Prove that

(a) the only nonzero commutation relations for the annihilation and creation operators of a charged scalar field \(\left(\hat{f}_{\mathbf{p}}, \hat{f}_{\mathbf{p}}^{\dagger}, \hat{g}_{\mathbf{p}}, \hat{g}_{\mathbf{p}}^{\dagger}\right)\) are given by Eq. (6.2.180), i.e., \(\left[\hat{f}_{\mathbf{p}}, \hat{f}_{\mathbf{p}^{\prime}}^{\dagger}\right]=\) \(\left[\hat{g}_{\mathbf{p}}, \hat{g}_{\mathbf{p}^{\prime}}^{\dagger}\right]=(2 \pi)^{3} \delta^{3}\left(\mathbf{p}-\mathbf{p}^{\prime}\right)\), and

(b) the only nonzero equal-time canonical commutation relations for the complex field operator \(\hat{\phi}\) and its canonical momentum density operator \(\hat{\pi}\) are given by Eq. (6.2.181), i.e., \(\left[\hat{\phi}(t, \mathbf{x}), \hat{\pi}\left(t, \mathbf{x}^{\prime}\right)\right]=\left[\hat{\phi}^{\dagger}(t, \mathbf{x}), \hat{\pi}^{\dagger}\left(t, \mathbf{x}^{\prime}\right)\right]=i \delta^{3}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)\).