GNS representation $\pi_\psi\circ\alpha$ on $\Hil_\psi$. \item \textbf{Reflection.} Let $r$ be an affine reflection of $\mathbb R^2$ with $r\Gamma=\Gamma$, and let $\beta_r\in\operatorname{Aut}(\A)$ satisfy $\beta_r(\A_X)=\A_{rX}$ f…

3. Lemma 3.1: Cones absorb thickenings
   PF:2026.000008/v1/LEM-3.1

   Let $\Lambda$ be a cone with opening angle $\varphi\in(0,2\pi)$, let $R\ge0$, and put \[ \kappa_R(\varphi):=\frac{R}{\sin(\varphi/2)}. \] Then \Lambda^{+R}\subset\Lambda-\kappa_R(\varphi)e_\Lambda \qquad\text{and}\qquad \bigl(\Lambda+\kappa_R(\varphi)e_\Lambda\bigr)^{+R}\subset\Lambda.

4. Lemma 3.2: Spread
   PF:2026.000008/v1/LEM-3.2

   Let $\alpha$ satisfy the spread condition (equation reference) of Theorem 2.1(i), and let $K,S\subset\mathbb R^2$. If $K^{+R}\subset S$, then $\alpha^{\pm1}(\A_K)\subset\A_S$. In particular, $\alpha^{\pm1}(\A_{(S^{+R})^c})\subset\A_{S^c}$ for every $S$.

5. Corollary 4.1: Comparison of the two nets
   PF:2026.000008/v1/COR-4.1

   Under the spread condition (equation reference), Lemmas 3.1 and 3.2 imply that, for every cone $\Lambda$, \alpha^{\pm1}(\A_{\Lambda^c})\subset\A_{(\Lambda+\kappa_\Lambda e_\Lambda)^c}, \\ \R_{\psi_\alpha}(\Lambda)\subset\R_\psi(\Lambda-\kappa_\Lambda e_\Lambda), \qquad \R_\psi(\Lambda)\subset\R_{\psi_\alpha}(\Lambda-\kappa_\Lambda e_\Lambda), \\ \R_\psi\bigl((\Lambda+\kappa_\Lambda e_\Lambda)^c\bigr)' \subset\R_{\psi_\alpha}(\Lambda^c)' \subset\R_\psi\bigl((\Lambda-\kappa_\Lambda e_\Lambda)^c\bigr)'.

6. Proposition 4.2
   PF:2026.000008/v1/PROP-4.2

   If $\psi$ satisfies AHD as in Definition 1.1 and $\alpha$ satisfies the spread condition (equation reference) of Theorem 2.1(i), then $\psi_\alpha$ satisfies AHD with R'_{\varphi,\varepsilon} &:=\kappa_R(\varphi)+R_{\varphi,\varepsilon}+\kappa_R(\varphi+2\varepsilon), \\ f'_{\varphi,\varepsilon,\delta}(t) &:= 2, & 0\le t<C,\\ f_{\varphi,\varepsilon,\delta}(t-C), & t\ge C, \qquad C:=\kappa_R(\varphi)+\kappa_R(\varphi+2\varepsilon+2\delta). In particular, if the functions $f$ for $\psi$ decay exponentially (or faster than any power), so do the functions $f'$.