Studying the scalar field and Klein-Gordon equation in quantum field theory I came across this definition for the inner product in the space of the solutions of the K.G. equation:
$\langle \Phi_1 | \Phi_2 \rangle = \int \mathrm{d}\vec{x} \Phi_1 ^* i \overleftrightarrow{\partial_0}\Phi_2 = \int (\mathrm{d}\vec{x} \Phi_1 ^* i \partial_0\Phi_2 - \Phi_2 i \partial_0\Phi_1^* $
I see that this definition should be invariant under Poincaré transformations, but I couldn't prove it. Do you know some references about this?
Moreover I couldn't find the reason why such a scalar product is introduced? Aren't there other possible scalar products? Why choose this one?
