The LSZ scattering approach starts with initial and final asymptotic momentum states. But we know that $\langle k' | k\rangle = \delta^3(k'-k)$, which means that it is not a properly normalizable state. But we still talk in scattering experiment of initial state being in $|k\rangle$ and final state being in $|k'\rangle$ and compute scattering amplitude. Even if we restrict ourselves to realistic scenario where a wavepacket has a finite "width" or variance so that it is not sharply localized in one momentum state, this does not deflect the question of normalization.
My question thus is, what are the reasons that allow us to deflect this concern?
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