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Lorentz Invariance of Propagator for Complex Scalar Field
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[QUOTE="vanhees71, post: 4518733, member: 260864"] You are integrating over [itex]\vec{k}[/itex]. Thanks to the invariance of [itex]\mathrm{d}^3 \vec{k}/\omega[/itex] where [itex]\omega=\sqrt{m^2+\vec{k}^2}[/itex] you know that the result is a Lorentz scalar [tex]F(x_1-x_2)[/tex]. That means that under Lorentz transformations [tex]F'(x')=F(x)=F(\Lambda^{-1} x).[/tex] Thus you only need to transform [itex]x=x_1-x_2[/itex]. It's only crucial to show that for a spacelike [itex]x[/itex] you can always find a Lorentz transformation such that [itex]\Lambda x=-x[/itex]. Since you can orient your coordinate system always such that [itex]x=(0,\xi,0,0)[/itex] you just need to find a Lorentz transformation which makes out of this [itex]\Lambda x=(0,-\xi,0,0)[/itex]. [/QUOTE]
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Lorentz Invariance of Propagator for Complex Scalar Field
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