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"Lagrangian",
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"scope_note": "将薛定谔绘景场的具体使用挂到绘景入口，保留场算符与有限自由度量子力学的区别。",
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"Woit/Szekeres：绘景中的态与算符的时间依赖。",
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"Schrödinger picture fields",
"$\\hat{\\phi}(\\vec{x})$"
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{
"concept_id": "urn:uuid:bed044e3-0bd6-48d8-8aab-253bd7096faa",
"scope_note": "推论讨论基本群与第一同调群的关系，涉及基本群概念。",
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"quote": "The creation and annihilation operators for spinors must anticommute by the spin-statistics theorem:",
"citation": "by the spin-statistics theorem",
"target_marker": "spin-statistics theorem",
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"quote": "Proof. Both sides are groups according to lemma 3.1. Define a map $\\varphi : G_1 / \\ker f \\to \\operatorname{im} f$ by $\\varphi([x]) = f(x)$ . This map is well defined since for $x' \\in [x]$ , there exists $h \\in \\ker f$ such that $x' = x + h$ and $f(x') = f(x + h) = f(x) + f(h) = f(x)$ . Now we show that $\\varphi$ is an isomorphism. First, $\\varphi$ is a homomorphism,",
"citation": "according to lemma 3.1",
"target_marker": "Lemma 3.1.",
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"quote": "Example 3.2. Let $f: \\mathbb{Z} \\to \\mathbb{Z}_2$ be defined by $f(2n) = 0$ and $f(2n + 1) = 1$ . Then $\\ker f = 2\\mathbb{Z}$ and $\\operatorname{im} f = \\mathbb{Z}_2$ are groups. Theorem 3.1 states that $\\mathbb{Z}/2\\mathbb{Z} \\cong \\mathbb{Z}_2$ , in agreement with example 3.1.",
"citation": "Theorem 3.1 states",
"target_marker": "Theorem 3.1.",
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"quote": "From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r}(K)$ are subgroups of $C_{r}(K)$ . We now prove an important relation between $Z_{r}(K)$ and $B_{r}(K)$ , which is crucial in the definition of homology groups.",
"citation": "From lemma 3.1",
"target_marker": "Lemma 3.1.",
"review_status": "codex-source-reviewed; pending-independent-human-review"
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"quote": "Proof. This is obvious from lemma 3.3. Any element $c$ of $B_r(K)$ is written as $c = \\partial_{r+1}d$ for some $d \\in C_{r+1}(K)$ . Then we find $\\partial_r c = \\partial_r(\\partial_{r+1}d) = 0$ , that is, $c \\in Z_r(K)$ . This implies $Z_r(K) \\supset B_r(K)$ .",
"citation": "from lemma 3.3",
"target_marker": "Lemma 3.3.",
"review_status": "codex-source-reviewed; pending-independent-human-review"
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"quote": "Now we can use the Gaussian integral in Eq. (14.6), $\\int dp\\exp \\left(-\\frac{1}{2} ap^2 +Jp\\right) = \\sqrt{\\frac{2\\pi}{a}}\\exp \\left(\\frac{J^2}{2a}\\right)$ , with $a = i\\frac{\\delta t}{m}$ and $J = i(x_{j + 1} - x_j)$ to get",
"citation": "Eq. (14.6)",
"target_marker": "14.6",
"review_status": "codex-source-reviewed; pending-independent-human-review"
},
{
"id": "dependency-7f3c4b0192d0aa5ba72681eb",
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"from_statement": "statement-3c6ad35017afb105cdf7e24e",
"to_statement": "statement-0993b4cf0a986537ff39c920",
"quote": "Using Eq. (14.12), each term in Eq. (14.10) becomes just a number and the product reduces to",
"citation": "Eq. (14.12)",
"target_marker": "14.12",
"review_status": "codex-source-reviewed; pending-independent-human-review"
},
{
"id": "dependency-90164fa13753544dd9251387",
"relation": "uses-author-statement",
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"from_statement": "statement-3c6ad35017afb105cdf7e24e",
"to_statement": "statement-9dc83328caf7b27d5fc2f15e",
"quote": "Using Eq. (14.12), each term in Eq. (14.10) becomes just a number and the product reduces to",
"citation": "Eq. (14.10)",
"target_marker": "14.10",
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{
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"basis": "author-transition",
"quote": "Since $B_r(K)$ is a subgroup of $Z_r(K)$ , $H_r(K)$ is well defined.",
"citation": null,
"target_marker": "Theorem 3.3. Let $Z_{r}(K)$ and $B_{r}(K)$ be the ",
"reason": "作者的 Since 承接使用了定理3.3的包含关系，保证分母是分子中的子群；阿贝尔条件另由链群定义提供。",
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"from_statement": "statement-ae16e6e7da0a84f365502acf",
"to_statement": "statement-ddc45b084c0fae9befd3525f",
"basis": "construction-use",
"quote": "$$\nH _ {r} (K) \\equiv Z _ {r} (K) / B _ {r} (K). \\tag {3.28}\n$$",
"citation": null,
"target_marker": "Let $H$ be a subgroup of $G$ . We say $x, y \\in G$ ",
"reason": "这里实际使用商群构造；本章阿贝尔条件保证无需另加一般群的正规性前提。此边是来源构造使用候选，不是编号回指。",
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"from_statement": "statement-ae16e6e7da0a84f365502acf",
"to_statement": "statement-a9109552ffd06f4f8a886bfa",
"basis": "construction-use",
"quote": "Since $B_r(K)$ is a subgroup of $Z_r(K)$ , $H_r(K)$ is well defined.",
"citation": null,
"target_marker": "Definition 3.2. The $r$ -chain group $C_r(K)$ of a s",
"reason": "链群的阿贝尔结构经子群继承，说明本处的商群条件；这不是把一般群的子群都当成正规子群。",
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"from_statement": "statement-ae16e6e7da0a84f365502acf",
"to_statement": "statement-ac7699973242d9783a9bb1ca",
"basis": "construction-use",
"quote": "$$\nH _ {r} (K) \\equiv Z _ {r} (K) / B _ {r} (K). \\tag {3.28}\n$$",
"citation": null,
"target_marker": "Definition 3.3. If $c \\in C_r(K)$ satisfies",
"reason": "商群分子 Zr 必须按作者的核定义理解，而非把所有链都当作循环。",
"evidence_status": "codex-source-reviewed-candidate",
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"from_statement": "statement-ae16e6e7da0a84f365502acf",
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"basis": "construction-use",
"quote": "$$\nH _ {r} (K) \\equiv Z _ {r} (K) / B _ {r} (K). \\tag {3.28}\n$$",
"citation": null,
"target_marker": "Definition 3.4. Let $K$ be an $n$ -dimensional simpl",
"reason": "商群分母 Br 必须按作者的像定义理解，商掉的是边界循环。",
"evidence_status": "codex-source-reviewed-candidate",
"review_status": "codex-source-reviewed; pending-independent-human-review"
},
{
"id": "dependency-5ab4c184178868d2f11eed23",
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"from_statement": "statement-ae16e6e7da0a84f365502acf",
"to_statement": "statement-45381cbdd77285069b293927",
"basis": "construction-use",
"quote": "Since $B_r(K)$ is a subgroup of $Z_r(K)$ , $H_r(K)$ is well defined.",
"citation": null,
"target_marker": "From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r",
"reason": "先确认 Zr、Br 的群资格，再结合定理3.3的包含关系组成商群。该前置结论单独不证明 Br 包含于 Zr。",
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"id": "dependency-fd0436467a9a3a8bd6545786",
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"from_statement": "statement-45381cbdd77285069b293927",
"to_statement": "statement-ac7699973242d9783a9bb1ca",
"basis": "construction-use",
"quote": "From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r}(K)$ are subgroups of $C_{r}(K)$ . We now prove an important relation between $Z_{r}(K)$ and $B_{r}(K)$ , which is crucial in the definition of homology groups.",
"citation": null,
"target_marker": "Definition 3.3. If $c \\in C_r(K)$ satisfies",
"reason": "引理3.1应用到核时，核正是定义3.3的循环群。",
"evidence_status": "codex-source-reviewed-candidate",
"review_status": "codex-source-reviewed; pending-independent-human-review"
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"relation": "uses-author-statement",
"from_statement": "statement-45381cbdd77285069b293927",
"to_statement": "statement-e9cdbaabe0ae1a11cb9cc9b3",
"basis": "construction-use",
"quote": "From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r}(K)$ are subgroups of $C_{r}(K)$ . We now prove an important relation between $Z_{r}(K)$ and $B_{r}(K)$ , which is crucial in the definition of homology groups.",
"citation": null,
"target_marker": "Definition 3.4. Let $K$ be an $n$ -dimensional simpl",
"reason": "引理3.1应用到像时，像正是定义3.4的边界群。",
"evidence_status": "codex-source-reviewed-candidate",
"review_status": "codex-source-reviewed; pending-independent-human-review"
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"id": "dependency-897207a3e13dbddb26b2a837",
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"from_statement": "statement-ac7699973242d9783a9bb1ca",
"to_statement": "statement-27db493e412102426ffd26d7",
"basis": "construction-use",
"quote": "$c$ is called an $r$ -cycle. The set of $r$ -cycles $Z_{r}(K)$ is a subgroup of $C_{r}(K)$ and is called the $r$ -cycle group. Note that $Z_{r}(K) = \\ker \\partial_{r}$ . [Remark: If $r = 0$ , $\\partial_0 c$ vanishes identically and $Z_{0}(K) = C_{0}(K)$ , see (3.23).]",
"citation": null,
"target_marker": "Let $\\sigma_r(p_0\\ldots p_r)$ ( $r > 0$ ) be an orien",
"reason": "循环条件和核需要此前定义的边界同态。",
"evidence_status": "codex-source-reviewed-candidate",
"review_status": "codex-source-reviewed; pending-independent-human-review"
},
{
"id": "dependency-cb72658754eea40f65bd416a",
"relation": "uses-author-statement",
"from_statement": "statement-e9cdbaabe0ae1a11cb9cc9b3",
"to_statement": "statement-27db493e412102426ffd26d7",
"basis": "construction-use",
"quote": "then $c$ is called an $r$ -boundary. The set of $r$ -boundaries $B_r(K)$ is a subgroup of $C_r(K)$ and is called the $r$ -boundary group. Note that $B_r(K) = \\operatorname{im} \\partial_{r+1}$ . [Remark: $B_n(K)$ is defined to be 0.]",
"citation": null,
"target_marker": "Let $\\sigma_r(p_0\\ldots p_r)$ ( $r > 0$ ) be an orien",
"reason": "边界及像需要此前从 r+1 链到 r 链的边界同态。",
"evidence_status": "codex-source-reviewed-candidate",
"review_status": "codex-source-reviewed; pending-independent-human-review"
},
{
"id": "dependency-e997bfe83e609355c7530f3b",
"relation": "uses-author-statement",
"from_statement": "statement-27db493e412102426ffd26d7",
"to_statement": "statement-a9109552ffd06f4f8a886bfa",
"basis": "author-transition",
"quote": "The RHS of (3.21) is an element of $C_{r-1}(K)$ . Accordingly, $\\partial_r$ defines a map",
"citation": null,
"target_marker": "Definition 3.2. The $r$ -chain group $C_r(K)$ of a s",
"reason": "作者将边界线性延拓到链，因此映射的定义域、值域取自定义3.2的链群。",
"evidence_status": "codex-source-reviewed-candidate",
"review_status": "codex-source-reviewed; pending-independent-human-review"
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{
"id": "dependency-dc78aa117f41159a20c5afd3",
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"from_statement": "statement-dd7721e0c9adf5e7b891bfcd",
"to_statement": "statement-27db493e412102426ffd26d7",
"basis": "construction-use",
"quote": "Lemma 3.3. The composite map $\\partial_r\\circ \\partial_{r + 1}:C_{r + 1}(K)\\to C_{r - 1}(K)$ is a zero map; that is, $\\partial_r(\\partial_{r + 1}c) = 0$ for any $c\\in C_{r + 1}(K)$ .",
"citation": null,
"target_marker": "Let $\\sigma_r(p_0\\ldots p_r)$ ( $r > 0$ ) be an orien",
"reason": "连续取边界的零映射结论使用此前的边界算子；这条边不替代引理3.3的证明。",
"evidence_status": "codex-source-reviewed-candidate",
"review_status": "codex-source-reviewed; pending-independent-human-review"
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"from_statement": "statement-68ebd9974e3ebfba6471dbb4",
"to_statement": "statement-8d8d2177e831615076629fce",
"basis": "construction-use",
"quote": "Example 4.11. From $\\pi_1(\\text{Klein bottle}) \\cong (x, y; xyxy^{-1})$ , we have",
"target_marker": "Theorem 4.9. Let $K$ be a connected simplicial complex. Then $H_{1}(K)$ is isomorphic to $\\pi_1(|K|)/F$",
"citation": null,
"reason": "例 4.11 的计算过程正是定理 4.9 所述构造的具体实施：取基本群模去换位子群。",
"review_status": "codex-source-reviewed; pending-independent-human-review",
"evidence_status": "codex-source-reviewed-candidate"
},
{
"id": "dependency-368799583a78ce7e483543d8",
"relation": "uses-author-statement",
"from_statement": "statement-e147ac6a63afbf8aa27b13b9",
"to_statement": "statement-8d8d2177e831615076629fce",
"basis": "explicit-author-reference",
"quote": "[Use theorem 4.9.]",
"target_marker": "Theorem 4.9. Let $K$ be a connected simplicial complex.",
"citation": "theorem 4.9",
"reason": "作者明确指示使用定理 4.9 来推导此推论。",
"review_status": "codex-source-reviewed; pending-independent-human-review"
},
{
"id": "dependency-faa1666301a7c6eb4731bb1f",
"relation": "uses-author-statement",
"from_statement": "statement-b9954f1c50ffc2105adda9f2",
"to_statement": "statement-8d8d2177e831615076629fce",
"basis": "explicit-author-reference",
"quote": "[Use theorems 4.9 and 4.3.]",
"target_marker": "Theorem 4.9. Let $K$ be a connected simplicial complex.",
"citation": "theorems 4.9",
"reason": "作者明确指示使用定理 4.9 作为推导依据之一。",
"review_status": "codex-source-reviewed; pending-independent-human-review"
},
{
"id": "dependency-d6aa06fe2b4be3c8aa50649d",
"relation": "uses-author-statement",
"from_statement": "statement-445d27e174a7c2fa11b79bb7",
"to_statement": "statement-8d8d2177e831615076629fce",
"basis": "author-transition",
"quote": "The theorem states that if $\\pi_1(|K|) = (x_i : r_m)$ , then $H_1(K) \\cong (x_i : r_m, x_i x_j x_i^{-1} x_j^{-1})$ .",
"target_marker": "Theorem 4.9. Let $K$ be a connected simplicial complex.",
"citation": null,
"reason": "作者以 'The theorem states' 回指定理 4.9 的结论来推导 bouquet 的同调群。",
"review_status": "codex-source-reviewed; pending-independent-human-review",
"evidence_status": "codex-source-reviewed-candidate"
}
],
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"source_kind": "ribosome-atoms",
"scope": {
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"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"chapter": "nakahara-3",
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"quote": "The mathematical structures underlying homology groups are finitely generated Abelian groups. Throughout this chapter, the group operation is denoted by $+$ since all the groups considered here are Abelian (commutative). The unit element is denoted by 0."
