ラグランジュの運動方程式(複素場の理論)

\begin{align}\partial_\mu\left(\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi}\right)-\frac{\partial \mathscr{L}}{\partial\varPhi}=0\\\partial_\mu\left(\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi^*}\right)-\frac{\partial \mathscr{L}}{\partial\varPhi^*}=0\end{align}

\begin{align*}X[\phi]=\int d^4x\ \mathscr X(\phi( x))\end{align*}

\begin{align*}\frac{\delta f( x)}{\delta f( y)}=\delta^4( x- y)\end{align*}

\begin{align*}\frac{\delta X[\phi]}{\delta \phi( y)}&=\frac{\partial \mathscr X}{\partial \phi(y)}\tag{5}\end{align*}

\begin{align*}\varPhi&=\phi_1+i\phi_2\\\varPhi^*&=\phi_1-i\phi_2\end{align*}

\begin{align}\varPhi(x)^A{}|_{\scriptsize {x^\mu\rightarrow\pm\infty}}&=\varPhi(x)^B{}|_{\scriptsize {x^\mu\rightarrow\pm\infty}}=\cdots=0\tag{1}\\\varPhi^*(x)^A{}|_{\scriptsize {x^\mu\rightarrow\pm\infty}}&=\varPhi^*(x)^B{}|_{\scriptsize {x^\mu\rightarrow\pm\infty}}=\cdots=0\tag{2}\end{align}

\begin{align}\delta \varPhi(x)|_{\scriptsize {x^\mu\rightarrow\pm\infty}}=\varPhi(x)|_{\scriptsize {x^\mu\rightarrow\pm\infty}}&=0\tag{3}\\\delta \varPhi^*(x)|_{\scriptsize {x^\mu\rightarrow\pm\infty}}=\varPhi^*(x)|_{\scriptsize {x^\mu\rightarrow\pm\infty}}&=0\tag{4}\end{align}

\begin{align}\delta S=0\tag{5}\end{align}

\begin{align}\delta S&=S[\varPhi+\delta \varPhi,\partial_\mu\varPhi+\partial_\mu\delta\varPhi,\varPhi^*+\delta \varPhi^*,\partial_\mu\varPhi^*+\partial_\mu\delta\varPhi^*]-S[\varPhi,\partial_\mu\varPhi,\varPhi^*,\partial_\mu\varPhi^*]\\&=\int {d}^4x\ \mathscr{L}(\varPhi+\delta \varPhi,\partial_\mu\varPhi+\partial_\mu\delta\varPhi,\varPhi^*+\delta \varPhi^*,\partial_\mu\varPhi^*+\partial_\mu\delta\varPhi^*)-\int {d}^4x\ \mathscr{L}(\varPhi,\partial_\mu\varPhi,\varPhi^*,\partial_\mu\varPhi^*)\\&=0\tag{6}\end{align}

\begin{align}&\mathscr{L}(\varPhi+\delta \varPhi,\partial_\mu\varPhi+\partial_\mu\delta\varPhi,\varPhi^*+\delta \varPhi^*,\partial_\mu\varPhi^*+\partial_\mu\delta\varPhi^*)\\&=\mathscr{L}(\varPhi,\partial_\mu\varPhi,\varPhi^*,\partial_\mu\varPhi^*)+\left\{\delta \varPhi\frac{\partial \mathscr{L}}{\partial \varPhi}+\partial_\mu\delta\varPhi\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi}+\delta \varPhi^*\frac{\partial \mathscr{L}}{\partial \varPhi^*}+\partial_\mu\delta\varPhi\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi^*}\right\}\tag{7}\end{align}

\begin{align}\delta S&=\int{d}^4x\ \left\{\delta \varPhi\frac{\partial \mathscr{L}}{\partial \varPhi}+\partial_\mu\delta\varPhi\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi}+\delta \varPhi^*\frac{\partial \mathscr{L}}{\partial \varPhi^*}+\partial_\mu\delta\varPhi^*\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi^*}\right\}\\&=0\tag{8}\end{align}

\begin{align}\partial_\mu\delta\varPhi\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi}&=\partial_\mu\left(\delta \varPhi\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi}\right)-\delta \varPhi\partial_\mu\left(\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi}\right)\tag{9}\\\partial_\mu\delta\varPhi^*\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi^*}&=\partial_\mu\left(\delta \varPhi^*\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi^*}\right)-\delta \varPhi^*\partial_\mu\left(\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi^*}\right)\tag{10}\end{align}

\begin{align}\delta S&=\int{d}^4x\ \left\{\delta\varPhi\left[\frac{\partial \mathscr{L}}{\partial\varPhi}-\partial_\mu\left(\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi}\right)\right]+\partial_\mu\left(\delta \varPhi\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi}\right)+\delta\varPhi^*\left[\frac{\partial \mathscr{L}}{\partial\varPhi^*}-\partial_\mu\left(\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi^*}\right)\right]+\partial_\mu\left(\delta \varPhi^*\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi^*}\right)\right\}\\&=\int{d}^4x\ \left\{\delta\phi\left[\frac{\partial \mathscr{L}}{\partial\varPhi}-\partial_\mu\left(\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi}\right)\right]+\delta\varPhi^*\left[\frac{\partial \mathscr{L}}{\partial\varPhi^*}-\partial_\mu\left(\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi^*}\right)\right]\right\}+\int{d}^4x\left\{\partial_\mu\left(\delta \varPhi\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi}\right)+\partial_\mu\left(\delta \varPhi^*\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi^*}\right)\right\}\\&=\int{d}^4x\ \left\{\delta\varPhi\left[\frac{\partial \mathscr{L}}{\partial\varPhi}-\partial_\mu\left(\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi}\right)\right]+\delta\varPhi^*\left[\frac{\partial \mathscr{L}}{\partial\varPhi^*}-\partial_\mu\left(\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi^*}\right)\right]\right\}+\int{d}\sigma_\mu\left\{\left(\delta \varPhi\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi}\right)+\left(\delta \varPhi^*\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi^*}\right)\right\}\\&=\int{d}^4x\ \left\{\delta\varPhi\left[\frac{\partial \mathscr{L}}{\partial\varPhi}-\partial_\mu\left(\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi}\right)\right]+\delta\varPhi^*\left[\frac{\partial \mathscr{L}}{\partial\varPhi^*}-\partial_\mu\left(\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi^*}\right)\right]\right\}\\&=0\tag{11}\end{align}

\begin{align}\int{d}^4x\ \partial_\mu X^\mu=\int_{\partial V} {d}\sigma_\mu X^\mu\tag{12}\end{align}

\begin{align}\partial_\mu\left(\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi}\right)-\frac{\partial \mathscr{L}}{\partial\varPhi}=0\tag{13}\\\partial_\mu\left(\frac{\partial \mathscr{L}}{\partial \partial_\mu\varPhi^*}\right)-\frac{\partial \mathscr{L}}{\partial\varPhi^*}=0\tag{14}\end{align}