@Slereah Basically, yes. You take the single-particle space $L^2(X)$ where $X$ is the positive mass-shell (i.e. all orthochronous four-momenta whose square is the mass of the field) and the measure is the standard Lorentz invariant measure, build the Fock space out of it and define the quantum field as the Fourier transform of the creation/annihilation operators.
By the properties of the Fourier transform and because the functions are on-shell, it then turns out that this field is a (distributional) solution to KG
@ACuriousMind Eh, I'd argue things like cosmology, which try to explain the Big Bang or the past in general, attempt to answer why the Universe ended up the way it did
@0celo7 I wanted to thank you again for the book advice earlier, I've only had time to read the introduction, but I really like the structure it is laying out and I am really excited to read through it (slowly as I unfortunately do not have that much time :< )