{"id":3586,"date":"2013-07-31T21:22:55","date_gmt":"2013-07-31T13:22:55","guid":{"rendered":"http:\/\/lttt.blog.ustc.edu.cn\/?p=3586"},"modified":"2013-07-31T21:22:55","modified_gmt":"2013-07-31T13:22:55","slug":"clifford%e4%bb%a3%e6%95%b0","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=3586","title":{"rendered":"Clifford\u4ee3\u6570"},"content":{"rendered":"

\u6700\u8fd1\u5b66\u4e60X.N. Ma\u7684Index theory, \u9700\u8981\u70b9Clifford\u4ee3\u6570\u7684\u77e5\u8bc6, \u5728\u8fd9\u91cc\u8bb0\u5f55\u4e0b.<\/p>\n

Clifford\u4ee3\u6570\u7684\u5386\u53f2<\/h4>\n

200\u5e74\u6765\uff0c\u8bb8\u591a\u4eba\u4ece\u4e0d\u540c\u65b9\u9762\u7814\u7a76\u8fc7Clifford\u4ee3\u6570\u548c\u81ea\u65cb\u8868\u793a\uff0c\u4f46\u8ba4\u8bc6\u5230\u5b83\u4eec\u5904\u5728\u6574\u4e2a\u6570\u5b66\/\u6570\u5b66\u7269\u7406\u7684\u6838\u5fc3\u5219\u662f\u665a\u8fd1\u5f97\u591a\u7684\u4e8b\u3002\u4e0b\u9762\u662f\u4e00\u4efd(\u4e0d\u5b8c\u5907\u7684)\u5386\u53f2\u56de\u987e\u3002<\/p>\n\n\n\n\n\n\n\n\n
Clifford\u4ee3\u6570\u7684\u521d\u7565\u5386\u53f2\u56de\u987e<\/th>\n<\/th>\n<\/tr>\n<\/thead>\n
\u8d85\u590d\u6570\u7cfb<\/td>\nHamilton, Grassmann, Clifford<\/td>\n<\/tr>\n
\u8868\u793a\u8bba\uff0cLie\u7fa4\uff0cBott\u5468\u671f\u6027<\/td>\nE.Cartan, Weyl, Chevalley, Bott<\/td>\n<\/tr>\n
Riemann\u51e0\u4f55<\/td>\nE. Cartan, Berger<\/td>\n<\/tr>\n
Dirac\u7b97\u5b50\uff0c\u91cf\u5b50\u573a\u8bba\uff0c\u8d85\u5bf9\u79f0<\/td>\nDirac, Atiyah, Singer, Witten<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u56de\u5fc6\u7ed9\u5b9a\u4e00\u4e2a$F$-\u5411\u91cf\u7a7a\u95f4$V$, \u5176\u4e0a\u53ef\u5b9a\u4e49\u53d6\u503c\u4e8e\u6570\u57df$F$\u7684\u4e8c\u6b21\u578b$q:V\\times V\\to F$\u662f$V\\times V$\u4e0a\u7684\u53cc\u7ebf\u6027\u51fd\u6570. \u7531\u6b64, \u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u5ef6\u62d3\u5230\u5f20\u91cf\u7a7a\u95f4(\u5173\u4e8e\u5f20\u91cf\u7a7a\u95f4\u7684\u5b9a\u4e49, \u6211\u8fd9\u91ccrefer to 1<\/a>, P54-55, \u5b83\u662f\u4e00\u7ed3\u5408\u4ee3\u6570)$T(V)=V\\otimes V$\u4e0a\u53bb, \u4ecd\u8bb0\u4e3a$q$.<\/p>\n

