测试：MathJax

发表于 2022-11-05 更新于 2022-12-15

这篇文章的目的是用一些$\LaTeX$公式测试Hexo+NexT/MathJax这一组合的有效性与可靠性。也可以欣赏一下这些公式。 \[ K(\phi,t;\phi_0,t_0)=\int\mathcal{D}\phi e^{iS}. \]

概要

这个网站基于Hexo框架与NexT主题搭建。因为可预期地我需要在这里输入一些公式，没有公式插入和渲染这一功能就很麻烦了，虽然插入图片或多或少可以作为一种代替方案，但终究不妙。虽然Hexo原生不支持公式输入，但是好在只要加入一些插件，典型的如KaTex插件或MathJax插件，就可支持$\LaTeX$公式输入。这里NexT主题非常善良地集成了MathJax插件，只需要将原生渲染器hexo-renderer-marked替换为hexo-renderer-pandoc，并在主题配置中启用MathJax即可。这一页用于测试Hexo+NexT/MathJax这一组合的有效性与可靠性。也可以欣赏一下这些公式。

基本

行内公式：$\vec{F}=m\vec{a}$（牛顿第二定律）。

1	\vec{F}=m\vec{a}

行间公式（旋量场）： \[ \mathcal{L}=\bar{\psi}(i\!\!\not\!{\partial}-m)\psi. \]

1	\mathcal{L}=\bar{\psi}(i\!\!\not\!{\partial}-m)\psi. %无\slashed，用\!\!\not\!曲线救国。

环境：equation

经典作用量原理： \[ \begin{equation} \delta S=0. \end{equation} \]

1
2
3

\begin{equation}
    \delta S=0.
\end{equation}

量子作用量原理： \[ \begin{equation} \langle x|e^{-iH(x^0-x^0_0)}|x_0\rangle=\int\mathcal{D}xe^{iS}, \end{equation} \]

1
2
3

\begin{equation}
    \langle x|e^{-iH(x^0-x^0_0)}|x_0\rangle=\int\mathcal{D}xe^{iS},
\end{equation}

量子作用量原理（标量场）： \[ \begin{equation} K(\phi,t;\phi_0,t_0)=\int\mathcal{D}\phi e^{iS}. \end{equation} \]

1
2
3

\begin{equation}
    K(\phi,t;\phi_0,t_0)=\int\mathcal{D}\phi e^{iS}.
\end{equation}

环境：align

Noether定理（标量场）： \[ \begin{align} \delta S&=\int\mathrm{d}^4x\partial_\mu j^\mu,\\ j^\mu&=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\bar{\delta}\phi+\mathcal{L}\delta x^\mu. \end{align} \]

\begin{align}
    \delta S&=\int\mathrm{d}^4x\partial_\mu j^\mu,\\
    j^\mu&=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\bar{\delta}\phi+\mathcal{L}\delta x^\mu.
\end{align}

环境：aligned

Maxwell方程组（SI）： \[ \begin{equation} \begin{aligned} \nabla\cdot\boldsymbol{E}&=\frac{\rho}{\varepsilon_0}&&(电场有源),\\ \nabla\cdot\boldsymbol{B}&=0&&(磁场无源),\\ \nabla\times\boldsymbol{B}&=\mu_0\boldsymbol{J}+\frac{1}{c^2}\frac{\partial\boldsymbol{E}}{\partial t}&&(电动生磁),\\ \nabla\times\boldsymbol{E}&=-\frac{\partial\boldsymbol{B}}{\partial t}&&(磁动生电).\\ \end{aligned} \end{equation} \]

\begin{equation}
    \begin{aligned}
        \nabla\cdot\boldsymbol{E}&=\frac{\rho}{\varepsilon_0}&&(电场有源)\\
        \nabla\cdot\boldsymbol{B}&=0&&(磁场无源)\\
        \nabla\times\boldsymbol{B}&=\mu_0\boldsymbol{J}+\frac{1}{c^2}\frac{\partial\boldsymbol{E}}{\partial t}&&(电动生磁)\\
        \nabla\times\boldsymbol{E}&=-\frac{\partial\boldsymbol{B}}{\partial t}&&(磁动生电)\\
    \end{aligned}
\end{equation}

环境：矩阵

Minkowski度规： \[ g= \begin{bmatrix} 1&&&\\ &-1&&\\ &&-1&\\ &&&-1\\ \end{bmatrix}. \]

g=
\begin{bmatrix}
    1&&&\\
    &-1&&\\
    &&-1&\\
    &&&-1\\
\end{bmatrix}.

