论文标题
谎言组的标准子空间网
Nets of standard subspaces on Lie groups
论文作者
论文摘要
让G成为一个谎言组，带有Lie代数$ \ Mathfrak {g} $，$ h \ in \ frak {g} $是一个元素，该元素为此定义了$ \ mathfrak {g} $和$τ_g$ ndection的3级级别的3年级，$τ_g$ ndiencation o g n of g n right of g n Mathip odution ro. ad（h）} $。我们考虑Lie组的反统一表示$g_τ= g \ rtimes \ {在$ exp（\ mathbb {r} h）$和一个开放子集$ o \ subseteq g $下不变的分布向量的实际子空间e，我们将真实的子空间$ h_e（o）\ subseteq h $，由子空间$ u（φ）是$ o $的实值测试功能。然后，对于每个非空开放子集$ o \ subseteq g $（reeh-schlider属性），$ h_e（o）$在$ h_e（g）$中是密度的。 For the real standard subspace $V \subseteq H$, for which $J_V = U(τ_G)$ is the modular conjugation and $Δ_V^{-it/2π} = U(\exp th)$ is the modular group, we obtain sufficient conditions to be of the form $H_E(S)$ for an open subsemigroup $S \subseteq G$.如果$ \ mathfrak {g} $是半岛型管类型的简单理想，我们会验证这些标准并获得环环子锅的网$ h_e（o）$，$ o \ o \ subseteq g $ Cayley类型。通过第二个量化，这些网络从哈格（Haag）的意义上导致了自由量子场 - 卡斯特勒（Kastler）在因果均质空间上，其组是由模块化组和结合产生的。
Let G be a Lie group with Lie algebra $\mathfrak{g}$, $h \in \frak{g}$ an element for which the derivation ad(h) defines a 3-grading of $\mathfrak{g}$ and $τ_G$ an involutive automorphism of G inducing on $\mathfrak{g}$ the involution $e^{πi ad(h)}$. We consider antiunitary representations $U$ of the Lie group $G_τ= G \rtimes \{e,τ_G\}$ for which the positive cone $C_U = \{ x \in \mathfrak{g} : -i \partial U(x) \geq 0\}$ and $h$ span $\mathfrak{g}$. To a real subspace E of distribution vectors invariant under $exp(\mathbb{R} h)$ and an open subset $O \subseteq G$, we associate the real subspace $H_E(O) \subseteq H$, generated by the subspaces $U(φ)E$, where $φ\in C^\infty_c(O,\mathbb{R})$ is a real-valued test function on $O$. Then $H_E(O)$ is dense in $H_E(G)$ for every non-empty open subset $O \subseteq G$ (Reeh--Schlider property). For the real standard subspace $V \subseteq H$, for which $J_V = U(τ_G)$ is the modular conjugation and $Δ_V^{-it/2π} = U(\exp th)$ is the modular group, we obtain sufficient conditions to be of the form $H_E(S)$ for an open subsemigroup $S \subseteq G$. If $\mathfrak{g}$ is semisimple with simple hermitian ideals of tube type, we verify these criteria and obtain nets of cyclic subspacs $H_E(O)$, $O \subseteq G$, satisfying the Bisognano--Wichman property for some domains O. Our construction also yields such nets on simple Jordan space-times and compactly causal symmetric spaces of Cayley type. By second quantization, these nets lead to free quantum fields in the sense of Haag--Kastler on causal homogeneous spaces whose groups are generated by modular groups and conjugations.
