Quantization on Cotangent Bundles

PI：Brian C. Hall，University of Notre DameDMS-0200649摘要：PI的研究涉及某些特殊的经典系统的量子化，这些系统的经典配置空间是一个紧凑的对称空间，如球体。这种系统最简单的物理例子是刚体（的旋转自由度），其位形空间是旋转群SO（3）。任何这样的系统的相空间，即紧对称空间的余切丛，都是使相空间成为Kahler流形的自然复结构。因此，这样一个系统的量子化可以用两种方法来完成，一种是使用通常的位置希尔伯特空间，另一种是使用全纯函数的希尔伯特空间。后者是经典Segal-Bargmann空间的推广。这两个可能的量子希尔伯特空间通过一个酉变换联系起来，即由PI和M发展的广义Segal-Bargmann变换。斯坦泽尔这种变换的么正性可以重新表述为对相关“相干态”恒等式的分解，正如PI和J. Mitchell所详细说明的，这些结果已被应用于二维Yang-Mills理论的量子化和Thiemann量子引力理论的经典极限。PI继续研究该理论的几个方面，包括相干态的半经典局域化性质，相关量子化方案（广义Wick，反Wick和Weyl量子化）的性质，以及该理论与几何量子化的关系。广义地说，PI的研究是在经典力学和量子力学之间的边界区域。量子力学是在原子尺度上统治世界的理论。尽管经典力学（牛顿力学）在宏观现象上工作得很好，但它不能解释原子和分子的结构--在这个层次上，量子理论占据了主导地位。为了使这两种理论相互一致，量子力学的预测必须随着尺度从微观到宏观的转变而平稳地过渡到经典力学的预测。另一方面，这两种理论的数学结构是非常不同的，所以理解这种量子到经典的转变是如何发生的是具有挑战性的。PI的研究涉及到量子力学的重新表述，它与通常的量子力学等价，但它使量子力学的描述更接近经典力学。具体来说，PI的工作采用了量子力学的一个标准重新表述，Segal-Bargmann变换，并将其扩展到具有更复杂自由度的系统，例如旋转。这一工作已应用于粒子物理中强相互作用的简化模型和T.蒂曼和他的合作者发展了引力的量子理论。

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.