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        "quote": "类似地，我们也可以把[<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-algebra-representation\" data-concept-identity=\"urn:uuid:5bad1df2-281a-4c11-8362-128d54fb523a\" data-concept-route=\"/concept-identities#concept-5bad1df2-281a-4c11-8362-128d54fb523a\" id=\"first-7fce2cf1162463ce211a-zh\">李代数表示(Lie algebra representation)</u>](../../../../../concepts/lie-theory/lie-algebra-representation)定义为保持李括号的李代数同态：\n<span id=\"first-ed25c65fa4aa1a6c7f32-zh\" class=\"block-anchor\"></span>",
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        "quote": "Similarly, we can define a [<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-algebra-representation\" data-concept-identity=\"urn:uuid:5bad1df2-281a-4c11-8362-128d54fb523a\" data-concept-route=\"/concept-identities#concept-5bad1df2-281a-4c11-8362-128d54fb523a\" id=\"first-7fce2cf1162463ce211a-en\">Lie algebra representation</u>](../../../../../concepts/lie-theory/lie-algebra-representation) as a map of Lie algebras that preserves the Lie bracket:",
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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[<u id=\"concept-7-quantum-observable\" class=\"concept-term\" data-concept-identity=\"urn:uuid:1f30c262-f238-4dfd-acec-55532f758890\" data-concept-route=\"/concept-identities#concept-1f30c262-f238-4dfd-acec-55532f758890\">可观测量(quantum observable)</u>](../../../../../concepts/quantum-mechanics/quantum-observable)对应于相应的[<u id=\"concept-7-lie-algebra-representation\" class=\"concept-term\" data-concept-identity=\"urn:uuid:5bad1df2-281a-4c11-8362-128d54fb523a\" data-concept-route=\"/concept-identities#concept-5bad1df2-281a-4c11-8362-128d54fb523a\">李代数表示(Lie algebra representation)</u>](../../../../../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[<u id=\"concept-7-spin\" class=\"concept-term\" data-concept-identity=\"urn:uuid:3e11a671-c004-44f2-8d6e-bf7933e08bbb\" data-concept-route=\"/concept-identities#concept-3e11a671-c004-44f2-8d6e-bf7933e08bbb\">自旋(spin)</u>](../../../../../concepts/quantum-mechanics/spin)；特别是，二维[<u id=\"concept-7-irreducible-representation\" class=\"concept-term\" data-concept-identity=\"urn:uuid:4e9e2c89-0422-42cf-8762-6b5e59837ef0\" data-concept-route=\"/concept-identities#concept-4e9e2c89-0422-42cf-8762-6b5e59837ef0\">不可约表示(irreducible representation)</u>](../../../../../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": "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[<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)。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[<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-algebra-representation\" data-concept-identity=\"urn:uuid:5bad1df2-281a-4c11-8362-128d54fb523a\" data-concept-route=\"/concept-identities#concept-5bad1df2-281a-4c11-8362-128d54fb523a\">李代数表示(Lie algebra representation)</u>](../../../../../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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        "quote": "This method turns unitary symmetries of classical phase space into [<u class=\"concept-term\" data-concept-reference=\"concepts/representation-theory/unitary-representation\" data-concept-identity=\"urn:uuid:74059dab-b498-4887-9978-fad6081fc63d\" data-concept-route=\"/concept-identities#concept-74059dab-b498-4887-9978-fad6081fc63d\" id=\"first-79b3af86386f7fc2ae7b-en\">unitary representations</u>](../../../../../concepts/representation-theory/unitary-representation) of symmetry groups on quantum state spaces. It is explained in detail here because it applies to simple systems such as the $d$-dimensional harmonic oscillator and will also be fundamental to our later study of quantum field theory. In that setting, the observables of interest are operators in a [<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-algebra-representation\" data-concept-identity=\"urn:uuid:5bad1df2-281a-4c11-8362-128d54fb523a\" data-concept-route=\"/concept-identities#concept-5bad1df2-281a-4c11-8362-128d54fb523a\">Lie algebra representation</u>](../../../../../concepts/lie-theory/lie-algebra-representation), built from quadratic combinations of annihilation and creation operators. They arise from the constructions in this chapter applied to unitary group actions on phase space, which is infinite-dimensional in quantum field theory.",
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