{"id":1039,"date":"2022-04-22T03:54:53","date_gmt":"2022-04-22T03:54:53","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=1039"},"modified":"2023-04-28T13:28:25","modified_gmt":"2023-04-28T05:28:25","slug":"clifforddaishuyuxuanliangqun","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=1039","title":{"rendered":"Clifford\u4ee3\u6570\u4e0e\u65cb\u91cf\u7fa4"},"content":{"rendered":"

1. \u4ee3\u6570\u7684\u5b9a\u4e49<\/a><\/span>\n\n

\u5b9a\u4e49 1<\/span>.<\/span> \u5047\u8bbe$K$\u662f\u4e00\u4e2a\u57df\uff0c$V$\u662f$K$\u4e0a\u7684\u4e00\u4e2a\u7ebf\u6027\u7a7a\u95f4\u3002\u82e5\u5b58\u5728\u4e8c\u5143\u8fd0\u7b97$\\tau:V\\times V\\to V$, $(x,y)\\mapsto xy$, \u4f7f\u5f97\u5bf9\u4efb\u610f\u7684$x,y,z\\in V$, \u4ee5\u53ca\u4efb\u610f\u7684$a,b\\in K$, \u90fd\u6709
\n
  1. \u53f3\u5206\u914d\u5f8b: $(x+y)z=xz+yz$;<\/li>
  2. \u5de6\u5206\u914d\u5f8b\uff1a$z(x+y)=zx+zy$;<\/li>
  3. \u6570\u4e58\u76f8\u5bb9\u6027\uff1a$(ax)(by)=(ab)(xy)$.<\/li><\/ol>\u5219\u79f0$(V,\\tau)$\u4e3a\u6570\u57df$K$\u4e0a\u7684\u4e00\u4e2a\u4ee3\u6570<\/span>\u3002
    \n<\/div><\/p>\n

    <\/p>\n

    \u6ce8\u8bb0 1<\/span>.<\/span>
    1. \u6211\u4eec\u6ca1\u6709\u8981\u6c42$\\tau$\u6ee1\u8db3\u7ed3\u5408\u6027\uff1a\u5373$(xy)z=x(yz)$\u4e0d\u4e00\u5b9a\u6210\u7acb\u3002\u6ee1\u8db3\u7ed3\u5408\u6027\u7684\u4ee3\u6570\u4e5f\u79f0\u4e3a\u7ed3\u5408\u4ee3\u6570\u3002\u4f8b\u5982\u5168\u4f53$n$\u9636\u65b9\u9635\u6784\u6210\u4e00\u4e2a\u7ed3\u5408\u4ee3\u6570\u3002\u975e\u7ed3\u5408\u4ee3\u6570\u7684\u4f8b\u5b50\u662f\uff1a$\\mathbb{R}^3$\u4e2d\u5411\u91cf\u7684\u53c9\u79ef\u3002<\/li>
    2. \u82e5\u4ee3\u6570$(V,\\tau)$\u542b\u6709\u6052\u7b49\u5143\u7d20$e\\in V$, \u5373$ex=xe=x$\u5bf9\u4efb\u610f\u7684$x\\in V$\u90fd\u6210\u7acb, \u7684\u4ee3\u6570\u79f0\u4e3a\u9149\u4ee3\u6570<\/span>\uff08unitary algebra).<\/li>
    3. \u5f53\u4e58\u6cd5\u8fd0\u7b97$\\tau$\u4ea4\u6362\u65f6\uff0c\u5de6\u5206\u914d\u5f8b\u548c\u53f3\u5206\u914d\u5f8b\u662f\u7b49\u4ef7\u7684\uff0c\u4ece\u800c\u5728\u5b9a\u4e49\u4e2d\u53ea\u9700\u8981\u6c42\u4e00\u6761\u6ee1\u8db3\u5373\u53ef\u3002\u5f97\u5230\u7684\u4ee3\u6570\u901a\u5e38\u79f0\u4e3a\u4ea4\u6362\u4ee3\u6570<\/span>\u3002<\/li>
    4. \u82e5\u5c06\u6570\u57df$K$\u6362\u6210\u4ea4\u6362\u9149\u73af$R$\uff0c\u5219\u5728\u4e0a\u8ff0\u5b9a\u4e49\u4e2d\u53ea\u9700\u5c06\u5411\u91cf\u7a7a\u95f4$V$\u6362\u6210$R$-\u6a21\u5373\u53ef\u5f97\u5230\u66f4\u5e7f\u7684\u4ea4\u6362\u4ee3\u6570\u7684\u5b9a\u4e49\u3002<\/li><\/ol><\/div>
      \n\u4e0b\u9762\uff0c\u6211\u4eec\u6765\u770b\u4ee3\u6570\u7684\u540c\u6001(homomorphism)\u3002
      \n
      \u5b9a\u4e49 2<\/span>.<\/span> \u5047\u8bbe$(V_1,\\tau_1)$, $(V_2,\\tau_2)$\u662f\u6570\u57df$K$\u4e0a\u7684\u4e24\u4e2a\u4ee3\u6570\u3002\u82e5$K$-\u7ebf\u6027\u6620\u5c04$f:V_1\\to V_2$\u6ee1\u8db3\uff1a\u5bf9\u6240\u6709\u7684$x,y\\in V_1$\u6709$f(xy)=f(x)f(y)$, \u5219\u79f0$f$\u4e3a\u4ee3\u6570$(V_1,\\tau_1)$\u5230$(V_2,\\tau_2)$\u7684\u540c\u6001<\/span>\u3002<\/p>\n

