How would you describe, in hand-wavey heuristic words, why the K-G Hamiltonian is of the form
$\int [(\partial_0\psi^*)(\partial^0 \psi) + (\nabla \psi^*)(\nabla \psi) + m^2 \psi^* \psi)d^3x$, it has to be a bilinear function of $\psi$ and $\psi^*$ because all operators are, I can wave my hands and get it from $T + V$ but how about something even more hand-wavey? e.g. the energy has to contain a term like $\psi^* \psi$ because...