[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki47\/archives\/1032#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2jp\/wiki47\/archives\/1032","headline":"Liei Algebra-\u30a6\u30a3\u30ad\u30da\u30c7\u30a3\u30a2\u3001\u7121\u6599\u200b\u200b\u767e\u79d1\u4e8b\u5178","name":"Liei Algebra-\u30a6\u30a3\u30ad\u30da\u30c7\u30a3\u30a2\u3001\u7121\u6599\u200b\u200b\u767e\u79d1\u4e8b\u5178","description":"before-content-x4 \u4ee3\u6570\u30ea\u30a8\u30b4 – \u3053\u308c\u306f\u3001\u5b9f\u969b\u306e\u6570\u5b57\u307e\u305f\u306f\u7d71\u4e00\u3055\u308c\u305f\u6570\u5b57\u306e\u672c\u4f53\u3068\u540c\u6642\u306b\u4ee3\u6570\u306e\u30d9\u30af\u30c8\u30eb\u7a7a\u9593\u3067\u3042\u308a\u3001\u547c\u3070\u308c\u308b\u8981\u7d20\u306e\u4e57\u7b97\u3092\u5b9a\u7fa9\u3057\u307e\u3057\u305f \u30dc\u30f3\u30cd\u30c3\u30c8\u306b\u3088\u3063\u3066 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\u3002\u540c\u6642\u306b\u3001\u4ee3\u6570\u3068\u305d\u306e\u8868\u73fe\u306e\u9593\u306b\u306f\u660e\u78ba\u306a\u5bfe\u5fdc\u304c\u3042\u308a\u307e\u3059","datePublished":"2022-05-02","dateModified":"2022-05-02","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki47\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2jp\/wiki47\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?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:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2b76fce82a62ed5461908f0dc8f037de4e3686b0","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2b76fce82a62ed5461908f0dc8f037de4e3686b0","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/all2jp\/wiki47\/archives\/1032","wordCount":13641,"articleBody":" 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\u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u4ee3\u6570Heisenberga H3\uff08R\uff09 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] 3\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u4ee3\u6570\u7ffb\u8a33 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u901f\u5ea6\u30b0\u30eb\u30fc\u30d7\u306e\u4ee3\u6570SO\uff083\uff09 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u4ee3\u6570 su\uff08 2 \uff09\uff09 {displaystyle {text {su}}\uff082\uff09} [      \u305d\u308c\u306f   \uff08  2  \uff09\uff09    {displaystyle {text {su}}\uff082\uff09}   \u300c>\u7de8\u96c6 |      \u305d\u308c\u306f   \uff08  2  \uff09\uff09    {displaystyle {text {su}}\uff082\uff09}   \u300c>\u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30d9\u30af\u30c8\u30eb\u7a7a\u9593 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30b9\u30da\u30fc\u30b9\u30d9\u30af\u30c8\u30eb Rn{displaystyle mathbf {r} ^{n}} \u5618\u306e\u30d0\u30e9\u30f3\u30b9\u304c\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u308b\u305f\u3081\u3001\u4efb\u610f\u306e\u30d9\u30af\u30c8\u30eb\u306b\u3064\u3044\u3066\u306f\u3001\u58ca\u308c\u3066\u3044\u307e\u3059\u3002 [ a \u3001 b ] = 0 {displaystyle [a\u3001b] = 0} \u5f7c\u3089\u306f\u5618\u306e\u4ee3\u6570\u3092\u5f62\u6210\u3057\u307e\u3059\u3002 \u8a3c\u62e0 \uff1a\u4e57\u7b97\u3092\u6e80\u305f\u3059\u5404\u6761\u4ef6\u3067\u306f\u3001\u767a\u751f\u3059\u308b\u3059\u3079\u3066\u306e\u62ec\u5f27\u304c\u30bc\u30ed\u3067\u3042\u308b\u305f\u3081\u3001\u30a2\u30a4\u30c7\u30f3\u30c6\u30a3\u30c6\u30a3\u3092\u53d7\u3051\u53d6\u308a\u307e\u3059\u3002 \u6b63\u65b9\u884c\u5217 n \u00d7 n {displaystyle n} \u30de\u30c8\u30ea\u30c3\u30af\u30b9\u6574\u6d41\u5b50\u3067\u3042\u308b\u5618\u306e\u30d0\u30e9\u30f3\u30b9\u3092\u6301\u3064\u5b9f\u969b\u306e\u8981\u7d20\u306b\u3064\u3044\u3066\u3001\u3064\u307e\u308a [\u521d\u3081] [A1,A2]\u2261A1A2\u2212A2A1{displaystyle left [a_ {1}\u3001a_ {2}\u53f3] equiv a_ {1} a_ {2} -a_ {2} a_ {1}} \u30d5\u30eb\u30e9\u30a4\u30f3\u30b0\u30eb\u30fc\u30d7\u306e\u5618\u4ee3\u6570\u3092\u5f62\u6210\u3057\u307e\u3059 g l \uff08 n \u3001 r \uff09\uff09 {displaystyle gl\uff08n\u3001mathbf {r}\uff09}        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u53ef\u9006\u7684\u306a\u554f\u984c\uff08\u3059\u306a\u308f\u3061\u3001\u6c7a\u5b9a\u56e0\u5b50\u306e\u554f\u984c \u2260 0 {displaystyle neq 0} \uff09\uff09 Antihermit Matrices\u5bf8\u6cd5 n \u00d7 n {displaystyle n} \u5618\u306e\u672c\u5f53\u306e\u4ee3\u6570\u3092\u4f5c\u6210\u3057\u307e\u3059 \u306e \uff08 n \uff09\uff09 {displaystyle u\uff08n\uff09} comutator\u306b\u3088\u3063\u3066\u6f0f\u308c\u305f\u5618\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u3067\u3001\u30e6\u30cb\u30bf\u30ea\u30fc\u306e\u30b0\u30eb\u30fc\u30d7\u306e\u305f\u3081\u306b\u672c\u5f53\u306e\u5618\u4ee3\u6570\u3092\u4f5c\u6210\u3057\u307e\u3059 \u306e \uff08 n \uff09\uff09 \u3002 {displaystyle u\uff08n\uff09\u3002} \u518d\u751f [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u4e00\u822c\u7684\u306a\u7dda\u5f62\u4ee3\u6570\u3067 g l \uff08 n \u3001 f \uff09\uff09 {displaystyle gl\uff08n\u3001f\uff09} \u542b\u307e\u308c\u3066\u3044\u307e\u3059 \u7279\u5225  \u7dda\u5f62\u4ee3\u6570 \u5618 s l \uff08 n \u3001 f \uff09\uff09 {displaystyle sl\uff08n\u3001f\uff09} \u30bc\u30ed\u306e\u30c8\u30ec\u30fc\u30b9\u3092\u6301\u3064\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u3067\u69cb\u6210\u3055\u308c\u3066\u3044\u307e\u3059\u3002 \u5b9f\u969b\u306e\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u30b0\u30eb\u30fc\u30d7 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u5404\u30b0\u30eb\u30fc\u30d7 g {displaystyle g} \u5618\u306e\u30ea\u30f3\u30af\u3055\u308c\u305f\u4ee3\u6570\u3092\u5b9a\u7fa9\u3057\u307e\u3059        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4g = l \u79c1 \u305d\u3046\u3067\u3059 \uff08 g \uff09\uff09 \u3002 {displaystyle g = lie\uff08g\uff09\u3002} \u5168\u4f53\u7684\u306a\u4f9d\u5b58\u6027\u306f\u3084\u3084\u8907\u96d1\u3067\u3059\u304c\u3001\u5b9f\u969b\u306e \/\u63d0\u51fa\u3055\u308c\u305f\u4e8b\u9805\u306e\u5834\u5408\u3001\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306e\u51fa\u5c55\u8005\uff1a\u4ee3\u6570\u5618\u306b\u3088\u3063\u3066\u7b56\u5b9a\u3067\u304d\u307e\u3059 g {displaystyle g} \u305d\u308c\u3089\u306f\u3053\u308c\u3089\u306e\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u3092\u5f62\u6210\u3057\u307e\u3059 \u30d0\u30c4 \u3001 {displaystyle x\u3001} \u3069\u308c\u306e exp \u2061 \uff08 t \u30d0\u30c4 \uff09\uff09 {displaystyle exp\uff08TX\uff09} \u30b0\u30eb\u30fc\u30d7\u306b\u5c5e\u3059\u308b\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u3067\u3059        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4g {displaystyle g} \u3059\u3079\u3066\u306e\u6570\u5b57\u306b\u3064\u3044\u3066 t {displaystylet} \u30ea\u30a2\u30eb \/\u30b3\u30f3\u30d7\u30ec\u30c3\u30af\u30b9\u3002 \u30bc\u30ed\u306b\u7b49\u3057\u3044\u30d6\u30e9\u30b1\u30c3\u30c8 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u3059\u3079\u3066\u306e\u8981\u7d20\u306e\u30d0\u30e9\u30f3\u30b9\u3092\u30bc\u30ed\u306b\u7b49\u3057\u3044\u3068\u5b9a\u7fa9\u3059\u308b\u30d9\u30af\u30c8\u30eb\u7a7a\u9593\u3001\u3064\u307e\u308a [ a \u3001 b ] = 0 {displaystyle [a\u3001b] = 0} \u5f7c\u306f\u5618\u306e\u4ee3\u6570\u3067\u3059\u3002\u305d\u306e\u3088\u3046\u306a\u4ee3\u6570\u306e\u5618\u306f\u305d\u3046\u3067\u3059 \u4ea4\u4e92\uff08\u30a2\u30d9\u30ed\u30ef\uff09 \u3002 \u30d9\u30af\u30c8\u30eb\u88fd\u54c1 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30d9\u30af\u30c8\u30eb\u7a7a\u9593\u306e\u30dc\u30f3\u30cd\u30c3\u30c8\u3068\u3057\u3066 r 3 {displaystyle mathbb {r} ^{3}} \u30d9\u30af\u30c8\u30eb\u8981\u7d20\u306e\u7a4d\u3001\u3064\u307e\u308a [ a \u3001 b ] = a \u00d7 b \u3002 {displaystyle [a\u3001b] = atimes b\u3002} \u30d9\u30af\u30bf\u30fc\u88fd\u54c1\u304c\u5618\u62ec\u5f27\u306e\u5b9a\u7fa9\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3057\u3066\u3044\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3059\u308b\u306e\u306f\u7c21\u5358\u3067\u3059\u3002 \u6574\u6d41\u5b50 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u5618\u306e\u4ee3\u6570\u306f\u3001\u63a5\u7d9a\u4ee3\u6570\u3068\u3001\u5618\u306e\u30d6\u30e9\u30b1\u30c3\u30c8\u304c\u6574\u6d41\u5b50\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u308b\u3082\u306e\u3001\u3064\u307e\u308a [ a \u3001 b ] = a b  –  b a \u3002 {displaystyle [a\u3001b] = ab-ba\u3002} \u6574\u6d41\u5b50\u306f\u3001\u5618\u306e\u62ec\u5f27\u306e\u5b9a\u7fa9\u306e\u305f\u3081\u306e\u3059\u3079\u3066\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3057\u3066\u3044\u307e\u3059\u3002 \u5b9f\u969b\u306e\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u30b0\u30eb\u30fc\u30d7\u306e\u4ee3\u6570 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] Matrix\u30b0\u30eb\u30fc\u30d7\u3067\u6307\u5b9a\u3055\u308c\u305fLii\u4ee3\u6570\u306e\u5834\u5408\u3001\u663c\u5bdd [ A1\u3001 A2] = a \u521d\u3081 a 2  –  a 2 a \u521d\u3081 \u3002 {displaystyle left [a_ {1}\u3001a_ {2}\u53f3] = a_ {1} a_ {2} -a_ {2} a_ {1}\u3002} \u6574\u6d41\u5b50\u306f\u5bf8\u6cd5n\u306e\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u3067\u3082\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044\u3002 \u4ee3\u6570\u3092\u5f62\u6210\u3059\u308b\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u30b0\u30eb\u30fc\u30d7\uff1a 1\uff09\u4ee3\u6570 l\uff08 n \u3001 c \uff09\uff09 {displaystyle {mathfrak {l}}\uff08n\u3001mathbb {c}\uff09}  – \u5bf8\u6cd5\u306e\u3059\u3079\u3066\u306e\u6b63\u65b9\u884c\u5217\u3092\u53ce\u7a6b\u3057\u307e\u3059 n \u00d7 n {displaystyle n} \u8907\u96d1\u306a\u8981\u7d20\u306b\u3064\u3044\u3066\u3001 2\uff09\u4ee3\u6570 sl\uff08 n \u3001 c \uff09\uff09 {displaystyle {mathfrak {sl}}\uff08n\u3001mathbb {c}\uff09}  – \u30bc\u30ed\u306b\u7b49\u3057\u3044\u30c8\u30ec\u30a4\u30eb\u3092\u5099\u3048\u305f\u8907\u96d1\u306a\u554f\u984c\u306e\u30bb\u30c3\u30c8\u3002\u30dd\u30c0\u30eb\u30e9\u4ee3\u8868 l\uff08 n \u3001 c \uff09\uff09 \u3001 {displaystyle {mathfrak {l}}\uff08n\u3001mathbb {c}\uff09\u3001} 3\uff09\u4ee3\u6570 u\uff08 n \u3001 c \uff09\uff09 {displaystyle {mathfrak {u}}\uff08n\u3001mathbb {c}\uff09} -anthermit Matrix\u30b3\u30ec\u30af\u30b7\u30e7\u30f3\u3002\u30dd\u30c0\u30eb\u30e9\u4ee3\u8868 l\uff08 n \u3001 c \uff09\uff09 \u3001 {displaystyle {mathfrak {l}}\uff08n\u3001mathbb {c}\uff09\u3001} 4\uff09\u4ee3\u6570 su\uff08 n \u3001 c \uff09\uff09 {displaystyle {mathfrak {su}}\uff08n\u3001mathbb {c}\uff09} -Podalgebra Algebry l\uff08 n \u3001 c \uff09\uff09 \u3001 {displaystyle {mathfrak {l}}\uff08n\u3001mathbb {c}\uff09\u3001} \u3053\u308c\u306f\u4e0a\u8a18\u306e\u4ea4\u5dee\u70b9\u3067\u3059\u3001 5\uff09\u4ee3\u6570 so\uff08 n \u3001 r \uff09\uff09 {displaystyle {mathfrak {so}}\uff08n\u3001mathbb {r}\uff09}  – \u6297\u30b8\u30e1\u30c8\u30ea\u30c3\u30af\u306a\u56db\u89d2\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u5bf8\u6cd5\u306e\u4ee3\u6570 n {displaystyle n} \u5b9f\u969b\u306e\u8981\u7d20\u3001\u7279\u306b\u6297\u4f53\u6e2c\u5b9a\u306f\u3001\u3053\u308c\u3089\u306e\u554f\u984c\u306e\u75d5\u8de1\u304c\u30bc\u30ed\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u307e\u3059\u3002 \u4ee3\u6570\u30b8\u30a7\u30cd\u30ec\u30fc\u30bf\u30fc\u3068\u305d\u306e\u5bf8\u6cd5\u3002\u6c38\u4e45\u69cb\u9020 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \uff08 \u521d\u3081 \uff09\uff09 Generatorami Algebry \u5618\u306f\u3001\u76f4\u7dda\u7684\u306b\u72ec\u7acb\u3057\u305f\u8981\u7d20\u306e\u30e9\u30a4\u30f3\u3068\u547c\u3070\u308c\u307e\u3059 g a \u3001 a = \u521d\u3081 \u3001 2 \u3001 … \u3001 n \u3001 {displaystyle g_ {a}\u3001a = 1,2\u3001dots\u3001n\u3001} \u3059\u3079\u3066\u306e\u8981\u7d20\u306e\u3088\u3046\u306b \u30d0\u30c4 {displaystyle x} \u30a2\u30eb\u30b2\u30d6\u30ea\u30fc\u306f\u3001\u767a\u96fb\u6a5f\u306e\u7dda\u5f62\u7d50\u5408\u3001\u3064\u307e\u308a \u30d0\u30c4 = \u2211 a = \u521d\u3081 n \u30d0\u30c4 a g a \u3002 {displaystyle x = sum _ {a = 1}^{n} x^{a} g_ {a}\u3002} \u767a\u96fb\u6a5f\u306f\u7dda\u5f62\u7a7a\u9593\u30d9\u30fc\u30b9\u3092\u4f5c\u6210\u3057\u307e\u3059\u3002\u30b8\u30a7\u30cd\u30ec\u30fc\u30bf\u30fc\u306f\u3001\u5c55\u793a\u4f1a\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u5618\u306e\u4ee3\u6570\u306b\u95a2\u9023\u3059\u308b\u5618\u30b0\u30eb\u30fc\u30d7\u306e\u4efb\u610f\u306e\u8981\u7d20\u3092\u4f5c\u6210\u3059\u308b\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002 \u305d\u3046\u3067\u3059 ix= \u305d\u3046\u3067\u3059 i\u2211a=1nxaGa{displaystyle e^{ix} = e^{isum _ {a = 1}^{n} x^{a} g_ {a}}}} \uff08 2 \uff09\uff09 \u5bf8\u6cd5 Liei\u4ee3\u6570\u306f\u3001\u76f4\u7dda\u7684\u306b\u72ec\u7acb\u3057\u305f\u767a\u96fb\u6a5f\u306e\u6700\u5927\u6570\u306b\u7b49\u3057\u304f\u306a\u308a\u307e\u3059\u3002 \uff08 3 \uff09\u767a\u96fb\u6a5f\u306e\u30bb\u30c3\u30c8\u306f\u7279\u5fb4\u3065\u3051\u3089\u308c\u307e\u3059 \u5f57\u661f\u306e\u6761\u4ef6\u3001 \u3001\u3064\u307e\u308a\u3001\u4efb\u610f\u306e2\u3064\u306e\u30b8\u30a7\u30cd\u30ec\u30fc\u30bf\u30fc\u306e\u6574\u6d41\u5b50\u306f\u3001\u3059\u3079\u3066\u306e\u767a\u96fb\u6a5f\u306e\u7dda\u5f62\u306e\u7d44\u307f\u5408\u308f\u305b\u3067\u3059 [ g a\u3001 g b] = \u2211 c=1nf abcg c\u3001 a \u3001 b = \u521d\u3081 \u3001 2 \u3001 … \u3001 n {displaystyle [g_ {a}\u3001g_ {b}] = sum _ {c = 1}^{n} f_ {abc}\u3001g_ {c}\u3001quad a\u3001b = 1,2\u3001dots\u3001n} \u4fc2\u6570 f \u79c1 j k {displaystyle f_ {ijk}} \u5f7c\u3089\u306f\u6570\u5b57\u3067\u3059 – \u5f7c\u3089\u306f\u547c\u3070\u308c\u3066\u3044\u307e\u3059 \u6c38\u4e45\u69cb\u9020 \u30ea\u30fc\u306e\u4ee3\u6570\u3002\u6b21\u5143\u306e\u4ee3\u6570 n {displaystyle n} \u3068 n 3 {displaystyle n^{3}} \u6c38\u4e45\u69cb\u9020\u3002 \uff08 4 \uff09\u767a\u96fb\u6a5f\u306e\u9078\u629e\u306f\u4e00\u610f\u3067\u306f\u3042\u308a\u307e\u305b\u3093\uff08\u30d9\u30af\u30c8\u30eb\u7a7a\u9593\u30d9\u30fc\u30b9\u306e\u9078\u629e\u3068\u540c\u3058\u3067\u3059\uff09\u3002\u7279\u5b9a\u306e\u4ee3\u6570\u306e\u72ec\u81ea\u6027\u306f\u3001\u6c38\u4e45\u69cb\u9020\u306b\u3088\u3063\u3066\u660e\u78ba\u306b\u7279\u5fb4\u4ed8\u3051\u3089\u308c\u307e\u3059\u3002 \uff08 5 \uff09\u3059\u3079\u3066\u306e\u6574\u6d41\u5b50\u304c\u30bc\u30ed\u306e\u5834\u5408\u3001\u4ee3\u6570\u306f\u30b0\u30eb\u30fc\u30d7\u3067\u3059 \u30a2\u30d9\u30ed\u30ef \uff08 \u4ea4\u4e92\uff09 \u3002 \u4ee3\u6570\u306e\u5618\u3068\u305d\u306e\u767a\u96fb\u6a5f\u306e\u4f8b [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u4ee3\u6570Heisenberga H3\uff08R\uff09 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \uff08a\uff09\u30b8\u30a7\u30cd\u30ec\u30fc\u30bf\u30fc\u3092\u5099\u3048\u305f3\u6b21\u5143\u306eliei algera\u3067\u3059 \u30d0\u30c4 \u3001 \u3068 \u3001 \u3068 {displaystyle x\u3001y\u3001z} \u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u308b\u5618\u306e\u62ec\u5f27\uff1a [ \u30d0\u30c4 \u3001 \u3068 ] = \u3068 \u3001 [ \u30d0\u30c4 \u3001 \u3068 ] = 0 \u3001 [ \u3068 \u3001 \u3068 ] = 0\u3002 {displaystyle [x\u3001y] = z\u3001quad [x\u3001z] = 0\u3001quad [y\u3001z] = 0.} \uff08b\uff093\u00d73\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u7a7a\u9593\u3067\u306f\u3001\u4ee3\u6570\u304c\u3053\u308c\u3092\u8868\u3057\u3066\u3044\u307e\u3059 \u30a2\u30c3\u30d1\u30fc\u30af\u30e9\u30f3\u30d4\u30fc\u30de\u30de\u30fc \uff1a \u30d0\u30c4 = \uff08 010000000\uff09\uff09 \u3001 \u3068 = \uff08 000001000\uff09\uff09 \u3001 \u3068 = \uff08 001000000\uff09\uff09 \u3002 {displaystyle x = left\uff08{begin {array} {ccc} 0\uff061\uff060 \\ 0\uff060\uff060\uff060\uff060end {array}}\u53f3\uff09\u3001quad y = left\uff08{begin {array} {ccc} 0\uff060\uff060\uff060 \\ 0\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060\uff060 {ccc} 0\uff060\uff061 \\ 0\uff060\uff060\uff060 \\ 0\uff060\uff060end {array}}\u53f3\uff09\u3002}} \u305d\u3057\u3066\u3001\u5618\u306e\u62ec\u5f27\u306f\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u6574\u6d41\u5b50\u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002 \u4ee3\u6570\u306e\u3053\u306e\u8868\u73fe\u306b\u5bfe\u5fdc\u3059\u308b\u5618\u30b0\u30eb\u30fc\u30d7\u306e\u8981\u7d20\u5618 \uff08 1ac01b001\uff09\uff09 {displaystyle left\uff08{begin {array} {ccc} 1\uff06a\uff06c \\ 0\uff061\uff06b \\ 0\uff060\uff061end {array}}}}  – \u3053\u306e\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306f\u3001\u30b8\u30a7\u30cd\u30ec\u30fc\u30bf\u30fc\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306e\u5c55\u793a\u3092\u639b\u3051\u308b\u3053\u3068\u3067\u53d6\u5f97\u3067\u304d\u307e\u3059 a \u30d0\u30c4 \u3001 {displaystyle ax\u3001}  b \u3068 {displaystyle by} \u3068 c \u3068 \u3001 {disspastyle cz\u3001} TJ\u3002 \uff08 1ac01b001\uff09\uff09 = \u305d\u3046\u3067\u3059 by\u305d\u3046\u3067\u3059 