Expressing Feynman Green's function as a 4-momentum integral

[realanswers]

Homework Statement

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Relevant Equations

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I am a bit confused on how we can just say that (z',p) form a 4-vector. In my head, four vectors are sacred objects that are Lorentz covariant, but now we introduced some new variable and say it forms a 4-vector with momentum. I understand that these are just integration variables but I still do not see how this is okay. The interpretation of z' now is different.

 

[strangerep]

I am a bit confused on how we can just say that (z',p) form a 4-vector. In my head, four vectors are sacred objects that are Lorentz covariant, but now we introduced some new variable and say it forms a 4-vector with momentum. I understand that these are just integration variables but I still do not see how this is okay. The interpretation of z' now is different.

But where did it come from? $\theta(x^0 - y^0)$, right? The restricted Lorentz group (identity-connected part) preserves time orientation, so it's ok to use $\theta(x^0 - y^0)$ in that circumstance. The rest just involves expressing it in momentum space.