Woit · §1.3 Unitary group representations
英文原文
原文相关讨论摘录
If the set has a finite number of elements, this is called a “finite group”. The theory of these and their use in quantum mechanics is a well-developed subject, but one we mostly will bypass in favor of the study of “Lie groups”, which have an infinite number of elements. The elements of a Lie group make up a geometrical space of some dimension, and choosing local coordinates on the space, the group operations are given by differentiable functions. Most of the Lie groups we will consider are “matrix groups”, meaning subgroups of the group of by invertible matrices (with real or complex matrix entries). The group multiplication in this case is matrix multiplication. An example we will consider in great detail is the group of all rotations about a point in three dimensional space, in which case such rotations can be identified with 3 by 3 matrices, with composition of rotations corresponding to multiplication of matrices.
Digression. A standard definition of a Lie group is as a smooth manifold, with group laws given by smooth (infinitely differentiable) maps. More generally, one might consider topological manifolds and continuous maps, but this gives nothing new (by the solution to Hilbert’s Fifth problem). Most of the finite dimensional Lie groups of interest are matrix Lie groups, which can be defined as closed subgroups of the group of invertible matrices of some fixed dimension. One particular group of importance in quantum mechanics (the metaplectic group, see chapter 20) is not a matrix group, so the more general definition is needed to include this case.
来源与版本
正文：英文原文
来源版本：2025-10-20
原文位置：§1.3；PDF 页 23；印刷页 6
处理记录：PDF 文字层提取；MinerU OCR；AI 辅助转录对照
核对状态：AI 辅助转录核对，未作人工审阅或数学审稿
来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837
中文译文
译文摘录
如果集合只有有限个元素，则称为“有限群”。这类群的理论及其在量子力学中的应用是一个发展完善的课题，但我们大多会绕过它，转而研究“李群”，李群具有无限多个元素。李群的元素构成某个维数的几何空间，选取该空间上的局部坐标后，群运算由可微函数给出。我们将考虑的大多数李群是“矩阵群”，即由乘可逆矩阵（矩阵元素为实数或复数）构成的群的子群。在这种情况下，群乘法就是矩阵乘法。我们将详细考虑的一个例子是三维空间中绕一个点的所有旋转构成的群，此时这类旋转可以等同于3乘3矩阵，旋转的复合对应于矩阵的乘法。
题外话。李群的一个标准定义是：一个光滑流形，其群运算由光滑（无穷次可微）映射给出。更一般地，人们可以考虑拓扑流形和连续映射，但这不会带来任何新东西（根据希尔伯特第五问题的解答）。大多数令人感兴趣的有限维李群都是矩阵李群，它们可以定义为某个固定维数的可逆矩阵群的闭子群。量子力学中一个特别重要的群（metaplectic 群，见第 20 章）不是矩阵群，因此需要更一般的定义来包含这种情况。
来源与版本
正文：中文译文
来源版本：2025-10-20
原文位置：§1.3；PDF 页 23；印刷页 6
处理记录：machine-translation/deepseek-v4.1-flash
核对状态：机器译稿 · 待校对
英文原文
Woit calls this the standard definition: a smooth manifold with smooth (C∞) group laws; the nearby matrix-Lie-group sentence is a separate characterization.
原文定义摘录
Digression. A standard definition of a Lie group is as a smooth manifold, with group laws given by smooth (infinitely differentiable) maps. More generally, one might consider topological manifolds and continuous maps, but this gives nothing new (by the solution to Hilbert’s Fifth problem). Most of the finite dimensional Lie groups of interest are matrix Lie groups, which can be defined as closed subgroups of the group of invertible matrices of some fixed dimension. One particular group of importance in quantum mechanics (the metaplectic group, see chapter 20) is not a matrix group, so the more general definition is needed to include this case.
来源与版本
正文：英文原文
来源版本：2025-10-20
原文位置：§1.3；PDF 页 23；印刷页 6
处理记录：PDF 文字层提取；MinerU OCR；AI 辅助转录对照
核对状态：AI 辅助转录核对，未作人工审阅或数学审稿
来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837
中文译文
Woit 称此为标准定义：带有光滑（C∞）群运算的光滑流形；相邻的矩阵李群句子是另一种刻画。
译文摘录
题外话。李群的一个标准定义是：一个光滑流形，其群运算由光滑（无穷次可微）映射给出。更一般地，人们可以考虑拓扑流形和连续映射，但这不会带来任何新东西（根据希尔伯特第五问题的解答）。大多数令人感兴趣的有限维李群都是矩阵李群，它们可以定义为某个固定维数的可逆矩阵群的闭子群。量子力学中一个特别重要的群（metaplectic 群，见第 20 章）不是矩阵群，因此需要更一般的定义来包含这种情况。
来源与版本
正文：中文译文
来源版本：2025-10-20
原文位置：§1.3；PDF 页 23；印刷页 6
处理记录：machine-translation/deepseek-v4.1-flash
核对状态：机器译稿 · 待校对