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"excerpt": "A group $G$ is called abelian if the law of composition is commutative,\n\n$$\ng h = h g \\quad \\text { for all } g, h \\in G.\n$$\n\nThe notation gh for the product of two elements is the default notation. Other possibilities are $a \\cdot b , a \\times b , a + b , a \\circ b$ , etc. When the law of composition is written as an addition $g + h$ , we will always assume that the commutative law holds, $g + h = h + g$ . In this case the identity element is usually written as 0, so that (Gp2) reads $g + 0 = 0 + g = g$ . The inverse is then written $-g$, with (Gp3) reading $g + ( - g ) = 0$ or, more simply, $g - g = 0$ Again, the associative law means we never have to worry about parentheses in expressions such as $a + b + c + \\cdots + f .$\n\n",
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"excerpt": "\n\nSaying the map $\\pi$ is a homomorphism means\n\n\n\n$$\n\\pi (g _ {1}) \\pi (g _ {2}) = \\pi (g _ {1} g _ {2})\n$$\n\n\n\nfor all $g _ { 1 } , g _ { 2 } \\in G$ , i.e., that it satisfies the property needed to get a group action. We will mostly be interested in the case of complex representations, where $V$ is a complex vector space, so one should assume from now on that a representation is complex unless otherwise specified (there will be cases where the representations are real).\n\n",
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"summary": "Woit 在表示映射的语境中说明保持乘法；通用群同态定义回指 Szekeres §2.4。",
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"summary": "The author states that the Spin(3)-to-SO(3) map is a group homomorphism. OCR and translation remain machine-draft-not-reviewed.",
"conditions": "原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。",
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"summary": "The author states that the covering map is a group homomorphism. OCR and translation remain machine-draft-not-reviewed.",
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"start_marker": "The spin representation of $S U ( 2 )$",
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"chapter": "nakahara-3",
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"label": "§3.1.1 商群 (3.2)–(3.3)",
"role": "definition",
"summary": "以模子群的等价类定义商，并验证代表元选择不影响加法。",
"conditions": "Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。 本章每个子群均正规，不能省去一般群商构造中的正规性条件。",
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"quote": "\n$$\nx - y \\in H \\tag {3.2}\n$$\n"
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"quote": "and write $x \\sim y$ . Clearly $\\sim$ is an equivalence relation. The equivalence class to which $x$ belongs is denoted by $[x]$ . Let $G / H$ be the quotient space. The group operation $+$ in $G$ naturally induces the group operation $+$ in $G / H$ by"
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"quote": "\n$$\n[ x ] + [ y ] = [ x + y ]. \\tag {3.3}\n$$\n"
},
{
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"quote": "Note that $+$ on the LHS is an operation in $G / H$ while $+$ on the RHS is that in $G$ . The operation in $G / H$ should be independent of the choice of representatives. In fact, if $[x'] = [x]$ , $[y'] = [y]$ , then $x - x' = h$ , $y - y' = g$ for some $h, g \\in H$ and we find that"
},
{
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"quote": "\n$$\nx ^ {\\prime} + y ^ {\\prime} = x + y - (h + g) \\in [ x + y ]\n$$\n"
},
{
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"quote": "Furthermore, $G / H$ becomes a group with this operation, since $H$ is always a normal subgroup of $G$ ; see example 2.6. The unit element of $G / H$ is $[0] = [h]$ ,"
},
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"quote": "$h \\in H$ . If $H = G$ , $0 - x \\in G$ for any $x \\in G$ and $G / G$ has just one element [0]. If $H = \\{0\\}$ , $G / H$ is $G$ itself since $x - y = 0$ if and only if $x = y$ ."
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},
"chapter": "library-szekeres-2",
"chapter_number": 2,
"author_label": "Szekeres §2.5",
"label": "一般群的商群构造",
"role": "definition",
"summary": "正规子群的陪集在诱导乘法下形成商群。",
"conditions": "Szekeres 要求 $N$ 正规；不是任意子群都可形成群商。",
"symbols": "保留原文符号。",
"location": {
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"excerpt": "When we multiply left cosets of a subgroup $H$ together, for example\n\n$$\ng H g ^ {\\prime} H = \\{g h g ^ {\\prime} h ^ {\\prime} \\mid h, h ^ {\\prime} \\in H \\},\n$$\n\nthe result is not in general another coset. On the other hand, the product ofcosets ofa norma subgroup $N$ is always another coset,\n\n$$\ng N g ^ {\\prime} N = g g ^ {\\prime} N N = (g g ^ {\\prime}) N,\n$$\n\n\nand satisfies the associative law,\n\n$$\n(g N g ^ {\\prime} N) g ^ {\\prime \\prime} N = (g g ^ {\\prime} g ^ {\\prime \\prime}) N = g N (g ^ {\\prime} N g ^ {\\prime \\prime} N).\n$$\n\nFurthermore, the coset $e N = N$ plays the role of an identity element, while every coset has an inverse $( g N ) ^ { - 1 } = g ^ { - 1 } N$ . Hence the cosets of a normal subgroup $N$ form a group called the factor group of $G$ by $N$, denoted $G / N$\n\n",
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"chapter": "nakahara-3",
"chapter_number": 3,
"author_label": "Nakahara 第3章",
"label": "定义3.5 同调群",
"role": "definition",
"summary": "对 n 维单纯复形，以循环群模边界群定义第 r 同调群。",
"conditions": "本章主要采用整数系数；$0\\le r\\le n$。不能与实系数同调空间按定义完全等同。",
"symbols": "$H_r(K)=Z_r(K)/B_r(K)$",
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"quote": "Definition 3.5. Let $K$ be an $n$ -dimensional simplicial complex. The $r$ th homology group $H_{r}(K)$ , $0 \\leq r \\leq n$ , associated with $K$ is defined by"
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"quote": "\n$$\nH _ {r} (K) \\equiv Z _ {r} (K) / B _ {r} (K). \\tag {3.28}\n$$\n"
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"author_label": "Szekeres §17.4",
"label": "实系数同调空间与同调群称呼",
"role": "definition",
"summary": "作者把同调空间也称作同调群，此处只使用阿贝尔群性质。",
"conditions": "Szekeres 用光滑奇异单形的实系数链；整数系数的挠信息不能由该入口抹去。",
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"excerpt": "We define the $p$th homology space to be the factor space\n\n$$\nH _ {p} (M) = Z _ {p} (M) / B _ {p} (M).\n$$\n\nCommonly this is called the $p$th homology group, only the abelian group property being relevant. Two cycles $C _ { 1 }$ and $C _ { 2 }$ are said to be homologous if they belong to the same homology class – that is, if there exists a chain $C$ such that $C _ { 1 } - C _ { 2 } = \\partial C$ . ",
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"author_label": "Nakahara 第3章",
"label": "定义3.6 贝蒂数",
"role": "definition",
"summary": "贝蒂数取实系数同调群的维数，也等于整数同调自由部分的秩。",
"conditions": "不得把整数同调的挠部分计入维数。",
"symbols": "$b_r(K)=\\dim H_r(K;\\mathbb R)$",
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"quote": "Definition 3.6. Let $K$ be a simplicial complex. The $r$ th Betti number $b_{r}(K)$ is defined by"
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"quote": "\n$$\nb _ {r} (K) \\equiv \\dim H _ {r} (K; \\mathbb {R}). \\tag {3.56}\n$$\n"
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"quote": "In other words, $b_{r}(K)$ is the rank of the free Abelian part of $H_{r}(K; \\mathbb{Z})$ ."
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"summary": "以实同调空间维数定义贝蒂数。",
"conditions": "实系数同调空间。",
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"summary": "作者陈述整数自旋态交换对称，半整数自旋态交换反对称。",
"conditions": "相对论与量子力学结合的语境；仅作者陈述，不等于已核验证明。",
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"quote": "One of the most profound consequences of merging special relativity with quantum mechanics is the spin-statistics theorem: states with identical particles of integer spin are symmetric under the interchange of the particles, while states with identical particles of half-integer spin are antisymmetric under the interchange of the particles. This is equivalent to the statement that the creation and annihilation operators for integer spin particles satisfy canonical commutation relations, while creation and annihilation operators for half-integer spin particles satisfy canonical anticommutation relations. Particles quantized with canonical commutation relations are called bosons, and satisfy Bose-Einstein statistics, and particles quantized with canonical anticommutation relations are called fermions, and satisfy Fermi-Dirac statistics."
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"label": "§13 自旋–统计回顾与反对易关系",
"role": "recall",
"summary": "作者据自旋–统计定理要求旋量产生和湮灭算符反对易。",
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"quote": "The creation and annihilation operators for spinors must anticommute by the spin-statistics theorem:"
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"quote": "\n$$\n\\left\\{a _ {p} ^ {s \\dagger}, a _ {q} ^ {s ^ {\\prime} \\dagger} \\right\\} = \\left\\{a _ {p} ^ {s}, a _ {q} ^ {s ^ {\\prime}} \\right\\} = \\left\\{b _ {p} ^ {s \\dagger}, b _ {q} ^ {s ^ {\\prime} \\dagger} \\right\\} = \\left\\{b _ {p} ^ {s}, b _ {q} ^ {s ^ {\\prime}} \\right\\} = 0 \\tag {13.4}\n$$\n"
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"quote": "\n$$\n\\left\\{a _ {p} ^ {s}, a _ {q} ^ {s ^ {\\prime} \\dagger} \\right\\} = \\left\\{b _ {p} ^ {s}, b _ {q} ^ {s ^ {\\prime} \\dagger} \\right\\} = \\delta_ {s s ^ {\\prime}} (2 \\pi) ^ {3} \\delta^ {3} (p - q). \\tag {13.5}\n$$\n"
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"label": "§13 费米子负号的自旋–统计回指",
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"summary": "额外负号归因于自旋统计，作者说稍后解释。",
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"quote": "The extra minus sign is due to spin-statistics, as will be explained shortly. Contracting all the spinor indices and replacing $p_1^\\mu$ by $p^\\mu + k^\\mu$ and $p_2^\\mu$ by $k^\\mu$ :"
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"excerpt": "实验观察到的一个事实是，自旋为 $\\frac { 1 } { 2 }$ 的基本粒子表现为费米子，并由费米子域描述。在非相对论量子场论中，这种自旋为 $\\textstyle { \\frac { 1 } { 2 } }$ 的基本粒子原则上可以是玻色子，由第 38.3.3 节中的玻色子域描述。然而，在相对论量子场论中存在一个 a “自旋-统计定理”，它指出自旋为 $\\begin{array} { l } { { \\frac { 1 } { 2 } } } \\end{array}$ 的域必须用反对易子量子化。这为观察到的自旋值与粒子统计之间的关联提供了解释，原因在于描述基本粒子的非相对论理论应当是相对论理论的低能极限。",
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"quote": "In particular, our interaction $\\mathcal{M}_t$ has three $\\gamma$ -matrices, so it couples $RR$ and $LL$ . Thus,"
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"quote": "which is helicity conserving. So, $u(p_2)$ and $\\bar{u}(p_3)$ should be either both right-handed or both left-handed. This is consistent with a general property of QED, that in the limit of a massless electron, the left- and right-handed states completely decouple."
