[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki21\/2022\/04\/27\/%e3%83%95%e3%83%93%e3%83%8b%e3%83%bb%e3%82%b9%e3%82%bf%e3%83%87%e3%82%a3%e8%a8%88%e9%87%8f-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki21\/2022\/04\/27\/%e3%83%95%e3%83%93%e3%83%8b%e3%83%bb%e3%82%b9%e3%82%bf%e3%83%87%e3%82%a3%e8%a8%88%e9%87%8f-wikipedia\/","headline":"\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf – Wikipedia","name":"\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf – 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\u304c\u30a8\u30eb\u30df\u30fc\u30c8\u5f62\u5f0f\u3092\u6301\u3064\u3053\u3068\u3092\u8a00\u3046\u3002\u3053\u306e\u8a08\u91cf\u306f\u3001\u3082\u3068\u3082\u3068\u306f1904\u5e74\u30681905\u5e74\u306b\u30b0\u30a4\u30c9\u30fb\u30d5\u30d3\u30cb\uff08\u82f1\u8a9e\u7248\uff09(Guido Fubini)\u3068\u30a8\u30c9\u30ef\u30fc\u30c9\u30fb\u30b9\u30bf\u30c7\u30a3\uff08\u82f1\u8a9e\u7248\uff09(Eduard Study)\u304c\u8a18\u8ff0\u3057\u305f\u3082\u306e\u3067\u3042\u3063\u305f\u3002 \u30d9\u30af\u30c8\u30eb\u7a7a\u9593 Cn+1 \u306e\u30a8\u30eb\u30df\u30fc\u30c8\u5f62\u5f0f\u306f\u3001GL(n+1,C) \u306e\u4e2d\u306e\u30e6\u30cb\u30bf\u30ea\u90e8\u5206\u7fa4 U(n+1) \u3092\u5b9a\u7fa9\u3059\u308b\u3002\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u306f\u3001U(n+1) \u4f5c\u7528\u306e\u4e0b\u3067\u306e\u4e0d\u5909\u6027\uff08\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u306b\u5bfe\u3057\u3066\uff09\u306b\u3088\u308a\u5dee\u7570\u3092\u540c\u4e00\u8996\u3059\u308b\u3068\u6c7a\u5b9a\u3057\u3001\u7b49\u8cea\u6027\u3092\u6301\u3064\u3002\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u3092\u6301\u3064 CPn \u306f\u3001\uff08\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u3092\u6e21\u308b\uff09\u5bfe\u79f0\u7a7a\u9593\uff08\u82f1\u8a9e\u7248\uff09(symmetric space)\u3067\u3042\u308b\u3002\u7279\u306b\u3001\u8a08\u91cf\u306e\u6b63\u898f\u5316\u306f\u3001\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u306e\u9069\u7528\u306b\u4f9d\u5b58\u3059\u308b\u3002\u30ea\u30fc\u30de\u30f3\u5e7e\u4f55\u5b66\u306b\u304a\u3044\u3066\u306f\u3001\u6b63\u898f\u5316\u3055\u308c\u305f\u8a08\u91cf\u3092\u4f7f\u3046\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\u3001(2n + 1) \u6b21\u5143\u7403\u9762\u4e0a\u306e\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u306f\u3001\u5358\u7d14\u306b\u6a19\u6e96\u306e\u8a08\u91cf\u3068\u95a2\u9023\u4ed8\u3051\u3089\u308c\u308b\u3002\u4ee3\u6570\u5e7e\u4f55\u5b66\u3067\u306f\u3001\u6b63\u898f\u5316\u3092\u4f7f\u3044\u3001CPn","datePublished":"2022-04-27","dateModified":"2022-04-27","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki21\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki21\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/b\/b2\/Blue_question_mark.svg\/30px-Blue_question_mark.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/b\/b2\/Blue_question_mark.svg\/30px-Blue_question_mark.svg.png","height":"30","width":"30"},"url":"https:\/\/wiki.edu.vn\/jp\/wiki21\/2022\/04\/27\/%e3%83%95%e3%83%93%e3%83%8b%e3%83%bb%e3%82%b9%e3%82%bf%e3%83%87%e3%82%a3%e8%a8%88%e9%87%8f-wikipedia\/","about":["Wiki"],"wordCount":12232,"articleBody":"\u539f\u6587\u3068\u6bd4\u3079\u305f\u7d50\u679c\u3001\u3053\u306e\u8a18\u4e8b\u306b\u306f\u591a\u6570\uff08\u5c11\u306a\u304f\u3068\u30825\u500b\u4ee5\u4e0a\uff09\u306e\u8aa4\u8a33\u304c\u3042\u308b\u3053\u3068\u304c\u5224\u660e\u3057\u3066\u3044\u307e\u3059\u3002\u60c5\u5831\u306e\u5229\u7528\u306b\u306f\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u6b63\u78ba\u306a\u8868\u73fe\u306b\u6539\u8a33\u3067\u304d\u308b\u65b9\u3092\u6c42\u3081\u3066\u3044\u307e\u3059\u3002\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf(Fubini\u2013Study metric)\u306f\u3001\u5c04\u5f71\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u4e0a\u306e\u30b1\u30fc\u30e9\u30fc\u8a08\u91cf\u3067\u3042\u308b\u3002\u3064\u307e\u308a\u3001\u8907\u7d20\u5c04\u5f71\u7a7a\u9593 CPn \u304c\u30a8\u30eb\u30df\u30fc\u30c8\u5f62\u5f0f\u3092\u6301\u3064\u3053\u3068\u3092\u8a00\u3046\u3002\u3053\u306e\u8a08\u91cf\u306f\u3001\u3082\u3068\u3082\u3068\u306f1904\u5e74\u30681905\u5e74\u306b\u30b0\u30a4\u30c9\u30fb\u30d5\u30d3\u30cb\uff08\u82f1\u8a9e\u7248\uff09(Guido Fubini)\u3068\u30a8\u30c9\u30ef\u30fc\u30c9\u30fb\u30b9\u30bf\u30c7\u30a3\uff08\u82f1\u8a9e\u7248\uff09(Eduard