Quantization on Cotangent Bundles

余切丛的量化

• 批准号: 0200649
• 负责人: Brian Hall
• 金额: $ 10.34万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200649&HistoricalAwards=false
• 关键词: Quantization; Cotangent; Bundles

PI: Brian C. Hall, University of Notre DameDMS-0200649Abstract:The PI's research concerns the quantization of certainspecial classical systems, those whose classical configurationspace is a compact symmetric space, such as a sphere. Thesimplest physical example of such a system is (the rotationaldegrees of freedom of) a rigid body, whose configuration space isthe rotation group SO(3). The phase space of any such system,namely, the cotangent bundle of the compact symmetric space, hasa natural complex structure that makes the phase space into aKahler manifold. Thus the quantization of such a system can bedone in two ways, one using the usual position Hilbert space andthe other using a Hilbert space of holomorphic functions. Thelatter space generalizes the classical Segal-Bargmann space. Thetwo possible quantum Hilbert spaces are related by a unitarytransform, the generalized Segal-Bargmann transform, developed bythe PI and M. Stenzel. The unitarity of this transform can bere-formulated as a resolution of the identity for the associated"coherent states," as shown in detail by the PI and J. Mitchell.These results have been applied to the quantization oftwo-dimensional Yang-Mills theory and to the classical limit ofThiemann's quantum gravity theory. The PI is continuing toinvestigate several aspects of the theory, including thesemiclassical localization properties of the coherent states,properties of the associated quantization schemes (generalizedWick, anti-Wick, and Weyl quantizations), and the relationship ofthe theory to geometric quantization. Broadly speaking the PI's research is in the boundaryregion between classical and quantum mechanics. Quantum mechanicsis the theory that governs the world at the atomic scale.Although classical (Newtonian) mechanics works well formacroscopic phenomena, it cannot account for the structure ofatoms and molecules--at this level the quantum theory takes over.For the two theories to be consistent with one another thepredictions of quantum mechanics must pass smoothly into those ofclassical mechanics as the scale passes from microscopic tomacroscopic. On the other hand, the mathematical structure of thetwo theories is very different, so it is challenging tounderstand how this quantum-to-classical transition takes place.The PI's research concerns a reformulation of quantum mechanicswhich is equivalent to the usual one but which brings thedescription of quantum mechanics closer to that of classicalmechanics. Specifically, the PI's work takes one standardreformulation of quantum mechanics, the Segal-Bargmann transform,and extends it to apply to systems with more complicated degreesof freedom, such as rotations. This work has been applied in asimplified model of the strong interaction in particle physicsand in an ambitious program of T. Thiemann and collaborators todevelop a quantum theory of gravity.

PI：Brian C. Hall，圣母大学 DMS-0200649 摘要：PI 的研究涉及某些特殊经典系统的量子化，这些系统的经典构型空间是紧致对称空间，例如球体。这种系统最简单的物理例子是刚体（的旋转自由度），其配置空间是旋转群SO(3)。任何这样的系统的相空间，即紧对称空间的余切丛，都具有自然的复结构，使得相空间成为卡勒流形。因此，这种系统的量化可以通过两种方式完成，一种使用通常的位置希尔伯特空间，另一种使用全纯函数的希尔伯特空间。后者空间概括了经典的 Segal-Bargmann 空间。两个可能的量子希尔伯特空间通过酉变换（由 PI 和 M. Stenzel 开发的广义 Segal-Bargmann 变换）关联起来。这种变换的幺正性可以被重新表述为相关“相干态”恒等式的解析，正如 PI 和 J. Mitchell 所详细展示的那样。这些结果已应用于二维杨-米尔斯理论的量子化和蒂曼量子引力理论的经典极限。 PI 正在继续研究该理论的几个方面，包括相干态的半经典局域化特性、相关量化方案（广义 Wick、反 Wick 和 Weyl 量化）的特性，以及该理论与几何量化的关系。从广义上讲，PI 的研究处于经典力学和量子力学之间的边界区域。量子力学是在原子尺度上统治世界的理论。虽然经典（牛顿）力学在宏观现象上表现良好，但它无法解释原子和分子的结构——在这个层面上，量子理论占据了主导地位。为了使这两种理论彼此一致，当尺度从微观过渡到宏观时，量子力学的预测必须顺利地过渡到经典力学的预测。另一方面，这两种理论的数学结构非常不同，因此理解这种量子到经典的转变是如何发生的具有挑战性。PI的研究涉及量子力学的重新表述，这与通常的力学相同，但使量子力学的描述更接近经典力学的描述。具体来说，PI 的工作采用了量子力学的一种标准重新表述，即 Segal-Bargmann 变换，并将其扩展到具有更复杂自由度（例如旋转）的系统。这项工作已应用于粒子物理学中强相互作用的简化模型，以及 T. Thiemann 及其合作者开发量子引力理论的雄心勃勃的计划。