When quantized, the neutral scalar field can be expanded as follows:
$\phi(\mathrm{x}, t)=\sum_{\mathbf{k}} c \sqrt{\frac{h}{2 \omega} V}\left(a_{k}(t) e^{l_{k} \cdot x}+a_{k}^{*}(t) e^{-i k^{-} x}\right)$
$\omega / c=\sqrt{\mathbf{k}^{2}+(m c / h)^{2}}, \quad a_{k}(t)=a_{k}(0) e^{-1 \omega t}$
$\left[a_{k,} a_{k^{k}}\right]=\left[a_{k}^{\prime}, a_{k}^{\prime} \cdot\right]=0, \quad\left[a_{k}, a_{k}^{t}\right]=\delta_{k k^{\prime}}$
a) Prove the equal-time commutation relation
$$
\left[\phi(x, t), \pi\left(x^{\prime}, t\right)\right]=i h \delta^{(3)}\left(x-x^{\prime}\right)
$$
where
$$
\pi(x, t)=\left(1 / c^{2}\right)(2 \phi / \partial t)
$$
b) Suppose we define the average fieldl operator $\bar{\phi}$ in the neighborhood of the origin by
$$
\bar{\phi}=\left[1 /\left(2 \pi b^{2}\right)^{3 / 2}\right] \int d^{3} x e^{-r^{1} / 2 s^{\prime}} \phi(x, t)
$$
Show that apart from a numerical factor the vacuum expectation value of $(\bar{\phi})^{2}$ is given bv $c h / b^{2}$ orovided $b \ll$ himc.