Clifford\u4ee3\u6570\u7684\u5b9a\u4e49<\/h4>\n

\\begin{defn} \u7ed9\u5b9a\u6570\u57df$F$\u4e0a\u7684(\u6709\u9650\u7ef4)\u5411\u91cf\u7a7a\u95f4$V$, \u4ee5\u53ca$V$\u4e0a\u4efb\u4e00\u4e8c\u6b21\u578b$q$\uff0c\u5219$V$\u5173\u4e8e$q$\u7684**Clifford\u4ee3\u6570**$Cl(V,q)$\u5b9a\u4e49\u4e3a$T(V)$\u6a21\u53bb\u53cc\u8fb9\u7406\u60f3$I=\\set{v\\in V|v\\otimes v+q(v,v)}$. \u7531\u4e8e\u7ebf\u6027\u7a7a\u95f4\u4e4b\u95f4\u4fdd\u6301\u4e8c\u6b21\u578b\u7684\u7ebf\u6027\u6620\u5c04\u53ef\u552f\u4e00\u6269\u5f20\u4e3a\u76f8\u5e94Clifford\u4ee3\u6570\u4e4b\u95f4\u7684\u540c\u6001, \u4e0a\u8ff0\u5b9a\u4e49\u7ed9\u51fa$(V,q)$\u5230\u7ed3\u5408\u4ee3\u6570\u7684**Clifford\u51fd\u5b50**. \\end{defn} \u6ce8\u610f\u5230\u5982\u679c$\\char F\\neq2$, \u5219$u\\cdot v+v\\cdot u=-2q(u,v)$, \u5176\u4e2d$2q(u,v)=q(u+v,u+v)-q(u,u)-q(v,v)$\u79f0\u4e3a\u6781\u5316\u6052\u7b49\u5f0f<\/strong>. \u4e8b\u5b9e\u4e0a, \\begin{align*} -2q(u,v)&=-(q(u+v,u+v)-q(u,u)-q(v,v))\\\\ &=(u+v)\\otimes (u+v)-u\\otimes u-v\\otimes v\\\\ &=u\\otimes v+v\\otimes u\\\\ &=u\\cdot v+v\\cdot u. \\end{align*} \u6211\u4eec\u4e8b\u5b9e\u4e0a\u4e5f\u53ef\u4ee5\u5c06\u8fd9\u4e00\u6027\u8d28\u4f5c\u4e3aClifford\u4ee3\u6570\u7684\u5b9a\u4e49, \u5373\u5728\u5f20\u91cf\u4ee3\u6570\u7684\u5b9a\u4e49\u4e2d\u591a\u4e86\u4e00\u4e2a\u53d8\u4e3a\u96f6\u7684”\u6cd5\u5219”.<\/p>\n

Clifford\u4ee3\u6570\u7684\u540c\u6784<\/h4>\n

Clifford\u4ee3\u6570\u9644\u5e26\u6709\u5982\u4e0b\u51e0\u79cd\u540c\u6784:<\/p>\n

    \n
  1. \u53cd\u5c04\u81ea\u540c\u6784$\\alpha$: \u7ebf\u6027\u6620\u5c04$\\alpha(v)=-v$\u7684\u6269\u5f20;<\/li>\n
  2. \u8f6c\u7f6e\u53cd\u81ea\u540c\u6784:$x\\mapsto x^t$, $x_1\\otimes \\cdots\\otimes x_k\\mapsto x_k\\otimes \\cdots x_1$;<\/li>\n
  3. \u5171\u8f6d\u53cd\u81ea\u540c\u6784: $\\bar x=\\alpha(x^t)$.<\/li>\n<\/ol>\n

    \u4e0b\u9762\u662f2\u4e2a\u6709\u6df1\u523b\u7269\u7406\u80cc\u666f\u7684\u6027\u8d28:<\/p>\n

