Obstructions in Quantization Theory

NSF奖摘要-DMS-0072434数学科学：量子化理论中的障碍抽象0072434哥泰从经典近似的知识构建系统的量子公式的过程称为“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，50多年前格罗内沃德和范霍夫发现了量子化的“障碍”，这就证明了这一点。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可观测对象，无论采用哪种量子化过程。在各种各样的情况下，类似的结果也是成立的。但不去定理并不是通用的；主要研究者和合作者最近构造了允许一致的完全量子化的相空间的例子。该项目的目标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。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将被用来改进现有的量子化过程，或设计新的量子化过程，以适应障碍并量化这些极大子代数。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。在物理上，这项研究将有助于澄清经典力学和一般量子力学之间的对应关系，特别是将增强我们对特定经典系统的量子化的理解。尽管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。因此，从经典近似的知识来构造系统的量子公式通常是可取的。这个过程被称为“量化”，并且已经开发了许多不同的量化方案。不幸的是，量子化并不是一个直截了当的命题，正如50多年前发现的对量子化的“障碍”所证明的那样：原则上不可能始终如一地量子化(非相对论)粒子。但现在人们知道，不去定理并不是普适的；有一些经典系统允许一致的量子化。该项目的目标是描述出现这种障碍的情况，并研究产生这些障碍的机制。这项研究将有助于澄清经典力学和量子力学之间的对应关系。

NSF Award Abstract - DMS-0072434Mathematical Sciences: Obstructions in Quantization TheoryAbstract0072434 GotayThe 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, over 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. Similar results hold under a wide variety of circumstances. But no-go theorems are not universal; the principal investigator and collaborators have recently constructed examples of phase spaces which admit consistent full quantizations. The goals of this project are to delineate the circumstances under which such obstructions will appear and to study the underlying mechanisms that 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 will be used to refine extant quantization procedures, or design new ones, to adapt to the obstruction and 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. Physically, this research will aid in clarifying the correspondence between classical and quantum mechanics in general, and in particular will enhance our understanding of quantizations of specific classical systems.Although the universe is quantum mechanical in nature, our perceptions of it are rooted in classical physics. Thus it is often desirable to construct a quantum formulation of a system from knowledge of a classical approximation to it. This process is called "quantization," and many different quantization schemes have been developed. Unfortunately, quantization is not a straightforward proposition, as evidenced by the discovery, over fifty years ago, of an "obstruction" to quantization: in principle it is impossible to consistently quantize a (nonrelativistic) particle. But it is now known that no-go theorems are not universal; there are classical systems which admit consistent quantizations. The goals of this project are to delineate the circumstances under which such obstructions appear and to study the mechanisms which produce them. This research will aid in clarifying the correspondence between classical and quantum mechanics.