Completeness of Hilbert space (on-shell states) is a very powerful concept in canonical quantization, for example, to study the nonperturbative characteristics of the S-matrix, like polology (pole and mass correspondence). However, it is not so clear how that concept of completeness is employed in functional integration.
How completeness of (on-shell states) Hilbert space is realized in functional integration? Let's take a concrete example. So, if one want to derive the polology of S-matrix, how that can be done in functional integration? How can an on-shell state be realize in the formalism?