In Peskin & Schroeder's book on page 297 in deriving the photon propagator the authors say that
(−k2gμν+(1−1ξ)kμkν)DνρF(k)=iδρμ
With the solution given in the next line in equation (9.58) as
DμνF(k)=−ik2+iϵ(gμν−(1−ξ)kμkνk2)
Which is the propagator. I can verify this equation by inserting DμνF(k) into the first equation, but I have no idea how to actually solve DνρF(k) from (9.57b). If anyone can help, it would be much appreciated.
Answer
Dμν=Agμν+Bkμkν with A and B two unknown functions of the scalar k^2. The two tensor after A and B are the only possible Lorentz invariant tensors . Simply plugin and calculate the unknown functions.
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