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.

该建议研究量化理论领域内的两个问题。在PI早期工作的基础上，第一个问题是对称空间的全体形态量化。将注意力集中在非恰当情况下，在这种情况下，在紧凑型情况下没有看到奇异性。 PI基于PI与J. Mitchell的工作以及B. Krotz，G。Olafsson和R. Stanton的工作，打算找到取消这些奇异性的适当方法。他打算在非2型对称空间上为Segal-Bargmann变换开发反转和等轴测公式，以使这些公式与双重紧凑型情况中的结果尽可能平行。 PI希望将他的作品与Mitchell的方法与Krotz，Olafsson和Stanton的Shift-Sperator方法相结合。第二个问题涉及量化与减少的关系。在与W. Kirwin的合作中，PI将研究Guillemin-sternberg地图的单位性（或缺乏其单位性）之间的“首先量化”，然后减少“首先减少空间，然后降低然后量化空间”。我们希望证明这张地图不是统一的，即使是在普朗克恒定的领导下，但其中的一半校正的收益率产生了统一性。 该提议涉及量化量子力学理论的构建，该理论与给定的经典理论相对应。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。该建议试图理解这种几何形状如何在量子理论中表现出来。该提案的第一部分是试图将量子理论中的标准工具扩展到Segal-Bargmann变换（与一致态的无处不在概念密切相关），到存在的几何有趣的情况。 PI在该领域的早期工作已经通过循环量子重力的工人以几种不同的方式应用。该提案的第二部分涉及对称与量子力学相互作用的方式。大多数现代的物理理论（例如重力，电磁，相对论）以必不可少的方式使用对称性。 PI的工作解决了在量化之前是否施加对称性的微妙但重要的问题，其结果与在量化后施加相同的结果。