流形上的旋量和旋量场（自旋系数和克氏符）
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上两篇，引入旋量概念以及旋量代数，以及旋量的一个直观的几何解释。
自旋系数，是和克氏符对标的概念。引入了新的记号，复用度规符号 \(\textcolor{red}{g_\mu^{\ \ a}}\) 来表示
矢量基底； \(\textcolor{red}{g_\mu^{\ \ AA'}}\) 表示泡利矩阵，复用旋量度规符号 \(\textcolor{red}{\epsilon_\Sigma^{\ \ A}}\) 来表示旋量基底;张量指标和旋量指标混用。
本篇虽然看起来公式“巨复杂”，但是掌握了指标的提升、降低、置换、缩并，完全可以无脑写出。
度规符号的复用【新记号】
首先，我们引入新记号【旧瓶新装】:
\[ \begin{aligned}\textcolor{red}{g_{ \mu}^{\ \ a}}=g^a_{\ \ \mu}\textcolor{blue}{\overset{\Delta}{=}}&\textcolor{red}{(e_\mu)^a}\\ \textcolor{red}{g^\mu_{\ \ a}}=g_a^{\ \ \mu}=&g^{\mu\upsilon}g_{ab}g_{\upsilon}^{\ \ b}=\textcolor{red}{(e^\mu)_a}\end{aligned} \]对应 \(g_{ab}\) 是度规张量， \(g_{\mu\upsilon}\) 是度规分量， \(\textcolor{red}{g_{ \mu}^{\ \ a}}\) 就是张量和分量之间建立联系的标架（基底）,比如：
\[ \begin{aligned}\theta^a=g_\mu^{\ \ a}\theta^\mu\quad \textcolor{red}{矢量按基底展开} \\ \theta^\mu=g^\mu_{\ \ a}\theta^a \quad \textcolor{red}{矢量分量的表示}\end{aligned} \]由于可见， \(\textcolor{red}{g_{ \mu}^{\ \ a}}\) 不仅仅表示标架，还可简单理解成抽象指标和具体指标互换的算符。 相对应的有： \(g^\mu_{\ \ \upsilon}=g^{\ \ \mu}_{\upsilon}=\delta^\mu_\upsilon\) 可将具体指标互换， \(g^a_{\ \ b}=g^{\ \ a}_{b}=\delta^a_b\) 则可将抽象指标互换。【所以，通过将 \(g\) 复用来表示标架（基底）是很合理的】
这个约定（复用），虽然没有本质上的区别，但是好处是：方便无脑进行指标的提升和下降，再加上水平置换的处理。
理解克氏符
【注意】当涉及标架的“缩并”时，导数算符和此种“缩并”是不可交换的。 其它情况，导数算符和缩并是可交换的。
考察 \(\textcolor{blue}{\nabla_a\theta^b}\) 的分量 \(\boxed{g_\mu^{\ \ a}g^\upsilon_{\ \ b}(\textcolor{blue}{\nabla_a\theta^b})}\ne \nabla_\mu\theta^\upsilon=\partial_\mu\theta^\upsilon\)
\[ \begin{aligned}g_\mu^{\ \ a}g^\upsilon_{\ \ b}(\textcolor{blue}{\nabla_a\theta^b})=&g_\mu^{\ \ a}g^\upsilon_{\ \ b}\nabla_a(\theta^\sigma g_\sigma^{\ \ b}) \quad \textcolor{red}{将矢量按基底展开}\\ =&g_\mu^{\ \ a}g^\upsilon_{\ \ b}g_\sigma^{\ \ b}\nabla_a\theta^\sigma +g_\mu^{\ \ a}g^\upsilon_{\ \ b}\theta^\sigma\nabla_a g_\sigma^{\ \ b}\quad \textcolor{red}{莱布尼茨律}\\ =&g_\mu^{\ \ a}g^\upsilon_{\ \ \sigma}\boxed{\partial_a\theta^\sigma} +\theta^\sigma (\textcolor{green}{g^\upsilon_{\ \ b}\nabla_\mu g_\sigma^{\ \ b}})\quad \textcolor{red}{缩并,标量场导数算符}\\ =&\boxed{\partial_\mu\theta^\upsilon +\theta^\sigma \textcolor{green}{\Gamma^\upsilon_{\ \ \mu\sigma}}}\end{aligned} \]其中
