论文标题
扭曲的Araki-Woods代数的模块化结构和夹杂物
Modular Structure and Inclusions of Twisted Araki-Woods Algebras
论文作者
论文摘要
在扭曲的第二个量化的一般环境中（包括Bose/Fermi第二量化，$ s $ -Smmore-Symmortric Fock空间以及特殊情况下的自由概率中的完整Fock空间），分析了扭曲的Fock空间上的von Neumann代数。这些扭曲的Araki-Woods代数$ \ MATHCAL {l} _ {t}（h）$取决于扭曲操作员$ t $和一个标准子空间$ h $在单粒子空间中。在$ t $和$ h $上的兼容性假设下，事实证明，当$ t $满足$ t $满足交叉对称的标准子空间版本和Yang-Baidter方程（BraidEquation）时，Fock真空是循环的，并以$ \ Mathcal {l} _ {l} _ {t}（t}（h）$分开。在这种情况下，明确确定了Tomita-Takesaki模块化数据。 包含$ \ MATHCAL {l} _ {t}（k）\ subset \ Mathcal {l} _ {t}（t}（h）$ twisted araki-woods代数的$在两种情况下进行分析：如果包含是半侧的，并且曲折满足扭曲的满足，则显示出符号，它显示出单明。如果包含基础标准子空间$ k \ subset H $满足$ l^2 $ - 核条件，则$ \ nathcal {l} _ {t}（t}（k）\ subset \ subset \ mathcal {l} _ {l} _ {t} _ {t}（t） 讨论了这些结果在代数量子场理论中可观察到的定位的应用。
In the general setting of twisted second quantization (including Bose/Fermi second quantization, $S$-symmetric Fock spaces, and full Fock spaces from free probability as special cases), von Neumann algebras on twisted Fock spaces are analyzed. These twisted Araki-Woods algebras $\mathcal{L}_{T}(H)$ depend on the twist operator $T$ and a standard subspace $H$ in the one-particle space. Under a compatibility assumption on $T$ and $H$, it is proven that the Fock vacuum is cyclic and separating for $\mathcal{L}_{T}(H)$ if and only if $T$ satisfies a standard subspace version of crossing symmetry and the Yang-Baxter equation (braid equation). In this case, the Tomita-Takesaki modular data are explicitly determined. Inclusions $\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)$ of twisted Araki-Woods algebras are analyzed in two cases: If the inclusion is half-sided modular and the twist satisfies a norm bound, it is shown to be singular. If the inclusion of underlying standard subspaces $K\subset H$ satisfies an $L^2$-nuclearity condition, $\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)$ has type III relative commutant for suitable twists $T$. Applications of these results to localization of observables in algebraic quantum field theory are discussed.