      \n
    1. Clifford\u4ee3\u6570\u63a8\u5e7f\u4e86\u5916\u4ee3\u6570: \u5f53\u6211\u4eec\u53d6$q=0$\u65f6, \u6b64\u65f6Clifford\u4ee3\u6570\u5c31\u662f\u5916\u4ee3\u6570<\/strong><\/a>. \u66f4\u4e00\u822c\u5730, $T(V)$\u5728$Cl(V,q)$\u4e0a\u8bf1\u5bfc\u4e00\u4e2a\u6ee4\u8fc7\u4ee3\u6570<\/strong><\/a>\u7ed3\u6784\uff0c\u5916\u4ee3\u6570$\\wedge(V)$\u4f5c\u4e3a\u5176\u4f34\u968f\u5206\u6b21\u4ee3\u6570<\/strong>\u540c\u6784\u4e8e$Cl(V,q)$. Clifford\u4ee3\u6570\u662f\u63cf\u8ff0Fermi\u5b50\u7684\u5408\u9002\u4ee3\u6570\u7ed3\u6784\uff0c\u5c31\u8fd9\u4e2a\u610f\u4e49\u4e0a\u6765\u8bf4\u5b83\u662f\u5916\u4ee3\u6570\u7684\u201c\u91cf\u5b50\u5316\u201d.. Bose\u5b50\u7531Weyl\u4ee3\u6570<\/strong><\/a>\u63cf\u8ff0\uff0c\u5b83\u662f\u5bf9\u79f0\u4ee3\u6570<\/strong><\/a>\u7684\u201c\u91cf\u5b50\u5316\u201d.<\/li>\n
    2. \u9664\u6ee4\u8fc7\u4ee3\u6570\u7ed3\u6784\u5916, $T(V)$\u8fd8\u5728$Cl(V,q)$\u4e0a\u8bf1\u5bfc\u4e00\u4e2a$\\Z_2$\u5206\u6b21\u4ee3\u6570\u7ed3\u6784\u3002\u5076\u90e8\u5206\u8bb0\u4e3a$Cl^0(V,q)$(\u5b83\u662f\u4e00\u4e2aClifford\u5b50\u4ee3\u6570)\uff0c\u5947\u90e8\u5206\u8bb0\u4e3a$Cl^1(V,q)$, \u5b83\u4eec\u5206\u522b\u5bf9\u5e94$\\alpha$\u7684\u6b63\u8d1f\u7279\u5f81\u5b50\u7a7a\u95f4.<\/li>\n<\/ol>\n

      Atiyah\u7b49\u4eba\u5f15\u5165\u5206\u6b21\u7ed3\u6784\u7684\u521d\u8877\u4e4b\u4e00\u662f\u57fa\u4e8e\u5982\u4e0b\u4f18\u7f8e<\/strong>\u6027\u8d28: \u8bbe$V=V_1\\oplus V_2$\u662f\u57fa\u4e8e$q$\u7684\u6b63\u4ea4\u5206\u89e3, $q_i=q|_{V_i}$, \u5219\u6709\u540c\u6784$Cl(V,q)\\cong Cl(V_1,q_1)\\hat\\otimes Cl(V_2,q_2)$, \u540c\u6784\u6620\u5c04\u7531$f:V_1\\oplus V_2\\to Cl(V_1,q_1)\\hat\\otimes Cl(V_2,q_2)$\u8bf1\u5bfc, \u5176\u4e2d$f(v_1,v_2)=v_1\\otimes 1+1\\otimes v_2$, \u8fd9\u91cc$\\hat\\otimes$\u4e3a$\\Z_2$\u5206\u6b21\u5f20\u91cf\u79ef. \u57fa\u4e8eWitten\u7b49\u4eba\u7684\u5de5\u4f5c, \u73b0\u5728\u719f\u77e5\u8fd9\u4e2a\u5206\u6b21\u7ed3\u6784\u5bf9\u5e94\u8d85\u5bf9\u79f0<\/strong><\/a>.<\/p>\n

      \u4e00\u4e2a\u5177\u4f53\u7684\u4f8b\u5b50<\/h4>\n

      \\begin{examp} \u8003\u5bdf\u5982\u4e0b\u7684$\\R^{p,q}$\u4e0a\u7684\u4e8c\u6b21\u578b $$ q(x,y)=\\sum_{i=1}^qx_iy_i-\\sum_{i=p+1}^{p+q}x_iy_i, $$ \u5176\u76f8\u5e94\u7684Clifford\u4ee3\u6570\u8bb0\u4e3a$Cl_{p,q}\\eqdef Cl(\\R^{p,q},q)$, $Cl_n\\eqdef Cl_{n,0}$, $Cl_n^*\\eqdef Cl_{0,n}$. \u6ce8\u610f$R^{3,1}$\u4e0a\u7684$q$\u6070\u597d\u662fMinkowski\u5185\u79ef. \u53ef\u4ee5\u53d1\u73b0$Cl_0=\\R$, $Cl_{1}=\\R\\oplus \\R$, $Cl_{1}^*=\\C$, $Cl_{2}=M_2(\\R)$, $Cl_{1,1}=M_2(\\R)$, $Cl_{2}^*=\\H$, $Cl_{3}=M_2(\\C)$, $Cl_3^*=\\H\\oplus \\H$. \\end{examp}<\/p>\n