      \u79f0\u4e24\u4e2a$K$-\u4ee3\u6570\u662f\u540c\u6784<\/span>\u7684\uff0c\u82e5\u5b58\u5728\u5b83\u4eec\u4e4b\u95f4\u7684\u4e00\u4e2a\u53cc\u5c04$f$, \u4f7f\u5f97$f,f^{-1}$\u90fd\u662f\u4ee3\u6570\u4e4b\u95f4\u7684\u540c\u6001\u3002
      \n<\/div>
      \n\u4e0b\u9762\uff0c\u6211\u4eec\u6765\u770b\u5b50\u4ee3\u6570(subalgebra)\u4e0e\u7406\u60f3(ideal)\u3002
      \n

      \u5b9a\u4e49 3<\/span>.<\/span> \u5047\u8bbe$(V,\\tau)$\u662f\u4e00\u4e2a\u6570\u57df$K$\u4e0a\u7684\u4ee3\u6570\u3002\u82e5$V’\\subset V$\u662f\u4e00\u4e2a\u7ebf\u6027\u5b50\u7a7a\u95f4\uff0c\u4e14\u5bf9\u4efb\u610f\u7684$x’,y’\\in V’$, \u90fd\u6709$x’y’\\in V’$\uff0c\u5219\u79f0$(V’,\\tau|_{V’\\times V’})$\u4e3a$(V,\\tau)$\u7684\u4e00\u4e2a\u5b50\u4ee3\u6570<\/span>\u3002
      \n<\/div>
      \n\u6362\u8a00\u4e4b\uff0c\u4e00\u4e2a\u5b50\u4ee3\u6570$(V’,\\tau’)\\subset (V,\\tau)$\u662f\u4e00\u4e2a\u975e\u7a7a\u96c6\u5408\uff0c\u6ee1\u8db3\u5b83\u5728\u52a0\u6cd5\u3001\u6570\u4e58\u4ee5\u53ca\u4e58\u6cd5\u8fd0\u7b97\u4e0b\u5c01\u95ed\u3002
      \n
      \u4f8b\u5b50 1<\/span>.<\/span> \u590d\u6570\u57df$\\mathbb{C}$\u53ef\u4ee5\u8ba4\u4e3a\u662f\u5b9e\u6570\u57df$\\mathbb{R}$\u4e0a\u7684\u4e8c\u7ef4$\\mathbb{R}$-\u4ee3\u6570\u3002\u4e00\u7ef4\u5b9e\u76f4\u7ebf$\\mathbb{R}$\u5c31\u662f$\\mathbb{C}$\u7684\u5b50\u4ee3\u6570\u3002
      \n<\/div>
      \n
      \u5b9a\u4e49 4<\/span>.<\/span> \u5047\u8bbe$(V,\\tau)$\u662f\u6570\u57df$K$\u4e0a\u4e00\u4e2a\u4ee3\u6570\u3002$I$\u662f$V$\u7684\u4e00\u4e2a\u7ebf\u6027\u5b50\u7a7a\u95f4\u3002\u82e5\u5bf9\u4efb\u610f\u7684$x’,y’\\in I$, \u4efb\u610f\u7684$z\\in V$, \u4ee5\u53ca\u4efb\u610f\u7684$c\\in K$, \u6211\u4eec\u6709
      \n
      1. \u52a0\u6cd5\u5c01\u95ed\u6027\uff1a$x’+y’\\in I$;<\/li>
      2. \u6570\u4e58\u5c01\u95ed\u6027\uff1a$cx’\\in I$;<\/li>
      3. \u4e58\u6cd5\u5bf9$V$\u5c01\u95ed\u6027\uff1a$zx’\\in I$\u4ee5\u53ca$x’z\\in I$.<\/li><\/ol>\u5219\u79f0$(I,\\tau|_{I\\times I})$\u4e3a$(V,\\tau)$\u7684\u4e00\u4e2a\uff08\u53cc\u8fb9\uff09\u7406\u60f3<\/span>\u3002
        \n<\/div>
        \n\u6ce8\u610f\uff0c\u4e0a\u8ff0\u5b9a\u4e49\u4e2d\u524d\u4e24\u6761\u8bf4\u660e\u7406\u60f3$I$\u662f\u4e00\u4e2a\u7ebf\u6027\u5b50\u7a7a\u95f4\u3002\u800c\u7b2c\u4e09\u6761\u8bf4\u660e\u7406\u60f3\u662f\u4e00\u4e2a\u5b50\u4ee3\u6570\u3002\u540c\u65f6\uff0c\u6ce8\u610f\u73af\u7684\u7406\u60f3\u4e0d\u8981\u6c42\u7b2c\u4e8c\u6761\uff0c\u548c\u4ee3\u6570\u7684\u7406\u60f3\u662f\u4e0d\u4e00\u6837\u7684\u3002<\/p>\n

