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" $\mathrm{i} \varepsilon$ 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?
In the meantime also posted and answered at MO.