\(\textcolor{green}{\Gamma^\upsilon_{\ \ \mu\sigma}}\)
就是克氏符
也可改写成：
\[ \boxed{\nabla_\mu g_\sigma^{\ \ a}=\Gamma^\upsilon_{\ \ \mu\sigma}g_\upsilon^{\ \ a}\\ \text{或} \\ \nabla_\mu(e_\sigma)^a =\Gamma^\upsilon_{\ \ \mu\sigma}(e_\upsilon)^a} \]由此得到，克氏符的理解：克氏符就是导数算符作用于标架场的系数。
利用标架，可将分量方程写成张量方程
\[ \nabla_a\theta^b=\partial_a\theta^b+\theta^c \Gamma^b_{\ \ ac} \]旋量度规符号的复用【新记号】
和前面类似，不妨复用旋量度规符号
\(\epsilon\)
，来表示旋量基底
其次，旋量张量加"撇"（ \('\) ）的映射记号【要和指标加“撇”区别开来】：
\[ \begin{aligned}&\omicron^A\mapsto(\omicron^A)'\textcolor{blue}{\overset{\Delta}{=}}i \iota^A\quad \iota^A\mapsto(\iota^A)'=i \omicron^A\\ &\bar{\omicron}^{A'}\mapsto(\bar{\omicron}^{A'})'=-i \bar{\iota}^{A'}\quad \bar{\iota}^{A'}\mapsto(\bar{\iota}^{A'})'=-i \bar{\omicron}^{A}\end{aligned} \]对一个旋量张量进行加"撇"（ \('\) ）映射，就是对其基底表示的所有基底进行加"撇"（ \('\) ）映射的结果。
自旋系数
有了旋量基底的方便表示，分量可用缩并的方式表示。
比如，旋量 \(\kappa^A\) 的分量：
\[ \kappa^\Sigma=\kappa^A\epsilon_{A}^{\ \ \Sigma} \]比如，导数算符 \(\nabla_{AA'}\) 的分量：
\[ \nabla_{\Sigma\Sigma'}=\epsilon_\Sigma^{\ \ A}\bar{\epsilon}_{\Sigma'}^{\ \ A'}\nabla_{AA'} \]要注意，***导数算符作用于旋量张量***的分量 \(\Large \textcolor{red}{不等于}\) 导数算符分量作用于旋量张量分量。
比如，导数算符 \(\nabla_{AA'}\) 作用于协变旋量 \(\kappa^A\) 的分量：
\[ \begin{aligned}\underbrace{\epsilon_{\Sigma}^{\ \ A}\bar{\epsilon}_{\Sigma'}^{\ \ A'}\epsilon_B^{\ \ \Omega}\textcolor{blue}{\nabla_{AA'}\kappa^B}}_{\textcolor{#228B22}{先作用后分量}}&=\epsilon_B^{\ \ \Omega}\nabla_{\Sigma\Sigma'}(\kappa^\Lambda\epsilon_\Lambda^{\ \ B})\\ &=\epsilon_B^{\ \ \Omega}\epsilon_\Lambda^{\ \ B}\nabla_{\Sigma\Sigma'}\kappa^\Lambda+\kappa^\Lambda\epsilon_B^{\ \ \Omega}\nabla_{\Sigma\Sigma'}\epsilon_\Lambda^{\ \ B}\\ &=\underbrace{\textcolor{blue}{\nabla_{\Sigma\Sigma'}\kappa^\Omega}}_{\textcolor{#228B22}{先分量后作用}}+\textcolor{blue}{\kappa^\Lambda}\textcolor{red}{\gamma_{\Sigma\Sigma'\Lambda}^{\ \ \quad \Omega}}\end{aligned} \]其中，红色标记的旋量张量分量，被称作自旋系数：
类似地，导数算符 \(\nabla_{AA'}\) 作用于逆变旋量 \(\mu^A\) 的分量：
