I'm a self-studying 14-year-old who has a passion for particle physics. I'm currently trying to calculate the QED photon propagator to one loop. However, in all the places I've looked, even with the same renormalization scheme, the results are different. I don't get it. Also, I need a step-by-step tutorial on how to calculate that trace in the numerator, specifically, the one with those gamma matrices.
This link has a very easy to follow derivation of the scalar field Feynman propagator. Once you understand that, you should be able to do it for vector particles like photons too. The correct answer is given at the end of the chart in this link.