形变量子化的GNS表示的局部性
Locality in GNS Representations of Deformation Quantization
摘要：
在形变量子化的框架下，我们应用形式GNS构造方法，在$\mathbb C[[λ]]$上的准Hilbert空间中找到了形变代数的表示，并在这些准希尔伯特空间中建立了局部算子的概念。局部算子中的交换子用于区分“热”和“纯”表示。本地交换子的计算举例说明在各种情况下，导致物理上合理的区别热表示和纯的。此外，在GNS表示关于KMS泛函和余切丛上的薛定谔表示的特殊情况下，证明了von Neumann的二次交换子定理的类似定理。最后证明了Tomita-Takesaki定理的一个形式化版本。
英文摘要：
In the framework of deformation quantization we apply the formal GNS construction to find representations of the deformed algebras in pre-Hilbert spaces over $\mathbb C[[λ]]$ and establish the notion of local operators in these pre-Hilbert spaces. The commutant within the local operators is used to distinguish `thermal' from `pure' representations. The computation of the local commutant is exemplified in various situations leading to the physically reasonable distinction between thermal representations and pure ones. Moreover, an analogue of von Neumann's double commutant theorem is proved in the particular situation of a GNS representation with respect to a KMS functional and for the Schrödinger representation on cotangent bundles. Finally we prove a formal version of the Tomita-Takesaki theorem.
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