\[ \epsilon_\Sigma^{\ \ A}\bar{\epsilon}_{\Sigma'}^{\ \ A'}\epsilon_\Omega^{\ \ B}\textcolor{blue}{\nabla_{AA'}\mu_B}=\textcolor{blue}{\nabla_{\Sigma\Sigma'}\mu_\Omega}-\textcolor{blue}{\mu_\Lambda}\textcolor{red}{\gamma_{\Sigma\Sigma'\Omega}^{\ \ \quad \Lambda}} \]另外两种旋量 \(\phi^{A'}\) 和 \(\zeta_{A'}\) 被作用后的分量：
\[ \epsilon_\Sigma^{\ \ A}\bar{\epsilon}_{\Sigma'}^{\ \ A'}\epsilon_{B'}^{\ \ \Omega'}\textcolor{blue}{\nabla_{AA'}\phi^{B'}}=\textcolor{blue}{\nabla_{\Sigma\Sigma'}\phi^{\Omega'}}+\textcolor{blue}{\phi^{\Lambda'}}\textcolor{red}{\bar{\gamma}_{\Sigma\Sigma'\Lambda'}^{\ \ \ \quad \Omega'}}\\ \epsilon_\Sigma^{\ \ A}\bar{\epsilon}_{\Sigma'}^{\ \ A'}\epsilon_{\Omega'}^{\ \ B'}\textcolor{blue}{\nabla_{AA'}\zeta_{B'}}=\textcolor{blue}{\nabla_{\Sigma\Sigma'}\zeta_{\Omega'}}-\textcolor{blue}{\zeta_{\Lambda'}}\textcolor{red}{\bar{\gamma}_{\Sigma\Sigma'\Omega'}^{\ \ \ \quad \Lambda'}} \]最后，给出任意旋量张量 \(T^{B_1\dots B'_{k'}}_{\quad \qquad C_1\dots C'_{l'}}\) 被 \(\nabla_{AA'}\) 作用后的分量：
\[ \boxed{\begin{aligned}&\epsilon_{\Sigma}^{\ \ A}\bar{\epsilon}_{\Sigma'}^{\ \ A'}\epsilon_{B_1}^{\ \ \Lambda_1}\dots\bar{\epsilon}_{B'_{k'}}^{\ \ \Lambda'_{k'}}\epsilon_{\Delta_1}^{\ \ C_1}\dots\bar{\epsilon}_{\Delta'_{l'}}^{\ \ C'_{l'}}\textcolor{blue}{\nabla_{AA'}T^{B_1\dots B'_{k'}}_{\quad \qquad C_1\dots C'_{l'}}}\\ = & \textcolor{blue}{\nabla_{\Sigma\Sigma'}T^{\Lambda_1\dots \Lambda'_{k'}}_{\quad \qquad \Delta_1\dots \Delta'_{l'}}}\\ \quad & +\sum_{i=1}^k{T^{\Lambda_1\dots \Gamma_i \dots \Lambda'_{k'}}_{\qquad \qquad \Delta_1\dots \Delta'_{l'}}\textcolor{red}{\gamma_{\Sigma\Sigma'\Gamma_i}^{\qquad \Lambda_i}}}+\sum_{i=1}^{k'}{T^{\Lambda_1\dots \Gamma'_i \dots \Lambda'_{k'}}_{\qquad \qquad \Delta_1\dots \Delta'_{l'}}\textcolor{red}{\bar{\gamma}_{\Sigma\Sigma'\Gamma'_i}^{\qquad \Lambda'_i}}} \\ \quad & -\sum_{i=1}^l{T^{\Lambda_1\dots \Lambda'_{k'}}_{\quad \qquad \Delta_1\dots \Gamma_i \dots \Delta'_{l'}}\textcolor{red}{\gamma_{\Sigma\Sigma'\Delta_i}^{\qquad \Gamma_i}}}-\sum_{i=1}^{l'}{T^{\Lambda_1\dots \Lambda'_{k'}}_{\quad \qquad \Delta_1\dots \Gamma'_i \dots \Delta'_{l'}}\textcolor{red}{\bar{\gamma}_{\Sigma\Sigma'\Delta'_i}^{\qquad \Gamma'_i}}} \end{aligned}} \]自旋系数的计算