        \u5229\u7528\u540c\u6001\u57fa\u672c\u5b9a\u7406\uff0c \u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u4ee3\u6570\u5546\u6389\u7406\u60f3\u6784\u6210\u4e00\u4e2a\u65b0\u7684\u4ee3\u6570\uff0c\u79f0\u4e3a\u5546\u4ee3\u6570\u3002<\/p>\n

        \u6700\u540e\uff0c\u6211\u4eec\u7ed9\u51fa\u7528\u6570\u57df$K$\u4e0a\u7684\u73af$(V,\\tau)$\u6765\u6784\u9020\u6570\u57df$K$\u4e0a\u7ed3\u5408\u9149\u4ee3\u6570\u7684\u65b9\u6cd5\u3002\u8003\u5bdf\u73af\u540c\u6001$\\eta:K\\to Z(V)$\uff0c\u5176\u4e2d$Z(V)$\u662f\u73af$(V,\\tau)$\u7684\u4e2d\u5fc3\u3002\u82e5$(V,\\tau)$\u4e0d\u662f\u96f6\u73af, \u5219$\\eta$\u662f\u5355\u5c04\u3002\u8fd9\u6837\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u6570\u4e58
        \n\\[
        \nK\\times V\\to V,\\quad (k,x)\\mapsto \\eta(k)x.
        \n\\]
        \n\u8fd9\u6837\uff0c\u73af$(V,\\tau)$\u6210\u4e3a\u4e00\u4e2a$K$-\u4ee3\u6570, \u800c\u4e14\u542b\u6709\u6052\u7b49\u5143$\\eta(1)\\in V$, \u5373\u8fd8\u662f\u4e00\u4e2a\u9149\u4ee3\u6570\u3002<\/p>\n