Pauli矩阵： \[ \sigma_a= \begin{pmatrix} \delta_{a3}&&\delta_{a1}-i\delta_{a2}\\ \delta_{a1}+i\delta_{a2}&&-\delta_{a3}\\ \end{pmatrix}. \]

\sigma_a=
\begin{pmatrix}
    \delta_{a3}&&\delta_{a1}-i\delta_{a2}\\
    \delta_{a1}+i\delta_{a2}&&-\delta_{a3}\\
\end{pmatrix}.

欧氏空间三维向量的2-范数： \[ \begin{Vmatrix} \boldsymbol{x} \end{Vmatrix} =x_1^2+x_2^2+x_3^2. \]

\begin{Vmatrix}
    \boldsymbol{x}
\end{Vmatrix}
=x_1^2+x_2^2+x_3^2.

颜色

标量QED： $$ \mathcal{L}=\textcolor{#00ff00}{-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}}+\textcolor{#00ffff}{D_\mu\phi^\dagger D^\mu\phi}-\textcolor{#0000ff}{m^2\phi^\dag\phi}. $$ （dag看来没法用，但dagger能用。）

1	\mathcal{L}=\textcolor{#00ff00}{-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}}+\textcolor{#00ffff}{D_\mu\phi^\dagger D^\mu\phi}-\textcolor{#0000ff}{m^2\phi^\dag\phi}.

旋量QED： \[ \mathcal{L}=\red{-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}}+\blue{\bar{\psi}(i\pink{\!\!\not\!\!{D}}-m)\psi}. \] （red、green、blue这些看来不能用。）

1	\mathcal{L}=\red{-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}}+\blue{\bar{\psi}(i\pink{\!\!\not\!\!{D}}-m)\psi}.

