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"quote": "对于 $U(1)$，我们在[定理 2.3](../chapter-02/02-the-group-u-1-and-its-representations#theorem-2-3) 中通过考察映射在单位元处的导数，分类了所有[不可约表示(irreducible representations)](../../../../../concepts/representation-theory/irreducible-representation)（即同态 $U(1)\\to GL(1,\\mathbf{C})=\\mathbf{C}^*$）。对于一般李群 $G$，同样可以通过在单位元处求导，证明 $G$ 的表示 $\\pi$ 会导出一个李代数表示，进而尝试对李代数表示进行分类。",
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"quote": "For $U(1)$, we classified all [irreducible representations](../../../../../concepts/representation-theory/irreducible-representation) (homomorphisms $U(1)\\to GL(1,\\mathbf{C})=\\mathbf{C}^*$) in [theorem 2.3](../chapter-02/02-the-group-u-1-and-its-representations#theorem-2-3) by examining the derivative of the map at the identity. For a general Lie group $G$, the same idea shows that a representation $\\pi$ of $G$ induces a representation of its Lie algebra by differentiation at the identity; one can then try to classify Lie algebra representations.",
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"quote": "这些构造给出了这两个群的两个有趣的[不可约表示(irreducible representation)](../../../../../concepts/representation-theory/irreducible-representation)：",
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"quote": "在[第3](../chapter-03/index#section-3)章中我们看到，一个具有$\\mathcal { H } = \\mathbf { C } ^ { 2 }$的一般量子系统可以通过$U ( 2 )$在$\\mathbf { C } ^ { 2 }$上的作用来理解。自伴算符所表示的[可观测量(quantum observable)](../../../../../concepts/quantum-mechanics/quantum-observable)对应于相应的[李代数表示(Lie algebra representation)](../../../../../concepts/lie-theory/lie-algebra-representation)（相差一个因子 $i$）。$U ( 1 ) \\subset U ( 2 )$子群与所有其他部分对易，可以单独分析，这里我们只考虑$S U ( 2 )$子群。对于任意这样的系统，群$S U ( 2 )$没有特别的几何意义。当它作为旋转群的双覆盖出现时，称该量子系统携带[自旋(spin)](../../../../../concepts/quantum-mechanics/spin)；特别是，二维[不可约表示(irreducible representation)](../../../../../concepts/representation-theory/irreducible-representation)对应自旋 $1/2$（在[第8](../chapter-08/index#section-8)章中我们将讨论更高自旋值的态空间）。",
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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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