Study)\u304c\u8a18\u8ff0\u3057\u305f\u3082\u306e\u3067\u3042\u3063\u305f\u3002\u30d9\u30af\u30c8\u30eb\u7a7a\u9593 Cn+1 \u306e\u30a8\u30eb\u30df\u30fc\u30c8\u5f62\u5f0f\u306f\u3001GL(n+1,C) \u306e\u4e2d\u306e\u30e6\u30cb\u30bf\u30ea\u90e8\u5206\u7fa4 U(n+1) \u3092\u5b9a\u7fa9\u3059\u308b\u3002\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u306f\u3001U(n+1) \u4f5c\u7528\u306e\u4e0b\u3067\u306e\u4e0d\u5909\u6027\uff08\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u306b\u5bfe\u3057\u3066\uff09\u306b\u3088\u308a\u5dee\u7570\u3092\u540c\u4e00\u8996\u3059\u308b\u3068\u6c7a\u5b9a\u3057\u3001\u7b49\u8cea\u6027\u3092\u6301\u3064\u3002\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u3092\u6301\u3064 CPn \u306f\u3001\uff08\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u3092\u6e21\u308b\uff09\u5bfe\u79f0\u7a7a\u9593\uff08\u82f1\u8a9e\u7248\uff09(symmetric space)\u3067\u3042\u308b\u3002\u7279\u306b\u3001\u8a08\u91cf\u306e\u6b63\u898f\u5316\u306f\u3001\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u306e\u9069\u7528\u306b\u4f9d\u5b58\u3059\u308b\u3002\u30ea\u30fc\u30de\u30f3\u5e7e\u4f55\u5b66\u306b\u304a\u3044\u3066\u306f\u3001\u6b63\u898f\u5316\u3055\u308c\u305f\u8a08\u91cf\u3092\u4f7f\u3046\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\u3001(2n + 1) \u6b21\u5143\u7403\u9762\u4e0a\u306e\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u306f\u3001\u5358\u7d14\u306b\u6a19\u6e96\u306e\u8a08\u91cf\u3068\u95a2\u9023\u4ed8\u3051\u3089\u308c\u308b\u3002\u4ee3\u6570\u5e7e\u4f55\u5b66\u3067\u306f\u3001\u6b63\u898f\u5316\u3092\u4f7f\u3044\u3001CPn \u3092\u30db\u30c3\u30b8\u591a\u69d8\u4f53\u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u306f\u8907\u7d20\u5c04\u5f71\u7a7a\u9593\u306e\u5546\u7a7a\u9593\u306e\u69cb\u6210\u306e\u4e2d\u3067\u81ea\u7136\u306b\u73fe\u308c\u308b\u3002\u7279\u306b\u3001CPn \u3092 Cn+1 \u306e\u4e2d\u306e\u3059\u3079\u3066\u306e\u8907\u7d20\u76f4\u7dda\u304b\u3089\u306a\u308b\u7a7a\u9593\u3068\u3057\u3066\u3001\u3064\u307e\u308a\u3001\u5404\u3005\u306e\u70b9\u306b\u8907\u7d20\u6570\u3092\u639b\u3051\u308b\u3053\u3068\uff08\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\uff09\u3092\u540c\u4e00\u8996\u3059\u308b\u3053\u3068\u306b\u3088\u308b Cn+1{0} \u306e\u5546\u7a7a\u9593\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u3053\u308c\u306f\u3001\u4e57\u6cd5\u7fa4 C*\u00a0=\u00a0C\u00a0\u00a0{0} \u306e\u5bfe\u89d2\u7684\u306a\u7fa4\u4f5c\u7528\u306b\u3088\u308b\u5546\u3068\u4e00\u81f4\u3059\u308b\u3002CPn={Z=[Z0,Z1,\u2026,Zn]\u2208Cn+1\u2216{0}}\/{Z\u223ccZ,c\u2208C\u2217}.{displaystyle mathbf {CP} ^{n}=left{mathbf {Z} =[Z_{0},Z_{1},ldots ,Z_{n}]in {mathbf {C} }^{n+1}setminus {0},right}\/{mathbf {Z} sim cmathbf {Z} ,cin mathbf {C} ^{*}}.}\u3053\u306e\u5546\u306f\u3001\u57fa\u790e\u7a7a\u9593 CPn \u4e0a\u306e\u8907\u7d20\u30e9\u30a4\u30f3\u30d0\u30f3\u30c9\u30eb\u3068\u3057\u3066 Cn+1{0} \u3068\u3057\u3066\u5b9f\u73fe\u3055\u308c\u308b\u3002\uff08\u5b9f\u969b\u3001\u3053\u306e\u5546\u306f CPn \u4e0a\u306e\u30c8\u30fc\u30c8\u30ed\u30b8\u30fc\u30d0\u30f3\u30c9\u30eb\uff08\u82f1\u8a9e\u7248\uff09(tautological bundle)\u3067\u3042\u308b\u3002\uff09\u3053\u306e\u3088\u3046\u306b\u3057\u3066\u3001CPn \u306f\u30010 \u3067\u306a\u3044\u8907\u7d20\u6570\u306b\u3088\u308b\u30ea\u30b9\u30b1\u30fc\u30eb\u3092 modulo \u3068\u3057\u305f (n + 1)-\u500b\u306e\u7d44 [Z0,…,Zn] \u306e\u540c\u5024\u985e\u3068\u540c\u4e00\u8996\u3055\u308c\u308b\u3002Zi \u3092\u305d\u306e\u70b9\u3067\u306e\u6589\u6b21\u5ea7\u6a19\uff08\u82f1\u8a9e\u7248\uff09(homogeneous coordinates)\u3068\u3044\u3046\u3002\u3055\u3089\u306b\u30012\u3064\u306e\u30b9\u30c6\u30c3\u30d7\u3092\u7d4c\u3066\u3001\u3053\u306e\u5546\u3092\u5f97\u308b\u30020 \u3067\u306a\u3044\u8907\u7d20\u30b9\u30ab\u30e9\u30fc z\u00a0=\u00a0R\u2009ei\u03b8 \u306b\u3088\u308b\u7a4d\u306f\u3001\u4e00\u610f\u7684\u306b\u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3057\u3066\u53cd\u6642\u8a08\u56de\u308a\u306e\u89d2\u5ea6 \u03b8{displaystyle theta } \u306e\u56de\u8ee2\u3092 modulus R \u306b\u3088\u308b\u9045\u308c\u306e\u5408\u6210\u3068\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u3001\u5546 Cn+1\u00a0\u2192\u00a0CPn \u306f\u3001\u6b21\u306e 2\u3064\u306e\u90e8\u5206\u3078\u3068\u5206\u89e3\u3059\u308b\u3002Cn+1\u2216{0}\u27f6(a)S2n+1\u27f6(b)CPn{displaystyle mathbf {C} ^{n+1}setminus {0}{stackrel {(a)}{longrightarrow }}S^{2n+1}{stackrel {(b)}{longrightarrow }}mathbf {CP} ^{n}}\u3053\u3053\u306b step (a) \u306f\u9045\u308c R\u00a0\u2208\u00a0R+\u3001\u3064\u307e\u308a\u3001\u6b63\u306e\u5b9f\u6570\u306b\u3088\u308b\u4e57\u6cd5\u306b\u5bfe\u3059\u308b\u5546 Z\u00a0~\u00a0RZ \u3067\u3042\u308a\u3001step (b) \u306f\u56de\u8ee2 Z\u00a0~\u00a0ei\u03b8Z \u306b\u3088\u308b\u5546\u3067\u3042\u308b\u3002(a) \u3067\u306e\u5546\u306e\u7d50\u679c\u306f\u3001\u65b9\u7a0b\u5f0f |Z|2 = |Z0|2\u00a0+\u00a0…\u00a0+\u00a0|Zn|2\u00a0=\u00a01 \u3067\u5b9a\u7fa9\u3055\u308c\u308b\u5b9f\u8d85\u7403\u9762 S2n+1 \u3067\u3042\u308b\u3002(b) \u306e\u5546\u306f CPn\u00a0=\u00a0S2n+1\/S1 \u304c\u5b9f\u73fe\u3055\u308c\u308b\u3002\u3053\u3053\u306b\u3001S1 \u306f\u56de\u8ee2\u7fa4\u3092\u8868\u73fe\u3059\u308b\u3002\u3053\u306e\u5546\u306f\u3001\u6709\u540d\u306a\u30db\u30c3\u30d7\u30d5\u30a1\u30a4\u30d0\u30fc\u69cb\u9020\uff08\u82f1\u8a9e\u7248\uff09(Hopf fibration) S1\u00a0\u2192\u00a0S2n+1\u00a0\u2192\u00a0CPn \u306b\u3088\u308a\u3001\u660e\u78ba\u306b\u5b9f\u73fe\u3055\u308c\u308b\u3002\u3053\u306e\u30d5\u30a1\u30a4\u30d0\u30fc\u306f S2n+1 \u306e\u5927\u5186\u306e\u4e2d\u306b\u3042\u308b\u3002Table of Contents\u8a08\u91cf\u306e\u5546\u3068\u3057\u3066[\u7de8\u96c6]\u5c40\u6240\u30a2\u30d5\u30a3\u30f3\u5ea7\u6a19\u306e\u4e2d\u3067\u306f[\u7de8\u96c6]\u6589\u6b21\u5ea7\u6a19[\u7de8\u96c6]n = 1 \u306e\u5834\u5408[\u7de8\u96c6]\u66f2\u7387\u306e\u6027\u8cea[\u7de8\u96c6]\u91cf\u5b50\u529b\u5b66\u3067\u306f[\u7de8\u96c6]\u53c2\u7167\u9805\u76ee[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u8a08\u91cf\u306e\u5546\u3068\u3057\u3066[\u7de8\u96c6]\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53\uff08\u3042\u308b\u3044\u306f\u3001\u4e00\u822c\u306b\u8a08\u91cf\u7a7a\u9593\u3067\u3082\u3088\u3044\uff09\u306e\u5546\u3092\u8003\u3048\u308b\u3068\u3001\u5546\u7a7a\u9593\u306f well-defined \u306a\u30ea\u30fc\u30de\u30f3\u8a08\u91cf\u3092\u6301\u3064\u3053\u3068\u3092\u78ba\u8a8d\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002\u305f\u3068\u3048\u3070\u3001\u7fa4 G \u304c\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53 (X,g) \u4e0a\u3078\u4f5c\u7528\u3057\u3066\u3044\u308b\u3068\u3001\u8ecc\u9053\u7a7a\u9593 X\/G \u304c\u8a98\u5c0e\u3055\u308c\u305f\u8a08\u91cf\u3092\u6301\u3064\u305f\u3081\u306b\u306f\u3001g{displaystyle g}   \u304c G-\u8ecc\u9053\u306b\u305d\u3063\u3066\u5b9a\u6570\u3067\u3042\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002\u3053\u306e\u305f\u3081\u306b\u306f\u3001\u4efb\u610f\u306e\u5143 h\u00a0\u2208\u00a0G \u3068\u30d9\u30af\u30c8\u30eb\u5834\u306e\u30da\u30a2 X,Y \u306b\u5bfe\u3057\u3001g(Xh,Yh)\u00a0=\u00a0g(X,Y) \u3067\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u3002Cn+1 \u4e0a\u306e\u6a19\u6e96\u30a8\u30eb\u30df\u30fc\u30c8\u8a08\u91cf\u306f\u3001ds2=dZ\u2297dZ\u00af=dZ0\u2297dZ0\u00af+\u22ef+dZn\u2297dZn\u00af{displaystyle ds^{2}=dmathbf {Z} otimes d{overline {mathbf {Z} }}=dZ_{0}otimes d{overline {Z_{0}}}+cdots +dZ_{n}otimes d{overline {Z_{n}}}}\u306b\u3088\u308a\u6a19\u6e96\u57fa\u5e95\u306e\u4e0a\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u3053\u306e\u30a8\u30eb\u30df\u30fc\u30c8\u8a08\u91cf\u306f\u3001R2n+2 \u4e0a\u306e\u6a19\u6e96\u306e\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u8a08\u91cf\u3068\u3057\u3066\u5b9f\u73fe\u3055\u308c\u308b\u3002\u3053\u306e\u8a08\u91cf\u306f\u3001C* \u4e0a\u306e\u5bfe\u89d2\u4f5c\u7528\u306e\u4e0b\u306b\u4e0d\u5909\u3067\u306f\u306a\u3044\u306e\u3067\u3001\u76f4\u63a5\u3001CPn \u306e\u4e2d\u306e\u5546\u3068\u3057\u3066\u843d\u3068\u3057\u8fbc\u3080\u3053\u3068\u306f\u4e0d\u53ef\u80fd\u3067\u3042\u308b\u3002\u3057\u304b\u3057\u3001\u3053\u306e\u8a08\u91cf\u306f S1\u00a0=\u00a0U(1) \u4e0a\u306e\u56de\u8ee2\u7fa4\u306e\u5bfe\u89d2\u4f5c\u7528\u306e\u4e0b\u3067\u306f\u4e0d\u5909\u3067\u3042\u308b\u306e\u3067\u3001\u4e0a\u306e\u69cb\u6210 step (a) \u304c\u5b8c\u4e86\u308c\u3070 step (b) \u304c\u53ef\u80fd\u3068\u306a\u308b\u3002  \u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf(Fubini\u2013Study metric)\u306f\u3001\u5546 CPn\u00a0=\u00a0S2n+1\/S1 \u4e0a\u306b\u8a98\u5c0e\u3055\u308c\u305f\u8a08\u91cf\u3067\u3042\u308a\u3001\u305d\u3053\u3067\u306f S2n+1{displaystyle S^{2n+1}} \u304c\u6a19\u6e96\u306e\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u8a08\u91cf\u306e\u5358\u4f4d\u8d85\u7403\u9762\u4e0a\u3078\u5236\u9650\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u3044\u308f\u3086\u308b\u300c\u5468\u56f2\u306e\u8a08\u91cf\u300d(round