大型公式

一个拉氏量（这在网络上被传为标准模型的拉氏量：可能是这并不是正确的标准模型拉氏量，很可能只有一部分和SM重合；另一部分是SM的扩展或者根本不具有太大意义。就算它确实是的话，那也是个被过度展开的版本，本可以合并到更加简洁。总之在任何地方传播这一形式都是无益甚至有害的。这里仅用作显示大型公式的测试，不要复制）： \[ \begin{equation} \begin{aligned} \mathcal{L}=&-\frac{1}{2} \partial_{\nu} g_{\mu}^{a} \partial_{\nu} g_{\mu}^{a}-g_{s} f^{a b c} \partial_{\mu} g_{\nu}^{a} g_{\mu}^{b} g_{\nu}^{c}-\frac{1}{4} g_{s}^{2} f^{a b c} f^{a d e} g_{\mu}^{b} g_{\nu}^{c} g_{\mu}^{d} g_{\nu}^{e}+\\ &\frac{1}{2} i g_{s}^{2}\left(\bar{q}_{i}^{\sigma} \gamma^{\mu} q_{j}^{\sigma}\right) g_{\mu}^{a}+\bar{G}^{a} \partial^{2} G^{a}+g_{s} f^{a b c} \partial_{\mu} \bar{G}^{a} G^{b} g_{\mu}^{c}-\partial_{\nu} W_{\mu}^{+} \partial_{\nu} W_{\mu}^{-}-\\ &M^{2} W_{\mu}^{+} W_{\mu}^{-}-\frac{1}{2} \partial_{\nu} Z_{\mu}^{0} \partial_{\nu} Z_{\mu}^{0}-\frac{1}{2 c_{w}^{2}} M^{2} Z_{\mu}^{0} Z_{\mu}^{0}-\frac{1}{2} \partial_{\mu} A_{\nu} \partial_{\mu} A_{\nu}-\frac{1}{2} \partial_{\mu} H \partial_{\mu} H-\\ &\frac{1}{2} m_{h}^{2} H^{2}-\partial_{\mu} \phi^{+} \partial_{\mu} \phi^{-}-M^{2} \phi^{+} \phi^{-}-\frac{1}{2} \partial_{\mu} \phi^{0} \partial_{\mu} \phi^{0}-\frac{1}{2 c_{w}^{2}} M \phi^{0} \phi^{0}-\beta_{h}\left[\frac{2 M^{2}}{g^{2}}+\right.\\ &\left.\frac{2 M}{g} H+\frac{1}{2}\left(H^{2}+\phi^{0} \phi^{0}+2 \phi^{+} \phi^{-}\right)\right]+\frac{2 M^{4}}{g^{2}} \alpha_{h}-i g c_{w}\left[\partial_{\nu} Z_{\mu}^{0}\left(W_{\mu}^{+} W_{\nu}^{-}-\right.\right.\\ &\left.W_{\nu}^{+} W_{\mu}^{-}\right)-Z_{\nu}^{0}\left(W_{\mu}^{+} \partial_{\nu} W_{\mu}^{-}-W_{\mu}^{-} \partial_{\nu} W_{\mu}^{+}\right)+Z_{\mu}^{0}\left(W_{\nu}^{+} \partial_{\nu} W_{\mu}^{-}-\right.\\ &\left.\left.W_{\nu}^{-} \partial_{\nu} W_{\mu}^{+}\right)\right]-i g s_{w}\left[\partial_{\nu} A_{\mu}\left(W_{\mu}^{+} W_{\nu}^{-}-W_{\nu}^{+} W_{\mu}^{-}\right)-A_{\nu}\left(W_{\mu}^{+} \partial_{\nu} W_{\mu}^{-}-\right.\right.\\ &\frac{1}{2} g^{2} W_{\mu}^{+} W_{\nu}^{-} W_{\mu}^{+} W_{\nu}^{-}+g^{2} c_{w}^{2}\left(Z_{\mu}^{0} W_{\mu}^{+} Z_{\nu}^{0} W_{\nu}^{-}-Z_{\mu}^{0} Z_{\mu}^{0} W_{\nu}^{+} W_{\nu}^{-}\right)+\\ &g^{2} s_{w}^{2}\left(A_{\mu} W_{\mu}^{+} A_{\nu} W_{\nu}^{-}-A_{\mu} A_{\mu} W_{\nu}^{+} W_{\nu}^{-}\right)+g^{2} s_{w} c_{w}\left[A_{\mu} Z_{\nu}^{0}\left(W_{\mu}^{+} W_{\nu}^{-}-\right.\right.\\ &\left.\left.W_{\nu}^{+} W_{\mu}^{-}\right)-2 A_{\mu} Z_{\mu}^{0} W_{\nu}^{+} W_{\nu}^{-}\right]-g \alpha\left[H^{3}+H \phi^{0} \phi^{0}+2 H \phi^{+} \phi^{-}\right]-\\ &\frac{1}{8} g^{2} \alpha_{h}\left[H^{4}+\left(\phi^{0}\right)^{4}+4\left(\phi^{+} \phi^{-}\right)^{2}+4\left(\phi^{0}\right)^{2} \phi^{+} \phi^{-}+4 H^{2} \phi^{+} \phi^{-}+2\left(\phi^{0}\right)^{2} H^{2}\right]-\\ &g M W_{\mu}^{+} W_{\mu}^{-} H-\frac{1}{2} g \frac{M}{c_{w}^{2}} Z_{\mu}^{0} Z_{\mu}^{0} H-\frac{1}{2} i g\left[W_{\mu}^{+}\left(\phi^{0} \partial_{\mu} \phi^{-}-\phi^{-} \partial_{\mu} \phi^{0}\right)-\right.