In chapter 2.2 of Feynman's book on QFT, he states that the probability amplitude of a particle going from a to b is the sum of contributions from all paths, and that each path contributes the same amplitude, but with a different phase. My question is, why does Feynman state that this is the rule? How is it known that the amplitude for each path is identical, or that the phase is determined by the action of the path? I have heard that using the same amplitude for each path leads to infinite summations, which I can see would be the case, and that this problem had to be solved with mathematical trickery. So how did these rules come about? Were they experimentally derived?