Photon self-energy in the finite temperature field theory (real -time)

[Tian]

TL;DR

I want to calculate photon selfenergy in the finite temperature field theory (real -time). There are two delta function. There may be some wrong in my calculation ,but I can not find it

 

[vanhees71]

Your formulae are hard to read, but as far as I can see, the initial expression looks correct (maybe up to a sign, because the +-vertex has the opposite sign of the --vertex; the additional - from the fermion loop is, of course, correct).

The only place, where you have trouble with the $\delta$ functions is in the evaluations of tadpole diagrams (i.e., loops beginning and ending at the same vertex).

 

[Tian]

Thank you very much , the note maybe hard to read . I should distinguish between three momentum and four momentum. Capital letters represent four momentum, and small letters represent three momentum. I hope you can see my note again. This probelm confused me long time .

 

[Tian]

And you are right, there is a sign problem, but I don't know the mean of “ the +-vertex has the opposite sign of the --vertex”

 

[vanhees71]

That's because one branch of the contour is in positive and the other in negative time direction. It depends on your convention which one you call the plus and which one the minus-branch.