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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?

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Something to think about: consider the current $J_\mu = i\Phi_1^* \partial_\mu \Phi_2 - i\Phi_2 \partial_\mu \Phi_1^*$ and maybe set $\mu = 0$... What can you say about $J_\mu$? – Vibert Jan 16 '13 at 22:46
up vote 4 down vote accepted

For a thorough treatment of that expression, consider this paper.

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