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"quote": "若 $G$ 是有限群，则事实上每个有限维表示都等价于内积空间上由酉变换给出的表示——称为 酉表示(unitary representation)。因为，设 $T$ 是任意有限维向量空间 $V ,$ 上的一个表示，并设 $\\{ e _ { i } \\}$ 是 $V$ 的任意一组基。通过令 $\\{ e _ { i } \\}$ 为正交归一集合，在 $V$ 上定义内积 $( u | v )$，",
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"quote": "外尔（Weyl）关于经典系统量子化的洞见——关键在于理解作用在经典相空间上的[李群(Lie group)](./03-unitary-group-representations#source-block-1062)，以及这些群的[酉表示(unitary representation)](./03-unitary-group-representations#concept-1-unitary-representation)——已被后来的发展所印证；这些发展极大拓展了相关思想的应用范围。利用表示论来发掘问题的对称性已成为一种强有力的工具，在科学的许多领域而不仅仅是量子力学中得到了应用。我希望那些主要兴趣在物理学的读者能够学会欣赏标准教科书计算背后的一些此类数学结构，帮助他们理解如何在其他情境中有效地利用它们。那些主要兴趣在数学的读者则有望获得对基础物理学的一些理解，同时看到群与表示的一些关键例子。这些应为欣赏该学科更抽象的表述——标准数学课程的一部分——提供良好的基础。任何对基础物理学与数学的关系、以及尤金·维格纳（Eugene Wigner）所称的“数学在自然科学中不合理的有效性”[\\[102\\]](../../../../../references/woit-2025-10-20.pdf#page=580)感到好奇的人，都应从接触这个处于两个学科交叉点的非凡故事中受益。",
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"quote": "量子力学与表示论之间的基本关系是：只要某个物理量子系统上有群 $G$ 作用，态空间 $\\mathcal{H}$ 上就有一个 $G$ 的[酉表示(unitary representation)](./03-unitary-group-representations#concept-1-unitary-representation)（这种表示允许相差一个相位因子）。对研究量子力学的物理学家而言，这意味着，只要 $G$ 作用于系统，表示论就能提供关于其量子态空间的信息。对研究表示论的数学家而言，这意味着物理学为酉表示的研究提供了丰富素材：任何有群 $G$ 作用的物理系统都会给出一个这样的表示。",
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"quote": "最简单的[李群(Lie group)](../../../../../concepts/group-theory/lie-group)例子是平面的旋转群，其元素由单个参数即旋转角$\\theta$来参数化。将这类群元素识别为复平面中的单位向量 $e^{i\\theta}$，会更方便。该群即[圆群(circle group)](../../../../../concepts/group-theory/circle-group)，记作 $U(1)$；这是因为这样的复数可视为 $1\\times1$ 的酉矩阵。我们将在本章中看到[第1](../chapter-01/index#section-1)章所描述的一般图景如何在这个简单情形中实现。状态空间将是 $U(1)$ 群的[酉表示(unitary representation)](../../../../../concepts/representation-theory/unitary-representation)，并且我们将看到任何这样的表示都分解为一系列一维表示之和。这些一维表示由整数 $q$ 标记；这些整数是自伴算符 $Q$ 的特征值，而 $Q$ 是量子理论中的一个可观测量。",
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"quote": "$\\mathrm{SL}(2,\\mathbf R)$ 是非紧群，其表示论与 $\\mathrm{SU}(2)$ 的情形很不相同。特别地，它的每个非平凡的[不可约表示(irreducible representation)](../../../../../concepts/representation-theory/irreducible-representation)，只要是[酉表示(unitary representation)](../../../../../concepts/representation-theory/unitary-representation)，就都是无限维的。这是数学中的重要课题，但超出了本书的讨论范围。我们只研究其中一个表示，即量子力学态空间给出的表示；它是 $\\mathrm{SL}(2,\\mathbf R)$ 的一个双覆盖群的表示，而不是 $\\mathrm{SL}(2,\\mathbf R)$ 本身的表示。",
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"quote": "这种方法把经典相空间的酉对称性转化为量子态空间上对称群的[酉表示(unitary representation)](../../../../../concepts/representation-theory/unitary-representation)。本章详细阐述这一方法，既因为它适用于 $d$ 维谐振子这样的简单量子系统，也因为它将成为后续量子场论研究的基础。在量子场论中，我们关注的是[李代数表示(Lie algebra representation)](../../../../../concepts/lie-theory/lie-algebra-representation)中的算符，它们由湮灭算符与产生算符的二次组合构成；这些算符来自本章的构造，而此时相空间上的酉群作用所作用的相空间是无限维的。",
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