Strict Quantization, Elliptic Operators, and E-Theory

摘要本课题旨在研究和深化Alain Connes和Nigel Higson的算子代数e理论、流形上椭圆微分算子的指标理论与量子物理之间的关系。首先，我们将研究（正）渐近态射（构成e理论中的基本环）与“渐近”投影值测度之间的对应关系。这将有助于理解e理论群作为严格（物理）量子化方案不变量的容器的基本用途。这也将给出新的公式，用于将渐近态射写成算子值积分，以及计算k -同调与k理论之间的配对，例如，计算耦合到规范连接的狄拉克型算子的指标。第二部分扩展了Erik Guentner的工作，利用e -理论群来理解椭圆微分算子与严格量化方案之间的关系，例如Kahler流形上的Dolbeault算子（e -理论元）与相关Bergmann-Fock空间上的Berezin-Toeplitz量化之间的关系。第三部分是一个长期项目，开发严格量化方案的e理论分类方法，与星积的形式代数（非物理）变形量化中的上同调分类平行。这个项目的目的是更彻底地研究量子物理学和运算代数e理论之间的关系，这些理论是由菲尔兹奖得主阿兰·康恩斯和奈吉尔·希格森提出的。这个数学理论把分类群和算子代数对联系起来。群元素由两个算子代数之间的渐近态射对象决定。它与量子理论的关系如下。量子理论中使用不同类型的结构来模拟亚原子和原子系统，例如量子场论中耦合到规范场的狄拉克算子，操作量子物理和量子计算中的算子值测度等。在适当的条件下，这些结构具有与它们相关的渐近态射（分别在经典可观测数和量子可观测数的代数之间）。因此，它们定义了e理论群中的元素。通过充分理解这种对应关系，我们想表明计算这些e理论不变量为分类和定义不同类型的量子力学系统的障碍提供了一个自然的过程。

AbstractTroutThis project is designed to investigate and deepen the relationship between the operator algebraic E-theory of Alain Connes and Nigel Higson, the index theory of elliptic differential operators on manifolds, and quantum physics. First, we will investigate the correspondence between (positive) asymptotic morphisms (which form the basic cycles in E-theory) and "asymptotic" projection-valued measures. This will help to understand the fundamental use of E-theory groups as receptacles for invariants of strict (physical) quantization schemes. This should also give novel formulas for writing asymptotic morphisms as operator-valued integrals, and for computing the pairing between K-homology and K-theory, e.g., in computing the index of a Dirac type operator coupled to a gauge connection. The second part deals with extending the work of Erik Guentner in using E-theory groups to understand the relationship between elliptic differential operators and strict quantization schemes, e.g., the relationship between the (E-theory elements of) the Dolbeault operator on a Kahler manifold and the Berezin-Toeplitz quantization on the associated Bergmann-Fock space. The third part is a long-term project to develop an E-theoretic classification method for strict quantization schemes that parallels the cohomological classification in the formally algebraic (nonphysical) deformation quantization of star products.The purpose of this project is to more thoroughly investigate the relationship between quantum physics and the operator algebraic E-theory of the Fields medalist Alain Connes and Nigel Higson. This mathematical theory associates classifying groups to pairs of operator algebras. Group elements are determined by objects called asymptotic morphisms between the two operator algebras. The relationship to quantum theory is as follows. Different types of structures are used in quantum theory to model subatomic and atomic systems, for example, Dirac operators coupled to a gauge field in quantum field theory, and operator-valued measures in operational quantum physics and quantum computing, etc. Under appropriate conditions, these structures have asymptotic morphisms associated to them (between the algebras of the classical observables and quantum observables, respectively). Hence, they define elements in an E-theory group. By fully understanding this correspondence, we want to show that computing these E-theory invariants provides a natural procedure for classifying, and defining obstructions for, diverse types of quantum-mechanical systems.