I have already asked this at [MSE](https://math.stackexchange.com/questions/4524317/propagators-and-pdes) but did not get an answer.


In quantum field theory one encounters the retarded, advanced and Feynman propagators as certain solutions to a wave equation. Mathematically, these derivations are somewhat magical (typically one inserts an "infinitesimally small" iε
term, and then the different propagators result from different integration contours around certain poles). On the other hand, there is a mathematically rigorous theory of fundamental solutions to PDEs, but I have never seen anything analogous to these propagators in a PDE book. Can somebody recommend a source (book/lecture notes/paper), where the retarded, advanced and Feynman propagators are treated in a mathematically rigorous way, so that I can see the connection between QFT and PDE theory?



Propagators and PDEs
You'll find some more info about the fundamental solutions of the wave equation in chapter 5.D of [Folland](https://press.princeton.edu/books/hardcover/9780691043616/introduction-to-partial-differential-equations) and chapter I.7 of [Trèves](https://store.doverpublications.com/0486453464.html).
The trick you use is the idea that (tempered) distributions, even ones with non-trivial singular support, can have well-defined Fourier transforms.
So it'll help if you know a bit about distribution theory, but if not, the chapter on that in Folland is quite good.
I like what Folland has to say about propagators:



> 
> ...there are a number of natural but quite different fundamental solutions of the wave equation. (In other words, there are a number of natural but quite different distributions that agree with the function $[4\pi^2(|\xi|^2 — \tau^2)]^{-1}$ *away from the light cone!*)
> 
> 
> 


(Emphasis mine.)
Hopefully this helps!


