[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki23\/archives\/282663#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki23\/archives\/282663","headline":"\u30b2\u30eb\u30de\u30f3\u884c\u5217 &#8211; Wikipedia","name":"\u30b2\u30eb\u30de\u30f3\u884c\u5217 &#8211; Wikipedia","description":"\u30b2\u30eb\u30de\u30f3\u884c\u5217\uff08\u30b2\u30eb\u30de\u30f3\u304e\u3087\u3046\u308c\u3064, \u82f1: Gell-Mann matrices\uff09\u3068\u306f\u30013\u6b21\u7279\u6b8a\u30e6\u30cb\u30bf\u30ea\u7fa4SU(3) \u306e\u7121\u9650\u5c0f\u5909\u63db\u306e\u751f\u6210\u5b50\u3092\u306a\u30598\u3064\u306e\u8907\u7d20\u884c\u5217\u306e\u7d44[1][2]\u3002SU(3) \u306b\u4ed8\u968f\u3059\u308b\u30ea\u30fc\u4ee3\u6570\u306e\u6a19\u6e96\u7684\u306a\u57fa\u5e95\u3068\u3057\u3066\u3001\u7528\u3044\u3089\u308c\u308b\u3002\u30b2\u30eb\u30de\u30f3\u884c\u5217\u306f\u30cf\u30c9\u30ed\u30f3\u306e\u5206\u985e\u306b\u304a\u3044\u3066\u3001SU(3)\u5bfe\u79f0\u6027\u306b\u57fa\u3065\u304f\u516b\u9053\u8aac\uff08\u82f1\u8a9e\u7248\uff09\u3092\u63d0\u5531\u3057\u305f\u7c73\u56fd\u306e\u7269\u7406\u5b66\u8005\u30de\u30ec\u30fc\u30fb\u30b2\u30eb\u30de\u30f3\u306b\u3088\u3063\u3066\u3001\u5c0e\u5165\u3055\u308c\u305f[3]\u3002 Table of Contents \u5b9a\u7fa9\u3068\u57fa\u672c\u7684\u306a\u6027\u8cea[\u7de8\u96c6]\u4ea4\u63db\u95a2\u4fc2\u30fb\u53cd\u4ea4\u63db\u95a2\u4fc2[\u7de8\u96c6]SU(3)\u306e\u751f\u6210\u5b50[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6] \u5b9a\u7fa9\u3068\u57fa\u672c\u7684\u306a\u6027\u8cea[\u7de8\u96c6] \u6b21\u5f0f\u3067\u5b9a\u7fa9\u3055\u308c\u308b8\u500b\u306e3\u00d73\u8907\u7d20\u884c\u5217\u306e\u7d44\u3092\u30b2\u30eb\u30de\u30f3\u884c\u5217\u3068\u3044\u3046\u3002 \u03bb1=[010100000]\u03bb2=[0\u2212i0i00000]{displaystyle lambda _{1}={begin{bmatrix}0&#038;1&#038;0\\1&#038;0&#038;0\\0&#038;0&#038;0\\end{bmatrix}}quad lambda _{2}={begin{bmatrix}0&#038;-i&#038;0\\i&#038;0&#038;0\\0&#038;0&#038;0\\end{bmatrix}}} \u03bb3=[1000\u221210000]\u03bb4=[001000100]{displaystyle lambda _{3}={begin{bmatrix}1&#038;0&#038;0\\0&#038;-1&#038;0\\0&#038;0&#038;0\\end{bmatrix}}quad","datePublished":"2022-04-26","dateModified":"2022-04-26","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki23\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki23\/archives\/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:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8937739bfd9642f8584229032e1a5475c33170eb","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8937739bfd9642f8584229032e1a5475c33170eb","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/jp\/wiki23\/archives\/282663","about":["Wiki"],"wordCount":5939,"articleBody":"\u30b2\u30eb\u30de\u30f3\u884c\u5217\uff08\u30b2\u30eb\u30de\u30f3\u304e\u3087\u3046\u308c\u3064, \u82f1: Gell-Mann matrices\uff09\u3068\u306f\u30013\u6b21\u7279\u6b8a\u30e6\u30cb\u30bf\u30ea\u7fa4SU(3) \u306e\u7121\u9650\u5c0f\u5909\u63db\u306e\u751f\u6210\u5b50\u3092\u306a\u30598\u3064\u306e\u8907\u7d20\u884c\u5217\u306e\u7d44[1][2]\u3002SU(3) \u306b\u4ed8\u968f\u3059\u308b\u30ea\u30fc\u4ee3\u6570\u306e\u6a19\u6e96\u7684\u306a\u57fa\u5e95\u3068\u3057\u3066\u3001\u7528\u3044\u3089\u308c\u308b\u3002\u30b2\u30eb\u30de\u30f3\u884c\u5217\u306f\u30cf\u30c9\u30ed\u30f3\u306e\u5206\u985e\u306b\u304a\u3044\u3066\u3001SU(3)\u5bfe\u79f0\u6027\u306b\u57fa\u3065\u304f\u516b\u9053\u8aac\uff08\u82f1\u8a9e\u7248\uff09\u3092\u63d0\u5531\u3057\u305f\u7c73\u56fd\u306e\u7269\u7406\u5b66\u8005\u30de\u30ec\u30fc\u30fb\u30b2\u30eb\u30de\u30f3\u306b\u3088\u3063\u3066\u3001\u5c0e\u5165\u3055\u308c\u305f[3]\u3002  Table of Contents\u5b9a\u7fa9\u3068\u57fa\u672c\u7684\u306a\u6027\u8cea[\u7de8\u96c6]\u4ea4\u63db\u95a2\u4fc2\u30fb\u53cd\u4ea4\u63db\u95a2\u4fc2[\u7de8\u96c6]SU(3)\u306e\u751f\u6210\u5b50[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u5b9a\u7fa9\u3068\u57fa\u672c\u7684\u306a\u6027\u8cea[\u7de8\u96c6]\u6b21\u5f0f\u3067\u5b9a\u7fa9\u3055\u308c\u308b8\u500b\u306e3\u00d73\u8907\u7d20\u884c\u5217\u306e\u7d44\u3092\u30b2\u30eb\u30de\u30f3\u884c\u5217\u3068\u3044\u3046\u3002  \u03bb1=[010100000]\u03bb2=[0\u2212i0i00000]{displaystyle lambda _{1}={begin{bmatrix}0&#038;1&#038;0\\1&#038;0&#038;0\\0&#038;0&#038;0\\end{bmatrix}}quad lambda _{2}={begin{bmatrix}0&#038;-i&#038;0\\i&#038;0&#038;0\\0&#038;0&#038;0\\end{bmatrix}}}\u03bb3=[1000\u221210000]\u03bb4=[001000100]{displaystyle lambda _{3}={begin{bmatrix}1&#038;0&#038;0\\0&#038;-1&#038;0\\0&#038;0&#038;0\\end{bmatrix}}quad lambda _{4}={begin{bmatrix}0&#038;0&#038;1\\0&#038;0&#038;0\\1&#038;0&#038;0\\end{bmatrix}}}\u03bb5=[00\u2212i000i00]\u03bb6=[000001010]{displaystyle lambda _{5}={begin{bmatrix}0&#038;0&#038;-i\\0&#038;0&#038;0\\i&#038;0&#038;0\\end{bmatrix}}quad lambda _{6}={begin{bmatrix}0&#038;0&#038;0\\0&#038;0&#038;1\\0&#038;1&#038;0\\end{bmatrix}}}\u03bb7=[00000\u2212i0i0]\u03bb8=13[10001000\u22122]{displaystyle lambda _{7}={begin{bmatrix}0&#038;0&#038;0\\0&#038;0&#038;-i\\0&#038;i&#038;0\\end{bmatrix}}quad lambda _{8}={frac {1}{sqrt {3}}}{begin{bmatrix}1&#038;0&#038;0\\0&#038;1&#038;0\\0&#038;0&#038;-2\\end{bmatrix}}}\u3053\u3053\u3067\u3001\u03bb1, \u03bb2, \u03bb3 \u306f\u90e8\u5206\u7a7a\u9593\u306b\u4f5c\u7528\u3059\u308b\u30d1\u30a6\u30ea\u884c\u5217 \u03c31, \u03c32, \u03c33 \u3092  \u03bba=[\u03c3a000](a=1,2,3){displaystyle lambda _{a}={begin{bmatrix}sigma _{a}&#038;0\\0&#038;0end{bmatrix}}quad (a=1,2,3)}\u306e\u5f62\u3067\u542b\u3093\u3067\u304a\u308a\u3001\u30b2\u30eb\u30de\u30f3\u884c\u5217\u306f\u30d1\u30a6\u30ea\u884c\u5217\u306e\u4e00\u822c\u5316\u3068\u306a\u3063\u3066\u3044\u308b[4]\u3002\u30b2\u30eb\u30de\u30f3\u884c\u5217 \u03bba (a=1,\u2026,8) \u306f\u30a8\u30eb\u30df\u30fc\u30c8\u884c\u5217\u304b\u3064\u30c8\u30ec\u30fc\u30b9\u306f\u30bc\u30ed\u3068\u306a\u308b\u3002\u03bba\u2020=\u03bba{displaystyle lambda _{a}^{,dagger }=lambda _{a}}Tr\u2061(\u03bba)=0{displaystyle operatorname {Tr} (lambda _{a})=0}\u307e\u305f\u3001\u4e8c\u3064\u306e\u30b2\u30eb\u30de\u30f3\u884c\u5217\u306e\u7a4d\u306e\u30c8\u30ec\u30fc\u30b9\u306f\u6b63\u898f\u5316\u3055\u308c\u3066\u304a\u308a\u3001\u6b21\u306e\u95a2\u4fc2\u5f0f\u3092\u6e80\u305f\u3059[5]\u3002Tr\u2061(\u03bba\u03bbb)=2\u03b4ab{displaystyle operatorname {Tr} (lambda _{a}lambda _{b})=2delta _{ab}}\u4f46\u3057\u3001\u03b4ab\u306f\u30af\u30ed\u30cd\u30c3\u30ab\u30fc\u306e\u30c7\u30eb\u30bf\u3067\u3042\u308b\u3002\u4ea4\u63db\u95a2\u4fc2\u30fb\u53cd\u4ea4\u63db\u95a2\u4fc2[\u7de8\u96c6]\u30b2\u30eb\u30de\u30f3\u884c\u5217\u306e\u4ea4\u63db\u95a2\u4fc2 [\u03bba, \u03bbb]=\u03bba \u03bbb&#8211;\u03bbb \u03bba \u306f\u6b21\u306e\u3088\u3046\u306a\u30b2\u30eb\u30de\u30f3\u884c\u5217\u306e\u7dda\u5f62\u7d50\u5408\u3067\u8868\u3055\u308c\u308b\u3002[\u03bba,\u03bbb]=2i\u2211c=18fabc\u03bbc{displaystyle [lambda _{a},lambda _{b}]=2isum _{c=1}^{8}f_{abc}lambda _{c}}\u3053\u3053\u3067\u3001fabc \u306f\u6dfb\u3048\u5b57 a, b, c \u306b\u3064\u3044\u3066\u3001\u5b8c\u5168\u53cd\u5bfe\u79f0\u306a\u5b9f\u4fc2\u6570\u3067\u3042\u308b\u3002fabc \u306e\u3046\u3061\u3001\u30bc\u30ed\u3067\u306a\u3044\u3082\u306e\u306f\u3001a < b < c \u3092\u6e80\u305f\u3059\u3082\u306e\u3067\u4ee3\u8868\u3055\u305b\u3066\u8868\u3059\u3068\u3001\u6b21\u306e\u3088\u3046\u306b\u306a\u308b\u3002f123=1{displaystyle f_{123}=1}f147=f246=f257=f345=12{displaystyle f_{147}=f_{246}=f_{257}=f_{345}={frac {1}{2}}}f156=f367=\u221212{displaystyle f_{156}=f_{367}=-{frac {1}{2}}}f458=f678=32{displaystyle f_{458}=f_{678}={frac {sqrt {3}}{2}}}\u4e00\u65b9\u3001\u53cd\u4ea4\u63db\u95a2\u4fc2 {\u03bba, \u03bbb}=\u03bba \u03bbb+\u03bbb \u03bba \u306f\u6b21\u306e\u5f62\u3092\u3068\u308b\u3002{\u03bba,\u03bbb}=43\u03b4ab+2\u2211i=18dabc\u03bbc{displaystyle {lambda _{a},lambda _{b}}={frac {4}{3}}delta _{ab}+2sum _{i=1}^{8}d_{abc}lambda _{c}}\u3053\u3053\u3067\u3001dabc \u306f\u6dfb\u3048\u5b57a, b, c \u306b\u3064\u3044\u3066\u3001\u5b8c\u5168\u5bfe\u79f0\u306a\u5b9f\u4fc2\u6570\u3067\u3042\u308b\u3002dabc \u306e\u3046\u3061\u3001\u30bc\u30ed\u3067\u306a\u3044\u3082\u306e\u3092a < b < c \u3092\u6e80\u305f\u3059\u3082\u306e\u3067\u4ee3\u8868\u3055\u305b\u3066\u8868\u3059\u3068\u3001d118=d228=d338=13{displaystyle d_{118}=d_{228}=d_{338}={frac {1}{sqrt {3}}}}d888=\u221213{displaystyle d_{888}=-{frac {1}{sqrt {3}}}}d146=d157=d256=d344=d355=12{displaystyle d_{146}=d_{157}=d_{256}=d_{344}=d_{355}={frac {1}{2}}}d247=d366=d377=\u221212{displaystyle d_{247}=d_{366}=d_{377}=-{frac {1}{2}}}d448=d558=d668=d778=\u2212123{displaystyle d_{448}=d_{558}=d_{668}=d_{778}=-{frac {1}{2{sqrt {3}}}}}SU(3)\u306e\u751f\u6210\u5b50[\u7de8\u96c6]3\u6b21\u7279\u6b8a\u30e6\u30cb\u30bf\u30ea\u7fa4SU(3) \u306f\u884c\u5217\u5f0f\u304c\uff11\u3068\u306a\u308b3\u00d73\u30e6\u30cb\u30bf\u30ea\u884c\u5217\u304b\u3089\u69cb\u6210\u3055\u308c\u308b\u3002SU(3) \u306f\u7dda\u5f62\u30ea\u30fc\u7fa4\u3067\u3042\u308a\u30018\u500b\u306e\u30b2\u30eb\u30de\u30f3\u884c\u5217\u306f\u305d\u306e\u4e00\u6b21\u72ec\u7acb\u306a\u751f\u6210\u5b50\u3067\u3042\u308b\u3002\u4f46\u3057\u3001\u7269\u7406\u5b66\u306e\u6163\u7fd2\u306b\u3088\u308a\u3001\u751f\u6210\u5b50\u306f\u30a8\u30eb\u30df\u30fc\u30c8\u884c\u5217\u306b\u306a\u308b\u3088\u3046\u306b\u3068\u308b\u305f\u3081\u3001\u30b2\u30eb\u30de\u30f3\u884c\u5217\u306f\u305d\u308c\u81ea\u8eab\u30ea\u30fc\u4ee3\u6570 \ud835\udd30\ud835\udd32(3) \u306e\u5143\u3067\u306f\u306a\u304f\u3001\u30b2\u30eb\u30de\u30f3\u884c\u5217\u306b i=\u221a-1 \u3092\u4e57\u3058\u305f\u3082\u306e\u304c \ud835\udd30\ud835\udd32(3) \u306e\u5143\u3068\u306a\u308b\u3002\u901a\u5e38\u3001SU(3)\u306e\u751f\u6210\u5b50\u3068\u3057\u3066\u306f\u3001\u03bba \u306e\u4ee3\u308f\u308a\u306b1\/2 \u3092\u4e57\u3058\u305f Ta \u304c\u7528\u3044\u3089\u308c\u308b\u3002Ta=\u03bba2{displaystyle T_{a}={frac {lambda _{a}}{2}}}\u30b3\u30f3\u30d1\u30af\u30c8\u3067\u9023\u7d50\u306a\u30ea\u30fc\u7fa4SU(3)\u306e\u4efb\u610f\u306e\u5143\u306f\u30ea\u30fc\u74b0\u306e\u6307\u6570\u5199\u50cf\u306b\u3088\u3063\u3066ei\u2211a=18\u03b8aTa(\u03b8a\u2208R,a=1,\u22ef,8){displaystyle e^{isum _{a=1}^{8}theta _{a}T_{a}}quad (theta _{a}in mathbb {R} ,,a=1,cdots ,8)}\u306e\u5f62\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u30b2\u30eb\u30de\u30f3\u884c\u5217 \u03bba\u3001 \u307e\u305f\u306f Ta \u306e\u7dda\u5f62\u7d50\u5408\u3067\u5f35\u3089\u308c\u308b\u7dda\u5f62\u7a7a\u9593\u306f\u4ea4\u63db\u5b50\u7a4d[Ta,Tb]=TaTb\u2212TbTa{displaystyle [T_{a},T_{b}]=T_{a}T_{b}-T_{b}T_{a}}\u306b\u3088\u308a\u3001\u30ea\u30fc\u4ee3\u6570\u3068\u306a\u308a\u3001\u305d\u306e\u69cb\u9020\u306f[Ta,Tb]=i\u2211i=18fabcTc{displaystyle [T_{a},T_{b}]=isum _{i=1}^{8}f_{abc}T_{c}}\u3067\u5b9a\u307e\u308b\u69cb\u9020\u5b9a\u6570 fabc \u3067\u898f\u5b9a\u3055\u308c\u308b[6]\u3002\u3053\u306e\u30ea\u30fc\u4ee3\u6570\u306f\u30b3\u30f3\u30d1\u30af\u30c8\u30fb\u30ea\u30fc\u4ee3\u6570\u3067\u3042\u308b\u305f\u3081\u3001fabc \u306f\u6dfb\u3048\u5b57a, b, c \u306b\u3064\u3044\u3066\u3001\u5b8c\u5168\u53cd\u5bfe\u79f0\u3067\u3042\u308b\u3002{T1,T2,T3}\u306e\u7d44\u306f\u3001[T1,T2]=iT3,[T2,T3]=iT1,[T3,T1]=iT2{displaystyle [T_{1},T_{2}]=iT_{3},quad [T_{2},T_{3}]=iT_{1},quad [T_{3},T_{1}]=iT_{2}}\u3068\u4ea4\u63db\u5b50\u7a4d\u306b\u3064\u3044\u3066\u9589\u3058\u3066\u304a\u308a\u3001SU(2)\u306b\u5bfe\u5fdc\u3059\u308b\u90e8\u5206\u30ea\u30fc\u4ee3\u6570\u3092\u306a\u3059\u3002\u3053\u308c\u4ee5\u5916\u306b\u3082\u3044\u304f\u3064\u304b\u306e\u7d44\u306fSU(2)\u306b\u5bfe\u5fdc\u3059\u308b\u90e8\u5206\u30ea\u30fc\u4ee3\u6570\u3092\u306a\u3059\u3002\u3053\u306e\u30ea\u30fc\u4ee3\u6570\u306e\u5168\u3066\u306e\u5143\u3068\u53ef\u63db\u306b\u306a\u308b\u30ab\u30b7\u30df\u30e4\u6f14\u7b97\u5b50\u306fC1=\u2211aTa2{displaystyle C_{1}=sum _{a}T_{a}^{,2}}C2=\u2211adabcTaTbTc{displaystyle C_{2}=sum _{a}d_{abc}T_{a}T_{b}T_{c}}\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002^ G.B. Arfken, H.J. Weber and F.E. Harris (2012), chapter.4^ H. Georgi (1999), chapter.7-9^ Murray Gell-Mann,&#8221;Symmetries of Baryons and Mesons&#8221;, Phys. Rev. 125, 1067 (1962) doi:10.1103\/PhysRev.125.1067^ \u30d1\u30a6\u30ea\u884c\u5217\u306f SU(2) \u306e\u751f\u6210\u5b50\u3067\u3042\u308a\u3001\u30b2\u30eb\u30de\u30f3\u884c\u5217\u306f SU(3) \u306e\u751f\u6210\u5b50\u3067\u3042\u308b\u3002^ \u30ea\u30fc\u4ee3\u6570\u306b\u304a\u3051\u308b\u30ab\u30eb\u30bf\u30f3\u8a08\u91cf\u306b\u5bfe\u5fdc\u3059\u308b\u3002^ fabc \u306f\u4ea4\u63db\u95a2\u4fc2\u30fb\u53cd\u4ea4\u63db\u95a2\u4fc2\u306e\u7bc0\u3067\u8ff0\u3079\u305f\u3082\u306e\u3068\u540c\u4e00\u3067\u3042\u308b\u3002\u53c2\u8003\u6587\u732e[\u7de8\u96c6]George B. Arfken, Hans J. Weber and Frank E. Harris, Mathematical Methods for Physicists (7th ed.)\u00a0: Academic Press (2012). ISBN 978-0123846549H. Georgi, Lie Algebras in Particle Physics: from Isospin To Unified Theories (2nd ed.), Westview Press (1999). ISBN 978-0738202334.\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki23\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki23\/archives\/282663#breadcrumbitem","name":"\u30b2\u30eb\u30de\u30f3\u884c\u5217 &#8211; Wikipedia"}}]}]