论文标题
量化：历史和问题
Quantization: History and Problems
论文作者
论文摘要
在这项工作中，我探讨了量化的概念，作为从经典相位空间函数到量子运算符的映射。我讨论了这种量化概念的早期历史，重点是Schrödinger和Dirac的作品，以及量化如何适合他们对1920年代量子理论的整体理解。特别是狄拉克（Dirac）提出了一个应满足某些属性的量化图，包括量子换向器应以特定方式与经典泊松支架相关的属性。但是，在1946年，Groenewold证明了Dirac的映射是不一致的，这使得定义严格的量化图的问题比最初预期的要难以捉摸。该结果被称为Groenewold-Van Hove定理，在物理文本中并不经常讨论，但是在这里我将对定理及其对DIRAC计划的潜在“校正”的含义进行说明。多年来，其他有关量化的建议已经出现，第一个主要的提案是1927年的Weyl，后来由包括Groenewold在内的许多人提出，此后在数学文献中已被称为Weyl量化。另一种称为几何量化的，通过吸引经典相位空间作为符号歧管的特征，以差分几何的术语进行了量化。这种方法始于1960年代的Souriau，Kostant和Kirillov的工作。我将描述这些建议，以量化它们与Dirac的原始程序的关系。一路上，将描述操作员排序和量化曲线坐标中量化的问题，因为这些是自然的问题，它们在考虑量化时立即出现了自己。
In this work, I explore the concept of quantization as a mapping from classical phase space functions to quantum operators. I discuss the early history of this notion of quantization with emphasis on the works of Schrödinger and Dirac, and how quantization fit into their overall understanding of quantum theory in the 1920's. Dirac, in particular, proposed a quantization map which should satisfy certain properties, including the property that quantum commutators should be related to classical Poisson brackets in a particular way. However, in 1946, Groenewold proved that Dirac's mapping was inconsistent, making the problem of defining a rigorous quantization map more elusive than originally expected. This result, known as the Groenewold-Van Hove theorem, is not often discussed in physics texts, but here I will give an account of the theorem and what it means for potential "corrections" to Dirac's scheme. Other proposals for quantization have arisen over the years, the first major one being that of Weyl in 1927, which was later developed by many, including Groenewold, and which has since become known as Weyl Quantization in the mathematical literature. Another, known as Geometric Quantization, formulates quantization in differential-geometric terms by appealing to the character of classical phase spaces as symplectic manifolds; this approach began with the work of Souriau, Kostant, and Kirillov in the 1960's. I will describe these proposals for quantization and comment on their relation to Dirac's original program. Along the way, the problem of operator ordering and of quantizing in curvilinear coordinates will be described, since these are natural questions that immediately present themselves when thinking about quantization.