仔细观察自旋系数的定义【有16个分量】：
\[ \boxed{\gamma_{\Sigma\Sigma'\Lambda}^{\ \ \quad \Omega}=\textcolor{blue}{\epsilon_A^{\ \ \Omega}}\textcolor{red}{\nabla_{\Sigma\Sigma'}}\textcolor{#228B22}{\epsilon_\Lambda^{\ \ A}}=-\epsilon_\Lambda^{\ \ A}\nabla_{\Sigma\Sigma'}\epsilon_A^{\ \ \Omega}}\\ \text{可改写成} \\ \textcolor{red}{\nabla_{\Sigma\Sigma'}}\epsilon_\Lambda^{\ \ A}=\textcolor{red}{\gamma_{\Sigma\Sigma'\Lambda}^{\ \ \quad \Omega}}\epsilon_\Omega^{\ \ \Lambda} \quad \textcolor{blue}{对标}\textcolor{red}{克氏符} \]其中， \(\textcolor{blue}{\epsilon_A^{\ \ \Omega}}=(-\iota_A,\omicron_A)\) 是协变旋量基底， \(\textcolor{#228B22}{\epsilon_\Lambda^{\ \ A}}=(\omicron^A,\iota^A)\) 是逆变旋量基底。为 \(\textcolor{red}{\nabla_{\Sigma\Sigma'}}\) 引入如下记号：
\[ \begin{aligned}\textcolor{red}{D}\textcolor{blue}{\overset{\Delta}{=}}&\nabla_{00'}=\omicron^A\bar{\omicron}^{A'}\nabla_{AA'}=-\textcolor{blue}{l^a}\nabla_a=\bar{D} \\ \textcolor{red}{\delta}\textcolor{blue}{\overset{\Delta}{=}}&\nabla_{01'}=\omicron^A\bar{\iota}^{A'}\nabla_{AA'}=-\textcolor{blue}{m^a}\nabla_a=\bar{\delta}' \\ \textcolor{red}{\delta'}=&\nabla_{10'}=\iota^A\bar{\omicron}^{A'}\nabla_{AA'}=-\textcolor{blue}{\bar{m}^a}\nabla_a=\bar{\delta} \\ \textcolor{red}{D'}=&\nabla_{11'}=\iota^A\bar{\iota}^{A'}\nabla_{AA'}=-\textcolor{blue}{n^a}\nabla_a=\bar{D}'\end{aligned} \]其中蓝色部分
\[ \textcolor{blue}{l^{AA'}}=\omicron^A\bar{\omicron}^{A'}\quad \textcolor{blue}{m^{AA'}}=\omicron^A\bar{\iota}^{A'}\quad \textcolor{blue}{n^{AA'}}=\iota^A\bar{\iota}^{A'}\\ \quad \\ (l^{AA'})'=n^{AA'}\quad (m^{AA'})'=\bar{m}^{AA'}\quad (\bar{m}^{AA'})'=m^{AA'}\quad(n^{AA'})'=l^{AA'} \]利用上面的记号，自旋系数可表示成
\[ \gamma_{\textcolor{red}{\Sigma\Sigma'}\textcolor{blue}{\Lambda}}^{\ \ \quad \textcolor{blue}{\Omega}}=\textcolor{red}{\begin{matrix} 00' \\ 01' \\ 10' \\11' \end{matrix}}\overbrace{\begin{pmatrix} \omicron^A D \iota_A & -\omicron^A D \omicron_A & \iota^A D \iota_A & -\iota^A D \omicron_A \\ \omicron^A\delta\iota_A & -\omicron^A \delta \omicron_A & \iota^A\delta\iota_A & -\iota^A \delta \omicron_A \\ \omicron^A\delta'\iota_A & -\omicron^A \delta' \omicron_A & \iota^A\delta'\iota_A & -\iota^A \delta' \omicron_A \\ \omicron^A D' \iota_A & -\omicron^A D'\omicron_A & \iota^A D'\iota_A & -\iota^A D' \omicron_A \end{pmatrix}}^{\textcolor{blue}{\begin{pmatrix} & 0 \\ 0 & \end{pmatrix}\qquad\begin{pmatrix} & 1 \\ 0 & \end{pmatrix}\qquad\begin{pmatrix} & 0 \\ 1 & \end{pmatrix}\qquad\begin{pmatrix} & 1 \\ 1 & \end{pmatrix}}} \]有了旋量张量加"撇"（ \('\) ）的映射记号，这个自旋系数（16个分量），可以只用8个分量表示：
\[ \gamma_{\textcolor{red}{\Sigma\Sigma'}\textcolor{blue}{\Lambda}}^{\ \ \quad \textcolor{blue}{\Omega}}=\textcolor{red}{\begin{matrix} 00' \\ 01' \\ 10' \\11' \end{matrix}} \overbrace{\begin{pmatrix} \varepsilon & -\kappa & -\tau' & \gamma' \\ \alpha & -\rho & -\sigma' & \beta' \\ \beta & -\sigma & -\rho' & \alpha' \\ \gamma & -\tau & -\kappa' & \varepsilon' \end{pmatrix}}^{\textcolor{blue}{\begin{pmatrix} & 0 \\ 0 & \end{pmatrix}\begin{pmatrix} & 1 \\ 0 & \end{pmatrix}\begin{pmatrix} & 0 \\ 1 & \end{pmatrix}\begin{pmatrix} & 1 \\ 1 & \end{pmatrix}}} \]其中， \(\varepsilon,\alpha,\beta,\gamma,\kappa,\rho,\sigma,\tau\) 和前面的系数对应相等，加"撇"（ \('\) ）的是前面定义的映射。
进而可以写出 \(\gamma_{\textcolor{red}{\Sigma\Sigma'}\textcolor{blue}{\Lambda\Omega}}\) 的分量：
\[ \gamma_{\textcolor{red}{\Sigma\Sigma'}\textcolor{blue}{\Lambda\Omega}}=\textcolor{red}{\begin{matrix} 00' \\ 01' \\ 10' \\11' \end{matrix}} \overbrace{\begin{pmatrix} \kappa & \varepsilon & \varepsilon & -\tau' \\ \rho & \alpha & \alpha & -\sigma' \\ \sigma & \beta & \beta & -\rho' \\ \tau & \gamma & \gamma & -\kappa' \end{pmatrix}}^{\textcolor{blue}{00\qquad 01\qquad 10 \qquad 11}} \]由于我们选择的都是归一化旋量基底，部分系数之间还有关系：
\[ \boxed{\begin{aligned}\alpha=-\beta'\quad \beta=-\alpha' \\ \varepsilon=-\gamma'\quad \gamma=-\varepsilon' \end{aligned}} \]旋量指标和张量指标混合
要允许旋量指标和张量指标的混合，就离不开将旋量和张量联系在一起的泡利矩阵【含系数
\(1/\sqrt{2}\)
】：
由此可见，这个泡利矩阵，也完全可以用度规符号 \(g\) 代替表示
\[ \boxed{\begin{aligned}\textcolor{blue}{g_\mu^{\ \ AA'}}\overset{\Delta}{=}\sigma_\mu^{\ \ AA'} \quad \textcolor{blue}{g_a^{\ \ AA'}}=\sigma_a^{\ \ AA'} \\ \quad \\ \textcolor{green}{g缩并新规：}\textcolor{red}{指标对（AA'）缩并加负号}\end{aligned}} \]在前面的一系列约定下，特别式这个缩并新规，我将很容易将张量指标和旋量指标混合在一起。