      \u5b9eClifford\u4ee3\u6570\u7684\u5206\u7c7b<\/h4>\n

      \u7531\u4e8e\u4efb\u4f55\u4e00\u4e2a\u5b9e\u4e8c\u6b21\u578b\u90fd\u53ef\u7ea6\u5316\u5230\u4f8b\u5b50\u4e2d\u7684\u60c5\u5f62, \u4e8e\u662f\u6240\u6709\u6709\u9650\u7ef4\u5b9eClifford\u4ee3\u6570\u5728\u540c\u6784\u610f\u4e49\u4e0b\u5c31\u53ea\u6709\u5f62\u5982$Cl_{p,q}$\u7684\u5f62\u5f0f. \u95ee\u9898\u662f$Cl_{p,q}$\u662f\u4ec0\u4e48\u5462? \u66f4\u4e00\u822c\u5730, \u6211\u4eec\u6709
      \n\\begin{prop}[C.f. [^3],P175] \u5bf9\u4efb\u610f\u7684$p,q\\in\\N$, \u6709 \\begin{align*} Cl_{p+1,p+1}&=Cl_{p,q}\\otimes_\\R M_2(\\R),\\\\ Cl_{p+2,q}&=Cl_{q,p}\\otimes_\\R M_2(\\R),\\\\ Cl_{p,q+2}&=Cl_{q,p}\\otimes_\\R \\H,\\\\ Cl_{p+4,q}&=Cl_{p,q}\\otimes_\\R M_2(\\H)=Cl_{p,q+4},\\\\ \\end{align*} \\end{prop}
      \n\u7531\u6b64, \u53ef\u5f97\u5982\u4e0b\u63a8\u8bba
      \n\\begin{cor} \u5bf9\u4efb\u610f\u7684$p,q\\in\\N$, \u6211\u4eec\u6709 $$ Cl_{p+8,q}=Cl_{p,q}\\otimes_\\R M_{16}(\\R)=Cl_{p,q+8}. $$ \\end{cor}
      \n\u7531\u6b64, \u53ef\u89c1\u6211\u4eec\u53ea\u9700\u77e5\u9053$Cl_{n,0}$, $n=0,1,\\ldots,7$\u4fbf\u53ef\u77e5\u9053\u6240\u6709\u7684$Cl_{p,q}$. \u5229\u7528\u524d\u9762\u7684\u547d\u9898\u548c\u63a8\u8bba\u4ee5\u53ca\u4f8b\u5b50\u5e76\u6ce8\u610f\u5230$H\\otimes_\\R\\C=M_2(\\C)$, \u4ee5\u53ca$\\H\\otimes_\\R\\H=M_4(\\R)$\u53ef\u5f97\u5982\u4e0b\u5217\u8868:<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n
      n<\/th>\n$Cl_{n,0}$<\/th>\n$Cl_{n,0}^0$<\/th>\n<\/tr>\n<\/thead>\n
      1<\/td>\n$\\R\\otimes \\R$<\/td>\n$\\R$<\/td>\n<\/tr>\n
      2<\/td>\n$M_2(\\R)$<\/td>\n$\\C$<\/td>\n<\/tr>\n
      3<\/td>\n$M_2(\\C)$<\/td>\n$\\H$<\/td>\n<\/tr>\n
      4<\/td>\n$M_2(\\H)$<\/td>\n$\\H\\oplus \\H$<\/td>\n<\/tr>\n
      5<\/td>\n$M_2(\\H\\oplus \\H)$<\/td>\n$M_2(\\H)$<\/td>\n<\/tr>\n
      6<\/td>\n$M_4(\\H)$<\/td>\n$M_4(\\C)$<\/td>\n<\/tr>\n
      7<\/td>\n$M_8(\\C)$<\/td>\n$M_8(\\R)$<\/td>\n<\/tr>\n
      8<\/td>\n$M_16(\\R)$<\/td>\n$M_8(\\R\\oplus\\R)$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      \u8fd9\u6837, \u7ed3\u5408\u5982\u4e0b\u7684Spinorial clock, \u6211\u4eec\u53ef\u4ee5\u7ed9\u51fa\u5b9eClifford\u4ee3\u6570\u7684\u5206\u7c7b: \"**Spinorial \\begin{thm}[\u5b9eClifford\u4ee3\u6570\u7684\u5206\u7c7b, C.f. 2<\/a>P177]\u6240\u6709\u7684\u6709\u9650\u7ef4\u5b9eClifford\u4ee3\u6570\u90fd\u7531Spinorial clock\u5b8c\u5168\u5206\u7c7b. \u5206\u7c7b\u6b65\u9aa4\u5982\u4e0b:<\/p>\n

