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        "quote": "在<a href=\"../chapter-10/08-topological-groups#section-10-8\" class=\"cross-reference\" id=\"xref-01-lie-groups-1\" data-reference=\"section-10-8\">第 10.8 节</a>中，我们将拓扑群定义为一个群 $G$，它同时也是一个拓扑空间，使得映射 $\\psi : ( g , h ) \\mapsto g h ^ { - 1 }$ 连续。如果 $G$ 具有可微流形的结构且 $\\psi$ 是光滑映射，则称其为 <u id=\"concept-19-lie-group\" class=\"concept-term\" data-concept-identity=\"urn:uuid:8c364498-0e05-4c91-8fd7-1730d6e8656e\" data-concept-route=\"/concept-identities#concept-8c364498-0e05-4c91-8fd7-1730d6e8656e\">李群(Lie group)</u>。<a href=\"../chapter-10/08-topological-groups#section-10-8\" class=\"cross-reference\" id=\"xref-01-lie-groups-2\" data-reference=\"section-10-8\">第 10.8 节</a>中给出的用以证明映射 $\\phi : ( g , h ) \\mapsto g h$ 和 $\\tau : g \\mapsto g ^ { - 1 }$ 均连续的论证，可以容易地推广以证明在李群的情形下二者均可微。映射 $\\tau$ 显然是 $G$ 的微分同胚，因为 $\\tau ^ { - 1 } = \\tau$。李群理论中证明的细节有时会变得相当技术化。当完整证明不具过多启发性时，我们常常 采用 概述性的证明。细节可在许多文献中找到，例如 [1–7]。沃纳 [6] 在这方面特别有用。",
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        "quote": "In <a href=\"../chapter-10/08-topological-groups#section-10-8\" class=\"cross-reference\" id=\"xref-01-lie-groups-1\" data-reference=\"section-10-8\">Section 10.8</a> we defined a topological group as a group $G$ that is also a topological space such that the map $\\psi : ( g , h ) \\mapsto g h ^ { - 1 }$ is continuous. If $G$ has the structure of a differentiable manifold and $\\psi$ is a smooth map it is said to be a <u class=\"concept-term\" data-concept-identity=\"urn:uuid:8c364498-0e05-4c91-8fd7-1730d6e8656e\" data-concept-route=\"/concept-identities#concept-8c364498-0e05-4c91-8fd7-1730d6e8656e\" id=\"first-5b8eee65f4e6910b5a7b-en\">Lie group</u>. The arguments given in <a href=\"../chapter-10/08-topological-groups#section-10-8\" class=\"cross-reference\" id=\"xref-01-lie-groups-2\" data-reference=\"section-10-8\">Section 10.8</a> to show that the maps $\\phi : ( g , h ) \\mapsto g h$ and $\\tau : g \\mapsto g ^ { - 1 }$ are both continuous are easily extended to show that both are differentiable in the case of a Lie group. The map $\\tau$ is clearly a diffeomorphism of $G$ since $\\tau ^ { - 1 } = \\tau$ . Details of proofs in Lie group theory can sometimes become rather technical. We will often resort to outline proofs, when the full proof is not overly instructive. Details can be found in a number of texts, such as [1–7]. Warner [6] is particularly useful in this respect.",
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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[<u class=\"concept-term\" data-concept-reference=\"source-block-1062\" data-concept-identity=\"urn:uuid:8c364498-0e05-4c91-8fd7-1730d6e8656e\" data-concept-route=\"/concept-identities#concept-8c364498-0e05-4c91-8fd7-1730d6e8656e\" id=\"first-e952a0fbafb6add3a4d3-zh\">李群(Lie group)</u>](./03-unitary-group-representations#source-block-1062)，以及这些群的[<span id=\"first-851e0548d24298e19cdf-zh\" class=\"block-anchor\"></span><u class=\"concept-term\" data-concept-reference=\"concept-1-unitary-representation\" data-concept-identity=\"urn:uuid:8adb7f10-55df-4db3-bbdc-c44765232a11\" data-concept-route=\"/concept-identities#concept-8adb7f10-55df-4db3-bbdc-c44765232a11\" id=\"first-851e0548d24298e19cdf-group-zh\">酉表示(unitary representation)</u>](./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": "若群的元素只有有限多个，就称为[<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/finite-group\" data-concept-identity=\"urn:uuid:91c6b7de-78fd-4b5a-9d1e-2b69ee97ba8e\" data-concept-route=\"/concept-identities#concept-91c6b7de-78fd-4b5a-9d1e-2b69ee97ba8e\" id=\"first-bbdbf409a34a3551c1be-zh\">有限群(finite group)</u>](../../../../../concepts/group-theory/finite-group)。有限群理论及其在量子力学中的应用已相当成熟，但本书大多略过这一主题，转而研究元素个数无限的[<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/lie-group\" data-concept-identity=\"urn:uuid:8c364498-0e05-4c91-8fd7-1730d6e8656e\" data-concept-route=\"/concept-identities#concept-8c364498-0e05-4c91-8fd7-1730d6e8656e\">李群(Lie group)</u>](../../../../../concepts/group-theory/lie-group)。