Mathematical Sciences: Studies in Quantization Theory

9623083 Gotay根据经典近似的知识构建一个系统的量子公式的过程被称为“量子化”，多年来已经发展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，正如格罗内沃德和范·霍夫在整整50年前发现的对量子化的“障碍”所证明的那样。他们的‘不去定理’断言，原则上不可能一致地量子化欧几里得相空间上的每一个经典观测，无论采用哪种量子化过程。就在过去的一年里，首席研究员证明了类似的结果也适用于该球体。但不去定理并不是普遍有效的；首席研究者最近证明，环面允许一致的完全量子化。这项提案的目的是描述在什么情况下会出现这种障碍，并研究产生这些障碍的基本机制。当障碍物确实存在时，另一个问题是确定可以一致量子化的可观量的极大子代数。这些问题的解决方案可能被用来改进现有的量子化过程，或设计新的量子化过程，以适应障碍，因为它们将能够量子化这些极大子代数。从数学的角度，这项研究将导致对经典系统的泊松代数及其表示的结构的洞察。虽然宇宙本质上是量子力学的，但我们对它的认识植根于经典物理学。因此，人们经常面临这样的问题，即如何根据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“量化”，多年来已经发展了许多不同的量化方案。不幸的是，量化并不是一个简单的命题，正如整整50年前发现的对量化的“障碍”所证明的那样。这个“不去定理”断言，无论采用哪种量子化程序，原则上都不可能始终如一地量子化一个(非相对论)粒子。就在过去的一年里，首席研究员对一个旋转的粒子证明了类似的结果。但不去定理并不是普遍有效的；首席研究员最近发现了一个经典系统，它允许一致的量子化。这项提案的目的是描述在什么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能被用来改进现有的量化程序，或者设计新的量化程序，这些程序是“最佳的”，因为它们将能够在障碍允许的范围内对系统进行量化。***

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. ***