\\ &\left.W_{\mu}^{-}\left(\phi^{0} \partial_{\mu} \phi^{+}-\phi^{+} \partial_{\mu} \phi^{0}\right)\right]+\frac{1}{2} g\left[W_{\mu}^{+}\left(H \partial_{\mu} \phi^{-}-\phi^{-} \partial_{\mu} H\right)-W_{\mu}^{-}\left(H \partial_{\mu} \phi^{+}-\right.\right.\\ &\left.\left.\phi^{+} \partial_{\mu} H\right)\right]+\frac{1}{2} g \frac{1}{c_{w}}\left(Z_{\mu}^{0}\left(H \partial_{\mu} \phi^{0}-\phi^{0} \partial_{\mu} H\right)-i g_{c_{w}}^{s_{w}^{2}} M Z_{\mu}^{0}\left(W_{\mu}^{+} \phi^{-}-W_{\mu}^{-} \phi^{+}\right)+\right.\\ &i g s_{w} M A_{\mu}\left(W_{\mu}^{+} \phi^{-}-W_{\mu}^{-} \phi^{+}\right)-i g \frac{1-2 c_{w}^{2}}{2 c_{w}} Z_{\mu}^{0}\left(\phi^{+} \partial_{\mu} \phi^{-}-\phi^{-} \partial_{\mu} \phi^{+}\right)+\\ &igs_{w} A_{\mu}\left(\phi^{+} \partial_{\mu} \phi^{-}-\phi^{-} \partial_{\mu} \phi^{+}\right)-\frac{1}{4} g^{2} W_{\mu}^{+} W_{\mu}^{-}\left[H^{2}+\left(\phi^{0}\right)^{2}+2 \phi^{+} \phi^{-}\right]-\\ &\frac{1}{4} g^{2} \frac{1}{c_{w}^{2}} Z_{\mu}^{0} Z_{\mu}^{0}\left[H^{2}+\left(\phi^{0}\right)^{2}+2\left(2 s_{w}^{2}-1\right)^{2} \phi^{+} \phi^{-}\right]-\frac{1}{2} g^{2} \frac{s_{w}^{2}}{c_{w}} Z_{\mu}^{0} \phi^{0}\left(W_{\mu}^{+} \phi^{-}+\right.\\ &\left.W_{\mu}^{-} \phi^{+}\right)-\frac{1}{2} i g^{2} \frac{s_{w}^{2}}{c_{w}} Z_{\mu}^{0} H\left(W_{\mu}^{+} \phi^{-}-W_{\mu}^{-} \phi^{+}\right)+\frac{1}{2} g^{2} s_{w} A_{\mu} \phi^{0}\left(W_{\mu}^{+} \phi^{-}+\right.\\ &\left.W_{\mu}^{-} \phi^{+}\right)+\frac{1}{2} i g^{2} s_{w} A_{\mu} H\left(W_{\mu}^{+} \phi^{-}-W_{\mu}^{-} \phi^{+}\right)-g^{2} \frac{s_{w}}{c_{w}}\left(2 c_{w}^{2}-1\right) Z_{\mu}^{0} A_{\mu} \phi^{+} \phi^{-}-\\ &g^{1} s_{w}^{2} A_{\mu} A_{\mu} \phi^{+} \phi^{-}-\bar{e}^{\lambda}\left(\gamma \partial+m_{e}^{\lambda}\right) e^{\lambda}-\bar{\nu}^{\lambda} \gamma \partial \nu^{\lambda}-\bar{u}_{j}^{\lambda}\left(\gamma \partial+m_{u}^{\lambda}\right) u_{j}^{\lambda}-\\ &\bar{d}_{j}^{\lambda}\left(\gamma \partial+m_{d}^{\lambda}\right) d_{j}^{\lambda}+i g s_{w} A_{\mu}\left[-\left(\bar{e}^{\lambda} \gamma^{\mu} e^{\lambda}\right)+\frac{2}{3}\left(\bar{u}_{j}^{\lambda} \gamma^{\mu} u_{j}^{\lambda}\right)-\frac{1}{3}\left(\bar{d}_{j}^{\lambda} \gamma^{\mu} d_{j}^{\lambda}\right)\right]+\\ &\frac{i g}{4 c_{w}} Z_{\mu}^{0}\left[\left(\bar{\nu}^{\lambda} \gamma^{\mu}\left(1+\gamma^{5}\right) \nu^{\lambda}\right)+\left(\bar{e}^{\lambda} \gamma^{\mu}\left(4 s_{w}^{2}-1-\gamma^{5}\right) e^{\lambda}\right)+\left(\bar{u}_{j}^{\lambda} \gamma^{\mu}\left(\frac{4}{3} s_{w}^{2}-\right.\right.\right.\\ &\left.\left.\left.1-\gamma^{5}\right) u_{j}^{\lambda}\right)+\left(\bar{d}_{j}^{\lambda} \gamma^{\mu}\left(1-\frac{8}{3} s_{w}^{2}-\gamma^{5}\right) d_{j}^{\lambda}\right)\right]+\frac{i g}{2 \sqrt{2}} W_{\mu}^{+}\left[\left(\bar{\nu}^{\lambda} \gamma^{\mu}\left(1+\gamma^{5}\right) e^{\lambda}\right)+\right.\\ &\left.\left(\bar{u}_{j}^{\lambda} \gamma^{\mu}\left(1+\gamma^{5}\right) C_{\lambda \kappa} d_{j}^{\kappa}\right)\right]+\frac{i g}{2 \sqrt{2}} W_{\mu}^{-}\left[\left(\bar{e}^{\lambda} \gamma^{\mu}\left(1+\gamma^{5}\right) \nu^{\lambda}\right)+\left(\bar{d}_{j}^{\kappa} C_{\lambda \kappa}^{\dagger} \gamma^{\mu}(1+\right.