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"excerpt": "so we see that our solutions have helicity (eigenvalue of J · P divided by the square root of the eigenvalue of $| \\mathbf { P } | ^ { 2 } )$ values $\\pm { \\frac { 1 } { 2 } } .$ , as opposed to the integral helicity values discussed in chapter 19, where $E ( 3 { \\bar { ) } }$ appeared and not its double cover. These two representations on the spaces of solutions $\\widetilde { \\psi } _ { E , \\pm } ( \\mathbf { p } )$ are thus the $E ( 3 )$ representations described in section 19.3, the ones labeled by the helicity $\\pm \\frac { 1 } { 2 }$ representations of the stabilizer group SO(2).",
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"quote": "So far, we have studied quantum field theory using the canonical quantization approach, which is based on creation and annihilation operators. There is a completely different way to do quantum field theory called the path integral formulation. It says"
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"quote": "Finally, taking the limit $\\delta t\\to 0$ , the exponentials combine into an integral over $dt$ and we get"
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"quote": "\n$$\n\\langle f | i \\rangle = N \\int_ {x \\left(t _ {i}\\right) = x _ {i}} ^ {x \\left(t _ {f}\\right) = x _ {f}} \\mathcal {D} x (t) e ^ {i S [ x ]}, \\tag {14.15}\n$$\n"
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"quote": "where $\\mathcal{D}x$ means sum over all paths $x(t)$ with the correct boundary conditions and the action is $S[x] = \\int dt\\mathcal{L}[x(t),\\dot{x} (t)]$ . Note that $N$ has been redefined and is now formally infinite, but it will drop out of any physical quantities, as we will see in the path integral case."
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"excerpt": "There is however a very diferent approach to relating classical and quantum theories, which completely bypasses the Hamiltonian formalism, just using the Lagrangian. This is the path integral formalism, which is based upon a method for calculating matrix elements of the time evolution operator\n\n$$\n\\langle q _ {T} | e ^ {- \\frac {i}{\\hbar} H T} | q _ {0} \\rangle\n$$\n\nin the position eigenstate basis in terms of an integral over the space of paths that $_ \\mathrm { g o }$ from position $q _ { 0 }$ to position $q _ { T }$ in time $T$ (we will here only treat the $d = 1 { \\mathrm { ~ c a s e ) } }$ . Here $\\left| q _ { 0 } \\right.$ is an eigenstate of $Q$ with eigenvalue $q _ { 0 }$ (a delta-function at $q _ { 0 }$ in the position space representation), and $\\left| q _ { T } \\right.$ has $Q$ eigenvalue $q _ { T }$ . This matrix element has a physical interpretation as the amplitude for a particle starting at $q _ { 0 }$ at $t = 0$ to have position $q _ { T }$ at time $T ,$ with its norm-squared giving the probability density for observing the particle at position $q _ { T }$ . It is also the kernel function that allows one to determine the wavefunction $\\psi ( q , T )$ at any time $t = T$ in terms of its initial value at $t = 0$ , by calculating\n\n$$\n\\psi (q _ {T}, T) = \\int_ {- \\infty} ^ {\\infty} \\langle q _ {T} | e ^ {- \\frac {i}{\\hbar} H T} | q _ {0} \\rangle \\psi (q _ {0}, 0) d q _ {0}\n$$",
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"excerpt": "然而，还有一种截然不同的方法将经典理论与量子理论联系起来，它完全绕过了哈密顿算子形式，仅使用拉格朗日量。这就是路径积分形式，它基于一种计算时间演化算符的矩阵元的方法\n\n$$\n\\langle q _ {T} | e ^ {- \\frac {i}{\\hbar} H T} | q _ {0} \\rangle\n$$\n\n在位置本征态基下，用对路径空间的积分表示 $_ \\mathrm { g o }$ 在时间 $T$ 内从位置 $q _ { 0 }$ 到位置 $q _ { T }$（这里我们只处理 $d = 1 { \\mathrm { ~ c a s e ) } }$。这里 $\\left| q _ { 0 } \\right.$ 是 $Q$ 的一个本征态，特征值为 $q _ { 0 }$（在位置空间表示中为位于 $q _ { 0 }$ 的 delta 函数），而 $\\left| q _ { T } \\right.$ 具有 $Q$ 特征值 $q _ { T }$。该矩阵元有一个物理解释：它是粒子在 $t = 0$ 时从 $q _ { 0 }$ 出发、在时间 $T ,$ 时具有位置 $q _ { T }$ 的振幅，其范数平方给出在位置 $q _ { T }$ 观测到该粒子的概率密度。它也是核函数，使得人们能够通过计算来确定任意时间 $t = T$ 的波函数 $\\psi ( q , T )$，以其在 $t = 0$ 的初值为依据。\n\n$$\n\\psi (q _ {T}, T) = \\int_ {- \\infty} ^ {\\infty} \\langle q _ {T} | e ^ {- \\frac {i}{\\hbar} H T} | q _ {0} \\rangle \\psi (q _ {0}, 0) d q _ {0}\n$$",
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"quote": "A Grassmann algebra is a set of objects $\\mathcal{G}$ that are generated by a basis $\\{\\theta_i\\}$ . These $\\theta_i$ are Grassmann numbers, which anticommute with each other, $\\theta_i\\theta_j = -\\theta_j\\theta_i$ , add commutatively, $\\theta_i + \\theta_j = \\theta_j + \\theta_i$ , and can be multiplied by complex numbers, $a\\theta \\in \\mathcal{G}$ for $\\theta \\in \\mathcal{G}$ and $a \\in \\mathbb{C}$ . The algebra must also have an element 0 so that $\\theta_i + 0 = \\theta_i$ ."
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"quote": "and so on. Elements of the algebra that have an even number of $\\theta_{i}$ commute with all elements of the algebra, so they compose the even-graded or bosonic subalgebra. Similarly, the odd-graded or fermionic subalgebra has an odd number of $\\theta_{i}$ and anticommutes within itself (but commutes with the bosonic subalgebra). The fermionic subalgebra is not closed, since $\\theta_{1}\\theta_{2}$ is bosonic."
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"excerpt": "> [!theorem] Definition (Grassmann algebra)\n> The algebra over the real numbers generated\n\nby $\\xi _ { j } , j = 1 , \\ldots , n$ , satisfying the relations\n\n$$\n\\xi_ {j} \\xi_ {k} + \\xi_ {k} \\xi_ {j} = 0\n$$\n\nis called the Grassmann algebra.",
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"excerpt": "> [!definition] 定义（格拉斯曼代数）\n> 由实数域上生成的代数\n\n由$\\xi _ { j } , j = 1 , \\ldots , n$，满足这些关系\n\n$$\n\\xi_ {j} \\xi_ {k} + \\xi_ {k} \\xi_ {j} = 0\n$$\n\n称为格拉斯曼代数。",
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"conditions": "形式路径积分与 $\\hbar\to0$；这里不新增严密测度存在性或渐近误差证明。",
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"quote": "As a first check on the path integral, we can take the classical limit. To do that, we need to put back $\\hbar$ , which can be done by dimensional analysis. Since $\\hbar$ has dimensions of action, it appears as"
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"quote": "\n$$\n\\langle 0; t _ {f} | 0; t _ {i} \\rangle = N \\int \\mathcal {D} \\Phi (\\vec {x}, t) e ^ {\\frac {i}{\\hbar} S [ \\Phi ]}. \\tag {14.31}\n$$\n"
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"quote": "Using the method of stationary phase we see that, in the limit $\\hbar \\rightarrow 0$ , this integral is dominated by the value of $\\Phi$ for which $S[\\Phi]$ has an extremum. But $\\delta S = 0$ is precisely the condition that determines the Euler-Lagrange equations which a classical field satisfies. Therefore, the only configuration that contributes in the classical limit is the classical solution to the equations of motion."
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"sha256": "610a081bfa8af6f32b46a9b041d1df2c14e95fe9e0eeaa411fdfab3f34111074"
},
"range": [
0,
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],
"quote": "Theorem 3.1. (Fundamental theorem of homomorphism) Let $f: G_1 \\to G_2$ be a homomorphism. Then"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-d84a0ba425020125c15354d8",
"rev": 1,
"sha256": "45efa1d8e63394d51871de3cd1e6db85b3493e92167991f296ce0a3149ba1054"
},
"range": [
0,
64
],
"quote": "\n$$\nG _ {1} / \\ker f \\cong \\operatorname {i m} f. \\tag {3.6}\n$$\n"
}
],
"available_after": 1648,
"units": [
"nakahara-3:u0029"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "frozen candidate atom source; Codex selected range"
},
{
"id": "statement-247c80347353b6759a829348",
"source_kind": "ribosome-atoms",
"scope": {
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"chapter": "nakahara-3",
"chapter_number": 3,
"author_label": "Nakahara 第3章",
"label": "定理3.1 的证明",
"role": "proof",
"summary": "先据引理3.1确定两边为群，再构造并验证同构。",
"conditions": "Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。",
"symbols": "",
"passages": [
{
"atom_ref": {
"kind": "atom",
"id": "a-1af06f5197e8c4a2154f53b7",
"rev": 1,
"sha256": "d75df69df9febb435ad302330eecd9cff79cae4117c22a773b63f025a73c5480"
},
"range": [
0,
383
],
"quote": "Proof. Both sides are groups according to lemma 3.1. Define a map $\\varphi : G_1 / \\ker f \\to \\operatorname{im} f$ by $\\varphi([x]) = f(x)$ . This map is well defined since for $x' \\in [x]$ , there exists $h \\in \\ker f$ such that $x' = x + h$ and $f(x') = f(x + h) = f(x) + f(h) = f(x)$ . Now we show that $\\varphi$ is an isomorphism. First, $\\varphi$ is a homomorphism,"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-e1f773f1c7c69199c0293310",
"rev": 1,
"sha256": "12f1a0d2d6dd17b1b30bfdfb5702964a5819200a8db68a83dc59913bc5ef9102"
},
"range": [
0,
153
],
"quote": "\n$$\n\\begin{array}{l} \\varphi ([ x ] + [ y ]) = \\varphi ([ x + y ]) = f (x + y) \\\\ = f (x) + f (y) = \\varphi ([ x ]) + \\varphi ([ y ]). \\\\ \\end{array}\n$$\n"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-f590124acb99157c79ee0e74",
"rev": 1,
"sha256": "ec972aead1fc893bc0363222cbc012338f568240aad3961ccd38d7cf95cdd5eb"
},
"range": [
0,
312
],
"quote": "Second, $\\varphi$ is one to one: if $\\varphi([x]) = \\varphi([y])$ , then $f(x) = f(y)$ or $f(x) - f(y) = f(x - y) = 0$ . This shows that $x - y \\in \\ker f$ and $[x] = [y]$ . Finally, $\\varphi$ is onto: if $y \\in \\operatorname{im} f$ , there exists $x \\in G_1$ such that $f(x) = y = \\varphi([x])$ ."
}
],
"available_after": 1651,
"units": [
"nakahara-3:u0029"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "frozen candidate atom source; Codex selected range"
},
{
"id": "statement-a6ea5696ed85e50e73dbbcb4",
"source_kind": "ribosome-atoms",
"scope": {
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"chapter": "nakahara-3",
"chapter_number": 3,
"author_label": "Nakahara 第3章",
"label": "例3.2 对定理3.1的应用",
"role": "use",
"summary": "对整数到二元循环群的映射使用同态基本定理。",
"conditions": "Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。",
"symbols": "",
"passages": [
{
"atom_ref": {
"kind": "atom",
"id": "a-0a4e53b3e74fb3e1a6583eae",
"rev": 1,
"sha256": "8d66d11ec0dedd63368716766acbd0fcd5ce937af618e0fb173b3c6ab8cf7d3a"
},
"range": [
0,
289
],
"quote": "Example 3.2. Let $f: \\mathbb{Z} \\to \\mathbb{Z}_2$ be defined by $f(2n) = 0$ and $f(2n + 1) = 1$ . Then $\\ker f = 2\\mathbb{Z}$ and $\\operatorname{im} f = \\mathbb{Z}_2$ are groups. Theorem 3.1 states that $\\mathbb{Z}/2\\mathbb{Z} \\cong \\mathbb{Z}_2$ , in agreement with example 3.1."
}
],
"available_after": 1652,
"units": [
"nakahara-3:u0029"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "frozen candidate atom source; Codex selected range"
},
{
"id": "statement-45381cbdd77285069b293927",
"source_kind": "ribosome-atoms",
"scope": {
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"chapter": "nakahara-3",
"chapter_number": 3,
"author_label": "Nakahara 第3章",
"label": "循环群和边界群的子群资格",
"role": "use",
"summary": "作者明确从引理3.1得到 Zr 与 Br 是链群的子群。",
"conditions": "已引入边界同态，尚不把 Br 包含于 Zr 当作这里已证明。",
"symbols": "",
"passages": [
{
"atom_ref": {
"kind": "atom",
"id": "a-19671c7e478fcddd158d5517",
"rev": 1,
"sha256": "4531bea9194acbb7b745a05f750186d615b8439623d025e5227a1c990c19ac25"
},
"range": [
0,
221
],
"quote": "From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r}(K)$ are subgroups of $C_{r}(K)$ . We now prove an important relation between $Z_{r}(K)$ and $B_{r}(K)$ , which is crucial in the definition of homology groups."