首先，导数算符 \(\nabla_a\) 作为一个“协变矢量”，也可用旋量导数算符 \(\nabla_{AA'}\) 表示【按缩并新规也可】：
\[ \nabla_a=-g_a^{\ \ AA'}\nabla_{AA'}\\\nabla_\mu=-g_\mu^{\ \ AA'}\nabla_{AA'} \]在缩并新规下， \(\textcolor{blue}{\nabla_{AA'}\kappa^B}\) 对应的更简单形式 \(\textcolor{red}{\nabla_{a}\kappa^B}=-g_a^{AA'}\textcolor{blue}{\nabla_{AA'}\kappa^B}\) 的分量是：
\[ g^{\ \ a}_{\mu}\epsilon_B^{\ \ \Omega}\textcolor{red}{\nabla_a\kappa^B}=\nabla_\mu\kappa^\Omega+\kappa^\Lambda \textcolor{blue}{\gamma_{\mu\Lambda}^{\quad \Omega}} \]其中【也符合缩并新规】
\[ \boxed{\gamma_{\mu\Lambda}^{\quad \Omega}\overset{\Delta}{=}-g_\mu^{\ \ \Sigma\Sigma'}\gamma_{\Sigma\Sigma'\Lambda}^{\ \ \quad \Omega}}=\epsilon_A^{\ \ \Omega}\nabla_\mu\epsilon_{\Lambda}^{\ \ A} \]类似地，还有：
\[ g^{\ \ a}_{\mu}\epsilon_\Omega^{\ \ B}\textcolor{red}{\nabla_a\omega_B}=\nabla_\mu\omega_\Omega-\omega_\Lambda \textcolor{blue}{\gamma_{\mu\Omega}^{\quad \Lambda}} \]最后给一个混合张量导数 \(\textcolor{red}{\nabla_a\theta_B^{\ \ c}}\) 的情况
\[ g_\mu^{\ \ a}\epsilon_\Omega^{\ \ B}g_c^{\ \ \upsilon}\textcolor{red}{\nabla_a\theta_B^{\ \ c}}=\nabla_\mu\theta_{\textcolor{red}{\Omega}}^{\ \ \textcolor{blue}{\upsilon}} -\theta_\Lambda^{\ \ \upsilon}\textcolor{red}{\gamma_{\mu\Omega}^{\quad \Lambda}}+\theta_\Omega^{\ \ \rho}\textcolor{blue}{\Gamma^\upsilon_{\ \ \mu\rho}} \]红色的指标对应红色的 \(\textcolor{red}{自旋系数}\) ，蓝色的指标对应蓝色的 \(\textcolor{blue}{克氏符}\) 。
克氏符的纯旋量表示
将克氏符按旋量表示展开【利用莱布尼茨律，以及升降指标、缩并等一系列操作】
\[ \begin{aligned}\Gamma^\sigma_{\ \ \mu\upsilon}=&g^\sigma_{\ \ b}\nabla_\mu g_\upsilon^{\ \ b}=-g^\sigma_{\ \ AA'}\nabla_\mu g_\upsilon^{\ \ AA'}\\ = & g^\sigma_{\ \ AA'}\nabla_\mu (g_\upsilon^{\ \ \Sigma\Sigma'}g_{\Sigma\Sigma'}^{\ \quad AA'})= -g^\sigma_{\ \ AA'}\nabla_\mu (g_\upsilon^{\ \ \Sigma\Sigma'}\epsilon_{\Sigma}^{\ \ A}\bar{\epsilon}_{\Sigma'}^{\ \ A'})\\ =&-g^\sigma_{\ \ AA'}\epsilon_{\Sigma}^{\ \ A}\bar{\epsilon}_{\Sigma'}^{\ \ A'}\nabla_\mu g_\upsilon^{\ \ \Sigma\Sigma'}-g_\upsilon^{\ \ \Sigma\Sigma'}g^\sigma_{\ \ AA'}\nabla_\mu (\epsilon_{\Sigma}^{\ \ A}\bar{\epsilon}_{\Sigma'}^{\ \ A'})\\ =&-g^\sigma_{\ \ \Sigma\Sigma'}\nabla_\mu g_\upsilon^{\ \ \Sigma\Sigma'}-g_\upsilon^{\ \ \Sigma\Sigma'}g^\sigma_{\ \ \Omega\Omega'}\epsilon_A^{\ \ \Omega}\bar{\epsilon}_{A'}^{\ \ \Omega'}(\bar{\epsilon}_{\Sigma'}^{\ \ A'}\nabla_\mu \epsilon_{\Sigma}^{\ \ A}+\epsilon_{\Sigma}^{\ \ A}\nabla_\mu \bar{\epsilon}_{\Sigma'}^{\ \ A'}) \end{aligned} \]进而有：
\[ \begin{aligned}g^\upsilon_{\ \ \Omega\Omega'}g_\sigma^{\ \ \Lambda\Lambda'}\Gamma^\sigma_{\ \ \mu\upsilon}=&g^\upsilon_{\ \ \Omega\Omega'}\nabla_\mu g_\upsilon^{\ \ \Lambda\Lambda'}-(\bar{\epsilon}_{\Omega'}^{\ \ \Lambda'}\epsilon_A^{\ \ \Lambda}\nabla_\mu \epsilon_{\Omega}^{\ \ A}+\epsilon_{\Omega}^{\ \ \Lambda}\bar{\epsilon}_{A'}^{\ \ \Lambda'}\nabla_\mu \bar{\epsilon}_{\Omega'}^{\ \ A'})\\ =&g^\upsilon_{\ \ \Omega\Omega'}\nabla_\mu g_\upsilon^{\ \ \Lambda\Lambda'}-\bar{\epsilon}_{\Omega'}^{\ \ \Lambda'}\gamma_{\mu\Omega}^{\quad \Lambda}-\epsilon_{\Omega}^{\ \ \Lambda}\bar{\gamma}_{\mu\Omega'}^{\quad \Lambda'}\end{aligned} \]将 \(\Lambda'\) 和 \(\Omega'\) 缩并得 \[ \gamma_{\mu\Omega}^{\quad \Lambda}=-\frac{1}{2}g^\upsilon_{\ \ \Omega\Omega'}g_\sigma^{\ \ \Lambda\Omega'}\Gamma^\sigma_{\ \ \mu\upsilon}+\frac{1}{2}g^\upsilon_{\ \ \Omega\Omega'}\nabla_\mu g_\upsilon^{\ \ \Lambda\Omega'} \]
指标操作规则梳理
【注意】涉及旋量的规则，依赖度规 (-1，1，1，1) 或 (1，1，1)
规则一：【兼容】兼容张量指标操作：指标缩并、指标提升、指标下降、指标替换。
规则二：【缩并】张量指标要求一上一下即可；旋量指标要求左上右下，否则加负号，比如： \[ \theta^a=g^{ab}\theta_b=g^{ba}\theta_b\\ \theta^A=\epsilon^{AB}\theta_B=-\epsilon^{BA}\theta_B \] 规则三：【张量和分量】张量抽象指标和具体指标替换，比如： \[ \begin{aligned}\theta^a=g_\mu^{\ \ a}\theta^\mu\quad \textcolor{red}{矢量按基底展开} \\ \theta^\mu=g^\mu_{\ \ a}\theta^a \quad \textcolor{red}{矢量分量的表示}\\ \quad \\ \theta^A=\theta^\Sigma\epsilon_\Sigma^{\ \ A}\quad \textcolor{blue}{旋量按基底展开} \\ \theta^\Sigma=\theta^A\epsilon_A^{\ \ \Sigma}\quad \textcolor{blue}{旋量分量的表示}\end{aligned} \] 规则四：旋量指标对缩并或拆分加负号。比如 \[ \theta^\mu=-g^\mu_{\ \ AA'}\theta^{AA'} \quad \theta^a\omega_a=-\theta^{AA'}\omega_{AA'}\\ g_{AA'BB'}=-\epsilon_{AB}\bar{\epsilon}_{A'B'}\quad g_{AA'}^{\quad BB'}=-\epsilon_A^{\ \ B}\bar{\epsilon}_{A'}^{\ \ B'} \]