metric)\u3068\u3057\u3066\u4e0e\u3048\u3089\u308c\u308b\u3002\u5c40\u6240\u30a2\u30d5\u30a3\u30f3\u5ea7\u6a19\u306e\u4e2d\u3067\u306f[\u7de8\u96c6]CPn \u306e\u4e2d\u3067\u540c\u6b21\u5ea7\u6a19 (Z0,…,Zn) \u3092\u6301\u3064\u70b9\u306b\u5bfe\u3057\u3066\u3001Z0\u00a0\u2260\u00a00 \u3067\u3042\u308a\u3001\u7279\u306b\u3001zj\u00a0=\u00a0Zj\/Z0 \u3068\u3059\u308b\u3068\u3001\u4e00\u610f\u306b n \u500b\u306e\u5ea7\u6a19\u306e\u7d44 (z1,\u2026,zn) \u304c\u5b58\u5728\u3057\u3001  [Z0,\u2026,Zn]\u223c[1,z1,\u2026,zn],{displaystyle [Z_{0},dots ,Z_{n}]{sim }[1,z_{1},dots ,z_{n}],}\u3068\u306a\u308b\u3002\u3059\u308b\u3068\u3001(z1,\u2026,zn) \u306f\u3001\u5ea7\u6a19\u306e\u8cbc\u308a\u3042\u308f\u305b U0 = {Z0\u00a0\u2260\u00a00} \u3067\u306e CPn \u306e\u30a2\u30d5\u30a3\u30f3\u5ea7\u6a19\u7cfb\uff08\u82f1\u8a9e\u7248\uff09(affine coordinate system)\u3092\u5f62\u6210\u3059\u308b\u3002\u30a2\u30d5\u30a3\u30f3\u5ea7\u6a19\u306f\u3001\u660e\u3089\u304b\u306b\u3001\u4ee3\u308f\u308a\u306b Zi \u3067\u5272\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u4efb\u610f\u306e\u5ea7\u6a19\u7cfb\u3067\u306e\u8cbc\u308a\u5408\u308f\u305b\u3067\u306e Ui\u00a0=\u00a0{Zi\u00a0\u2260\u00a00} \u3068\u3057\u3066\u30a2\u30d5\u30a3\u30f3\u5ea7\u6a19\u7cfb\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002n + 1 \u500b\u306e\u5ea7\u6a19\u306f\u3001CPn \u3092\u8986\u3046\u88ab\u8986 Ui \u3092\u8cbc\u308a\u5408\u308f\u305b\u3001Ui \u4e0a\u306e\u30a2\u30d5\u30a3\u30f3\u5ea7\u6a19 (z1,\u2026,zn) \u306e\u9805\u3068\u3057\u3066\u660e\u78ba\u306b\u8a08\u91cf\u3092\u4e0e\u3048\u308b\u3053\u3068\u304c\u53ef\u80fd\u3068\u306a\u308b\u3002\u3053\u306e\u5ea7\u6a19\u306e\u5fae\u5206\u306f\u3001CPn \u306e\u6b63\u5247\u63a5\u30d0\u30f3\u30c9\u30eb\u306e\u6a19\u69cb {\u22021,\u2026,\u2202n}{displaystyle {partial _{1},ldots ,partial _{n}}} \u3092\u5b9a\u7fa9\u3057\u3001\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u306f\u3001\u30a8\u30eb\u30df\u30fc\u30c8\u6210\u5206hij\u00af=h(\u2202i,\u2202\u00afj)=(1+|z|2)\u03b4ij\u00af\u2212z\u00afizj(1+|z|2)2{displaystyle h_{i{bar {j}}}=h(partial _{i},{bar {partial }}_{j})={frac {(1+|mathbf {z} |^{2})delta _{i{bar {j}}}-{bar {z}}_{i}z_{j}}{(1+|mathbf {z} |^{2})^{2}}}}\u3068\u3057\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u3053\u306b |z|2\u00a0=\u00a0z12+…+zn2 \u3067\u3042\u308b\u3002\u3064\u307e\u308a\u3001\u3053\u306e\u6a19\u69cb\u3067\u306e\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u306e\u30a8\u30eb\u30df\u30fc\u30c8\u884c\u5217\u306f\u3001(hij\u00af)=1(1+|z|2)2[1+|z|2\u2212|z1|2\u2212z\u00af1z2\u22ef\u2212z\u00af1zn\u2212z\u00af2z11+|z|2\u2212|z2|2\u22ef\u2212z\u00af2zn\u22ee\u22ee\u22f1\u22ee\u2212z\u00afnz1\u2212z\u00afnz2\u22ef1+|z|2\u2212|zn|2]{displaystyle {bigl (}h_{i{bar {j}}}{bigr )}={frac {1}{(1+|mathbf {z} |^{2})^{2}}}left[{begin{array}{cccc}1+|mathbf {z} |^{2}-|z_{1}|^{2}&-{bar {z}}_{1}z_{2}&cdots &-{bar {z}}_{1}z_{n}\\-{bar {z}}_{2}z_{1}&1+|mathbf {z} |^{2}-|z_{2}|^{2}&cdots &-{bar {z}}_{2}z_{n}\\vdots &vdots &ddots &vdots \\-{bar {z}}_{n}z_{1}&-{bar {z}}_{n}z_{2}&cdots &1+|mathbf {z} |^{2}-|z_{n}|^{2}end{array}}right]}\u3067\u3042\u308b\u3002\u5404\u3005\u306e\u884c\u5217\u8981\u7d20\u306f\u30e6\u30cb\u30bf\u30ea\u4e0d\u5909\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u3068\u3001\u5bfe\u89d2\u4f5c\u7528 z\u21a6ei\u03b8z{displaystyle mathbf {z} mapsto e^{itheta }mathbf {z} } \u306f\u3053\u306e\u884c\u5217\u3092\u4e0d\u5909\u3068\u3059\u308b\u3002\u6589\u6b21\u5ea7\u6a19[\u7de8\u96c6]\u6589\u6b21\u5ea7\u6a19 Z\u00a0=\u00a0[Z0,…,Zn] \u306b\u3088\u308b\u8868\u73fe\u3082\u53ef\u80fd\u3067\u3042\u308b\u3002\u8868\u73fe\u306e\u610f\u5473\u3092\u3046\u307e\u304f\u89e3\u91c8\u3059\u308b\u3068\u3001ds2=|Z|2|dZ|2\u2212(Z\u00af\u22c5dZ)(Z\u22c5dZ\u00af)|Z|4=Z\u03b1Z\u00af\u03b1dZ\u03b2dZ\u00af\u03b2\u2212Z\u00af\u03b1Z\u03b2dZ\u03b1dZ\u00af\u03b2(Z\u03b1Z\u00af\u03b1)2=2Z[\u03b1dZ\u03b2]Z\u00af[\u03b1dZ\u00af\u03b2](Z\u03b1Z\u00af\u03b1)2.