}
],
"available_after": 1776,
"units": [
"nakahara-3:u0171"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "frozen candidate atom source; Codex selected range"
},
{
"id": "statement-dd7721e0c9adf5e7b891bfcd",
"source_kind": "ribosome-atoms",
"scope": {
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"chapter": "nakahara-3",
"chapter_number": 3,
"author_label": "Nakahara 第3章",
"label": "引理3.3 连续两次取边界为零",
"role": "theorem",
"summary": "边界算子的复合为零。",
"conditions": "r 链与边界算子按本书有向单形约定。",
"symbols": "",
"passages": [
{
"atom_ref": {
"kind": "atom",
"id": "a-f3a208445d37f7d51029a7ee",
"rev": 1,
"sha256": "e49c829f55bdb4635209f6c67681744b1b0056784243aa4e72a938feaa869a2f"
},
"range": [
0,
187
],
"quote": "Lemma 3.3. The composite map $\\partial_r\\circ \\partial_{r + 1}:C_{r + 1}(K)\\to C_{r - 1}(K)$ is a zero map; that is, $\\partial_r(\\partial_{r + 1}c) = 0$ for any $c\\in C_{r + 1}(K)$ ."
}
],
"available_after": 1777,
"units": [
"nakahara-3:u0171"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "frozen candidate atom source; Codex selected range"
},
{
"id": "statement-cf70f6e882eed0df163c05a1",
"source_kind": "ribosome-atoms",
"scope": {
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"chapter": "nakahara-3",
"chapter_number": 3,
"author_label": "Nakahara 第3章",
"label": "定理3.3 边界群包含于循环群",
"role": "proof",
"summary": "证明明确使用引理3.3。",
"conditions": "这一步为同调商群提供子群包含关系。",
"symbols": "",
"passages": [
{
"atom_ref": {
"kind": "atom",
"id": "a-a8cf5ae8a234eb8eb4969495",
"rev": 1,
"sha256": "bba0905956bd059c1e1db5cf3815ecba424accb4c1c6d7e01c25b1e89bd52ab1"
},
"range": [
0,
122
],
"quote": "Theorem 3.3. Let $Z_{r}(K)$ and $B_{r}(K)$ be the $r$ -cycle group and the $r$ -boundary group of $C_{r}(K)$ , then"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-a3cd75f8665a2ea53bde99b9",
"rev": 1,
"sha256": "6237d87fc9059e4e4bb16a255cee05f2f598f4dd2996b01118c152140c50537d"
},
"range": [
0,
80
],
"quote": "\n$$\nB _ {r} (K) \\subset Z _ {r} (K) \\quad (\\subset C _ {r} (K)). \\tag {3.27}\n$$\n"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-a4156793e1419ae34a1c08dc",
"rev": 1,
"sha256": "3247ed5336c990c53f51ab15bd0829c5d9e2deb7ec71b65e875823d1f20eed71"
},
"range": [
0,
271
],
"quote": "Proof. This is obvious from lemma 3.3. Any element $c$ of $B_r(K)$ is written as $c = \\partial_{r+1}d$ for some $d \\in C_{r+1}(K)$ . Then we find $\\partial_r c = \\partial_r(\\partial_{r+1}d) = 0$ , that is, $c \\in Z_r(K)$ . This implies $Z_r(K) \\supset B_r(K)$ ."
}
],
"available_after": 1783,
"units": [
"nakahara-3:u0171"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "frozen candidate atom source; Codex selected range"
},
{
"id": "statement-eb6efb25730232282cb5cf47",
"source_kind": "ribosome-atoms",
"scope": {
"book_id": "schwartz-qft",
"edition_id": "local-experiment",
"source_id": "schwartz-gauge-mineru-20261012"
},
"chapter": "schwartz-14",
"chapter_number": 14,
"author_label": "Schwartz 第14章",
"label": "式(14.6) 一维高斯积分结果",
"role": "derivation",
"summary": "平方配方和积分变换后的高斯结果。",
"conditions": "复参数延拓与收敛条件不由此记录补充证明。",
"symbols": "",
"passages": [
{
"atom_ref": {
"kind": "atom",
"id": "a-abb6ec17b84ddf7b4388a5c8",
"rev": 1,
"sha256": "a6b3b883f70f5d91c804652f634625d8b6c7ada524c742f920a006f2d309b273"
},
"range": [
0,
143
],
"quote": "\n$$\n\\int_ {- \\infty} ^ {\\infty} d p e ^ {- \\frac {1}{2} a p ^ {2} + J p} = \\sqrt {\\frac {2 \\pi}{a}} e ^ {\\frac {J ^ {2}}{2 a}}. \\tag {14.6}\n$$\n"
}
],
"available_after": 2332,
"units": [
"schwartz-14:u0025"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "frozen candidate atom source; Codex selected range"
},
{
"id": "statement-0993b4cf0a986537ff39c920",
"source_kind": "ribosome-atoms",
"scope": {
"book_id": "schwartz-qft",
"edition_id": "local-experiment",
"source_id": "schwartz-gauge-mineru-20261012"
},
"chapter": "schwartz-14",
"chapter_number": 14,
"author_label": "Schwartz 第14章",
"label": "用式(14.6)计算短时间矩阵元",
"role": "use",
"summary": "作者明确代入高斯公式得到短时间矩阵元。",
"conditions": "非相对论二次动量哈密顿量与时间片近似。",
"symbols": "",
"passages": [
{
"atom_ref": {
"kind": "atom",
"id": "a-ac6d85775630f770d011581c",
"rev": 1,
"sha256": "10ab6fa2ca557682fec3c2ebf79628ad380466c7f6010728ea69b44a8a2314bb"
},
"range": [
0,
230
],
"quote": "Now we can use the Gaussian integral in Eq. (14.6), $\\int dp\\exp \\left(-\\frac{1}{2} ap^2 +Jp\\right) = \\sqrt{\\frac{2\\pi}{a}}\\exp \\left(\\frac{J^2}{2a}\\right)$ , with $a = i\\frac{\\delta t}{m}$ and $J = i(x_{j + 1} - x_j)$ to get"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-3969e9582ae4e4d1ea7346c9",
"rev": 1,
"sha256": "5545933f6b9d708d951b3a1c5d23709eb7cdfe5388d0a2656fdc5c0d0173647c"
},
"range": [
0,
249
],
"quote": "\n$$\n\\langle x _ {j + 1} | e ^ {- i H \\delta t} | x _ {j} \\rangle = N e ^ {- i V (x _ {j}, t _ {j}) \\delta t} e ^ {i \\frac {m}{2} \\delta t \\frac {(x _ {j + 1} - x _ {j}) ^ {2}}{(\\delta t) ^ {2}}} = N e ^ {i L (x, \\dot {x}) \\delta t}, \\tag {14.12}\n$$\n"
}
],
"available_after": 2349,
"units": [
"schwartz-14:u0047"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "frozen candidate atom source; Codex selected range"
},
{
"id": "statement-9dc83328caf7b27d5fc2f15e",
"source_kind": "ribosome-atoms",
"scope": {
"book_id": "schwartz-qft",
"edition_id": "local-experiment",
"source_id": "schwartz-gauge-mineru-20261012"
},
"chapter": "schwartz-14",
"chapter_number": 14,
"author_label": "Schwartz 第14章",
"label": "式(14.10) 时间切片",
"role": "derivation",
"summary": "在小时间片上分解振幅。",
"conditions": "平滑时间依赖及无穷小区间近似。",
"symbols": "",
"passages": [
{
"atom_ref": {
"kind": "atom",
"id": "a-57da11c3619b2026d9451a15",
"rev": 1,
"sha256": "52a21a4333488ec5c1bbbf7e8d5de643e113e6aeda78794120fa6cab6f6983f5"
},
"range": [
0,
296
],
"quote": "If instead we only assume $\\tilde{H}(t)$ is a smooth function of $t$ , then we can only solve for the matrix element this way for infinitesimal time intervals. So, let us break this down into $n$ small time intervals $\\delta t$ and define $t_j = t_i + j\\delta t$ and $t_n = t_f$ . Then,"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-62e29b1a8e0ff4ece58551c0",
"rev": 1,
"sha256": "1340979be6a91632537d3ff178af549c3dc7ba780d5ebd23e0998bd2880987b3"
},
"range": [
0,
441
],
"quote": "\n$$\n\\langle f | i \\rangle = \\int d x _ {n} \\dots d x _ {1} \\left\\langle x _ {f} \\right| e ^ {- i H \\left(t _ {f}\\right) \\delta t} \\left| x _ {n} \\right\\rangle \\left\\langle x _ {n} \\right| \\dots \\left| x _ {2} \\right\\rangle \\left\\langle x _ {2} \\right| e ^ {- i H \\left(t _ {2}\\right) \\delta t} \\left| x _ {1} \\right\\rangle \\left\\langle x _ {1} \\right| e ^ {- i H \\left(t _ {1}\\right) \\delta t} \\left| x _ {i} \\right\\rangle . \\tag {14.10}\n$$\n"
}
],
"available_after": 2343,
"units": [
"schwartz-14:u0036"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "frozen candidate atom source; Codex selected range"
},
{
"id": "statement-3c6ad35017afb105cdf7e24e",
"source_kind": "ribosome-atoms",
"scope": {
"book_id": "schwartz-qft",
"edition_id": "local-experiment",
"source_id": "schwartz-gauge-mineru-20261012"
},
"chapter": "schwartz-14",
"chapter_number": 14,
"author_label": "Schwartz 第14章",
"label": "式(14.14) 路径积分前的时间片乘积",
"role": "use",
"summary": "作者把(14.12)代入(14.10)，得到拉格朗日量指数乘积。",
"conditions": "依赖已求得的短时间矩阵元；还未取连续极限。",
"symbols": "",
"passages": [
{
"atom_ref": {
"kind": "atom",
"id": "a-d6b09c2819ae36fd14db89ed",
"rev": 1,
"sha256": "2421db4b85dc37262f8bd2436a0884d70c9396e199fbaed24f8b682099fa8f09"
},
"range": [
0,
92
],
"quote": "Using Eq. (14.12), each term in Eq. (14.10) becomes just a number and the product reduces to"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-65423adb6e38c3e9f07020c7",
"rev": 1,
"sha256": "14a5281448397ebaf1d87aa7fd61f800fbafe260aaf3fc56d3f9e0191304a89d"
},
"range": [
0,
202
],
"quote": "\n$$\n\\langle f | i \\rangle = N ^ {n} \\int d x _ {n} \\dots d x _ {1} e ^ {i L \\left(x _ {n}, \\dot {x} _ {n}\\right) \\delta t} \\dots e ^ {i L \\left(x _ {1}, \\dot {x} _ {1}\\right) \\delta t}. \\tag {14.14}\n$$\n"
}
],
"available_after": 2354,
"units": [
"schwartz-14:u0047"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "frozen candidate atom source; Codex selected range"
},
{
"id": "statement-6bcd38923f6cdde20f7532b3",
"source_kind": "ribosome-atoms",
"scope": {
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"chapter": "nakahara-3",
"chapter_number": 3,
"author_label": "Nakahara 第3章",
"label": "定理3.7证明中的维数分解",
"role": "proof",
"summary": "作者对实系数边界映射应用此前定理2.1。",
"conditions": "该前置定理尚未绑定，本批不能用别的书替代。",
"symbols": "",
"passages": [
{
"atom_ref": {
"kind": "atom",
"id": "a-3f165b3b0b7ee2a212af11d9",
"rev": 1,
"sha256": "55481304c8d50561c93035ce327ad554f8f5896c23d8d84d00c96b198be69167"
},
"range": [
0,
42
],
"quote": "Proof. Consider the boundary homomorphism,"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-476a5c539ddf7d4918fcc8d7",
"rev": 1,
"sha256": "9668f230798ed385c3c3fa24ffa17cfb4d73b85bf506a204cc293b82957c2e0a"
},
"range": [
0,
80
],
"quote": "\n$$\n\\partial_ {r}: C _ {r} (K; \\mathbb {R}) \\to C _ {r - 1} (K; \\mathbb {R})\n$$\n"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-83bfbbe64e214ef4452c8660",
"rev": 1,
"sha256": "8b38c432111f5bdf2bdf1d5ba0fa6a7dcce3077c00e0290c0bdbfb2240682932"
},
"range": [
0,
180
],
"quote": "where $C_{-1}(K; \\mathbb{R})$ is defined to be $\\{0\\}$ . Since both $C_{r - 1}(K; \\mathbb{R})$ and $C_r(K;\\mathbb{R})$ are vector spaces, theorem 2.1 can be applied to yield"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-05903e6740f20dca6c8068d1",
"rev": 1,
"sha256": "0fb075941745a54830bff5dabd0b563d70aab5ed40fea2a55ea26373f5cb6888"
},
"range": [
0,
221
],
"quote": "\n$$\n\\begin{array}{l} I _ {r} = \\dim C _ {r} (K; \\mathbb {R}) = \\dim (\\ker \\partial_ {r}) + \\dim (\\operatorname {i m} \\partial_ {r}) \\\\ = \\dim Z _ {r} (K; \\mathbb {R}) + \\dim B _ {r - 1} (K; \\mathbb {R}) \\\\ \\end{array}\n$$\n"
}
],
"available_after": 1969,
"units": [
"nakahara-3:u0361"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "frozen candidate atom source; Codex selected range"
},
{
"id": "statement-a9109552ffd06f4f8a886bfa",
"source_kind": "ribosome-atoms",
"scope": {
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"chapter": "nakahara-3",
"chapter_number": 3,
"author_label": "Nakahara 第3章",
"label": "定义3.2：整数系数链群",
"role": "definition",
"summary": "有向 r 单形生成自由阿贝尔链群，链的系数为整数。",
"conditions": "本章整数系数单纯链；链群为自由阿贝尔群。该路径以群、子群和有向单形知识为已具备背景。",
"symbols": "$C_r(K)$",
"passages": [
{
"atom_ref": {
"kind": "atom",
"id": "a-3edf946e15462cd329adccbb",
"rev": 1,
"sha256": "4d8fa60eb21c6f6175a3641d1047ae1b52c8ff9fe19a01568335ed8f47b04f09"
},
"range": [
0,
255
],
"quote": "Definition 3.2. The $r$ -chain group $C_r(K)$ of a simplicial complex $K$ is a free Abelian group generated by the oriented $r$ -simplexes of $K$ . If $r > \\dim K$ , $C_r(K)$ is defined to be 0. An element of $C_r(K)$ is called an $r$ -chain."