        \u5982\u6b64\u6784\u9020\u7684$K$-\u4ee3\u6570\u4e4b\u95f4\u7684\u540c\u6001$f:V_1\\to V_2$\u5c31\u662f\u6ee1\u8db3$f(kx)=kf(x)$\u7684\u73af\u540c\u6001\u3002
        \n2. Clifford\u4ee3\u6570<\/a><\/span>\n\n\u7ed9\u5b9a\u5b9e\u6570\u57df$\\mathbb{R}$\u4e0a\u7684\u7ebf\u6027\u7a7a\u95f4$V$, \u5176\u5f20\u91cf\u4ee3\u6570\u8bb0\u4e3a$T(V)=\\oplus_{k\\geq0}\\underbrace{V\\otimes\\cdots\\otimes V}_k$. \u82e5$V$\u4e0a\u6709\u975e\u9000\u5316\u7684\u4e8c\u6b21\u578b$\\langle \\cdot,\\cdot \\rangle$, \u5219\u53ef\u8003\u5bdf\u6240\u6709\u5f62\u5982$\\alpha(v):=v\\otimes v+\\langle v,v \\rangle$, $v\\in V$, \u751f\u6210\u7684\u7406\u60f3$I(V)$\u3002$(V,\\langle \\cdot,\\cdot \\rangle)$\u4e0a\u7684Clliford\u4ee3\u6570\u5b9a\u4e49\u4e3a
        \n\\[
        \n\\mathrm{Cl}(V):=T(V)\/I(V).
        \n\\]<\/p>\n

        \u5bb9\u6613\u9a8c\u8bc1\uff1a\u82e5$v,w\\in V$, \u5219
        \n\\[
        \nI(V)\\ni(v+w)\\otimes(v+w)+\\langle v+w,v+w \\rangle
        \n=v\\otimes w+w\\otimes v+v\\otimes v+w\\otimes w+\\langle v,v \\rangle+\\langle w,w \\rangle+2\\langle v,w \\rangle,
        \n\\]
        \n\u56e0\u4e3a$v\\otimes v+\\langle v,v \\rangle\\in I(V)$\u4ee5\u53ca$w\\otimes w+\\langle w,w \\rangle\\in I(V)$, \u6211\u4eec\u5f97\u5230
        \n\\[
        \nv\\otimes w+w\\otimes v+2\\langle v,w \\rangle\\in I(V).
        \n\\]
        \n\u4ece\u800c\u82e5$\\bar{v},\\bar{w}\\in \\mathrm{Cl}(V)$\uff0c\u5219
        \n\\[
        \n\\bar{v}+\\bar{w}=v+w+I(V),
        \n\\]
        \n\u4ee5\u53ca
        \n\\[
        \n\\bar{v}\\cdot\\bar{w}+\\bar{w}\\cdot\\bar{v}
        \n=v\\otimes w+w\\otimes v+I(V)=-2\\langle v,w \\rangle+I(V)=-2\\overline{\\langle v,w \\rangle}.
        \n\\]
        \n\u4e3a\u4e86\u65b9\u4fbf\uff0c\u6211\u4eec\u4e0d\u7528\u52a0\u9776\u6765\u533a\u5206\u4ee3\u8868\u5143\u3002\u5c06\u4e0a\u8ff0\u7b49\u5f0f\u8868\u793a\u4e3a
        \n\\[
        \nv\\cdot w+w\\cdot v=-2\\langle v,w \\rangle.
        \n\\]
        \n\u4ece\u800c\uff0c\u82e5\u9009\u53d6$\\left\\{ e_i \\right\\}_{i=1}^n$\u4e3a$(V,\\langle \\cdot,\\cdot \\rangle)$\u7684\u6807\u51c6\u6b63\u4ea4\u57fa\u5e95\u3002\u5219
        \n\\[
        \ne_i\\cdot e_i=-\\langle e_i,e_i \\rangle=-1,\\quad
        \ne_i\\cdot e_j=-e_j\\cdot e_i.
        \n\\]
        \n\u7531\u6b64\uff0c\u5bb9\u6613\u5f97\u5230$\\mathrm{Cl}(V)$\u4f5c\u4e3a\u4e00\u4e2a\u7ebf\u6027\u7a7a\u95f4\uff0c\u5176\u57fa\u5e95\u7531
        \n\\[
        \ne_0:=1,\\quad e_\\alpha:=e_{\\alpha_1}\\cdot e_{\\alpha_2}\\cdots e_{\\alpha_k}
        \n\\]
        \n\u7ed9\u51fa\uff0c\u5176\u4e2d$\\alpha=\\left\\{ \\alpha_1,\\ldots,\\alpha_k \\right\\}\\subset \\left\\{ 1,2,\\ldots,n \\right\\}$, \u4e14$\\alpha_1<\\alpha_2<\\cdots<\\alpha_k$. \u5bf9\u8fd9\u6837\u7684$\\alpha$, \u6211\u4eec\u4ee4$\\lvert \\alpha \\rvert:=k$\uff0c\u79f0\u4e3a$e_\\alpha$\u7684\u5ea6\u3002\u7279\u522b\u5730\uff0c$\\mathrm{Cl}(V)$\u4f5c\u4e3a\u7ebf\u6027\u7a7a\u95f4\u548c\u5916\u4ee3\u6570$\\Lambda^*(V)$\u540c\u6784\uff0c\u5176\u7ef4\u6570\u4e3a$2^n$. \u800c\u4e14\uff0c\u901a\u8fc7\u5b9a\u4e49\u8fd9\u4e9b\u57fa\u5e95\u4e3a\u6807\u51c6\u6b63\u4ea4\u57fa\u5e95\uff0c\u6211\u4eec\u5c06$V$\u7684\u5185\u79ef\uff0c\u6269\u5145\u5230$\\mathrm{Cl}(V)$.<\/p>\n

