标题： 谱非交换几何与量子化：一个简单例子 显示英文标题

标题： Spectral noncommutative geometry and quantization: a simple example

摘要： 我们探讨非交换几何（在谱三元组形式下）与量子力学之间的关系。 为此，我们考虑由谱三元组定义的非交换几何的动力学理论，并研究其量化。 特别是，我们考虑一个基于有限维谱三元组（A, H, D）的简单模型，该模型模仿了广义相对论谱形式的某些方面。 我们找到了物理相空间，即与A和H兼容的在壳狄拉克算子的空间。我们在该相空间上定义了一个自然的辛结构，并使用协变正则量化方法构建相应的量子理论。 我们证明，在经典非交换几何中，代数A上的某些两个态（两个“时空点”）之间的康尼斯距离是一个任意正数，在量子理论中却变为离散的，并计算了它的谱。 非交换几何的量子态形成一个希尔伯特空间K。D被提升为H和K的直积*H上的算子*D。三元组（A, *H, *D）可以看作是三元组（A, H, D）族的量化。 显示英文摘要

摘要： We explore the relation between noncommutative geometry, in the spectral triple formulation, and quantum mechanics. To this aim, we consider a dynamical theory of a noncommutative geometry defined by a spectral triple, and study its quantization. In particular, we consider a simple model based on a finite dimensional spectral triple (A, H, D), which mimics certain aspects of the spectral formulation of general relativity. We find the physical phase space, which is the space of the onshell Dirac operators compatible with A and H. We define a natural symplectic structure over this phase space and construct the corresponding quantum theory using a covariant canonical quantization approach. We show that the Connes distance between certain two states over the algebra A (two ``spacetime points''), which is an arbitrary positive number in the classical noncommutative geometry, turns out to be discrete in the quantum theory, and we compute its spectrum. The quantum states of the noncommutative geometry form a Hilbert space K. D is promoted to an operator *D on the direct product *H of H and K. The triple (A, *H, *D) can be viewed as the quantization of the family of the triples (A, H, D).