Woit · §1.5 Groups and symmetries
英文原文
原文相关讨论摘录
The subject we are considering is often described as the study of “symmetry groups”, since the groups may occur as groups of elements acting by transformations of a space preserving some particular structure (thus, a “symmetry transformation”). We would like to emphasize though that it is not necessary that the transformations under consideration preserve any particular structure. In the applications to physics, the term “symmetry” is best restricted to the case of groups acting on a physical system in a way that preserves the equations of motion (for example, by leaving the Hamiltonian function unchanged in the case of a classical mechanical system). For the case of groups of such symmetry transformations, the use of the representation theory of the group to derive implications for the behavior of a quantum mechanical system is an important application of the theory. We will see however that the role of representation theory in quantum mechanics is quite a bit deeper than this, with the overall structure of the theory determined by group actions that are not symmetries (in the sense of not preserving the Hamiltonian).
来源与版本
正文：英文原文
来源版本：2025-10-20
原文位置：§1.5；PDF 页 28；印刷页 11
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
我们正在考虑的主题常被描述为“对称群”的研究，因为这些群可能作为由空间 的变换作用的元素群出现，并保持某种特定结构（因此，是一种“对称变换”）。不过，我们想强调，所考虑的变换并不必须保持任何特定结构。在物理学的应用中，“对称”一词最好限于群以保持运动方程的方式作用于物理系统的情况（例如，在经典力学系统的情况下保持哈密顿算子函数不变）。对于这类对称变换的群，利用群的表示理论来推导对量子力学系统行为的蕴含，是该理论的一项重要应用。然而我们将看到，表示理论在量子力学中的作用远比这更为深刻，因为该理论的整体结构由并非对称（即不保持哈密顿算子）的群作用所决定。
来源与版本
正文：中文译文
来源版本：2025-10-20
原文位置：§1.5；PDF 页 28；印刷页 11
处理记录：machine-translation/deepseek-v4.1-flash
核对状态：机器译稿 · 待校对