{displaystyle {begin{aligned}ds^{2}&={frac {|mathbf {Z} |^{2}|dmathbf {Z} |^{2}-({bar {mathbf {Z} }}cdot dmathbf {Z} )(mathbf {Z} cdot d{bar {mathbf {Z} }})}{|mathbf {Z} |^{4}}}\\&={frac {Z_{alpha }{bar {Z}}^{alpha }dZ_{beta }d{bar {Z}}^{beta }-{bar {Z}}^{alpha }Z_{beta }dZ_{alpha }d{bar {Z}}^{beta }}{(Z_{alpha }{bar {Z}}^{alpha })^{2}}}\\&={frac {2Z_{[alpha }dZ_{beta ]}{overline {Z}}^{[alpha }{overline {dZ}}^{beta ]}}{left(Z_{alpha }{overline {Z}}^{alpha }right)^{2}}}.end{aligned}}}\u3092\u5f97\u308b\u3002\u3053\u3053\u306b\u548c\u306f\u3001\u30ae\u30ea\u30b7\u30e3\u6587\u5b57\u306e\u30a4\u30f3\u30c7\u30c3\u30af\u30b9 \u03b1 \u03b2 \u304c 0 \u304b\u3089 n \u307e\u3067\u3092\u6e21\u308b\u3088\u3046\u306b\u3068\u308a\u3001\u6700\u5f8c\u306e\u7b49\u5f0f\u306f\u6b21\u306e\u30c6\u30f3\u30bd\u30eb\u7a4d\u306e\u975e\u5bfe\u79f0\u90e8\u5206\u306e\u6a19\u6e96\u8a18\u6cd5\u304c\u4f7f\u308f\u308c\u308b\u3002Z[\u03b1W\u03b2]=12(Z\u03b1W\u03b2\u2212Z\u03b2W\u03b1).{displaystyle Z_{[alpha }W_{beta ]}={frac {1}{2}}left(Z_{alpha }W_{beta }-Z_{beta }W_{alpha }right).}\u3053\u306e ds2 \u306e\u8868\u73fe\u306f\u3001\u5168\u30c8\u30fc\u30c8\u30ed\u30b8\u30fc\u30d0\u30f3\u30c9\u30eb Cn+1\u2216{0}{displaystyle mathbb {C} ^{n+1}backslash {0}} \u306e\u5168\u7a7a\u9593\u4e0a\u306e\u30c6\u30f3\u30bd\u30eb\u3092\u5b9a\u7fa9\u3059\u308b\u3088\u3046\u306b\u4e00\u898b\u3001\u601d\u308f\u308c\u308b\u3002CPn \u306e\u30c8\u30fc\u30c8\u30ed\u30b8\u30fc\u30d0\u30f3\u30c9\u30eb\u306e\u6b63\u5247\u5207\u65ad \u03c3 \u306b\u305d\u3063\u3066\u5f15\u304d\u623b\u3059\u3053\u3068\u306b\u3088\u308a\u3001CPn \u4e0a\u306e\u30c6\u30f3\u30bd\u30eb\u3067\u3042\u308b\u3053\u3068\u304c\u5206\u304b\u308b\u3002\u5f93\u3063\u3066\u3001\u3053\u306e\u5024\u306f\u3001\u5f15\u304d\u623b\u3057\u306e\u5024\u304c\u5207\u65ad\u306e\u9078\u629e\u306b\u72ec\u7acb\u3067\u3042\u308b\u3053\u3068\u304c\u5224\u660e\u3057\u3001\u76f4\u63a5\u3001\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u306e\u8a08\u91cf\u306e\u30b1\u30fc\u30e9\u30fc\u5f62\u5f0f\u306f\u3001\u5168\u4f53\u6e21\u308b\u5b9a\u6570\u6b63\u898f\u5316\u3092\u3001\u03c9=i\u2202\u2202\u00aflog\u2061|Z|2{displaystyle omega =ipartial {overline {partial }}log |mathbf {Z} |^{2}}\u3068\u3059\u308b\u3068\u3001\u6b63\u5247\u5207\u65ad\u306e\u9078\u629e\u3068\u306f\u660e\u3089\u304b\u306b\u72ec\u7acb\u3067\u3042\u308b\u5f15\u304d\u623b\u3057\u3068\u306a\u308b\u3002log|Z|2 \u306e\u5024\u306f\u3001CPn \u306e\u30b1\u30fc\u30e9\u30fc\u30b9\u30ab\u30e9\u30fc\u3067\u3042\u308b\u3002n = 1 \u306e\u5834\u5408[\u7de8\u96c6]n = 1 \u306e\u5834\u5408\u306f\u3001\u7acb\u4f53\u5c04\u5f71\u306b\u3088\u308a\u5fae\u5206\u540c\u76f8 S2\u2245CP1{displaystyle S^{2}cong mathbb {CP} ^{1}} \u304c\u5b58\u5728\u3059\u308b\u3002\u3053\u306e\u540c\u76f8\u306f\u3001\u300c\u7279\u5225\u306a\u300d\u30db\u30c3\u30d7\u30d5\u30a1\u30a4\u30d0\u30fc S1\u00a0\u2192\u00a0S3\u00a0\u2192\u00a0S2 \u3092\u5c0e\u304f\u3002\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u304c CP1 \u4e0a\u306e\u5ea7\u6a19\u3067\u8a18\u8ff0\u3055\u308c\u308b\u3068\u3001\u5b9f\u63a5\u30d0\u30f3\u30c9\u30eb\u3078\u306e\u5236\u9650\u306f\u3001S2 \u4e0a\u306e\u534a\u5f84 1\/2 \uff08\u30ac\u30a6\u30b9\u66f2\u7387\u304c 4 \u3067\u3042\u308b\uff09\u901a\u5e38\u306e\u300c\u5468\u308a\u306e\u300d\uff08\u7403\u9762\u4e0a\u306e\uff09\u8a08\u91cf\u306e\u8868\u73fe\u3068\u306a\u308b\u3002\u3059\u306a\u308f\u3061\u3001z\u00a0=\u00a0x\u00a0+\u00a0iy \u3092\u30ea\u30fc\u30de\u30f3\u7403\u9762 CP1 \u4e0a\u306e\u6a19\u6e96\u7684\u30a2\u30d5\u30a3\u30f3\u5ea7\u6a19\u7cfb\u3068\u3057\u3001x\u00a0=\u00a0r\u2009cos\u03b8, y\u00a0=\u00a0r\u2009sin\u03b8 \u304c C \u4e0a\u306e\u6975\u5ea7\u6a19\u7cfb\u3068\u3059\u308b\u3068\u3001\u56de\u8ee2\u306e\u8a08\u7b97\u306f\u3001ds2=Re\u2061(dz\u2297dz\u00af)(1+|z|2)2=dx2+dy2(1+r2)2=14(d\u03d52+sin2\u2061\u03d5d\u03b82)=14dsus2{displaystyle ds^{2}={frac {operatorname {Re} (dzotimes d{overline {z}})}{left(1+|z|^{2}right)^{2}}}={frac {dx^{2}+dy^{2}}{left(1+r^{2}right)^{2}}}={frac {1}{4}}(dphi ^{2}+sin ^{2}phi ,dtheta ^{2})={frac {1}{4}}ds_{us}^{2}}\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3002\u3053\u3053\u306b\u3001dsus2{displaystyle ds_{us}^{2}} \u306f\u5358\u4f4d 2-\u7403\u9762\u306e\u4e0a\u306e\u56de\u8ee2\u3059\u308b\u8a08\u91cf\u3067\u3042\u308b\u3002\u3053\u3053\u306b \u03c6, \u03b8 \u306f\u6570\u5b66\u3067\u4f7f\u3046\u7acb\u4f53\u5c04\u5f71 r\u2009tan(\u03c6\/2)\u00a0=\u00a01, tan\u03b8\u00a0=\u00a0y\/x \u306b\u3088\u308b S2 \u4e0a\u306e\u7403\u9762\u5ea7\u6a19\u3067\u3042\u308b\uff08\u7269\u7406\u3067\u306f\u3001 \u03c6 \u3068 \u03b8 \u306e\u5f79\u5272\u304c\u5165\u308c\u66ff\u308f\u308b\u3053\u3068\u304c\u591a\u3044\uff09\u3002\u66f2\u7387\u306e\u6027\u8cea[\u7de8\u96c6]n = 1 \u306e\u7279\u5225\u306a\u5834\u5408\u306b\u306f\u3001\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u306f\u30012-\u7403\u9762\u306e\u4e0a\u306e\u8a08\u91cf\u3068\u306e\u540c\u4e00\u6027\u306b\u5f93\u3046\u3068\u30014 \u3067\u3042\u308b\u5b9a\u6570\u306e\u30b9\u30ab\u30e9\u30fc\u66f2\u7387\u3092\u6301\u3064\uff08\u3053\u306e\u3053\u3068\u306f\u4e0e\u3048\u3089\u308c\u305f\u534a\u5f84 R \u306e\u7403\u9762\u306f\u30b9\u30ab\u30e9\u30fc\u66f2\u7387 1\/R2{displaystyle 1\/R^{2}} \u3092\u6301\u3064\uff09\u3002\u3057\u304b\u3057\u3001n > 1 \u306b\u5bfe\u3057\u3066\u306f\u3001\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u306f\u5b9a\u6570\u66f2\u7387\u3092\u6301\u305f\u306a\u3044\u3002\u305d\u306e\u65ad\u9762\u66f2\u7387\u306f\u3001\u4ee3\u308f\u308a\u306b\u3001\u6b21\u306e\u7b49\u5f0f\u3067\u4e0e\u3048\u3089\u308c\u308b[1]\u3002K(\u03c3)=1+3\u27e8JX,Y\u27e92{displaystyle K(sigma )=1+3langle JX,Yrangle ^{2}}\u3053\u3053\u306b\u3001{X,Y}\u2208TpCPn{displaystyle {X,Y}in T_{p}mathbf {CP} ^{n}} \u306f 2-\u5e73\u9762 \u03c3 \u306e\u76f4\u4ea4\u57fa\u5e95\u3067\u3042\u308a\u3001J\u00a0:\u00a0TCPn\u00a0\u2192\u00a0TCPn \u306f CPn \u4e0a\u306e\u7dda\u578b\u8907\u7d20\u69cb\u9020\uff08\u82f1\u8a9e\u7248\uff09(linear complex structure)\u3067\u3042\u308a\u3001\u27e8\u22c5,\u22c5\u27e9{displaystyle langle cdot ,cdot rangle } \u306f\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u3067\u3042\u308b\u3002\u3053\u306e\u516c\u5f0f\u306e\u7d50\u679c\u3001\u65ad\u9762\u66f2\u7387\u306f\u3059\u3079\u3066\u306e 2-\u5e73\u9762 \u03c3{displaystyle sigma } \u306b\u5bfe\u3057 1\u2264K(\u03c3)\u22644{displaystyle 1leq K(sigma )leq 4} \u3092\u6e80\u305f\u3059\u3002\u6700\u5927\u65ad\u9762\u66f2\u7387 (4) \u306f\u6b63\u5247 2-\u5e73\u9762\u3067\u5230\u9054\u3055\u308c\u308b\u3002\u3064\u307e\u308a\u3001\u305d\u3053\u3067\u306f J(\u03c3) \u2282 \u03c3 \u3067\u3042\u308b\u3002\u4e00\u65b9\u3001\u6700\u5c0f\u65ad\u9762\u66f2\u7387 (1) \u306f J(\u03c3) \u304c \u03c3 \u306b\u76f4\u4ea4\u3067\u3042\u308b 2-\u5e73\u9762\u3067\u9054\u6210\u3055\u308c\u308b\u3002\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u304c 4 \u306b\u7b49\u3057\u3044\u300c\u5b9a\u6570\u300d\u6b63\u5247\u65ad\u9762\u66f2\u7387\u3067\u3042\u308b\u3068\u3088\u304f\u8a00\u308f\u308c\u308b\u7406\u7531\u3067\u3042\u308b\u3002\u3053\u306e\u3053\u3068\u306f\u3001CPn \u30921\/4\u30d4\u30f3\u30c1\u591a\u69d8\u4f53\uff08\u82f1\u8a9e\u7248\uff09(quarter pinched manifold)\u3067\u3042\u308b\u3002\u3053\u306e\u512a\u308c\u305f\u5b9a\u7406\u306f\u3001\u53b3\u5bc6\u306a 1\/4 \u3067\u306f\u3089\u308c\u308b\u5358\u9023\u7d50\u306a n \u6b21\u5143\u591a\u69d8\u4f53\u306f\u3001\u7403\u306b\u540c\u76f8\u3067\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3002\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u306f\u3001\u81ea\u5206\u81ea\u8eab\u306e\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb\u306b\u6bd4\u4f8b\u3059\u308b\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u8a08\u91cf\u3067\u3082\u3042\u308b\u3002\u3059\u306a\u308f\u3061\u3001\u5b9a\u6570 \u03bb \u304c\u5b58\u5728\u3057\u3066\u3001\u3059\u3079\u3066\u306e i, j \u306b\u5bfe\u3057\u3001Ricij=\u03bbgij{displaystyle Ric_{ij}=lambda g_{ij}}\u3067\u3042\u308b\u3002\u3053\u306e\u3053\u3068\u306f\u3001\u306a\u306b\u3088\u308a\u3082\u3001\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u304c\u30ea\u30c3\u30c1\u30d5\u30ed\u30fc\u306e\u30b9\u30ab\u30e9\u30fc\u500d\u306b\u5bfe\u3057\u3066\u306f\u4e0d\u5909\u306e\u307e\u307e\u3067\u3042\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002\u307e\u305f\u3001CPn \u306e\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u306f\u3001\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u306e\u5834\u306e\u65b9\u7a0b\u5f0f\u306e\u975e\u81ea\u660e\u306a\u771f\u7a7a\u89e3\u3068\u306a\u3063\u3066\u3044\u308b\u306e\u3067\u3001\u4e00\u822c\u76f8\u5bfe\u8ad6\u306b\u304a\u3044\u3066\u4e0d\u53ef\u6b20\u306a\u3082\u306e\u3068\u306a\u3063\u3066\u3044\u308b\u3002\u91cf\u5b50\u529b\u5b66\u3067\u306f[\u7de8\u96c6]\u91cf\u5b50\u529b\u5b66\u3067\u306f\u3001\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u306f\u3001\u30d3\u30e5\u30fc\u30ec\u30b9\u8a08\u91cf\uff08\u82f1\u8a9e\u7248\uff09(Bures