        \n
      1. \u9996\u5148\u8ba1\u7b97$m=p-q\\mod 8$, \u5e76\u5b9a\u4f4d\u5176\u5728Spinorial clock\u4e2d\u7684\u4f4d\u7f6e: $A\\xrightarrow{m}B$;<\/li>\n
      2. \u8ba1\u7b97\u6574\u6570$n$, \u4f7f\u5f97$\\dim_\\R M_n(B)=2^{p+q}$;<\/li>\n
      3. \u6700\u540e\u5f97\u5230: $Cl_{p,q}=M_n(B)$.<\/li>\n<\/ol>\n

        \u6b64\u5916, \u5f53$n$\u4e3a\u5947\u6570\u65f6, Clifford\u4ee3\u6570\u7684$\\Z_2$\u5206\u6b21\u7684\u5076\u90e8\u5206$Cl_{p,q}^0=M_n(A)$, \u5f53$n$\u4e3a\u5076\u6570\u65f6, $Cl_{p,q}^0=M_{n\/2}(A)$. \\end{thm}<\/p>\n

        \u53c2\u8003\u6587\u732e<\/h4>\n
          \n
        1. Fight with Infinity<\/a>\u7684\u535a\u5ba2, \u5176\u4e2d\u4ed6\u53c2\u8003\u4e86\u4e13\u84573<\/a>\u4ee5\u53ca\u4e00\u7bc7\u91cd\u8981\u7684\u8bba\u65874<\/a>.<\/li>\n
        2. Academic Cesspit<\/a>\u7684\u535a\u5ba2, \u4ed6\u4e3b\u8981\u53c2\u8003\u4e865<\/a>.<\/li>\n<\/ol>\n
          \n
          \n
            \n
          1. Warner, Frank W. Foundations of differentiable manifolds and Lie groups<\/em>. Vol. 94. Springer, 1971. \u21a9<\/a><\/li>\n
          2. Gracia-Bond\u00eda, Jos\u00e9 M., Joseph C. V\u00e1rilly, and H\u00e9ctor Figueroa. Elements of noncommutative geometry<\/em><\/a>. Springer, 2001. \u21a9<\/a><\/li>\n
          3. Lawson Jr, H. Blaine, and Marie-Louise Michelsohn. “Spin geometry<\/a>, volume 38 of Princeton Mathematical Series.” (1989). \u21a9<\/a><\/li>\n
          4. Atiyah, Michael F., Raoul Bott, and Arnold Shapiro. “Clifford modules<\/a>.” Topology 3 (1964): 3-38. \u21a9<\/a><\/li>\n
          5. Gracia-Bond\u00eda, Jos\u00e9 M., Joseph C. V\u00e1rilly, and H\u00e9ctor Figueroa. Elements of noncommutative geometry<\/em><\/a>. Springer, 2001. \u21a9<\/a><\/li>\n<\/ol>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

            \u6700\u8fd1\u5b66\u4e60X.N. Ma\u7684Index theory, \u9700\u8981\u70b9Clifford\u4ee3\u6570\u7684 …<\/p>\n

            Clifford\u4ee3\u6570<\/span> \u9605\u8bfb\u66f4\u591a »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[259,260],"class_list":["post-3586","post","type-post","status-publish","format-standard","hentry","category-mathnotes","tag-clifford-algebra","tag-clifford"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/3586","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3586"}],"version-history":[{"count":1,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/3586\/revisions"}],"predecessor-version":[{"id":5350,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/3586\/revisions\/5350"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3586"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3586"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3586"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}