论文标题
自旋系统的符号对应关系的渐近定位以及$ s^2 $的顺序量化
Asymptotic localization of symbol correspondences for spin systems and sequential quantizations of $S^2$
论文作者
论文摘要
$ su（2）$下的量子或经典机械系统对称称为自旋系统。 $ su（2）$ - 均等地图，从$（n+1）$ - 方形矩阵到$ 2 $ -Sphere s^2的功能，满足某些基本属性，称为spin- $ j $符号通信（$ j $ n = 2j \ in \ mathbb {n} $）。给定旋转$ j $符号对应，矩阵代数会引起符号的扭曲$ j $ - 代数。在本文的第一部分中，我们为$ s^2 $上的平滑函数的泊松代数（$ n \ to \ infty $）从扭曲的$ j $ -Elgebras中渐近出现时，建立了一个更直观的标准。现在，在许多情况下，在[20]中获得的数值标准在许多情况下等效于描述（反）泊松类型的符号对应序列，现在是根据某个家族中所有投影仪（量子纯状态）的符号定位（抗）的符号对应序列。对于某些重要的符号对应序列，这种经典定位条件等于泊松代数的渐近出现。但总的来说，经典的本地化条件比泊松出现强。因此，我们还考虑了投影仪符号渐近定位的一些较弱的概念。在本文的第二部分中，对于（抗）泊松类型的符号对应的每个序列，我们定义了$ s^2 $上的平滑函数的顺序量化及其作用于地面希尔伯特空间的渐近算子。然后，在提出了这些结构的一些具体示例之后，我们获得了符号对应序列的渐近定位与其在$ s^2 $上对平滑函数的顺序量化的渐近定位之间的某些关系。
Quantum or classical mechanical systems symmetric under $SU(2)$ are called spin systems. A $SU(2)$-equivariant map from $(n+1)$-square matrices to functions on the $2$-sphere S^2, satisfying some basic properties, is called a spin-$j$ symbol correspondence ($n = 2j \in \mathbb{N}$). Given a spin-$j$ symbol correspondence, the matrix algebra induces a twisted $j$-algebra of symbols. In the first part of this paper, we establish a more intuitive criterion for when the Poisson algebra of smooth functions on $S^2$ emerges asymptotically ($n \to \infty$) from the sequence of twisted $j$-algebras. This more geometric criterion, which in many cases is equivalent to the numerical criterion obtained in [20] for describing symbol correspondence sequences of (anti-)Poisson type, is now given in terms of a classical (asymptotic) localization of symbols of all projectors (quantum pure states) in a certain family. For some important kinds of symbol correspondence sequences, such a classical localization condition is equivalent to asymptotic emergence of the Poisson algebra. But in general, the classical localization condition is stronger than Poisson emergence. We thus also consider some weaker notions of asymptotic localization of projector-symbols. In the second part of this paper, for each sequence of symbol correspondences of (anti-)Poisson type, we define the sequential quantization of a smooth function on $S^2$ and its asymptotic operator acting on a ground Hilbert space. Then, after presenting some concrete examples of these constructions, we obtain some relations between asymptotic localization of a symbol correspondence sequence and the asymptotics of its sequential quantization of smooth functions on $S^2$.
