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[QUE/QFT-05001]

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Starting from the Lagrangian for a complex scalar field obtain the Hamiltonian for a free complex Klein Gordon field.
Write ETCR and for the quantized field prove that \begin{equation}\label{EQ01}
\big[H, \pi(x)\big] = - i \big( \nabla^2- \mu^2\big) \phi^*(x) .
\end{equation} Does relation,\eqref{EQ01}, hold only as equal time commutator or does it hold for  \(H\) and \(\pi(x)\) at arbitrary different times ?  Explain your answer.
Use \eqref{EQ01} to derive the usual Euler Lagrange equation of motion for the Klein Gordon field.

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