In path integral formulation we say that we are summing over all possible ways for the system get from initial to final state. Now if we just write the amplitude and then insert complete set of states, is it then true that this insertion of sets of states for every step is actually where we sum over all possible states? Of course, to get all possible paths we have to multiply, but is it in this moment that this can be seen as a sum over all paths?
To expand a little. In a book by A. Zee there is a fun derivation of path integral formulation where we start by observing the amplitude for a particle in some inital state I to be found in some final state F after some time T. Now, if these states are just eigenstates of a position vector then we can to this: we can split our time translation operator for T into small time intervals and the between those put for every interval a complete set of states. By doing so, are we explicitly doing integral over all possible paths and can it be seen as a result of combining, which is made, as usual, by multiplication of all possible realizations of position? I am pretty sure that this is it. But not completely sure.