九後汰一郎『ゲージ場の量子論 I』p15 から以下の計算をまとめる．

作用積分 $ S[\varphi(x)] $ が場 $ \varphi(x) $ の無限小変換

\begin{equation} \varphi(x)_i \to \varphi'_i(x) = \varphi_i(x) + \epsilon G_i(\varphi(x)) \end{equation}

について不変であるとする．

このとき，ネーターカレント $ j^\mu(x) $ から得られる保存電荷 $ Q=\int d^3x j^0(x) $ が，もとの無限小変換の生成子になることを $ G_i $ が $ \dot{\varphi} $ によらない場合について示そう．すなわち以下の同時刻のポアソン括弧と交換関係をそれぞれ示す．

\begin{align} &古典論 \qquad\{\varphi_i(x),Q\}_P = G_i(\varphi(x)) \\ &量子論 \qquad [iQ,\varphi_i(x)] =G_i(\varphi_i(x)) \end{align}

途中計算

場 $ \varphi_i(x) $ の正準運動量を $ \pi_i(x) := \frac{\partial \mathcal{L}}{\partial(\partial_0 \varphi_i(x))} $ と表す．

\begin{align} \{\varphi_i(x),Q\}_P &= \left \{\varphi_i(x),\int d^3y j^0 (y)\right \}_P \\ &= \Biggl \{\varphi_i(x),\int d^3y \Bigl (\underbrace{\frac{\partial \mathcal{L}}{\partial (\partial_0\varphi_j (y))}}_{=\pi_j(y)}G_j(\varphi(y)) - X^0(\varphi(y)) \Bigr )\Biggr \}_P \\ &= \int d^3z \sum_k \Biggl \{\underbrace{\frac{\delta\varphi_i(x)}{\delta\varphi_k(z)}}_{=\delta_{ik}\delta^3(x-z)}\frac{\delta}{\delta\pi_k(z)}\int d^3y \Bigl (\pi_j(y)G_j(\varphi(y)) - X^0(\varphi(y))\Bigr ) \\ &\qquad\qquad\qquad\qquad - \underbrace{\frac{\delta\varphi_i(x)}{\delta\pi_k(z)}}_{=0}\frac{\delta}{\delta\varphi_k(z)}\int d^3y \Bigl (\pi_j(y)G_j(\varphi(y)) - X^0(\varphi(y)\Bigr ) \Biggr \} \\ &= \int d^3z \sum_k \Biggl \{\delta_{ik}\delta^3(x-z)\underbrace{\int d^3y \Bigl ( \underbrace{\underbrace{\frac{\delta\pi_j(y)}{\delta\pi_k(z)}}_{=\delta_{jk}\delta^3(y-z)} G_j(\varphi(y))}_{\substack{jについて和をとる\\=\delta^3(y-z)G_k(\varphi(y))}} - \underbrace{\frac{\delta X^0(\varphi(y))}{\delta\pi_k(z)}}_{=0}\Bigr )}_{=G_k(\varphi(z))} \Biggr\} \\ &=G_i(x) \\ [iQ,\varphi_i(x)] &= [i\int d^3y j^0(y),\varphi_i(x)] = i\int d^3y [j^0(y),\varphi_i(x)] \\ &= i\int d^3y [\pi_j(y)G_j(\varphi(y))-X^0(\varphi(y)),\varphi_i(x)] \\ &= i\int d^3y \{\underbrace{[\pi_j(y)G_j(\varphi(y)),\varphi_i(x)]}_{=\underbrace{\underbrace{[\pi_j(y),\varphi_i(x)]}_{=-i\delta_{ij}\delta^3(x-y)}G_j(\varphi(y))}_{=-i\delta^3(x-y)G_i(\varphi(y))}}-\underbrace{[X^0(\varphi(y)),\varphi_i(x)]}_{=0} \} =G_i(\varphi(x)) \end{align}

参考文献