Quantization, Symmetric Spaces, and Symplectic Reduction

这一建议研究了量子化理论领域内的两个问题。第一个问题建立在PI早期工作的基础上，是对称空间的全纯量子化。我们将主要关注非紧致情况，在这种情况下会出现人们在紧致情况下看不到的奇点。在与J.Mitchell的PI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抵消这些奇点。他打算在非紧对称空间上建立西格尔-巴格曼变换的反演和等距公式，目的是使这些公式尽可能地与对偶紧情况下的结果平行。PI希望将他在与Mitchell的工作中使用的方法与Krotz、Olafsson和Stanton的移位算符方法相结合。第二个问题涉及量化与约化的关系。在与W.Kirwin的合作中，PI将调查吉列明-斯特恩伯格映射在“先量化后缩减”空间和“先缩减后量化空间”之间的统一性(或缺乏)。我们希望证明这个映射不是么正的，甚至不是普朗克常数中的导数阶的映射，但是包含半形修正得到的是导数阶么正性。这一建议涉及量子化，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，被量子化的相关理论往往具有有趣的几何性质，涉及各种对称性。这一提议试图理解这种几何如何在量子理论中表现出来。该提议的第一部分是试图将量子理论中的标准工具西格尔-巴格曼变换(与普遍存在的相干态概念密切相关)扩展到更几何上有趣的情况，其中存在良好的对称性。PI在这一领域的早期工作已经被环路量子重力的工作人员以几种不同的方式应用。提案的第二部分涉及对称性与量子力学的相互作用方式。大多数现代物理理论(例如，重力、电磁学、相对论)在本质上使用对称性。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会产生与量化后施加对称性相同的结果。

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.