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，Notre Damedms-0200649Abstract：PI的研究涉及某些特殊经典系统的量化，这些系统的经典配置空间是一个紧凑的对称空间，例如球体。这种系统的theimphest物理示例是（旋转自由的）刚体，其配置空间是旋转组SO（3）。任何此类系统的相空间，即紧凑型对称空间的cotangent束，Hasa天然复杂结构，使相位空间成为Akahler歧管。因此，这样的系统的量化可以通过两种方式进行，一种使用通常的位置希尔伯特空间，另一个使用Holmormormormorthic函数的希尔伯特空间。 Thelatter Space概括了经典的Segal-Bargmann空间。这可能是由Pi和M. Stenzel开发的一般性Segal-Bargmann变换的单位变换，可能的量子Hilbert空间与量子。如Pi和J. Mitchell所示，这种转换的单位性可以作为对相关“相干状态”的身份的分辨率进行构图。这些结果已应用于YANG-MILLS理论的量化和Thiemann量子量化理论的经典极限。 PI正在继续研究该理论的几个方面，包括相干状态的血统定位特性，相关的量化方案的特性（GeneralizedWick，Anti-Wick和Weyl量化）以及理论与几何量化的关系。从广义上讲，PI的研究是经典和量子力学之间的边界区域。量子力学是在原子量表上统治世界的理论。尽管古典（牛顿）力学效果很好地形成了刻骨现象，但它不能说明量子理论的结构 - 在这个级别上，量子理论接管了这两个理论，即与量子机械的量级机制相同，将两个理论与一个平稳的机制相关。另一方面，这种理论的数学结构是非常不同的，因此它具有挑战性的巨星，这是如何进行量子到经典的过渡。具体而言，PI的工作采用了量子力学，Segal-Bargmann变换的标准改革，并将其扩展到适用于自由度更复杂的系统，例如旋转。这项工作已被应用于T. Thiemann的雄心勃勃的颗粒物理学和粒子物理中强相互作用的模型，并合作者对重力的量子理论进行了开发。