metric)\u3068\u3057\u3066\u3082\u77e5\u3089\u308c\u3066\u3044\u308b[2]\u3002\u3057\u304b\u3057\u306a\u304c\u3089\u3001\u30d3\u30e5\u30fc\u30ec\u30b9\u8a08\u91cf\u306f\u3001\u5178\u578b\u7684\u306b\u306f\u6df7\u5408\u72b6\u614b\u306e\u8a18\u6cd5\u306e\u4e2d\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u4e00\u65b9\u3001\u4ee5\u4e0b\u306b\u793a\u3059\u3053\u3068\u306f\u7d14\u7c8b\u72b6\u614b\u306e\u9805\u3067\u8a18\u8ff0\u3055\u308c\u3066\u3044\u308b\u3002\u8a08\u91cf\u306e\u5b9f\u90e8\u306f\u3001\u30d5\u30a3\u30c3\u30b7\u30e3\u30fc\u60c5\u5831\u8a08\u91cf\uff08\u82f1\u8a9e\u7248\uff09(Fisher information metric)\uff08\u306e 4\u500d\uff09\u3067\u3042\u308b[2]\u3002\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u306f\u3001\u91cf\u5b50\u529b\u5b66\u3067\u5171\u901a\u3057\u3066\u4f7f\u308f\u308c\u3066\u3044\u308b\u30d6\u30e9\u3068\u30b1\u30c3\u30c8\u306e\u8a18\u6cd5\uff08\u82f1\u8a9e\u7248\uff09(bra\u2013ket notation)\u3092\u4f7f\u3044\u66f8\u304f\u3053\u3068\u3082\u3067\u304d\u308b\u3057\u3001\u4ee3\u6570\u5e7e\u4f55\u5b66\u306e\u5c04\u5f71\u591a\u69d8\u4f53\u306e\u8a18\u6cd5\u3092\u4f7f\u3063\u3066\u3082\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u308c\u3089 2\u3064\u306e\u3053\u3068\u3070\u304c\u660e\u3089\u304b\u306b\u540c\u3058\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059\u305f\u3081\u306b\u3001|\u03c8\u27e9=\u2211k=0nZk|ek\u27e9=[Z0:Z1:\u2026:Zn]{displaystyle vert psi rangle =sum _{k=0}^{n}Z_{k}vert e_{k}rangle =[Z_{0}:Z_{1}:ldots :Z_{n}]}\u3068\u3059\u308b\u3002\u3053\u3053\u306b\u3001{|ek\u27e9}{displaystyle {vert e_{k}rangle }} \u306f\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u306e\u76f4\u4ea4\u57fa\u5e95\u30d9\u30af\u30c8\u30eb\u306e\u96c6\u5408\u3067\u3042\u308a\u3001Zk{displaystyle Z_{k}} \u306f\u8907\u7d20\u6570\u3067\u3001Z\u03b1=[Z0:Z1:\u2026:Zn]{displaystyle Z_{alpha }=[Z_{0}:Z_{1}:ldots :Z_{n}]} \u306f\u6589\u6b21\u5ea7\u6a19\uff08\u82f1\u8a9e\u7248\uff09(homogenous coordinates)\u3067\u306e\u5c04\u5f71\u7a7a\u9593 CPn{displaystyle mathbb {C} P^{n}} \u306e\u6a19\u6e96\u7684\u8a18\u6cd5\u3067\u3042\u308b\u3002\u3059\u308b\u3068\u30012\u3064\u306e\u70b9 |\u03c8\u27e9=Z\u03b1{displaystyle vert psi rangle =Z_{alpha }} and |\u03d5\u27e9=W\u03b1{displaystyle vert phi rangle =W_{alpha }} \u304c\u7a7a\u9593\u5185\u306b\u4e0e\u3048\u3089\u308c\u308b\u3068\u3001\u3053\u308c\u3089\u306e\u9593\u306e\u8ddd\u96e2\u306f\u3001\u03b3(\u03c8,\u03d5)=arccos\u2061\u27e8\u03c8|\u03d5\u27e9\u27e8\u03d5|\u03c8\u27e9\u27e8\u03c8|\u03c8\u27e9\u27e8\u03d5|\u03d5\u27e9{displaystyle gamma (psi ,phi )=arccos {sqrt {frac {langle psi vert phi rangle ;langle phi vert psi rangle }{langle psi vert psi rangle ;langle phi vert phi rangle }}}}\u3042\u308b\u3044\u306f\u3001\u540c\u3058\u3053\u3068\u3067\u3042\u308b\u304c\u5c04\u5f71\u591a\u69d8\u4f53\u306e\u8a18\u6cd5\u3067\u306f\u3001\u03b3(\u03c8,\u03d5)=\u03b3(Z,W)=arccos\u2061Z\u03b1W\u00af\u03b1W\u03b2Z\u00af\u03b2Z\u03b1Z\u00af\u03b1W\u03b2W\u00af\u03b2.{displaystyle gamma (psi ,phi )=gamma (Z,W)=arccos {sqrt {frac {Z_{alpha }{overline {W}}^{alpha };W_{beta }{overline {Z}}^{beta }}{Z_{alpha }{overline {Z}}^{alpha };W_{beta }{overline {W}}^{beta }}}}.}\u3067\u3042\u308b\u3002\u3053\u3053\u306b\u3001Z\u00af\u03b1{displaystyle {overline {Z}}^{alpha }} \u306f Z\u03b1{displaystyle Z_{alpha }} \u306e\u8907\u7d20\u5171\u5f79\u3067\u3042\u308b\u3002\u5206\u6bcd\u306b \u27e8\u03c8|\u03c8\u27e9{displaystyle langle psi vert psi rangle } \u304c\u73fe\u308c\u305f\u3053\u3068\u306f\u3001|\u03c8\u27e9{displaystyle vert psi rangle } \u3068\u3001\u540c\u69d8\u306b |\u03d5\u27e9{displaystyle vert phi rangle } \u304c\u5358\u4f4d\u9577\u3078\u6b63\u898f\u5316\u3055\u308c\u3066\u3044\u306a\u3044\u306e\u3067\u6b63\u898f\u5316\u3059\u308b\u305f\u3081\u3067\u3042\u308b\u3002\u3053\u306e\u3088\u3046\u306b\u3001\u6b63\u898f\u5316\u306f\u660e\u78ba\u306b\u306a\u3055\u308c\u308b\u3002\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u3067\u306f\u3001\u8a08\u91cf\u306f 1\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u9593\u306e\u89d2\u5ea6\u3068\u3057\u3066\u3001\u3080\u3057\u308d\u5bb9\u6613\u306b\u89e3\u91c8\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u308c\u304c\u91cf\u5b50\u89d2\u5ea6(quantum angle)\u3068\u547c\u3070\u308c\u308b\u3082\u306e\u3067\u3042\u308b\u3002\u89d2\u5ea6\u306f\u5b9f\u6570\u5024\u3067 0 \u304b\u3089 \u03c0\/2{displaystyle pi \/2} \u307e\u3067\u5909\u5316\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u306e\u8a08\u91cf\u306e\u7121\u9650\u5c0f\u5f62\u5f0f\u306f\u3001\u03d5=\u03c8+\u03b4\u03c8{displaystyle phi =psi +delta psi }\u3001\u3042\u308b\u3044\u306f\u540c\u3058\u3053\u3068\u3067\u3042\u308b\u304c\u3001W\u03b1=Z\u03b1+dZ\u03b1{displaystyle W_{alpha }=Z_{alpha }+dZ_{alpha }} \u3092\u53d6\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u76f4\u3061\u306b\u306a\u3055\u308c\u3001ds2=\u27e8\u03b4\u03c8|\u03b4\u03c8\u27e9\u27e8\u03c8|\u03c8\u27e9\u2212\u27e8\u03b4\u03c8|\u03c8\u27e9\u27e8\u03c8|\u03b4\u03c8\u27e9\u27e8\u03c8|\u03c8\u27e92.