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I have already asked this at MSE 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?

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    $\begingroup$ the propagators are nothing but Green's function for the D'Lambertian, usually. But I was wondering the same as you, I could not find any rigorous description of these. $\endgroup$
    – L.F. Cavenaghi
    Sep 15, 2022 at 19:51

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You'll find some more info about the fundamental solutions of the wave equation in chapter 5.D of Folland and chapter I.7 of Trèves. 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!

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    $\begingroup$ I also recommend §III.2.5 in vol.1 of Gelfand & Shilov's treatise on Generalized functions. Their emphasis is on distributions in general and operations on them, with fundamental solutions appearing as a special application. That book, in particular, spends a lot of time showing how different $i\varepsilon$ prescriptions give rise to distributions with specific properties. $\endgroup$
    – Igor Khavkine
    Sep 16, 2022 at 8:18
  • $\begingroup$ Thank you both for your great references! $\endgroup$
    – Bettina
    Oct 2, 2022 at 10:43

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