Quantization, Symmetric Spaces, and Symplectic Reduction

这个建议研究量子化理论领域内的两个问题。第一个问题，建立在PI的早期工作，是对称空间的全纯量子化。注意力将主要集中在非紧的情况下，在那里出现的奇异性，在紧凑的情况下看不到。在PI与J. Mitchell的工作和B的工作的基础上。Krotz，G. Olafsson和R.斯坦顿，PI打算找到适当的方法来消除这些奇点。他打算在非紧对称空间上发展Segal-Bargmann变换的反演和等距公式，目的是使这些公式尽可能与对偶紧情况下的结果平行。PI希望联合收割机将他与Mitchell合作时使用的方法与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减少的关系。在与W。Kirwin，PI将研究Guillemin-Sternberg映射在“先约化后约化”空间和“先约化后约化”空间之间的酉性（或缺乏酉性）。“我们希望证明这张地图不是么正的，甚至在普朗克常数的领先阶，但包括半形式校正产生领先阶么正性。 这个建议涉及量子化，也就是说，构造一个对应于给定经典理论的量子力学理论。在现代物理学中，要量子化的相关理论通常具有涉及各种对称性的有趣几何性质。这个提议试图理解这种几何在量子理论中是如何表现出来的。该提案的第一部分是试图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，即存在良好对称性的情况。PI在这一领域的早期工作已经被圈量子引力的工作者以几种不同的方式应用。该计划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，引力、电磁学、相对论）以一种基本的方式使用对称性。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会产生与量化之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.