Quantization, Symmetric Spaces, and Symplectic Reduction

量化、对称空间和辛约简

• 批准号: 0555862
• 负责人: Brian Hall
• 金额: $ 10.64万
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555862&HistoricalAwards=false
• 关键词: Quantization; Symmetric; Spaces; Symplectic; Reduction

This proposal studies two issues within the area of quantization theory. The first issue, building on the PI's earlier work, is the holomorphic quantization of symmetric spaces. Attention will be given mainly to the noncompact case, where singularities arise that one does not see in the compact case. Building on work of the PI with J. Mitchell and work of B. Krotz, G. Olafsson, and R. Stanton, the PI intends to find appropriate ways to cancel out these singularities. He intends to develop inversion and isometry formulas for the Segal-Bargmann transform on noncompact symmetric spaces, with the goal of making these formulas as parallel as possible to the results in the dual compact case. The PI hopes to combine the approach used in his work with Mitchell with the shift-operator method of Krotz, Olafsson, and Stanton. The second issue concerns the relationship of quantization to reduction. In work with W. Kirwin, the PI will investigate the unitarity (or lack thereof) of the Guillemin-Sternberg map between the "first quantize then reduce" space and the "first reduce and then quantize space." We hope to demonstrate that this map is not unitary, even to leading order in Planck's constant, but that inclusion of the half-form correction yields leading-order unitarity. This proposal concerns quantization, namely, the construction of a quantum-mechanical theory corresponding to a given classical theory. In modern physics, the relevant theories to be quantized often have interesting geometric properties involving various sorts of symmetries. This proposal attempts to understand how this geometry manifests itself in the quantum theory. The first part of the proposal is an attempt to extend a standard tool in quantum theory, the Segal-Bargmann transform (closely related to the ubiquitous concept of coherent states), to more geometrically interesting situations where nice symmetries are present. The PI's earlier work in this area has already been applied in several different ways by workers in loop quantum gravity. The second part of the proposal concerns the way symmetries interact with quantum mechanics. Most modern physical theories (e.g., gravity, electromagnetism, relativity) use symmetry in an essential way. The PI's work addresses the subtle but important question of whether imposing the symmetries before quantizing gives the same result as imposing them after quantizing.

该提案研究了量化理论领域内的两个问题。第一个问题是基于 PI 早期工作的对称空间的全纯量化。我们将主要关注非紧致情况，其中出现在紧致情况下看不到的奇点。在 PI 与 J. Mitchell 的工作以及 B. Krotz、G. Olafsson 和 R. Stanton 的工作的基础上，PI 打算找到适当的方法来消除这些奇点。他打算开发非紧对称空间上 Segal-Bargmann 变换的反演和等距公式，目标是使这些公式尽可能与对偶紧致情况下的结果平行。 PI 希望将他与 Mitchell 合作中使用的方法与 Krotz、Olafsson 和 Stanton 的移位算子方法结合起来。第二个问题涉及量化与缩减的关系。在与 W. Kirwin 的合作中，PI 将研究“先量化然后减少”空间和“先减少然后量化空间”之间 Guillemin-Sternberg 映射的单一性（或缺乏单一性）。我们希望证明该映射不是酉的，即使对于普朗克常数的前导阶也是如此，但是包含半形式校正会产生前阶幺正性。 该提议涉及量子化，即构建与给定经典理论相对应的量子力学理论。在现代物理学中，要量化的相关理论通常具有涉及各种对称性的有趣几何特性。该提议试图理解这种几何学如何在量子理论中表现出来。该提案的第一部分是尝试将量子理论中的标准工具 Segal-Bargmann 变换（与普遍存在的相干态概念密切相关）扩展到几何上更有趣的情况，其中存在良好的对称性。 PI 早期在该领域的工作已经被圈量子引力研究人员以多种不同的方式应用。该提案的第二部分涉及对称性与量子力学相互作用的方式。大多数现代物理理论（例如重力、电磁学、相对论）都以基本方式使用对称性。 PI 的工作解决了一个微妙但重要的问题，即在量化之前施加对称性是否与在量化之后施加对称性产生相同的结果。