To compute $n$-point functions in quantum field theory we use Wick's theorem to reduce this problem to computing 2-point functions. In many textbooks, such as Peskin & Schroeder, the 2-point functions are computed directly and as a consequence it is shown that the solution ends up being a Green's function for the free theory. For example, this is how they introduce the Klein-Gordon propagator.
Another way to phrase this is that the 2-point function is the inverse operator to the free theory, i.e. if the free theory is given by a differential operator $L$ then the 2-point function is equivalent to a function (or distribution) $f$ such that $$Lf = \delta.$$
I have a feeling that there is something deeper going on here but I have been unable to find anything useful in any textbooks. Is there any intuitive reason why 2-point functions are inverse operators to the free Lagrangian? In other words, what allows/motives us to compute the 2-point function by simply "inverting the operator"?