        \u82e5\u5c06\u6240\u6709\u5ea6\u4e3a$k$\u7684\u5143\u7d20\u751f\u6210\u7684\u7ebf\u6027\u5b50\u7a7a\u95f4\u8bb0\u4f5c$\\mathrm{Cl}^k(V)$, \u5219\u5bb9\u6613\u9a8c\u8bc1\uff1a
        \n

        1. $\\mathrm{Cl}^0(V)=\\mathbb{R}$;<\/li>
        2. $\\mathrm{Cl}^1(V)=V$;<\/li>
        3. $\\mathrm{Cl}^{\\mathrm{ev}}(V)$\uff0c\u5373\u5168\u4f53\u5076\u6570\u5ea6\u5143\u7d20\u751f\u6210\u7684\u7ebf\u6027\u5b50\u7a7a\u95f4\uff0c\u662f$\\mathrm{Cl}(V)$\u7684\u4e00\u4e2a\u5b50\u4ee3\u6570\u3002<\/li><\/ol>3. Spin\u7fa4<\/a><\/span>\n\n\u6211\u4eec\u5b9a\u4e49$\\mathfrak{spin}(V):=\\mathrm{Cl}^2(V)$. \u53ef\u4ee5\u8bc1\u660e\u5b83\u662f\u4e00\u4e2a\u674e\u4ee3\u6570\uff0c\u5176\u4e2d\u62ec\u53f7\u79ef\u7531
          \n\\[
          \n[a,b]=a\\cdot b-b\\cdot a,\\quad a,b\\in \\mathrm{Cl}^2(V)
          \n\\]
          \n\u7ed9\u51fa\u3002\u4e8b\u5b9e\u4e0a\uff0c\u7531\u4e8e\u62ec\u53f7\u79ef\u4ee5\u53caClifford\u4e58\u6cd5\u7684\u7ebf\u6027\u6027\uff0c\u53ea\u9700\u5bf9\u57fa\u5e95\u8bc1\u660e\u5373\u53ef\u3002\u5047\u8bbe$a=e_i\\cdot e_j$, $b=e_k\\cdot e_l$\u662f$\\mathrm{Cl}^2(V)$\u4e2d\u4e24\u4e2a\u5143\u7d20\uff0c\u5219(\u4e0d\u59a8\u5047\u8bbe$i,j,k,l$\u4e92\u4e0d\u76f8\u540c\uff0c\u5176\u4ed6\u60c5\u5f62\u7c7b\u4f3c\uff09
          \n\\[
          \na\\cdot b-b\\cdot a=e_i\\cdot e_j\\cdot e_k\\cdot e_l-e_k\\cdot e_l\\cdot e_i\\cdot e_j
          \n=e_i\\cdot e_k\\cdot e_l\\cdot e_j-e_k\\cdot e_l\\cdot e_i\\cdot e_j
          \n=e_k\\cdot e_l\\cdot e_i\\cdot e_j-e_k\\cdot e_l\\cdot e_i\\cdot e_j
          \n=0.
          \n\\]<\/p>\n

