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        "quote": "In special relativity, spacetime is a real four-dimensional vector space with an indefinite inner product, of type $(r,s)=(1,3)$ or $(3,1)$ according to convention. Its preserving linear transformations form the [<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lorentz-group\" data-concept-identity=\"urn:uuid:937e2225-ecf0-4854-ab8b-d17f476f7021\" data-concept-route=\"/concept-identities#concept-937e2225-ecf0-4854-ab8b-d17f476f7021\" id=\"first-762a1e60efdf89af31a4-en\">Lorentz group</u>](../../../../../concepts/lie-theory/lorentz-group); the orientation-preserving part is denoted $SO(3,1)$ or $SO(1,3)$. Later chapters encounter the corresponding algebras $\\operatorname{Cliff}(3,1,\\mathbb R)$ or $\\operatorname{Cliff}(1,3,\\mathbb R)$. Their generators $\\gamma_j$ are known as “[<u class=\"concept-term\" data-concept-reference=\"concepts/quantum-mechanics/dirac-gamma-matrices\" data-concept-identity=\"urn:uuid:9445e0c4-8ed1-4567-844b-c66f13045539\" data-concept-route=\"/concept-identities#concept-9445e0c4-8ed1-4567-844b-c66f13045539\" id=\"first-73ede448c89ff1a9dee8-en\">Dirac $\\gamma$-matrices</u>](../../../../../concepts/quantum-mechanics/dirac-gamma-matrices)”.",
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            "text": "在 $SO(3,1)$ 的李代数中，三个生成元对应空间旋转，另三个对应沿空间方向的无穷小推动；把旋转与推动生成元作复线性组合后，复化的代数分成两个 $\\mathfrak{sl}(2,\\mathbf{C})$ 副本。",
            "evidence_ids": [
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        ],
        "author_views": [
          {
            "id": "woit-lorentz",
            "book_id": "woit-qtgr",
            "edition_id": "2025-10-20",
            "summary": "本章先从保持闵可夫斯基内积的矩阵群出发，再区分四个连通分支；随后用旋转与推动生成元说明李代数结构，并在复化后得到两个 $\\mathfrak{sl}(2,\\mathbf{C})$ 副本。",
            "conditions": "采用时间分量符号为负、三个空间分量为正的闵可夫斯基形式；这里描述的是 $O(3,1)$ 及其含单位元分支 $SO(3,1)$。",
            "notation": "记全群为 $O(3,1)$，含单位元分支为 $SO(3,1)$；空间反演与时间反演记为 $P,T$，无穷小旋转与推动生成元记为 $l_j,k_j$。",
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        "reading_suggestions": [
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            "id": "lorentz-reading-next",
            "text": "先读本节对闵可夫斯基度量及 $O(3,1)$ 的定义，再看旋转和推动生成元的对易关系；这条顺序只覆盖此处收录的第 40.2 节材料。",
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          "composition": "complete",
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          "whole_book_coverage": "not-exhausted"
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          "quote_sha256": "81240afe10598eefd5dab41f3f7a7a3144149ca56dad52eaf16633fd5836daa3"
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