* 柏，演習 場の量子論
** 例題1.2
$$\phi^*(t,\vec{x}) = \phi(t,\vec{x})$$
$$\iff \sum_{n=-\infty}^\infty q_n^*(t) f_n^*(\vec{x}) = \sum_{n=-\infty}^\infty q_n(t) f_n(\vec{x}) $$
$$=\sum_{n=-\infty}^\infty q_n^*(t) f_{-n}(\vec{x})$$
$$=\sum_{m=\infty}^{-\infty} q_{-m}^*(t) f_m(\vec{x})$$
$$=\sum_{n=\infty}^{-\infty} q_{-n}^*(t) f_n(\vec{x})$$
$$=\sum_{n=-\infty}^{\infty} q_{-n}^*(t) f_n(\vec{x})$$
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