李群的元素构成某个维数的几何空间；选取局部坐标后，群运算由可微函数给出。本书讨论的大多数李群都是<span id=\"concept-1-matrix-group\" class=\"block-anchor\"></span><u class=\"concept-term\" data-concept-id=\"concept-1-matrix-group\" data-concept-role=\"discussion\" data-concept-identity=\"urn:uuid:05f42bf3-d38b-49e0-bed7-1d0452d9ac3d\" data-concept-route=\"/concept-identities#concept-05f42bf3-d38b-49e0-bed7-1d0452d9ac3d\" id=\"first-fc558e266a7590e7bf39-zh\">矩阵群(matrix group)</u>，也就是由 $n\\times n$ 可逆实矩阵或复矩阵构成的矩阵群的子群。此时群乘法就是矩阵乘法。一个将详细讨论的例子是三维空间中绕某一点旋转所构成的群；这些旋转可以表示为 $3\\times3$ 矩阵，旋转的复合对应矩阵乘法。",
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        "quote": "最简单的[<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/lie-group\" data-concept-identity=\"urn:uuid:8c364498-0e05-4c91-8fd7-1730d6e8656e\" data-concept-route=\"/concept-identities#concept-8c364498-0e05-4c91-8fd7-1730d6e8656e\">李群(Lie group)</u>](../../../../../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[<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/circle-group\" data-concept-identity=\"urn:uuid:35b13cd0-fcbd-4d42-aa3a-04bf19d2fbb9\" data-concept-route=\"/concept-identities#concept-35b13cd0-fcbd-4d42-aa3a-04bf19d2fbb9\" id=\"first-c205b531a9556fbe5a3d-zh\">圆群(circle group)</u>](../../../../../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[<u class=\"concept-term\" data-concept-reference=\"concepts/representation-theory/unitary-representation\" data-concept-identity=\"urn:uuid:8adb7f10-55df-4db3-bbdc-c44765232a11\" data-concept-route=\"/concept-identities#concept-8adb7f10-55df-4db3-bbdc-c44765232a11\">酉表示(unitary representation)</u>](../../../../../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": "同理，$L$ 的矩阵的行向量也构成正交归一组。由于这类保持内积的线性变换可以复合且可逆，它们构成一个群；实向量空间与复向量空间上的这类群，是[<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/lie-group\" data-concept-identity=\"urn:uuid:8c364498-0e05-4c91-8fd7-1730d6e8656e\" data-concept-route=\"/concept-identities#concept-8c364498-0e05-4c91-8fd7-1730d6e8656e\">李群(Lie group)</u>](../../../../../concepts/group-theory/lie-group)的基本例子。",
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        "quote": "李代数 $\\mathfrak{sp}(2d,\\mathbf R)$ 中的元素 $L$ 与相空间 $M$ 上二次多项式 $\\mu_L$ 之间的对应，正是辛群 $\\mathrm{Sp}(2d,\\mathbf R)$ 作用于 $M$ 的矩映射。量子化这些二次函数，可得到与任意李子群 $G\\subset\\mathrm{Sp}(2d,\\mathbf R)$ 对应的量子可观测量；这里的李子群 $G$ 是指在线性作用于 $M$ 并保持其辛形式的[<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/lie-group\" data-concept-identity=\"urn:uuid:8c364498-0e05-4c91-8fd7-1730d6e8656e\" data-concept-route=\"/concept-identities#concept-8c364498-0e05-4c91-8fd7-1730d6e8656e\">李群(Lie group)</u>](../../../../../concepts/group-theory/lie-group)。这些可观测量未必都是对称性。此处“对称性”通常指由某个 $\\mu_L$ 量子化而来的可观测量，且该函数满足 $\\{\\mu_L,h\\}=0$，其中 $h$ 是哈密顿函数。",
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        "quote": "The correspondence between an element $L$ of $\\mathfrak{sp}(2d,\\mathbf R)$ and a quadratic polynomial $\\mu_L$ on phase space $M$ is the moment map for the action of $\\mathrm{Sp}(2d,\\mathbf R)$ on $M$. Quantizing these quadratic functions gives quantum observables associated with any Lie subgroup $G\\subset\\mathrm{Sp}(2d,\\mathbf R)$, that is, any [<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/lie-group\" data-concept-identity=\"urn:uuid:8c364498-0e05-4c91-8fd7-1730d6e8656e\" data-concept-route=\"/concept-identities#concept-8c364498-0e05-4c91-8fd7-1730d6e8656e\">Lie group</u>](../../../../../concepts/group-theory/lie-group) acting linearly on $M$ and preserving its symplectic form. These observables need not be symmetries. Here, “symmetry” usually means that the observable comes from quantizing some $\\mu_L$ satisfying $\\{\\mu_L,h\\}=0$, where $h$ is the Hamiltonian function.",
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