          \u4e8b\u5b9e\u4e0a\uff0c\u66f4\u8fdb\u4e00\u6b65\u5730\uff0c\u674e\u62ec\u53f7\u53ef\u4ee5\u5b9a\u4e49$\\mathrm{Cl}^2(V)$\u5728$\\mathrm{Cl}^1(V)$\u4e0a\u7684\u4e00\u4e2a\u4f5c\u7528$\\tau$:
          \n\\[
          \n\\tau(a)v:=[a,v]=a\\cdot v-v\\cdot a.
          \n\\]
          \n\u53ef\u4ee5\u8bc1\u660e\u82e5$a\\in \\mathrm{Cl}^2(V)$, $v\\in \\mathrm{Cl}^1(V)$, \u5219$[a,v]\\in \\mathrm{Cl}^1(V)$. \u6211\u4eec\u4ecd\u7136\u53ea\u9a8c\u8bc1\u4e24\u4e2a\u5178\u578b\u60c5\u5f62\uff1a$a=e_i\\cdot e_j$, $v=e_k$, \u5219
          \n\\[
          \n[a,v]=e_i\\cdot e_j\\cdot e_k-e_k\\cdot e_i\\cdot e_j=0\\in \\mathrm{Cl}^1(V);
          \n\\]
          \n\u82e5$v=e_j$, \u5219
          \n\\[
          \n[a,v]=e_i\\cdot e_j\\cdot e_j-e_j\\cdot e_i\\cdot e_j=-e_i-e_i=-2e_i\\in \\mathrm{Cl}^1(V).
          \n\\]
          \n

          \u5b9a\u4e49 5<\/span>.<\/span> $\\tau: \\mathfrak{spin}(V)\\to \\mathfrak{so}(V)$\u7ed9\u51fa\u4e86\u674e\u4ee3\u6570\u540c\u6784.
          \n<\/div>
          \n
          \u8bc1\u660e<\/span>.<\/span> \u9996\u5148\uff0c\u6ce8\u610f\u5229\u7528Jacobi\u6052\u7b49\u5f0f\uff0c
          \n\\[
          \n\\tau[a,b](v)=[[a,b],v]=-[[b,v],a]-[[v,a],b]=[a,[b,v]]-[b,[a,v]],\\quad
          \n[\\tau(a),\\tau(b)](v)=\\tau(a)\\tau(b)(v)-\\tau(b)\\tau(a)(v)=[a,[b,v]]-[b,[a,v]],
          \n\\]
          \n\u6545
          \n\\[
          \n\\tau[a,b]=[\\tau(a),\\tau(b)].
          \n\\]
          \n\u5373$\\tau$\u4fdd\u6301\u674e\u4ee3\u6570\u4e4b\u95f4\u7684\u4e58\u79ef\uff0c\u662f\u674e\u4ee3\u6570$\\mathfrak{spin}(V)$\u5230$\\mathfrak{gl}(V)$\u4e4b\u95f4\u7684\u540c\u6001\u3002\u4e0b\u9762\uff0c\u6211\u4eec\u6765\u9a8c\u8bc1$\\tau(a)\\in \\mathfrak{so}(V)$\u5bf9\u4efb\u610f\u7684$a\\in \\mathrm{Cl}^2(V)$\u6210\u7acb\u3002<\/p>\n