cz\u305d\u3046\u3067\u3059 ax{displaystyle left\uff08{begin {array} {ccc} 1\uff06a\uff06c \\ 0\uff061\uff06b \\ 0\uff060\uff061end {array}}\u53f3\uff09= e^{by} e^{cz} e^{ax}}}} \u6ce8\u610f\uff1a \u3053\u3053\u3067\u306f\u3001\u5c55\u793a\u7269\u306e\u4e57\u7b97\u306e\u9806\u5e8f\u304c\u91cd\u8981\u3067\u3059 \u305d\u3046\u3067\u3059 b \u3068 \u3001 \u305d\u3046\u3067\u3059 c \u3068 \u3001 \u305d\u3046\u3067\u3059 a \u30d0\u30c4 \u3001 {displaystyle e^{by}\u3001e^{cz}\u3001e^{ax}\u3001} \u5f7c\u3089\u306f\u4e00\u822c\u7684\u306b\u5909\u5316\u3057\u3066\u3044\u306a\u3044\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u3092\u63d0\u793a\u3059\u308b\u305f\u3081\u3067\u3059\u3002 3\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u4ee3\u6570\u7ffb\u8a33 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] 3\u6b21\u5143\u30b9\u30da\u30fc\u30b9\u306e\u7ffb\u8a33\u30b0\u30eb\u30fc\u30d7\u306b\u306f3\u3064\u306e\u30b8\u30a7\u30cd\u30ec\u30fc\u30bf\u30fc\u304c\u3042\u308a\u307e\u3059 \u30d0\u30c4 \u3001 \u3068 \u3001 \u3068 \u3001 {displaystyle x\u3001y\u3001z\u3001} \u8ef8\u306b\u5411\u304b\u3063\u3066\u9069\u5207\u306b\u7ffb\u8a33\u3092\u751f\u6210\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059 o \u30d0\u30c4 \u3001 {displaystyle ox\u3001}  o \u3068 {displaystyle oy} \u79c1 o \u3068 \u3002 {displaystyle oz\u3002} \u3053\u308c\u3089\u306e\u30b8\u30a7\u30cd\u30ec\u30fc\u30bf\u30fc\u306f\u3001\u6574\u6d41\u5b50\u3068\u4e00\u7dd2\u306b\u5618\u4ee3\u6570\u3092\u5f62\u6210\u3057\u307e\u3059\u3002 [ \u30d0\u30c4 \u3001 \u3068 ] = [ \u3068 \u3001 \u3068 ] = [ \u3068 \u3001 \u30d0\u30c4 ] = 0\u3002 {displaystyle [x\u3001y] = [y\u3001z] = [z\u3001x] = 0.} \u3057\u305f\u304c\u3063\u3066\u3001\u3053\u308c\u306f\u5909\u66f4\u3067\u3059\u3002 \u901f\u5ea6\u30b0\u30eb\u30fc\u30d7\u306e\u4ee3\u6570SO\uff083\uff09 [ \u7de8\u96c6 | \u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \uff08 \u521d\u3081 \uff09\u56de\u8ee2\u30b0\u30eb\u30fc\u30d7 s o \uff08 3 \uff09\uff09 {displaystyle so\uff083\uff09} 3\u3064\u306e\u6b21\u5143\u7a7a\u9593\u306b3\u3064\u306e\u767a\u96fb\u6a5f\u304c\u3042\u308a\u307e\u3059 t \u521d\u3081 \u3001 {displaystyle t^{1}\u3001}  t 2 \u3001 {displaystyle t^{2}\u3001}  t 3 \u3001 {displaystyle t^{3}\u3001} \u3053\u308c\u306b\u3088\u308a\u3001\u8ef8\u306e\u5468\u308a\u306e\u56de\u8ee2\u3092\u751f\u6210\u3067\u304d\u307e\u3059 o \u30d0\u30c4 \u3001 {displaystyle ox\u3001}  o \u3068 {displaystyle oy} \u79c1 o \u3068 \u3001 {displaystyle oz\u3001} TJ\u3002 t 1= [00000\u2212i0i0]\u3001  t 2= [00i000\u2212i00]\u3001  t 3= [0\u2212i0i00000]\u3002 {displaystyle t^{1} = {begin {bmatrix} 0\uff060\uff060 \\ 0\uff060\uff06-i \\ 0\uff06i\uff060end {bmatrix}}\u3001t^{2} = {begin {bmatrix} 0\uff060\uff060\uff06i \\ 0\uff060\uff060 \\ -i\uff060\uff060\uff060\uff060END}} {bmatrix} 0\uff06-i\uff060 \\ i\uff060\uff060 \\ 0\uff060\uff060end {bmatrix}}}} \u3053\u308c\u3089\u306e\u30b8\u30a7\u30cd\u30ec\u30fc\u30bf\u30fc\u306f\u3001\u4ee3\u6570\u5618\u306e\u30c7\u30fc\u30bf\u30d9\u30fc\u30b9\u3092\u5f62\u6210\u3057\u307e\u3059 s o \uff08 3 \uff09\uff09 \u3002 {displaystyle so\uff083\uff09\u3002}  \u767a\u96fb\u6a5f\u306e\u6574\u6d41\u5b50\u306b\u306f\u5024\u304c\u3042\u308a\u307e\u3059 [ t 1\u3001 t 2] = \u79c1 t 3\u3001 {displaystyle [t^{1}\u3001t^{2}] = it^{3}\u3001} [ t 2\u3001 t 3] = \u79c1 t 1\u3001 {displaystyle [t^{2}\u3001t^{3}] = it^{1}\u3001} [ t 3\u3001 t 1] = \u79c1 t 2\u3001 {displaystyle [t^{3}\u3001t^{1}] = it^{2}\u3001} TJ\u3002 [ t a\u3001 t b] = \u79c1 \u2211 c=13\u03f5 abct c\u3001 a \u3001 b = \u521d\u3081 \u3001 2 \u3001 3 {displaystyle [t^{a}\u3001t^{b}] = isum _ {c = 1}^{3} epsilon _ {abc}\u3001t^{c}\u3001quad a\u3001b = 1,2,3} \u3057\u305f\u304c\u3063\u3066\u3001\u4ee3\u6570\u306e\u6c38\u7d9a\u7684\u306a\u69cb\u9020\u306b\u7d9a\u304d\u307e\u3059 s o \uff08 3 \uff09\uff09 {displaystyle so\uff083\uff09} \u305d\u308c\u3089\u306f\u3001\u30ec\u30f4\u30a3\u30fc\u306e\u8c61\u5fb4\u306e\u30b7\u30f3\u30dc\u30eb\u306b\u3088\u3063\u3066\u6c7a\u5b9a\u3055\u308c\u307e\u3059 f abc\u559c\u3093\u3067 \u03f5 abc\u3001 a b c = \u521d\u3081 \u3001 2 \u3001 3 {displaystyle f_ {abc} equiv epsilon _ {abc}\u3001quad abc = 1,2,3} \uff08 2 \uff09\u901f\u5ea6\u30b0\u30eb\u30fc\u30d7\u306e\u4efb\u610f\u306e\u8981\u7d20 s o \uff08 3 \uff09\uff09 {displaystyle so\uff083\uff09} \u5c55\u793a\u7269\u3092\u4f7f\u7528\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\uff1a r \uff08 \u03d5 1\u3001 \u03d5 2\u3001 \u03d5 3\uff09\uff09 = exp \u2061 [i\u2211a=13\u03d5aTa]{displaystyle r\uff08phi _ {1}\u3001phi _ {2}\u3001phi _ {3}\uff09= exp {\u5de6[isum _ {a = 1}^{3} phi _ {a} t^{a}}}}} \u3069\u3053 \u03d5 \u521d\u3081 \u3001 \u03d5 2 \u3001 \u03d5 3 {displaystyle phi _ {1}\u3001phi _ {2}\u3001phi _ {3}}  – \u56de\u8ee2\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u3002 \u4ee3\u6570 su\uff08 2 \uff09\uff09 {displaystyle {text {su}}\uff082\uff09} [      \u305d\u308c\u306f   \uff08  2  \uff09\uff09    {displaystyle {text {su}}\uff082\uff09}   \u300c>\u7de8\u96c6 |      \u305d\u308c\u306f   \uff08  2  \uff09\uff09    {displaystyle {text {su}}\uff082\uff09}   \u300c>\u30b3\u30fc\u30c9\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \uff08 \u521d\u3081 \uff09\u4ee3\u6570 \u305d\u308c\u306f \uff08 2 \uff09\uff09 {displaystyle {text {su}}\uff082\uff09} \u3053\u308c\u306f\u3001\u63a1\u7b97\u3055\u308c\u3066\u3044\u306a\u3044\u96a0\u8005\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306e\u52a9\u3051\u3092\u501f\u308a\u3066\u8868\u3055\u308c\u308b3\u6b21\u5143\u4ee3\u6570\u3067\u3059\u3002 \uff08 2 \uff09\u30c7\u30a3\u30e1\u30f3\u30b7\u30e7\u30f3\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u3067\u8868\u73fe\u3055\u308c\u305f\u8868\u73fe\u306e\u30d9\u30fc\u30b9 2 \u00d7 2 {displaystyle 2Times 2} \u305d\u308c\u3089\u306f\u4f8b\u3048\u3070 t 1= 12[ 0110] \u3001 {displaystyle tau _ {1} = {tfrac {1} {2}} left [{begin {matrix} 0 && 1 \\ 1 && 0end {matrix}}\u53f3]\u3001}  t 2= 12[ 0\u2212ii0] \u3001 {displaystyle tau _ {2} = {tfrac {1} {2}} left [{begin {matrix} 0 && !!!-i \\ i && 0end {matrix}}\u53f3]\u3001}  t 3= 12[ 100\u22121] {displaystyle tau _ {3} = {tfrac {1} {2}}\u5de6[{begin {matrix} 1 && 0 \\ 0 && !!! -1END {matrix}}\u53f3]} \uff08\u3053\u308c\u3089\u306f\u30d1\u30a6\u30ea\u306e\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u30922\u3067\u5272\u3063\u305f\u3082\u306e\u3067\u3059\uff09\u3002 \uff08 3 \uff09\u767a\u96fb\u6a5f\u9593\u306e\u6574\u6d41\u95a2\u4fc2 [ t a\u3001 t b] = \u79c1 \u2211 c\u03f5 abct c{displaystyle [tau _ {a}\u3001tau _ {b}] = isum _ {c}\u3001epsilon _ {abc}\u3001tau _ {c}} \u6c38\u4e45\u4ee3\u6570\u69cb\u9020\u3092\u6c7a\u5b9a\u3057\u307e\u3059 \u305d\u308c\u306f \uff08 2 \uff09\uff09 {displaystyle {text {su}}\uff082\uff09}  f abc\u559c\u3093\u3067 \u03f5 abc\u3001 a b c = \u521d\u3081 \u3001 2 \u3001 3 {displaystyle f_ {abc} equiv epsilon _ {abc}\u3001quad abc = 1,2,3} \uff08 4 \uff09\u3053\u306e\u4ee3\u6570\u306f\u3001\u7279\u5225\u306a\u30e6\u30cb\u30bf\u30ea\u30fc\u306e\u5618su\uff082\uff09\u30b0\u30eb\u30fc\u30d7\u3092\u751f\u6210\u3057\u307e\u3059\u3002\u30dc\u30c7\u30a3\u30e6\u30cb\u30bf\u30eb\u30de\u30c8\u30ea\u30c3\u30af\u30b9 \u305d\u308c\u306f \uff08 2 \uff09\uff09 {displaystyle {text {su}}\uff082\uff09} \u5bf8\u6cd5 n \u00d7 n {displaystyle n} \u5c55\u793a\u3092\u4f7f\u7528\u3057\u3066\u53d6\u5f97\u3057\u307e\u3059\u3002 s \u306e \uff08 \u03d5 1\u3001 \u03d5 2\u3001 \u03d5 3\uff09\uff09 = exp \u2061 [i\u2211a=13\u03d5aGa]{displaystyle su\uff08phi _ _ _ _ _ _ _ \u3069\u3053\uff1a \u03d5 1\u3001 \u03d5 2\u3001 \u03d5 3{displaystyle phi _ {1}\u3001phi _ {2}\u3001phi _ {3}} – \u30d1\u30e9\u30e1\u30fc\u30bf\u30fc g 1\u3001 g 2\u3001 g 3{displaystyle g_ {1}\u3001g_ {2}\u3001g_ {3}}  – \u30b8\u30a7\u30cd\u30ec\u30fc\u30bf\u30fc\u3001\u5bf8\u6cd5\u30de\u30c8\u30ea\u30c3\u30af\u30b9 n \u00d7 n \u3001 {displaystyle n\u3001} Su Algebra\uff082\uff09\u306e\u30d9\u30fc\u30b9\u3092\u5f62\u6210\u3059\u308b;\u8868\u73fe\u306e1\u3064\u306f\u30012S+1 = n\u306b\u306a\u308b\u3088\u3046\u306b\u3001\u30b9\u30d4\u30f3\u6f14\u7b97\u5b50\u306e\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306b\u6bd4\u4f8b\u3057\u305f\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u3067\u69cb\u6210\u3055\u308c\u3066\u3044\u307e\u3059\u3002 \uff08 5 \uff09SU\uff082\uff09\u306e\u5b9a\u6570\u69cb\u9020\u306f\u30013\u6b21\u5143\u7a7a\u9593\u306eSO\uff083\uff09\u901f\u5ea6\u30b0\u30eb\u30fc\u30d7\u3068\u540c\u4e00\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u4ee3\u6570su\uff082\uff09\u306fSo -Called\u3067\u3059\u56de\u8ee2\u30b0\u30eb\u30fc\u30d7\u3092\u30ab\u30d0\u30fc\u3059\u308b\u4ee3\u6570\u3001\u56de\u8ee2\u306e\u5404\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306f\u3001SU\uff082\uff09\u306b\u3088\u3063\u3066\u751f\u6210\u3055\u308c\u305f2\u3064\u306e\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306b\u76f8\u4e92\u306b\u5bfe\u5fdc\u3057\u307e\u3059\u3002 J. Komorowski\u3001 \u30e6\u30cb\u30c3\u30c8\u6570\u304b\u3089\u30c6\u30f3\u30bd\u30eb\u3001\u30b9\u30d4\u30ca\u30fc\u3001\u30a2\u30fc\u30b8\u30dc\u30f3\u3001\u6b63\u65b9\u5f62\u307e\u3067 \u3001State Wydawnictwo naukowe\u30011978\u5e74\u30ef\u30eb\u30b7\u30e3\u30ef\u3002 J.\u30e2\u30ba\u30eb\u30b8\u30de\u30b9\u3001 \u73fe\u4ee3\u7269\u7406\u5b66\u306b\u304a\u3051\u308b\u30b0\u30eb\u30fc\u30d7\u7406\u8ad6\u306e\u9069\u7528 \u3001PWN\u3001\u30ef\u30eb\u30b7\u30e3\u30ef1967\u3002 \u30b8\u30e3\u30f3\u30d4\u30a8\u30fc\u30eb\u30bb\u30fc\u30eb\u3001 \u4ee3\u6570\u3068\u5618\u306e\u30b0\u30eb\u30fc\u30d7 \u3001\u7b2c2\u7248\u3001Springer\u30012006\u3001ISBN 3-540-55008-9 \u3002      (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki47\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki47\/archives\/1032#breadcrumbitem","name":"Liei Algebra-\u30a6\u30a3\u30ad\u30da\u30c7\u30a3\u30a2\u3001\u7121\u6599\u200b\u200b\u767e\u79d1\u4e8b\u5178"}}]}]