How does one write eigenstates of field operators in terms of particle states in scalar field theory?

Question

I am reading the first paper in Schwinger's QED anthology, where he discusses his action principle. In this, he writes down states that are simultaneous eigenkets of the field operators at all points of a space-like surface $\sigma $, $\lvert\Phi', \sigma\rangle$, which obey $\Phi(x)|\Phi', \sigma\rangle = \Phi'(x)|\Phi', \sigma\rangle$ for any $x \in \sigma$.

I am aware that $\Phi(x)$ will commute with $\Phi(y)$ if the separation between x and y is spacelike, and hence what he does is doable, at least in principle. But how does one write down such a state in terms of particle states? I have also seen this construction somewhere else (I think Peskin), where functional integrals are first introduced to deal with amplitudes of the form $\langle\Phi'_1, t_1\vert\Phi''_2,t_2\rangle$, i.e. the propagation amplitude between a simultaneous eigenket of all $\Phi(x,t_1)$ and a simultaneous eigenket of all $\Phi(x,t_2)$, and the formalism is then extended to deal with the vacuum in far past to vacuum in far future amplitude.