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Fourier expansions of Klein Gordon field not Lorentz invariant?

Physics Asked on July 6, 2021

I’m working in Peskin and Schroeders book on QFT and noticed that they expanded a solution to the Klein Gordon equation in a manner that seems to me not to be be Lorentz invariant even though the solutions to the KGE should be scalars…

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Am I correct that this combination is not Lorentz invariant?
Or is there a reason that this does not matter at this stage?

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