\right.\\ &\left.\left.\left.\gamma^{5}\right) u_{j}^{\lambda}\right)\right]+\frac{i g}{2 \sqrt{2}} \frac{m_{e}^{\lambda}}{M}\left[-\phi^{+}\left(\bar{\nu}^{\lambda}\left(1-\gamma^{5}\right) e^{\lambda}\right)+\phi^{-}\left(\bar{e}^{\lambda}\left(1+\gamma^{5}\right) \nu^{\lambda}\right)\right]-\\ &\frac{\partial}{2} \frac{m_{e}^{\lambda}}{M}\left[H\left(\bar{e}^{\lambda} e^{\lambda}\right)+i \phi^{0}\left(\bar{e}^{\lambda} \gamma^{5} e^{\lambda}\right)\right]+\frac{i g}{2 M \sqrt{2}} \phi^{+}\left[-m_{d}^{\kappa}\left(\bar{u}_{j}^{\lambda} C_{\lambda \kappa}\left(1-\gamma^{5}\right) d_{j}^{\kappa}\right)+\right.\\ &m_{u}^{\lambda}\left(\bar{u}_{j}^{\lambda} C_{\lambda \kappa}\left(1+\gamma^{5}\right) d_{j}^{\kappa}\right]+\frac{i g}{2 M \sqrt{2}} \phi^{-}\left[m_{d}^{\lambda}\left(\bar{d}_{j}^{\lambda} C_{\lambda \kappa}^{\dagger}\left(1+\gamma^{5}\right) u_{j}^{\kappa}\right)-m_{u}^{\kappa}\left(\bar{d}_{j}^{\lambda} C_{\lambda \kappa}^{\dagger}(1-\right.\right.\\ &\left.\left.\gamma^{5}\right) u_{j}^{\kappa}\right]-\frac{g}{2} \frac{m_{u}^{\lambda}}{M} H\left(\bar{u}_{j}^{\lambda} u_{j}^{\lambda}\right)-\frac{g}{2} \frac{m_{d}^{\lambda}}{M} H\left(\bar{d}_{j}^{\lambda} d_{j}^{\lambda}\right)+\frac{i g}{2} \frac{m_{u}^{\lambda}}{M} \phi^{0}\left(\bar{u}_{j}^{\lambda} \gamma^{5} u_{j}^{\lambda}\right)-\\ &\frac{i g}{2} \frac{m_{d}^{\lambda}}{M} \phi^{0}\left(\bar{d}_{j}^{\lambda} \gamma^{5} d_{j}^{\lambda}\right)+\bar{X}^{+}\left(\partial^{2}-M^{2}\right) X^{+}+\bar{X}^{-}\left(\partial^{2}-M^{2}\right) X^{-}+\bar{X}^{0}\left(\partial^{2}-\right.\\ &\left.\frac{M^{2}}{c_{w}^{2}}\right) X^{0}+\bar{Y} \partial^{2} Y+i g c_{w} W_{\mu}^{+}\left(\partial_{\mu} \bar{X}^{0} X^{-}-\partial_{\mu} \bar{X}^{+} X^{0}\right)+i g s_{w} W_{\mu}^{+}\left(\partial_{\mu} \bar{Y} X^{-}-\right.\\ &\left.\partial_{\mu} \bar{X}^{+} Y\right)+i g c_{w} W_{\mu}^{-}\left(\partial_{\mu} \bar{X}-X^{0}-\partial_{\mu} \bar{X}^{0} X^{+}\right)+i g s_{w} W_{\mu}^{-}\left(\partial_{\mu} \bar{X}-Y-\right.\\ &\left.\partial_{\mu} \bar{Y} X^{+}\right)+i g c_{w} Z_{\mu}^{0}\left(\partial_{\mu} \bar{X}^{+} X^{+}-\partial_{\mu} \bar{X}^{-} X^{-}\right)+i g s_{w} A_{\mu}\left(\partial_{\mu} \bar{X}^{+} X^{+}-\right.\\ &\left.\partial_{\mu} \bar{X}-X^{-}\right)-\frac{1}{2} g M\left[\bar{X}+X^{+} H+\bar{X}^{-} X^{-} H+\frac{1}{c_{w}^{2}} \bar{X}^{0} X^{0} H\right]+\\ &\frac{1-2 c_{w}^{2}}{2 c_{w}} i g M\left[\bar{X}^{+} X^{0} \phi^{+}-\bar{X}^{-} X^{0} \phi^{-}\right]+\frac{1}{2 c_{w}} i g M\left[\bar{X}^{0} X^{-} \phi^{+}-\bar{X}^{0} X^{+} \phi^{-}\right]+\\ &ig M s_{w}\left[\bar{X}^{0} X^{-} \phi^{+}-\bar{X}^{0} X^{+} \phi^{-}\right]+\frac{1}{2} i g M\left[\bar{X}^{+} X^{+} \phi^{0}-\bar{X}^{-} X^{-} \phi^{0}\right] \end{aligned} \end{equation} \]

目录

概要

基本

环境：equation

环境：align

环境：aligned

环境：矩阵

颜色

大型公式