},
{
"atom_ref": {
"kind": "atom",
"id": "a-ecff1a633cd6d60448e9c938",
"rev": 1,
"sha256": "ed16b288c91d58941fa4934cdaf32d778710f29e0a247a831a2dfd01072941c7"
},
"range": [
0,
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],
"quote": "Let there be $I_r$ $r$ -simplexes in $K$ . We denote each of them by $\\sigma_{r,i}$ ( $1 \\leq i \\leq I_r$ ). Then $c \\in C_r(K)$ is expressed as"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-cc3b70fafba12371ab14ef46",
"rev": 1,
"sha256": "256d41f9e1e03e6f8e9b8ca924321e503bebd981991217051319b5cb2cf8d9f2"
},
"range": [
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"quote": "\n$$\nc = \\sum_ {i = 1} ^ {I _ {r}} c _ {i} \\sigma_ {r, i} \\quad c _ {i} \\in \\mathbb {Z}. \\tag {3.15}\n$$\n"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-e842fe5b5ea68b869414a072",
"rev": 1,
"sha256": "8d1da5a833561bd54b0566751fc675325a9fefb80843d39d9f3e44cf045d2a87"
},
"range": [
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],
"quote": "{\"image_caption\": [], \"image_footnote\": [], \"img_path\": \"images/ddbc60f39a21b01109470e0abf58cca1ab547d5c55706fcee0682d813b959a74.jpg\"}"
},
{
"atom_ref": {
"kind": "atom",
"id": "cap-b3943ef98a840590fdbff357",
"rev": 1,
"sha256": "b9feb38d652db4014dee3e374f28a65f3ac0526657f381d8a4f51df5cbfef201"
},
"range": [
0,
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],
"quote": "Figure 3.7. (a) An oriented 1-simplex with a fictitious boundary $p_1$ . (b) A simplicial complex without a boundary."
},
{
"atom_ref": {
"kind": "atom",
"id": "a-cfd20991e561a6196f51cce7",
"rev": 1,
"sha256": "f6af5676476adf43145c79915616408fb0dfc8ed8b67995bf62183a7f9c2bf2c"
},
"range": [
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],
"quote": "The integers $c_{i}$ are called the coefficients of $c$ . The group structure is given as follows. The addition of two $r$ -chains, $c = \\sum_{i} c_{i} \\sigma_{r,i}$ and $c' = \\sum_{i} c_{i}' \\sigma_{r,i}$ , is"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-5fba62cb9204ca3706ad3c8f",
"rev": 1,
"sha256": "0a425c83fe0fc16a821b34e16cd17a0d1c47176f1f7addc6cf87bd7401e6b027"
},
"range": [
0,
106
],
"quote": "\n$$\nc + c ^ {\\prime} = \\sum_ {i} \\left(c _ {i} + c _ {i} ^ {\\prime}\\right) \\sigma_ {r, i}. \\tag {3.16}\n$$\n"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-d9b6f574b92081a0882c8574",
"rev": 1,
"sha256": "5fd6da9f24585c702b15f149ed6a993443239dffc1a822474cc87a125841ffac"
},
"range": [
0,
303
],
"quote": "The unit element is $0 = \\sum_{i} 0 \\cdot \\sigma_{r,i}$ , while the inverse element of $c$ is $-c = \\sum_{i} (-c_i) \\sigma_{r,i}$ . [Remark: An oppositely oriented $r$ -simplex $-\\sigma_r$ is identified with $(-1) \\sigma_r \\in C_r(K)$ .] Thus, $C_r(K)$ is a free Abelian group of rank $I_r$ ,"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-a7dc50f81684467a1a8dbf65",
"rev": 1,
"sha256": "28f8c99bea6da90522656da7f3ccf33700756e82ab0ef17aa705ec3b64122ce0"
},
"range": [
0,
128
],
"quote": "\n$$\nC _ {r} (K) \\cong \\underbrace {\\mathbb {Z} \\oplus \\mathbb {Z} \\oplus \\cdots \\oplus \\mathbb {Z}} _ {I _ {r}}. \\tag {3.17}\n$$\n"
}
],
"available_after": 1746,
"units": [
"nakahara-3:u0124",
"nakahara-3:u0136"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "Codex curated path completion from immutable v17 atoms; not automatic extraction"
},
{
"id": "statement-27db493e412102426ffd26d7",
"source_kind": "ribosome-atoms",
"scope": {
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"chapter": "nakahara-3",
"chapter_number": 3,
"author_label": "Nakahara 第3章",
"label": "边界同态：式(3.20)–(3.22)",
"role": "definition",
"summary": "按有向顶点的交错和定义边界，并线性延拓为相邻链群之间的同态。",
"conditions": "本章整数系数单纯链；链群为自由阿贝尔群。该路径以群、子群和有向单形知识为已具备背景。",
"symbols": "$\\\\partial_r:C_r(K)\\\\to C_{r-1}(K)$",
"passages": [
{
"atom_ref": {
"kind": "atom",
"id": "a-9efca508b7dc958f1465c4aa",
"rev": 1,
"sha256": "2b0b9f220551c96033003e34714c2f28130d31fef028ad9ecd92aa79eb4327eb"
},
"range": [
0,
160
],
"quote": "Let $\\sigma_r(p_0\\ldots p_r)$ ( $r > 0$ ) be an oriented $r$ -simplex. The boundary $\\partial_r\\sigma_r$ of $\\sigma_r$ is an $(r - 1)$ -chain defined by"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-d2d332422cab7e079e311cac",
"rev": 1,
"sha256": "61e4b25e4506b6cf02b4467435de0dabde5625e03db5e6a4f69259d9ad4ca91c"
},
"range": [
0,
148
],
"quote": "\n$$\n\\partial_ {r} \\sigma_ {r} \\equiv \\sum_ {i = 0} ^ {r} (- 1) ^ {i} \\left(p _ {0} p _ {1} \\dots \\hat {p} _ {i} \\dots p _ {r}\\right) \\tag {3.20}\n$$\n"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-aa44bf7990e60f4a817186cb",
"rev": 1,
"sha256": "25afe5f8175473370420d841394db23bb71581d1a5d81932aff2f1bdb8620a70"
},
"range": [
0,
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],
"quote": "where the point $p_i$ under $\\hat{\\mathbf{\\mu}}$ is omitted. For example,"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-1d228fa39972a355eeea215e",
"rev": 1,
"sha256": "24ffab37f498935157ef3e264836fbfe242815590d8b4bd3dfe2efa6de6e8cd3"
},
"range": [
0,
402
],
"quote": "\n$$\n\\begin{array}{l} \\partial_ {2} \\left(p _ {0} p _ {1} p _ {2}\\right) = \\left(p _ {1} p _ {2}\\right) - \\left(p _ {0} p _ {2}\\right) + \\left(p _ {0} p _ {1}\\right) \\\\ \\partial_ {3} \\left(p _ {0} p _ {1} p _ {2} p _ {3}\\right) = \\left(p _ {1} p _ {2} p _ {3}\\right) - \\left(p _ {0} p _ {2} p _ {3}\\right) + \\left(p _ {0} p _ {1} p _ {3}\\right) - \\left(p _ {0} p _ {1} p _ {2}\\right). \\\\ \\end{array}\n$$\n"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-86c071b59a16c0cb0df87de7",
"rev": 1,
"sha256": "581c3fd891d81cef7a36b5214338d778077a83327000ada5576563f974b3ae88"
},
"range": [
0,
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],
"quote": "We formally define $\\partial_0\\sigma_0 = 0$ for $r = 0$ ."
},
{
"atom_ref": {
"kind": "atom",
"id": "a-d567d6b09b2cfc657b9e7f46",
"rev": 1,
"sha256": "5e1b05d3604c2b339c65a31f49fa73a9384bc6d5cab7b7accc6ad7a06a270343"
},
"range": [
0,
104
],
"quote": "The operator $\\partial_r$ acts linearly on an element $c = \\sum_{i} c_i \\sigma_{r,i}$ of $C_r(K)$ ,"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-3ec53a7d592c742384e72ab6",
"rev": 1,
"sha256": "fa7cdcaa804c71fcfa90995d0d19ceb005a729ec43b3f933742ca80eef556de1"
},
"range": [
0,
85
],
"quote": "\n$$\n\\partial_ {r} c = \\sum_ {i} c _ {i} \\partial_ {r} \\sigma_ {r, i}. \\tag {3.21}\n$$\n"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-a03c11cc2a3989a861afae7b",
"rev": 1,
"sha256": "102180845957b9a732e6901a4902a8f2f04bfc7257805685cf055a8d7d33253c"
},
"range": [
0,
92
],
"quote": "The RHS of (3.21) is an element of $C_{r-1}(K)$ . Accordingly, $\\partial_r$ defines a map"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-0aef4ba7b1dad3f2c1410bab",
"rev": 1,
"sha256": "a9d094fc2f5d56dbdac0b163a79ec70f386caec00ddbaabfeff4badf3750482f"
},
"range": [
0,
75
],
"quote": "\n$$\n\\partial_ {r}: C _ {r} (K) \\rightarrow C _ {r - 1} (K). \\tag {3.22}\n$$\n"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-7e0e7bbecfd8a733a41dbb0c",
"rev": 1,
"sha256": "f578b17f37a1a6f8b7a6e286f1ac6b36161df13f9aba18f45ad07bb22fc40bf4"
},
"range": [
0,
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],
"quote": "$\\partial_r$ is called the boundary operator. It is easy to see that the boundary operator is a homomorphism."
}
],
"available_after": 1766,
"units": [
"nakahara-3:u0147",
"nakahara-3:u0159"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "Codex curated path completion from immutable v17 atoms; not automatic extraction"
},
{
"id": "statement-ac7699973242d9783a9bb1ca",
"source_kind": "ribosome-atoms",
"scope": {
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"chapter": "nakahara-3",
"chapter_number": 3,
"author_label": "Nakahara 第3章",
"label": "定义3.3：循环群是边界同态的核",
"role": "definition",
"summary": "r 循环的边界为零，循环群是边界同态的核。",
"conditions": "本章整数系数单纯链；链群为自由阿贝尔群。该路径以群、子群和有向单形知识为已具备背景。",
"symbols": "$Z_r(K)=\\\\ker\\\\partial_r$",
"passages": [
{
"atom_ref": {
"kind": "atom",
"id": "a-fe3f1f1204ca17761656a5d5",
"rev": 1,
"sha256": "e8c65c7b14ff1d38f9feb2f52ed221270adb8d6a6e49f1e8f1ab7403a8c106cb"
},
"range": [
0,
45
],
"quote": "Definition 3.3. If $c \\in C_r(K)$ satisfies"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-30e957ffc995d2ed5fa229c8",
"rev": 1,
"sha256": "5d82294f245b14ba3c9a13e8ae84e629114cd4d0db68d50da45fceb32092c610"
},
"range": [
0,
39
],
"quote": "\n$$\n\\partial_ {r} c = 0 \\tag {3.24}\n$$\n"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-bd745bddbef5433a0e0dd506",
"rev": 1,
"sha256": "683538972717766d42eee171f712a6eaf541ac7e85efd80c6264a687cacecc00"
},
"range": [
0,
281
],
"quote": "$c$ is called an $r$ -cycle. The set of $r$ -cycles $Z_{r}(K)$ is a subgroup of $C_{r}(K)$ and is called the $r$ -cycle group. Note that $Z_{r}(K) = \\ker \\partial_{r}$ . [Remark: If $r = 0$ , $\\partial_0 c$ vanishes identically and $Z_{0}(K) = C_{0}(K)$ , see (3.23).]"
}
],
"available_after": 1772,
"units": [
"nakahara-3:u0159"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "Codex curated path completion from immutable v17 atoms; not automatic extraction"
},
{
"id": "statement-e9cdbaabe0ae1a11cb9cc9b3",
"source_kind": "ribosome-atoms",
"scope": {
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"chapter": "nakahara-3",
"chapter_number": 3,
"author_label": "Nakahara 第3章",
"label": "定义3.4：边界群是前一边界同态的像",
"role": "definition",
"summary": "r 边界来自 r+1 链的边界，边界群是其像。",
"conditions": "本章整数系数单纯链；链群为自由阿贝尔群。该路径以群、子群和有向单形知识为已具备背景。",
"symbols": "$B_r(K)=\\\\operatorname{im}\\\\partial_{r+1}$",
"passages": [
{
"atom_ref": {
"kind": "atom",
"id": "a-075fd90cce3c868365101fbd",
"rev": 1,
"sha256": "b9f93c7ea8078d0c810d028c71b4bd2cb38871df94b9194228b8679eef7c0551"
},
"range": [
0,
154
],
"quote": "Definition 3.4. Let $K$ be an $n$ -dimensional simplicial complex and let $c \\in C_r(K)$ . If there exists an element $d \\in C_{r + 1}(K)$ such that"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-1e001c6774fe781a79ebe340",
"rev": 1,
"sha256": "e96127f02d183078691f487e58fe0044aae58f82acc8c6f8644aedeb97c57c82"
},
"range": [
0,
43
],
"quote": "\n$$\nc = \\partial_ {r + 1} d \\tag {3.25}\n$$\n"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-a640d82e944756100d483ec3",
"rev": 1,
"sha256": "9f524032ed9338dc18d9290c33b0ff11248a00d91fb508c6721db96dd2a26209"
},
"range": [
0,
244
],
"quote": "then $c$ is called an $r$ -boundary. The set of $r$ -boundaries $B_r(K)$ is a subgroup of $C_r(K)$ and is called the $r$ -boundary group. Note that $B_r(K) = \\operatorname{im} \\partial_{r+1}$ . [Remark: $B_n(K)$ is defined to be 0.]"
