{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:b0f1cb61-5cdc-49df-9d91-612c9a805bc9", "labels": { "zh-CN": "二阶酉群", "en": "unitary group in dimension two" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "bce990c04e876b88477e314fc1a9bc53e6af966ef36750beef27fb7a4d84ffc8", "occurrences": [ { "concept_id": "urn:uuid:b0f1cb61-5cdc-49df-9d91-612c9a805bc9", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-03/01-the-two-state-quantum-system", "anchor": "first-2e8b3d89d2c28f6a1e63-zh", "chapter": "3", "section": "3.1", "offset": 16652, "range": [ 29042, 29297 ], "source_revision": "45b988396e9d8b60042fff90b58ac157f2196275c5c7e86231f3f54857f6c921", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "二阶酉群(unitary group in dimension two)", "context": { "range": [ 28443, 29389 ], "quote": "因此，对自伴泡利矩阵的实线性组合乘以 $i$ 后取指数，会得到行列式为 1 的酉矩阵。它们可逆，并构成一个名为[特殊酉群(special unitary group)](../../../../../concepts/linear-algebra/special-unitary-group)（记作 $SU(2)$）的群，也就是由行列式为 1 的 $2\\times2$ 酉矩阵构成的群。若对 $i(\\phi\\mathbf 1+\\theta\\mathbf v\\cdot\\boldsymbol{\\sigma})$ 取指数，其中 $\\phi$ 为实数（当且仅当 $\\phi=0$ 时，该矩阵的迹为零），则会得到行列式为 $e^{i2\\phi}$ 的酉矩阵。全体 $2\\times2$ 酉矩阵构成二阶酉群(unitary group in dimension two)（记作 $U(2)$）。它包含 $SU(2)$ 以及本节开头介绍的 $U(1)$ 作为子群。$U(2)$ 与这两个子群的直积略有不同，因为 $-\\mathbf 1$ 同时属于两个子群：", "quote_sha256": "7faaa29ff3cd15621b015a7c00a4bf2f2458eceb65feca550f04a216a9f98c4f", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-03/01-the-two-state-quantum-system", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-5239907ba5cffce810430385" }, { "concept_id": "urn:uuid:b0f1cb61-5cdc-49df-9d91-612c9a805bc9", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-03/03-dynamics-of-a-two-state-system", "anchor": "first-c363a4bca209f00248fe-zh", "chapter": "3", "section": "3.3", "offset": 2927, "range": [ 10301, 10587 ], "source_revision": "bf7bed127428e26ee378fb9b0fdf940a9534adebb0b55c3a053bcb959f95722e", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "二阶酉群(unitary group in dimension two)", "context": { "range": [ 9490, 10704 ], "quote": "$H$ 中的 $h_0\\mathbf{1}$ 项贡献一个整体相位因子(overall phase factor) $e^{-ih_0t}$。提出这一相位后，$U(t)$ 的剩余部分属于[特殊酉群(special unitary group)](01-the-two-state-quantum-system#concept-3-special-unitary-group) $SU(2)$；它是更大的[二阶酉群(unitary group in dimension two)](01-the-two-state-quantum-system#concept-3-unitary-group-in-dimension-two) $U(2)$ 的子群，而 $U(2)$ 由所有 $2\\times2$ 酉矩阵构成。", "quote_sha256": "5150597860b9fbcf622ca989c25ec48bebbe9df953d5cdc31cb66e12d122a083", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-03/03-dynamics-of-a-two-state-system", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-235a529b910439dba805c887" }, { "concept_id": "urn:uuid:b0f1cb61-5cdc-49df-9d91-612c9a805bc9", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-03/03-dynamics-of-a-two-state-system", "anchor": "first-2e8b3d89d2c28f6a1e63-en", "chapter": "3", "section": "3.3", "offset": 3337, "range": [ 9954, 10269 ], "source_revision": "fa8dc6f1abe776e62ef8b602a476972f2cd33199732383a20aabeb58fb3bed8c", "source_category": "original_transcription", "review_status": "ai-transcription-compared", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "unitary group in dimension two", "context": { "range": [ 9077, 10381 ], "quote": "The $h_0\\mathbf{1}$ term in $H$ contributes an overall phase factor $e^{-ih_0t}$. 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