          \u4e8b\u5b9e\u4e0a\uff0c \u7531\u4e8e$\\mathfrak{so}(V)$\u7684\u674e\u4ee3\u6570\u4e3a$\\mathfrak{gl}(V)$\u4e2d\u7684\u53cd\u5bf9\u79f0\u77e9\u9635\uff0c\u6ee1\u8db3
          \n\\[
          \n\\langle \\tau(a)v,w \\rangle+\\langle v,\\tau(a)w \\rangle=0.
          \n\\]
          \n\u4e0a\u8ff0\u7b49\u5f0f\u53ef\u4ee5\u76f4\u63a5\u9a8c\u8bc1\u5982\u4e0b\uff1a
          \n\\begin{align*}
          \n\\langle \\tau(a)v,w \\rangle+\\langle v,\\tau(a)w \\rangle
          \n&=\\langle [a,v],w \\rangle+\\langle v,[a,w] \\rangle\\\\
          \n&=-\\frac{1}{2}\\left( [a,v]\\cdot w+w\\cdot[a,v]+v\\cdot[a,w]+[a,w]\\cdot v \\right)\\\\
          \n&=-\\frac{1}{2}\\big( a\\cdot v\\cdot w-v\\cdot a\\cdot w+w\\cdot a\\cdot v-w\\cdot v\\cdot a\\\\
          \n&\\qquad+v\\cdot a\\cdot w-v\\cdot w\\cdot a+a\\cdot w\\cdot v-w\\cdot a\\cdot v \\big)\\\\
          \n&= a\\cdot\\langle v,w \\rangle-\\langle v,w\\rangle\\cdot a=0.
          \n\\end{align*}
          \n\u6700\u540e\uff0c\u7531\u4e8e$\\tau: \\mathfrak{spin}(V)\\to \\mathfrak{gl}(V)$\u662f\u674e\u4ee3\u6570\u7684\u540c\u6001\uff0c\u6211\u4eec\u77e5\u9053
          \n\\[
          \n\\ker\\tau=\\left\\{ a\\in \\mathfrak{spin}(V)=\\mathrm{Cl}^2(V):\\tau(a)=0\\in \\mathfrak{gl}(V) \\right\\}
          \n\\]
          \n\u662f\u674e\u4ee3\u6570$\\mathfrak{spin}(V)$\u7684\u4e00\u4e2a\u7406\u60f3\u3002\u800c\u4e14
          \n\\[
          \n0=\\tau(a)v:=[a,v]=a \\cdot v-v\\cdot a,
          \n\\]
          \n\u4ee4$a=a_0+a_ie_i+a_{ij}e_i\\cdot e_j$, \u5219\u5f53$v=e_k\\in \\mathrm{Cl}^1(V)$\u65f6\uff0c
          \n\\[
          \na\\cdot v=v\\cdot a\\iff a_0 e_k+a_i e_i\\cdot e_k+a_{ij}e_{i}\\cdot e_j\\cdot e_k=a_0e_k+a_ie_k\\cdot e_i+a_{ij}e_k\\cdot e_i\\cdot e_j.
          \n\\]
          \n\u5373
          \n\\[
          \n\\sum_i a_i (e_i\\cdot e_k-e_k\\cdot e_i)=0,\\quad
          \n\\sum_{i,j}a_{ij}(e_i\\cdot e_j\\cdot e_k-e_k\\cdot e_i\\cdot e_j)=0.
          \n\\]
          \n\u6545\u5f53$i\\neq k$\u65f6\uff0c$a_i=0$; $i=k$\u65f6\uff0c$\\sum_j a_{kj}(e_j+e_j)=0$\uff0c \u6545$a_{kj}=0$. \u7531$k$\u7684\u4efb\u610f\u6027\uff0c\u6211\u4eec\u5f97\u5230$a=a_0\\in \\mathrm{Cl}^0(V)=\\mathbb{R}$.
          \n<\/div><\/p>\n","protected":false},"excerpt":{"rendered":"

          1. \u4ee3\u6570\u7684\u5b9a\u4e49 \u5b9a\u4e49 1. \u5047\u8bbe$K$\u662f\u4e00\u4e2a\u57df\uff0c$V$\u662f$K$\u4e0a\u7684\u4e00\u4e2a\u7ebf\u6027\u7a7a\u95f4\u3002\u82e5\u5b58\u5728\u4e8c\u5143\u8fd0\u7b97… \u7ee7\u7eed\u9605\u8bfbClifford\u4ee3\u6570\u4e0e\u65cb\u91cf\u7fa4<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[190,191,193,192],"class_list":["post-1039","post","type-post","status-publish","format-standard","hentry","category-math","tag-clifford","tag-daishu","tag-shang","tag-lixiang","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1039","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1039"}],"version-history":[{"count":33,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1039\/revisions"}],"predecessor-version":[{"id":1152,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1039\/revisions\/1152"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1039"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1039"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1039"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}