}
],
"available_after": 1775,
"units": [
"nakahara-3:u0171"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "Codex curated path completion from immutable v17 atoms; not automatic extraction"
},
{
"id": "statement-ae16e6e7da0a84f365502acf",
"source_kind": "ribosome-atoms",
"scope": {
"book_id": "nakahara-geometry-topology-physics",
"edition_id": "second-edition-local",
"source_id": "nakahara-mineru-bd76b709"
},
"chapter": "nakahara-3",
"chapter_number": 3,
"author_label": "Nakahara 第3章",
"label": "定义3.5：同调群及作者的良定义说明",
"role": "definition",
"summary": "以循环群模边界群定义同调群；作者随后用边界群包含于循环群说明商群良定义，并解释同调类。",
"conditions": "本章整数系数单纯链；链群为自由阿贝尔群。该路径以群、子群和有向单形知识为已具备背景。 可另行采用实数或模二系数；不将不同系数的定义自动等同。",
"symbols": "$H_r(K)=Z_r(K)/B_r(K)$",
"passages": [
{
"atom_ref": {
"kind": "atom",
"id": "a-3d7d62b31f2e0a390d34a6d5",
"rev": 1,
"sha256": "cd02bc175f06dea96a616dbba95fb07dda7251cc4e9f6572863db803a8192017"
},
"range": [
0,
167
],
"quote": "Definition 3.5. Let $K$ be an $n$ -dimensional simplicial complex. The $r$ th homology group $H_{r}(K)$ , $0 \\leq r \\leq n$ , associated with $K$ is defined by"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-793300deddbcb2469783200c",
"rev": 1,
"sha256": "ab37e963d53854dc076b180d01214d168a0c57582fb4111c1d3a85bdc55517f9"
},
"range": [
0,
65
],
"quote": "\n$$\nH _ {r} (K) \\equiv Z _ {r} (K) / B _ {r} (K). \\tag {3.28}\n$$\n"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-dd0378c077b051799af9749d",
"rev": 1,
"sha256": "1cbb65b11f9a10203950b3dbce40bdc8f194e5dbec3592c76a1423ee03d5a1a6"
},
"range": [
0,
163
],
"quote": "[Remarks: If necessary, we define $H_r(K) = 0$ for $r > n$ or $r < 0$ . If we want to stress that the group structure is defined with integer coefficients, we"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-e35cd83c9e78a858b033a18d",
"rev": 1,
"sha256": "41281932963e2e1012aa16ea249b985f9df050d119899e018865129e42e1eb12"
},
"range": [
0,
202
],
"quote": "write $H_{r}(K;\\mathbb{Z})$ . We may also define the homology groups with $\\mathbb{R}$ -coefficients, $H_{r}(K;\\mathbb{R})$ or those with $\\mathbb{Z}_2$ -coefficients, $H_{r}(K;\\mathbb{Z}_{2})$ .]"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-18a7166cf46acfa2be02028e",
"rev": 1,
"sha256": "84dfd468283e92a208efd56b17ddd6560495b09c0e6193cc78b2f457a98c6451"
},
"range": [
0,
145
],
"quote": "Since $B_r(K)$ is a subgroup of $Z_r(K)$ , $H_r(K)$ is well defined. The group $H_r(K)$ is the set of equivalence classes of $r$ -cycles,"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-ca56ecd8cc76231a154454df",
"rev": 1,
"sha256": "f810b8c7c8659a267b500bf33f5f2a1fa2e569b3e6bb5b7430480eb3dc7371f3"
},
"range": [
0,
69
],
"quote": "\n$$\nH _ {r} (K) \\equiv \\{[ z ] | z \\in Z _ {r} (K) \\} \\tag {3.29}\n$$\n"
},
{
"atom_ref": {
"kind": "atom",
"id": "a-8cc17079a61a4c4cf181c1dc",
"rev": 1,
"sha256": "80d78fd3d17b4d9c51af30a370fc20e12186ed813e2171d935949012c04ca373"
},
"range": [
0,
466
],
"quote": "where each equivalence class $[z]$ is called a homology class. Two $r$ -cycles $z$ and $z'$ are in the same equivalence class if and only if $z - z' \\in B_r(K)$ , in which case $z$ is said to be homologous to $z'$ and denoted by $z \\sim z'$ or $[z] = [z']$ . Geometrically $z - z'$ is a boundary of some space. By definition, any boundary $b \\in B_r(K)$ is homologous to 0 since $b - 0 \\in B_r(K)$ . We accept the following theorem without proof."
}
],
"available_after": 1797,
"units": [
"nakahara-3:u0183",
"nakahara-3:u0195"
],
"review_status": "codex-source-reviewed; pending-independent-human-review",
"mathematical_status": "author-stated-unverified",
"origin": "Codex curated path completion from immutable v17 atoms; not automatic extraction",
"extends_statement": "statement-8ae86ef7b666cb37cb3076ea"
},
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"author_label": "Nakahara §4.4.3 · 新单元候选",
"label": "定理 4.9",
"role": "theorem",
"summary": "对连通单纯复形 K，第一同调群 H_1(K) 同构于基本群 π_1(|K|) 模去其换位子群 F。",
"conditions": "K 为连通单纯复形；F 为 π_1(|K|) 的换位子群。",
"symbols": "$H_{1}(K) \\cong \\pi_1(|K|)/F$",
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"quote": "Theorem 4.9. Let $K$ be a connected simplicial complex. Then $H_{1}(K)$ is isomorphic to $\\pi_1(|K|)/F$ , where $F$ is the commutator subgroup (see later) of $\\pi_1(|K|)$ ."
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"translation": "定理 4.9。设 $K$ 为连通单纯复形。则 $H_{1}(K)$ 同构于 $\\pi_1(|K|)/F$，其中 $F$ 是 $\\pi_1(|K|)$ 的换位子群（见后文）。"
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"label": "换位子群定义",
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"summary": "对表示为 (x_i; r_m) 的群 G，其换位子群 F 由形如 x_i x_j x_i^{-1} x_j^{-1} 的元素生成；商群 G/F 由 {x_i} 生成，关系集为 {r_m} 并上所有换位子关系。",
"conditions": "G 有给定表示 (x_i; r_m)。",
"symbols": "$F$, $G/F$, $x_i x_j x_i^{-1} x_j^{-1}$",
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"quote": "Let $G$ be a group whose presentation is $(x_i; r_m)$ . The commutator subgroup $F$ of $G$ is a group generated by the elements of the form $x_i x_j x_i^{-1} x_j^{-1}$ . Thus, $G / F$ is a group generated by $\\{x_i\\}$ with the set of relations $\\{r_m\\}$ and $\\{x_i x_j x_i^{-1} x_j^{-1}\\}$ ."
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"translation": "设 $G$ 是一个表示为 $(x_i; r_m)$ 的群。$G$ 的换位子群 $F$ 是由形如 $x_i x_j x_i^{-1} x_j^{-1}$ 的元素生成的群。因此，$G / F$ 是由 $\\{x_i\\}$ 生成、关系集为 $\\{r_m\\}$ 和 $\\{x_i x_j x_i^{-1} x_j^{-1}\\}$ 的群。该定理表明，若 $\\pi_1(|K|) = (x_i : r_m)$，则 $H_1(K) \\cong (x_i : r_m, x_i x_j x_i^{-1} x_j^{-1})$。例如，由 $\\pi_1(2$-bouquet) $= (x, y : \\emptyset)$，我们得到"
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"label": "定理 4.9 应用于 2-bouquet",
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"summary": "由 π_1(2-bouquet) = (x, y : ∅) 及定理 4.9，得到 H_1(2-bouquet) ≅ (x, y : ∅, xyx^{-1}y^{-1}) ≅ Z ⊕ Z。",
"conditions": "π_1(2-bouquet) 的自由群表示已知。",
"symbols": "$\\pi_1(2$-bouquet)$, $H_{1}(2$-bouquet)$",
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"quote": "The theorem states that if $\\pi_1(|K|) = (x_i : r_m)$ , then $H_1(K) \\cong (x_i : r_m, x_i x_j x_i^{-1} x_j^{-1})$ . For example, from $\\pi_1(2$ -bouquet) $= (x, y : \\emptyset)$ , we find"
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"quote": "which is isomorphic to $H_{1}(2$ -bouquet)."
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"translation": "设 $G$ 是一个表示为 $(x_i; r_m)$ 的群。$G$ 的换位子群 $F$ 是由形如 $x_i x_j x_i^{-1} x_j^{-1}$ 的元素生成的群。因此，$G / F$ 是由 $\\{x_i\\}$ 生成、关系集为 $\\{r_m\\}$ 和 $\\{x_i x_j x_i^{-1} x_j^{-1}\\}$ 的群。该定理表明，若 $\\pi_1(|K|) = (x_i : r_m)$，则 $H_1(K) \\cong (x_i : r_m, x_i x_j x_i^{-1} x_j^{-1})$。例如，由 $\\pi_1(2$-bouquet) $= (x, y : \\emptyset)$，我们得到\n\n它同构于 $H_{1}(2$-bouquet)。"
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"label": "定理 4.9 证明的外部引用",
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"summary": "定理 4.9 的证明见于 Greenberg 和 Harper (1981)，并在 Croom (1978) 中给出概要；本书未提供完整证明。",
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"translation": "定理 4.9 的证明见于 Greenberg 和 Harper (1981)，并在 Croom (1978) 中给出概要。"
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"label": "例 4.11 Klein 瓶",
"role": "use",
"summary": "由 π_1(Klein bottle) ≅ (x, y; xyxy^{-1})，通过添加换位子关系 xyx^{-1}y^{-1}=1 并化简得 H_1(Klein bottle) ≅ Z ⊕ Z_2，其中 Z 由 y 生成，Z_2 由 x 生成。",
"conditions": "Klein 瓶基本群的表示已知。",
"symbols": "$\\pi_1(\\text{Klein bottle})$, $H_1(\\text{Klein bottle})$",
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"quote": "Example 4.11. From $\\pi_1(\\text{Klein bottle}) \\cong (x, y; xyxy^{-1})$ , we have"
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"quote": "Two relations are replaced by $x^{2} = 1$ and $xyx^{-1}y^{-1} = 1$ to yield"
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"quote": "where the factor $\\mathbb{Z}$ is generated by $y$ and $\\mathbb{Z}_2$ by $x$ ."
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"translation": "例 4.11。由 $\\pi_1(\\text{Klein bottle}) \\cong (x, y; xyxy^{-1})$，我们有\n\n两个关系被替换为 $x^{2} = 1$ 和 $xyx^{-1}y^{-1} = 1$，从而得到\n\n其中因子 $\\mathbb{Z}$ 由 $y$ 生成，$\\mathbb{Z}_2$ 由 $x$ 生成。"
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"author_label": "Nakahara §4.4.3 · 新单元候选",
"label": "推论 4.2",
"role": "theorem",
"summary": "对连通拓扑空间 X，π_1(X) 同构于 H_1(X) 当且仅当 π_1(X) 是交换的。特别地，若 π_1(X) 由单个生成元生成，则 π_1(X) 总同构于 H_1(X)。",
"conditions": "X 为连通拓扑空间。",
"symbols": "$\\pi_1(X)$, $H_1(X)$",
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"quote": "Corollary 4.2. Let $X$ be a connected topological space. Then $\\pi_1(X)$ is isomorphic to $H_1(X)$ if and only if $\\pi_1(X)$ is commutative. In particular, if $\\pi_1(X)$ is generated by one generator, $\\pi_1(X)$ is always isomorphic to $H_1(X)$ . [Use theorem 4.9.]"
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"translation": "推论 4.2。设 $X$ 为连通拓扑空间。则 $\\pi_1(X)$ 同构于 $H_1(X)$ 当且仅当 $\\pi_1(X)$ 是交换的。特别地，若 $\\pi_1(X)$ 由一个生成元生成，则 $\\pi_1(X)$ 总是同构于 $H_1(X)$。[使用定理 4.9。]"
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"label": "推论 4.3",
"role": "theorem",
"summary": "若 X 与 Y 具有相同的同伦型，则它们的第一同调群相同：H_1(X) = H_1(Y)。",
"conditions": "X 与 Y 具有相同的同伦型。",
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"quote": "Corollary 4.3. If $X$ and $Y$ are of the same homotopy type, their first homology groups are identical: $H_{1}(X) = H_{1}(Y)$ . [Use theorems 4.9 and 4.3.]"
