Strict Quantization, Elliptic Operators, and E-Theory

严格量化、椭圆算子和 E 理论

• 批准号: 0071120
• 负责人: John Trout
• 金额: $ 8.54万
• 依托单位: Dartmouth College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071120&HistoricalAwards=false
• 关键词: Strict; Quantization; Elliptic; Operators; Theory

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.

AbstractTrout该项目旨在研究和深化 Alain Connes 和 Nigel Higson 的算子代数 E 理论、流形上椭圆微分算子的指数理论以及量子物理之间的关系。首先，我们将研究（正）渐近态射（形成 E 理论中的基本循环）和“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）量化方案不变量容器的基本用途。这也应该给出新颖的公式，用于将渐近态射写为算子值积分，以及计算 K 同调和 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索引。第二部分涉及扩展 Erik Guentner 的工作，使用 E 理论群来理解椭圆微分算子和严格量化方案之间的关系，例如，卡勒流形上的 Dolbeault 算子（的 E 理论元素）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，旨在开发严格量化方案的E理论分类方法，该方法与明星产品的形式代数（非物理）形变量化中的上同调分类平行。该项目的目的是更彻底地研究量子物理学与菲尔兹奖得主阿兰·康尼斯和奈杰尔·希格森的算子代数E理论之间的关系。该数学理论将分类群与算子代数对相关联。群元素由两个算子代数之间称为渐近态射的对象确定。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，量子场论中与规范场耦合的狄拉克算子，以及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，这些结构具有与之相关的渐近态射（分别在经典可观测量和量子可观测量的代数之间）。因此，他们定义了 E 理论组中的元素。通过充分理解这种对应关系，我们希望证明计算这些 E 理论不变量提供了一种自然的过程，用于对不同类型的量子力学系统进行分类和定义障碍。