{displaystyle ds^{2}={frac {langle delta psi vert delta psi rangle }{langle psi vert psi rangle }}-{frac {langle delta psi vert psi rangle ;langle psi vert delta psi rangle }{{langle psi vert psi rangle }^{2}}}.}\u3092\u5f97\u308b\u3002\u91cf\u5b50\u529b\u5b66\u306e\u8108\u7d61\u3067\u306f\u3001CP1 \u306e\u3053\u3068\u3092\u30d6\u30ed\u30c3\u30db\u7403\u3068\u547c\u3076\u3002\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u306f\u3001\u91cf\u5b50\u529b\u5b66\u306e\u5e7e\u4f55\u5b66\u5316\u3078\u306e\u81ea\u7136\u306a\u8a08\u91cf\u3067\u3042\u308b\u3002\u91cf\u5b50\u30a8\u30f3\u30bf\u30f3\u30b0\u30eb\u30e1\u30f3\u30c8\u3084\u30d9\u30ea\u30fc\u4f4d\u76f8\u306a\u3069\u306e\u91cf\u5b50\u529b\u5b66\u3067\u306e\u7279\u5225\u306a\u632f\u308b\u821e\u3044\u306e\u591a\u304f\u306f\u3001\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u306e\u7279\u5225\u6027\u306b\u5e30\u7740\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u5206\u96e2\u6027\u306e\u5171\u901a\u306e\u8003\u3048\u65b9\u306f\u3001\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf\u306b\u3082\u9069\u7528\u3055\u308c\u308b\u3002\u3055\u3089\u306b\u8a73\u3057\u304f\u306f\u3001\u8a08\u91cf\u304c\u5c04\u5f71\u7a7a\u9593\u306e\u81ea\u7136\u306a\u7a4d\u3001\u30bb\u30b0\u30ec\u57cb\u3081\u8fbc\u307f\uff08\u82f1\u8a9e\u7248\uff09(Segre embedding)\u3067\u5206\u96e2\u7684\u3067\u3042\u308b\u3002\u3059\u306a\u308f\u3061\u3001|\u03c8\u27e9{displaystyle vert psi rangle } \u304c\u5206\u96e2\u7684\u72b6\u614b\uff08\u82f1\u8a9e\u7248\uff09(separable state)[3]\u3067\u3042\u308b\u3068\u304d\u3001\u5f93\u3063\u3066\u3001|\u03c8\u27e9=|\u03c8A\u27e9\u2297|\u03c8B\u27e9{displaystyle vert psi rangle =vert psi _{A}rangle otimes vert psi _{B}rangle } \u3068\u304b\u3051\u308b\u3068\u304d\u306b\u3001\u8a08\u91cf\u306f\u90e8\u5206\u7a7a\u9593\u306e\u8a08\u91cf\u306e\u548c\u3068\u3057\u3066\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002ds2=dsA2+dsB2\u00a0\u00a0.{displaystyle ds^{2}={ds_{A}}^{2}+{ds_{B}}^{2}  .}\u3053\u3053\u306b dsA2{displaystyle {ds_{A}}^{2}} \u3068 dsB2{displaystyle {ds_{B}}^{2}} \u306f\u305d\u308c\u305e\u308c\u90e8\u5206\u7a7a\u9593 A \u3068 B \u4e0a\u306e\u8a08\u91cf\u3068\u3059\u308b\u3002^ Sakai, T. Riemannian Geometry, Translations of Mathematical Monographs No. 149 (1995), American Mathematics Society.^ a b Paolo Facchi, Ravi Kulkarni, V. I. Man’ko, Giuseppe Marmo, E. C. G. Sudarshan, Franco Ventriglia “Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics” (2010), Physics Letters A 374 pp. 4801. DOI: 10.1016\/j.physleta.2010.10.005^ \u30a8\u30f3\u30bf\u30f3\u30b0\u30eb\u30e1\u30f3\u30c8\u3092\u6301\u305f\u306a\u3044\u72b6\u614b\u306e\u3053\u3068\u3092\u3044\u3046\u3002\u53c2\u7167\u9805\u76ee[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]Besse, Arthur L. (1987), Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Berlin, New York: Springer-Verlag, pp.\u00a0xii+510, ISBN\u00a0978-3-540-15279-8\u00a0Brody, D.C.; Hughston, L.P. (2001), \u201cGeometric Quantum Mechanics\u201d, Journal of Geometry and Physics 38:  19\u201353, arXiv:quant-ph\/9906086, Bibcode:\u00a02001JGP….38…19B, doi:10.1016\/S0393-0440(00)00052-8\u00a0Griffiths, P.; Harris, J. (1994), Principles of Algebraic Geometry, Wiley Classics Library, Wiley Interscience, pp.\u00a030\u201331, ISBN\u00a00-471-05059-8\u00a0Onishchik, A.L. (2001), “Fubini\u2013Study metric”,  in Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics, Springer, ISBN\u00a0978-1-55608-010-4\u3002."},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki21\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki21\/2022\/04\/27\/%e3%83%95%e3%83%93%e3%83%8b%e3%83%bb%e3%82%b9%e3%82%bf%e3%83%87%e3%82%a3%e8%a8%88%e9%87%8f-wikipedia\/#breadcrumbitem","name":"\u30d5\u30d3\u30cb\u30fb\u30b9\u30bf\u30c7\u30a3\u8a08\u91cf – Wikipedia"}}]}]