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"translation": "推论 4.3。若 $X$ 与 $Y$ 具有相同的同伦型，则它们的第一同调群相同：$H_{1}(X) = H_{1}(Y)$。[使用定理 4.9 和 4.3。]"
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"content": "for any $x, y \\in G_1$ . If $f$ is also a bijection, $f$ is called an isomorphism. If there exists an isomorphism $f: G_1 \\to G_2$ , $G_1$ is said to be isomorphic to $G_2$ , denoted by $G_1 \\cong G_2$ . For example, a map $f: \\mathbb{Z} \\to \\mathbb{Z}_2 = \\{0, 1\\}$ defined by",
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"content": "and write $x \\sim y$ . Clearly $\\sim$ is an equivalence relation. The equivalence class to which $x$ belongs is denoted by $[x]$ . Let $G / H$ be the quotient space. The group operation $+$ in $G$ naturally induces the group operation $+$ in $G / H$ by",
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"text": "and write $x \\sim y$ . Clearly $\\sim$ is an equivalence relation. The equivalence class to which $x$ belongs is denoted by $[x]$ . Let $G / H$ be the quotient space. The group operation $+$ in $G$ naturally induces the group operation $+$ in $G / H$ by",
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"content": "Note that $+$ on the LHS is an operation in $G / H$ while $+$ on the RHS is that in $G$ . The operation in $G / H$ should be independent of the choice of representatives. In fact, if $[x'] = [x]$ , $[y'] = [y]$ , then $x - x' = h$ , $y - y' = g$ for some $h, g \\in H$ and we find that",
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"content": "[Remark: The first equality defines the Euler characteristic of a general polyhedron $|K|$ . Note that this is the generalization of the Euler characteristic defined for surfaces in section 2.4.]",
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"content": "The extra minus sign is due to spin-statistics, as will be explained shortly. Contracting all the spinor indices and replacing $p_1^\\mu$ by $p^\\mu + k^\\mu$ and $p_2^\\mu$ by $k^\\mu$ :",
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"content": "which is helicity conserving. So, $u(p_2)$ and $\\bar{u}(p_3)$ should be either both right-handed or both left-handed. This is consistent with a general property of QED, that in the limit of a massless electron, the left- and right-handed states completely decouple.",
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"content": "A Grassmann algebra is a set of objects $\\mathcal{G}$ that are generated by a basis $\\{\\theta_i\\}$ . These $\\theta_i$ are Grassmann numbers, which anticommute with each other, $\\theta_i\\theta_j = -\\theta_j\\theta_i$ , add commutatively, $\\theta_i + \\theta_j = \\theta_j + \\theta_i$ , and can be multiplied by complex numbers, $a\\theta \\in \\mathcal{G}$ for $\\theta \\in \\mathcal{G}$ and $a \\in \\mathbb{C}$ . The algebra must also have an element 0 so that $\\theta_i + 0 = \\theta_i$ .",
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"content": "and so on. Elements of the algebra that have an even number of $\\theta_{i}$ commute with all elements of the algebra, so they compose the even-graded or bosonic subalgebra. Similarly, the odd-graded or fermionic subalgebra has an odd number of $\\theta_{i}$ and anticommutes within itself (but commutes with the bosonic subalgebra). The fermionic subalgebra is not closed, since $\\theta_{1}\\theta_{2}$ is bosonic.",
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"content": "Sometimes it is helpful to compare what we will do with fermions to an example of a Grassmann algebra that you might already be familiar with: the exterior algebra of differential forms. Two forms $A$ and $B$ form a Grassmann algebra with the product usually denoted with a wedge, so that $A \\wedge B = -B \\wedge A$ . So, for example, $\\mathbf{dx}$ and $\\mathbf{dy}$ would generate a two-dimensional Grassmann algebra.",
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"content": "(a) $\\ker f = \\{x|x\\in G_1,f(x) = 0\\}$ is a subgroup of $G_{1}$\n(b) $\\operatorname {im}f = \\{x|x\\in f(G_1)\\subset G_2\\}$ is a subgroup of $G_{2}$",
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"content": "\n$$\nG _ {1} / \\ker f \\cong \\operatorname {i m} f. \\tag {3.6}\n$$\n",
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"content": "Proof. Both sides are groups according to lemma 3.1. Define a map $\\varphi : G_1 / \\ker f \\to \\operatorname{im} f$ by $\\varphi([x]) = f(x)$ . This map is well defined since for $x' \\in [x]$ , there exists $h \\in \\ker f$ such that $x' = x + h$ and $f(x') = f(x + h) = f(x) + f(h) = f(x)$ . Now we show that $\\varphi$ is an isomorphism. First, $\\varphi$ is a homomorphism,",
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"text": "Proof. Both sides are groups according to lemma 3.1. Define a map $\\varphi : G_1 / \\ker f \\to \\operatorname{im} f$ by $\\varphi([x]) = f(x)$ . This map is well defined since for $x' \\in [x]$ , there exists $h \\in \\ker f$ such that $x' = x + h$ and $f(x') = f(x + h) = f(x) + f(h) = f(x)$ . Now we show that $\\varphi$ is an isomorphism. First, $\\varphi$ is a homomorphism,",
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"content": "\n$$\n\\begin{array}{l} \\varphi ([ x ] + [ y ]) = \\varphi ([ x + y ]) = f (x + y) \\\\ = f (x) + f (y) = \\varphi ([ x ]) + \\varphi ([ y ]). \\\\ \\end{array}\n$$\n",
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"content": "Second, $\\varphi$ is one to one: if $\\varphi([x]) = \\varphi([y])$ , then $f(x) = f(y)$ or $f(x) - f(y) = f(x - y) = 0$ . This shows that $x - y \\in \\ker f$ and $[x] = [y]$ . Finally, $\\varphi$ is onto: if $y \\in \\operatorname{im} f$ , there exists $x \\in G_1$ such that $f(x) = y = \\varphi([x])$ .",
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"content": "Example 3.2. Let $f: \\mathbb{Z} \\to \\mathbb{Z}_2$ be defined by $f(2n) = 0$ and $f(2n + 1) = 1$ . Then $\\ker f = 2\\mathbb{Z}$ and $\\operatorname{im} f = \\mathbb{Z}_2$ are groups. Theorem 3.1 states that $\\mathbb{Z}/2\\mathbb{Z} \\cong \\mathbb{Z}_2$ , in agreement with example 3.1.",
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"content": "From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r}(K)$ are subgroups of $C_{r}(K)$ . We now prove an important relation between $Z_{r}(K)$ and $B_{r}(K)$ , which is crucial in the definition of homology groups.",
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"content": "Lemma 3.3. The composite map $\\partial_r\\circ \\partial_{r + 1}:C_{r + 1}(K)\\to C_{r - 1}(K)$ is a zero map; that is, $\\partial_r(\\partial_{r + 1}c) = 0$ for any $c\\in C_{r + 1}(K)$ .",
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"content": "Theorem 3.3. Let $Z_{r}(K)$ and $B_{r}(K)$ be the $r$ -cycle group and the $r$ -boundary group of $C_{r}(K)$ , then",
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"content": "\n$$\nB _ {r} (K) \\subset Z _ {r} (K) \\quad (\\subset C _ {r} (K)). \\tag {3.27}\n$$\n",
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"content": "Proof. This is obvious from lemma 3.3. Any element $c$ of $B_r(K)$ is written as $c = \\partial_{r+1}d$ for some $d \\in C_{r+1}(K)$ . Then we find $\\partial_r c = \\partial_r(\\partial_{r+1}d) = 0$ , that is, $c \\in Z_r(K)$ . This implies $Z_r(K) \\supset B_r(K)$ .",
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"text": "Proof. This is obvious from lemma 3.3. Any element $c$ of $B_r(K)$ is written as $c = \\partial_{r+1}d$ for some $d \\in C_{r+1}(K)$ . Then we find $\\partial_r c = \\partial_r(\\partial_{r+1}d) = 0$ , that is, $c \\in Z_r(K)$ . This implies $Z_r(K) \\supset B_r(K)$ .",
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"content": "\n$$\n\\int_ {- \\infty} ^ {\\infty} d p e ^ {- \\frac {1}{2} a p ^ {2} + J p} = \\sqrt {\\frac {2 \\pi}{a}} e ^ {\\frac {J ^ {2}}{2 a}}. \\tag {14.6}\n$$\n",
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"content": "Now we can use the Gaussian integral in Eq. (14.6), $\\int dp\\exp \\left(-\\frac{1}{2} ap^2 +Jp\\right) = \\sqrt{\\frac{2\\pi}{a}}\\exp \\left(\\frac{J^2}{2a}\\right)$ , with $a = i\\frac{\\delta t}{m}$ and $J = i(x_{j + 1} - x_j)$ to get",
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"text": "Now we can use the Gaussian integral in Eq. (14.6), $\\int dp\\exp \\left(-\\frac{1}{2} ap^2 +Jp\\right) = \\sqrt{\\frac{2\\pi}{a}}\\exp \\left(\\frac{J^2}{2a}\\right)$ , with $a = i\\frac{\\delta t}{m}$ and $J = i(x_{j + 1} - x_j)$ to get",
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"content": "\n$$\n\\langle x _ {j + 1} | e ^ {- i H \\delta t} | x _ {j} \\rangle = N e ^ {- i V (x _ {j}, t _ {j}) \\delta t} e ^ {i \\frac {m}{2} \\delta t \\frac {(x _ {j + 1} - x _ {j}) ^ {2}}{(\\delta t) ^ {2}}} = N e ^ {i L (x, \\dot {x}) \\delta t}, \\tag {14.12}\n$$\n",
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"content": "If instead we only assume $\\tilde{H}(t)$ is a smooth function of $t$ , then we can only solve for the matrix element this way for infinitesimal time intervals. So, let us break this down into $n$ small time intervals $\\delta t$ and define $t_j = t_i + j\\delta t$ and $t_n = t_f$ . Then,",
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"content": "\n$$\n\\langle f | i \\rangle = \\int d x _ {n} \\dots d x _ {1} \\left\\langle x _ {f} \\right| e ^ {- i H \\left(t _ {f}\\right) \\delta t} \\left| x _ {n} \\right\\rangle \\left\\langle x _ {n} \\right| \\dots \\left| x _ {2} \\right\\rangle \\left\\langle x _ {2} \\right| e ^ {- i H \\left(t _ {2}\\right) \\delta t} \\left| x _ {1} \\right\\rangle \\left\\langle x _ {1} \\right| e ^ {- i H \\left(t _ {1}\\right) \\delta t} \\left| x _ {i} \\right\\rangle . \\tag {14.10}\n$$\n",
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"content": "\n$$\n\\langle f | i \\rangle = N ^ {n} \\int d x _ {n} \\dots d x _ {1} e ^ {i L \\left(x _ {n}, \\dot {x} _ {n}\\right) \\delta t} \\dots e ^ {i L \\left(x _ {1}, \\dot {x} _ {1}\\right) \\delta t}. \\tag {14.14}\n$$\n",
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"content": "\n$$\n\\partial_ {r}: C _ {r} (K; \\mathbb {R}) \\to C _ {r - 1} (K; \\mathbb {R})\n$$\n",
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"content": "where $C_{-1}(K; \\mathbb{R})$ is defined to be $\\{0\\}$ . Since both $C_{r - 1}(K; \\mathbb{R})$ and $C_r(K;\\mathbb{R})$ are vector spaces, theorem 2.1 can be applied to yield",
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"content": "\n$$\n\\begin{array}{l} I _ {r} = \\dim C _ {r} (K; \\mathbb {R}) = \\dim (\\ker \\partial_ {r}) + \\dim (\\operatorname {i m} \\partial_ {r}) \\\\ = \\dim Z _ {r} (K; \\mathbb {R}) + \\dim B _ {r - 1} (K; \\mathbb {R}) \\\\ \\end{array}\n$$\n",
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"content": "Definition 3.2. The $r$ -chain group $C_r(K)$ of a simplicial complex $K$ is a free Abelian group generated by the oriented $r$ -simplexes of $K$ . If $r > \\dim K$ , $C_r(K)$ is defined to be 0. An element of $C_r(K)$ is called an $r$ -chain.",
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"content": "Let there be $I_r$ $r$ -simplexes in $K$ . We denote each of them by $\\sigma_{r,i}$ ( $1 \\leq i \\leq I_r$ ). Then $c \\in C_r(K)$ is expressed as",
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"content": "The unit element is $0 = \\sum_{i} 0 \\cdot \\sigma_{r,i}$ , while the inverse element of $c$ is $-c = \\sum_{i} (-c_i) \\sigma_{r,i}$ . [Remark: An oppositely oriented $r$ -simplex $-\\sigma_r$ is identified with $(-1) \\sigma_r \\in C_r(K)$ .] Thus, $C_r(K)$ is a free Abelian group of rank $I_r$ ,",
