I would like to solve the Klein-Gordon equation $$ \partial_\mu \partial^\mu \phi + m^2\phi = f, \quad m>0 \qquad (*)$$ on flat Minkowski space, where $f$ is some test function (smooth and compactly supported, we might also admit Schwartz functions). If I look in physics books, the solution is given in terms of distributions (e.g. Itzykson, Zuber). People write down the advanced and retarded Green's function by $$ G_\pm(x) = - \frac{1}{\sqrt{(2\pi)^4}} \int_{\mathbb{R}^4} \frac{\text{e}^{-\text{i}\omega + \text{i} \langle\vec{p} , \vec{x}\rangle}}{(\omega\pm \text{i}\varepsilon)^2 - \vec{p}^2 -m^2}\,\text{d}p, \quad (**)$$ where $\omega$ is the zero component of $p$ and the arrows are used to denote spatial parts (assume that some coordinate system has been chosen, relativistic invariance is not the problem here). Obviously, this integral does not make too much sense (am I wrong?), but as a student of physics, I would be interested in understanding rigorously what is going on. As far as I have understood, a solution to $(*)$ is given by
$$ G_\pm[f](x) = - \frac{1}{\sqrt{(2\pi)^4}} \lim_{0<\varepsilon\to 0} \int_{\mathbb{R}^4} \frac{\text{e}^{-\text{i}\omega + \text{i} \langle\vec{p} , \vec{x}\rangle}f_* (\omega , \vec{k})}{(\omega\pm \text{i}\varepsilon)^2 - \vec{p}^2 -m^2}\,\text{d}p, \quad (***)$$
where $$ f_* (\omega ,\vec{k}) = \frac{1}{\sqrt{(2\pi)^4}} \int_{\mathbb{R}^4} f(s,\vec{y}) \text{e}^{\text{i} \omega s - \text{i} \langle\vec{k} , \vec{x}\rangle} \,\text{d} (s,\vec{y}) $$ is the Minkowski-Fourier transform. So the linear operators $G_\pm$ from the test functions into the smooth functions satisfy $$ \text{KG} \circ G_\pm = G_\pm \circ \text{KG} = \text{id}$$ on the space of space testfunctions. Here $\text{KG} = \partial_\mu \partial^\mu + m^2$ is the Klein-Gordon operator.
(1) Is this right so far?
(2) Given $(***)$, how can I show $$ \text{supp} G_\pm f \subset \text{J}_\pm (\text{supp} f),$$ where $\text{supp}$ denotes the support and $\text{J}_\pm(X)$ is the causal future/past of some subset $X$ of Minkowski space (the points that can be reached from $X$ by future/past oriented causal curves).
In physics books, you can find lots of arguments relating to (2), for instance invoking the residue theorem and so on, but all these arguments are formal calculations with expressions like $(**)$, where no testfunctions are inserted. I do not know how to compute these things, when test functions are inserted (a problem I see is that testfunctions or not analytic...). So I would love to see some direct argument using $(***)$, maybe using Lorentz invariance?
Is there any book I could use as a reference here, which is mathematically a little more rigorous that your average book on QFT?
Thanks in advance.
Sorry for the laconic answer but I should not disperse myself...:
(0) the integral (**) (in $\vec{p}$) defines a distribution in $x$ and indeed not a function (cf. oscillatory integrals in e.g. Reed and Simon 2; Zworski) cf. also two QFT books where (similar distributions) are expressed as a sum of a delta distribution + some modified Bessel function: Quantum Electrodynamics (2009), Walter Greiner, Joachim Reinhardt: exercise 2.5 p.68 forward and Introduction to the Theory of Quantized Fields (1976), N.N. Bogoliubov, D.V. Shirkov §16.1 p.147-148
(1) yes if one reckonizes a convolution product in (***) (by taking the inverse fourier transform of the formula $\mathcal{F}(f\ast g) = \mathcal{F}(f) \times \mathcal{F}(g)$, although there is a typo $\mathcal{F}f(\omega_{\vec{p}},\vec{p})$ instead of $k$)
(2) then follows from properties on the support of a convolution product: $$ \operatorname{supp}(f\ast g) \subseteq \operatorname{supp}(f) + \operatorname{supp}(g)$$ 2 accessible references in French on this subjet are: Cours d'analyse - Théorie des distributions et analyse de Fourier (2001), Jean-Michel Bony and Éléments de distributions et d’équations aux dérivées partielles, Claude Zuily
For the mathematics of QFT, I wish people told me about
General Principles of Quantum Field Theory (1990), N.N. Bogolubov, Anatoly A. Logunov, A.I. Oksak, I. Todorov
earlier, although it seems to require a lifetime to read...