Recently I have reviewed by reference books to get a better understanding of the fundamentals of QFT and there is one thing I still do not understand. In the QFT derivation of the path integral formula, it seems that every book and online resource makes the assumption that for the field operator f(x,t)_op there are "simultaenous eigenstates" |g(x,t)> such that f(x,t)_op |g(x,t)> = g(x,t) |g(x,t)> for every x and t. The same goes for the conjugate operator pi(x,t)_op. For sake of simplicity we can focus in the case of a real scalar field that obeys the Klein-Gordon equation. I see that by making that assumption it is straightforward to move from the path integral formulation in traditional quantum mechanics (with p and q operators) to the one used in QFT, leading to equations similar in form. However I have an issue with the existence of these eigenstates, they seem to come out of nowhere so to me they look like some sort of "input" to the theory, like an assumption and not a direct consequence of the formalism of QFT via "Action in terms of the Lagrangian -> Klein Gordon equation -> f(x,t)_op in terms of a(k)_op and a'(k)_op -> conmutation relations for the operators". Can somebody explain to me or give me a reference where I can see the mathematic reasoning behind these |g(x,t)> states, or otherwise just confirm that they are indeed a necessary input to the theory.