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"content": "\n$$\n\\partial_ {r} \\sigma_ {r} \\equiv \\sum_ {i = 0} ^ {r} (- 1) ^ {i} \\left(p _ {0} p _ {1} \\dots \\hat {p} _ {i} \\dots p _ {r}\\right) \\tag {3.20}\n$$\n",
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"content": "where the point $p_i$ under $\\hat{\\mathbf{\\mu}}$ is omitted. For example,",
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"content": "$c$ is called an $r$ -cycle. The set of $r$ -cycles $Z_{r}(K)$ is a subgroup of $C_{r}(K)$ and is called the $r$ -cycle group. Note that $Z_{r}(K) = \\ker \\partial_{r}$ . [Remark: If $r = 0$ , $\\partial_0 c$ vanishes identically and $Z_{0}(K) = C_{0}(K)$ , see (3.23).]",
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"text": "$c$ is called an $r$ -cycle. The set of $r$ -cycles $Z_{r}(K)$ is a subgroup of $C_{r}(K)$ and is called the $r$ -cycle group. Note that $Z_{r}(K) = \\ker \\partial_{r}$ . [Remark: If $r = 0$ , $\\partial_0 c$ vanishes identically and $Z_{0}(K) = C_{0}(K)$ , see (3.23).]",
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"content": "Definition 3.4. Let $K$ be an $n$ -dimensional simplicial complex and let $c \\in C_r(K)$ . If there exists an element $d \\in C_{r + 1}(K)$ such that",
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"content": "then $c$ is called an $r$ -boundary. The set of $r$ -boundaries $B_r(K)$ is a subgroup of $C_r(K)$ and is called the $r$ -boundary group. Note that $B_r(K) = \\operatorname{im} \\partial_{r+1}$ . [Remark: $B_n(K)$ is defined to be 0.]",
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"content": "[Remarks: If necessary, we define $H_r(K) = 0$ for $r > n$ or $r < 0$ . If we want to stress that the group structure is defined with integer coefficients, we",
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"content": "write $H_{r}(K;\\mathbb{Z})$ . We may also define the homology groups with $\\mathbb{R}$ -coefficients, $H_{r}(K;\\mathbb{R})$ or those with $\\mathbb{Z}_2$ -coefficients, $H_{r}(K;\\mathbb{Z}_{2})$ .]",
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"content": "Since $B_r(K)$ is a subgroup of $Z_r(K)$ , $H_r(K)$ is well defined. The group $H_r(K)$ is the set of equivalence classes of $r$ -cycles,",
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"content": "\n$$\nH _ {r} (K) \\equiv \\{[ z ] | z \\in Z _ {r} (K) \\} \\tag {3.29}\n$$\n",
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"content": "where each equivalence class $[z]$ is called a homology class. Two $r$ -cycles $z$ and $z'$ are in the same equivalence class if and only if $z - z' \\in B_r(K)$ , in which case $z$ is said to be homologous to $z'$ and denoted by $z \\sim z'$ or $[z] = [z']$ . Geometrically $z - z'$ is a boundary of some space. By definition, any boundary $b \\in B_r(K)$ is homologous to 0 since $b - 0 \\in B_r(K)$ . We accept the following theorem without proof.",
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"text": "where each equivalence class $[z]$ is called a homology class. Two $r$ -cycles $z$ and $z'$ are in the same equivalence class if and only if $z - z' \\in B_r(K)$ , in which case $z$ is said to be homologous to $z'$ and denoted by $z \\sim z'$ or $[z] = [z']$ . Geometrically $z - z'$ is a boundary of some space. By definition, any boundary $b \\in B_r(K)$ is homologous to 0 since $b - 0 \\in B_r(K)$ . We accept the following theorem without proof.",
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"content": "where the factor $\\mathbb{Z}$ is generated by $y$ and $\\mathbb{Z}_2$ by $x$ .",
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"content": "Corollary 4.2. Let $X$ be a connected topological space. Then $\\pi_1(X)$ is isomorphic to $H_1(X)$ if and only if $\\pi_1(X)$ is commutative. In particular, if $\\pi_1(X)$ is generated by one generator, $\\pi_1(X)$ is always isomorphic to $H_1(X)$ . [Use theorem 4.9.]",
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"text": "Corollary 4.2. Let $X$ be a connected topological space. Then $\\pi_1(X)$ is isomorphic to $H_1(X)$ if and only if $\\pi_1(X)$ is commutative. In particular, if $\\pi_1(X)$ is generated by one generator, $\\pi_1(X)$ is always isomorphic to $H_1(X)$ . [Use theorem 4.9.]",
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"content": "which is isomorphic to $H_{1}(2$ -bouquet).",
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"content": "Theorem 4.9. Let $K$ be a connected simplicial complex. Then $H_{1}(K)$ is isomorphic to $\\pi_1(|K|)/F$ , where $F$ is the commutator subgroup (see later) of $\\pi_1(|K|)$ .",
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"content": "The reader might have noticed that there is a certain similarity between the first homology group $H_{1}(K)$ and the fundamental group $\\pi_1(|K|)$ . For example, the fundamental groups of many spaces (circle, disc, $n$ -spheres, torus and many more) are identical to the corresponding first homology group. In some cases, however, they are different: $H_{1}(2$ -bouquet) $\\cong \\mathbb{Z} \\oplus \\mathbb{Z}$ and $\\pi_1(2$ -bouquet) $= (x, y : \\emptyset)$ , for",
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"content": "Let $G$ be a group whose presentation is $(x_i; r_m)$ . The commutator subgroup $F$ of $G$ is a group generated by the elements of the form $x_i x_j x_i^{-1} x_j^{-1}$ . Thus, $G / F$ is a group generated by $\\{x_i\\}$ with the set of relations $\\{r_m\\}$ and $\\{x_i x_j x_i^{-1} x_j^{-1}\\}$ . The theorem states that if $\\pi_1(|K|) = (x_i : r_m)$ , then $H_1(K) \\cong (x_i : r_m, x_i x_j x_i^{-1} x_j^{-1})$ . For example, from $\\pi_1(2$ -bouquet) $= (x, y : \\emptyset)$ , we find",
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"content": "4.4.3 Relations between $H_{1}(K)$ and $\\pi_1(|K|)$",
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"content": "example. Note that $H_{1}(2$ -bouquet) is a free Abelian group while $\\pi_{1}(2$ -bouquet) is a free group. The following theorem relates $\\pi_{1}(|K|)$ to $H_{1}(K)$ .",
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"text": "example. Note that $H_{1}(2$ -bouquet) is a free Abelian group while $\\pi_{1}(2$ -bouquet) is a free group. The following theorem relates $\\pi_{1}(|K|)$ to $H_{1}(K)$ .",
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"content": "Two relations are replaced by $x^{2} = 1$ and $xyx^{-1}y^{-1} = 1$ to yield",
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"content": "Corollary 4.3. If $X$ and $Y$ are of the same homotopy type, their first homology groups are identical: $H_{1}(X) = H_{1}(Y)$ . [Use theorems 4.9 and 4.3.]",
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"text": "Corollary 4.3. If $X$ and $Y$ are of the same homotopy type, their first homology groups are identical: $H_{1}(X) = H_{1}(Y)$ . [Use theorems 4.9 and 4.3.]",
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"content": "\n$$\n\\pi_ {1} (\\text {K l e i n b o t t l e}) / F \\cong (x, y; x y x y ^ {- 1}, x y x ^ {- 1} y ^ {- 1}).\n$$\n",
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"content": "Example 4.11. From $\\pi_1(\\text{Klein bottle}) \\cong (x, y; xyxy^{-1})$ , we have",
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"urn:uuid:02f7a498-5c17-4366-9479-8666d5f9bb2e": {
"kind": "prerequisite",
"explanation": "边界及像需要此前从 r+1 链到 r 链的边界同态。"
}
},
"dependency-e997bfe83e609355c7530f3b": {
"urn:uuid:02f7a498-5c17-4366-9479-8666d5f9bb2e": {
"kind": "prerequisite",
"explanation": "作者将边界线性延拓到链，因此映射的定义域、值域取自定义3.2的链群。"
}
},
"dependency-dc78aa117f41159a20c5afd3": {
"urn:uuid:02f7a498-5c17-4366-9479-8666d5f9bb2e": {
"kind": "prerequisite",
"explanation": "连续取边界的零映射结论使用此前的边界算子；这条边不替代引理3.3的证明。"
}
},
"dependency-581d033a4d3b64de8b484b60": {
"urn:uuid:02f7a498-5c17-4366-9479-8666d5f9bb2e": {
"kind": "prerequisite",
"explanation": "核与像为子群，保证循环群和边界群的群资格。"
},
"urn:uuid:e6dd2937-0598-4aa9-bbf2-317c9e35c0e6": {
"kind": "related-author-argument",
"explanation": "作者明确从引理3.1得到 Zr 与 Br 是链群的子群。 该段是本入口的相关作者论证，不自动成为概念定义的必经前提。"
},
"urn:uuid:3e749de6-9d6f-43ea-a6de-ddff7c549a00": {
"kind": "related-author-argument",
"explanation": "作者明确从引理3.1得到 Zr 与 Br 是链群的子群。 该段是本入口的相关作者论证，不自动成为概念定义的必经前提。"
},
"urn:uuid:34d7ee62-cceb-462f-a9c9-8307ed5c5244": {
"kind": "related-author-argument",
"explanation": "作者明确从引理3.1得到 Zr 与 Br 是链群的子群。 该段是本入口的相关作者论证，不自动成为概念定义的必经前提。"
}
},
"dependency-f504b265a21f6d412d13d8bb": {
"urn:uuid:02f7a498-5c17-4366-9479-8666d5f9bb2e": {
"kind": "prerequisite",
"explanation": "连续两次取边界为零，保证任意边界都是循环。"
},
"urn:uuid:e6dd2937-0598-4aa9-bbf2-317c9e35c0e6": {
"kind": "related-author-argument",
"explanation": "证明明确使用引理3.3。 该段是本入口的相关作者论证，不自动成为概念定义的必经前提。"
}
},
"dependency-0ec8fbcd4724a3ea6d6a737a": {
"urn:uuid:e6dd2937-0598-4aa9-bbf2-317c9e35c0e6": {
"kind": "related-author-argument",
"explanation": "先据引理3.1确定两边为群，再构造并验证同构。 该段是本入口的相关作者论证，不自动成为概念定义的必经前提。"
},
"urn:uuid:3e749de6-9d6f-43ea-a6de-ddff7c549a00": {
"kind": "related-author-argument",
"explanation": "先据引理3.1确定两边为群，再构造并验证同构。 该段是本入口的相关作者论证，不自动成为概念定义的必经前提。"
},
"urn:uuid:34d7ee62-cceb-462f-a9c9-8307ed5c5244": {
"kind": "related-author-argument",
"explanation": "先据引理3.1确定两边为群，再构造并验证同构。 该段是本入口的相关作者论证，不自动成为概念定义的必经前提。"
},
"urn:uuid:02f7a498-5c17-4366-9479-8666d5f9bb2e": {
"kind": "related-author-argument",
"explanation": "先据引理3.1确定两边为群，再构造并验证同构。 该段是本入口的相关作者论证，不自动成为概念定义的必经前提。"
}
},
"dependency-dcd0227cada583b0eb9d75da": {
"urn:uuid:3e749de6-9d6f-43ea-a6de-ddff7c549a00": {
"kind": "related-author-argument",
"explanation": "对整数到二元循环群的映射使用同态基本定理。 该段是本入口的相关作者论证，不自动成为概念定义的必经前提。"
},
"urn:uuid:34d7ee62-cceb-462f-a9c9-8307ed5c5244": {
"kind": "related-author-argument",
"explanation": "对整数到二元循环群的映射使用同态基本定理。 该段是本入口的相关作者论证，不自动成为概念定义的必经前提。"
}
},
"dependency-79d536f23a188cd3c9478bc0": {
"urn:uuid:02f7a498-5c17-4366-9479-8666d5f9bb2e": {
"kind": "related-author-argument",
"explanation": "例 4.11 的计算过程正是定理 4.9 所述构造的具体实施：取基本群模去换位子群。"
}
},
"dependency-368799583a78ce7e483543d8": {
"urn:uuid:02f7a498-5c17-4366-9479-8666d5f9bb2e": {
"kind": "related-author-argument",
"explanation": "作者明确指示使用定理 4.9 来推导此推论。"
}
},
"dependency-faa1666301a7c6eb4731bb1f": {
"urn:uuid:02f7a498-5c17-4366-9479-8666d5f9bb2e": {
"kind": "related-author-argument",
"explanation": "作者明确指示使用定理 4.9 作为推导依据之一。"
}
},
"dependency-d6aa06fe2b4be3c8aa50649d": {
"urn:uuid:02f7a498-5c17-4366-9479-8666d5f9bb2e": {
"kind": "related-author-argument",
"explanation": "作者以 'The theorem states' 回指定理 4.9 的结论来推导 bouquet 的同调群。"
}
}
},
"unmatched": [
{
"statement_id": "statement-c8e4552befe9bed2ee078e3a",
"label": "定理 4.9 证明的外部引用",
"status": "unmatched-in-provided-catalog",
"reason": "关于定理 4.9 证明出处的说明，不匹配已建立的数学概念条目。"
},
{
"statement_id": "statement-da2f78f48f8fed026b0bb222",
"label": "换位子群定义",
"status": "mapping-rejected-pending-identity-review",
"reason": "本次仅提供了“对易子 commutator”入口，不能据此把换位子群当作同一概念；全库未找到独立换位子群身份，保留未匹配并待人工定位。"
}
],
"unresolved_references": [
{
"source_refs": [
{
"atom_id": "a-caac149a628f1dc16c7f0bee",
"quote": "[Use theorems 4.9 and 4.3.]"
}
],
"citation": "theorems 4.9 and 4.3",
"reason": "推论 4.3 引用了定理 4.3，但该定理的内容不在当前源单元中，无法绑定其具体断言。",
"status": "source-not-bound"
}
],
"authority_extensions": [
"library/statements/homology-path.json",
"library/statements/nakahara-4-increment.json",
"library/statements/luna-concept-pilot-v1.json",
"library/statements/revision-10-20261016.json",
"library/statements/calibration-30-20261016.json"
]
}