\u901a\u8fc7\u8ba1\u7b97, \u597d\u50cfYang–Mills\u65b9\u7a0b\u5373\u4f7f\u5728\u5e93\u4f26\u89c4\u8303\u4e0b\u4e5f\u4e0d\u662f\u4e25\u683c\u692d\u5706\u7684\u554a?<\/p>\n
\u6211\u8bb0\u5f97Yang–Mills\u65b9\u7a0b\u4e3b\u9879\u662f$dd^*A+d^*dA$, \u5176\u4e2d$A=A_idx^i$. \u5219\u5176\u5f31\u5f62\u5f0f\u662f
\n$$
\n\\int\\langle d^*A,d^*B\\rangle+\\langle dA,dB\\rangle=0,\\quad\\forall B=B_jdx^j\\in C_0^\\infty.
\n$$
\n\u76f4\u63a5\u8ba1\u7b97, \u6211\u4eec\u77e5\u9053
\n\\begin{align*}
\ndA&=d(A_idx^i)=\\partial_jA_idx^j\\wedge dx^i\\\\
\ndB&=\\partial_k B_ld^k\\wedge dx^l\\\\
\nd^*A&=-*d*A=-*d\\left((-1)^{i-1}A_idx^1\\wedge\\cdots\\wedge\\widehat{dx^i}\\wedge\\cdots\\wedge dx^n\\right)\\\\
\n&=-*(\\partial_iA_idx^1\\wedge\\cdots dx^n)\\\\
\n&=-\\partial_iA_i\\\\
\nd^*B&=-\\partial_kB_k.
\n\\end{align*}<\/p>\n
\u56e0\u6b64,
\n\\begin{align*}
\n0&=\\int\\langle\\partial_j A_i,\\partial_kB_l\\rangle\\langle dx^j\\wedge dx^i,dx^k\\wedge dx^l\\rangle+\\langle\\partial_kA_k,\\partial_l B_l\\rangle\\\\
\n&=\\int\\langle\\partial_j A_i,\\partial_kB_l\\rangle(\\delta_{jk}\\delta_{il}-\\delta_{jl}\\delta_{ik})+\\langle\\partial_kA_k,\\partial_l B_l\\rangle\\\\
\n&=\\int\\langle\\partial_j A_i,\\partial_jB_i-\\partial_iB_j\\rangle+\\langle\\partial_kA_k,\\partial_l B_l\\rangle\\\\
\n&=\\int\\langle\\partial_jA_i-\\partial_iA_j,\\partial_jB_i\\rangle+\\langle\\partial_kA_k,\\partial_l B_l\\rangle\\\\
\n&=\\int\\langle\\partial_jA_i-\\partial_iA_j+\\delta_{ij}\\partial_mA_m,\\partial_jB_i\\rangle.
\n\\end{align*}
\n\u8fd9\u8868\u660e, \u82e5\u8bb0$\\xi_{j}^i=\\partial_jB_i$, $u^i=A_i$, $p_j^i=\\partial_jA_i$\u4ee5\u53ca$q_i^j(x,u,p)=p_j^i-p_i^j+\\delta_{ij}p_m^m$, \u5219\u5f31\u5f62\u5f0f\u53ef\u6539\u5199\u4e3a\u5982\u4e0b\u7684\u6807\u51c6\u5f62\u5f0f(\u65b9\u4fbf\u5e94\u7528Morrey\u7684\u5b9a\u74061.1.11):
\n$$
\n\\int q_i^j(x,A,\\nabla A)\\xi_j^i=0,
\n$$
\n\u6b64\u65f6, \u65b9\u7a0b\u7684\u692d\u5706\u6027\u53ef\u9a8c\u8bc1\u5982\u4e0b
\n\\begin{align*}
\n\\partial_{p_k^l}q_i^j\\pi_k^l\\pi_j^i&=\\pi_l^k\\pi_j^i(\\delta_{jk}\\delta_{il}-\\delta_{jl}\\delta_{ik}+\\delta_{ij}\\delta_{kl})\\\\
\n&=\\pi_j^i\\pi_j^i-\\pi_j^i\\pi_i^j+\\pi_i^i\\pi_j^j\\\\
\n&=\\sum_{i<j}\\pi_j^i\\pi_j^i+\\pi_i^j\\pi_i^j-\\pi_j^i\\pi_i^j-\\pi_i^j\\pi_j^i+(\\sum_i\\pi_i^i)^2\\\\
\n&=\\sum_{i<j}\\pi_j^i(\\pi_j^i-\\pi_i^j)+\\pi_i^j(\\pi_i^j-\\pi_j^i)+(\\sum_i\\pi_i^i)^2\\\\
\n&=\\sum_{i<j}(\\pi_j^i-\\pi_i^j)^2+(\\sum_i\\pi_i^i)^2
\n\\end{align*}
\n\u8fd9\u8868\u660e\u5b83\u5c06\u5728\u8ff9\u4e3a\u96f6\u7684\u5bf9\u79f0\u77e9\u9635\u4e0a\u9000\u5316.<\/p>\n","protected":false},"excerpt":{"rendered":"
\u901a\u8fc7\u8ba1\u7b97, \u597d\u50cfYang–Mills\u65b9\u7a0b\u5373\u4f7f\u5728\u5e93\u4f26\u89c4\u8303\u4e0b\u4e5f\u4e0d\u662f\u4e25\u683c\u692d\u5706\u7684\u554a? \u6211\u8bb0\u5f97Yang… \u7ee7\u7eed\u9605\u8bfbYang–Mills\u65b9\u7a0b\u7684\u692d\u5706\u6027\u9a8c\u8bc1<\/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":[81,82],"class_list":["post-397","post","type-post","status-publish","format-standard","hentry","category-math","tag-yang-mills","tag-tuoyuanxing","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/397","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=397"}],"version-history":[{"count":3,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/397\/revisions"}],"predecessor-version":[{"id":400,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/397\/revisions\/400"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=397"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=397"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=397"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}