Mathematical Sciences: Studies in Quantization Theory

数学科学：量子化理论研究

• 批准号: 9623083
• 负责人: Mark Gotay
• 金额: $ 7.35万
• 依托单位: University of Hawaii
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623083&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Studies; Quantization; Theory

9623083 Gotay The process of constructing a quantum formulation of a system from a knowledge of a classical approximation to it is called "quantization," and over the years many different quantization schemes have been developed. Unfortunately, quantization is not a straightforward proposition, as evidenced by the discovery, exactly fifty years ago, by Groenewold and Van Hove of an ``obstruction'' to quantization. Their ``no-go theorem'' asserts that in principle it is impossible to consistently quantize every classical observable on a Euclidean phase space, regardless of which quantization procedure is employed. Just this past year, the principal investigator proved that a similar result holds for the sphere. But no-go theorems are not universally valid; the principal investigator has recently shown that the torus admits a consistent full quantization. The goals of this proposal are to delineate the circumstances under which such obstructions will appear, and to study the underlying mechanisms which produce them. Another problem, when an obstruction does exist, is to determine the maximal subalgebras of observables that can be consistently quantized. Solutions to these problems might be used to refine extant quantization procedures, or design new ones, which are adapted to the obstruction in that they will be able to quantize these maximal subalgebras. From a mathematical standpoint, this research will lead to structural insights into the Poisson algebras of classical systems and their representations. %%% Although the universe is quantum mechanical in nature, our perceptions of it are rooted in classical physics. Thus one is often confronted with the problem of constructing a quantum formulation of a system from a knowledge of a classical approximation to it. This process is called "quantization,'' and over the years many different quantization schemes have been developed. Unfortunately, quantization is not a straightforward proposition, as ev idenced by the discovery, exactly fifty years ago, of an "obstruction'' to quantization. This "no-go theorem'' asserts that in principle it is impossible to consistently quantize a (nonrelativistic) particle, regardless of which quantization procedure is employed. Just this past year, the principal investigator proved a similar result for a spinning particle. But no-go theorems are not universally valid; the principal investigator recently found a classical system which admits a consistent quantization. The goals of this proposal are to delineate the circumstances under which such obstructions will appear, and to study the mechanisms which produce them. A solution to this problem might be used to refine extant quantization procedures, or design new ones, which are "optimal" in that they will be able to quantize systems to the extent permitted by the obstruction. ***

小行星9623083 从一个系统的经典近似的知识中构造一个系统的量子公式的过程被称为“量子化”，多年来已经开发了许多不同的量子化方案。 不幸的是，量子化并不是一个直截了当的命题，正如50年前格罗内沃尔德和货车霍韦发现量子化的“障碍”所证明的那样。他们的“不通过定理”断言，原则上不可能在欧几里得相空间上一致地将每个经典的可观测量进行量化，无论采用哪种量化过程。就在去年，首席研究员证明了类似的结果适用于球体。但是不通过定理并不是普遍有效的;主要研究者最近证明了环面允许一致的完全量子化。本建议的目的是描述在何种情况下会出现这种障碍，并研究产生这些障碍的基本机制。另一个问题，当障碍物确实存在时，是确定可以一致量化的观测量的最大子代数。这些问题的解决方案可能会被用来完善现有的量化程序，或设计新的，这是适应的障碍，因为他们将能够将这些极大子代数。从数学的角度来看，这项研究将导致结构的见解泊松代数的经典系统及其表示。 %%% 虽然宇宙在本质上是量子力学的，但我们对它的看法却植根于经典物理学。因此，人们经常会遇到这样的问题，即根据对一个系统的经典近似的知识来构造一个系统的量子公式。这个过程被称为“量子化”，多年来，人们开发了许多不同的量子化方案。不幸的是，量子化并不是一个直截了当的命题，正如五十年前发现量子化的“障碍”所证明的那样。这个“不通过定理”断言，原则上不可能一致地将一个（非相对论性的）粒子量子化，不管使用哪种量子化程序。就在去年，首席研究员证明了旋转粒子的类似结果。但是，不去定理并不是普遍有效的;主要研究人员最近发现了一个经典系统，其中承认一致的量化。这一建议的目的是描述在何种情况下会出现这种障碍，并研究产生这些障碍的机制。这个问题的解决方案可以用来改进现有的量化程序，或设计新的量化程序，这些程序是“最佳的”，因为它们能够在障碍物允许的范围内对系统进行量化。 ***