Dirac's remark that inspired Feynman when formulating path integral When Feynman was trying to formulate path integral of quantum mechanics, he was inspired by Dirac's remark which roughly states that
$e^{i\frac{S}{\hbar}}$corresponds to the transition amplitude, where $S$ is the action. 
I'm wondering what was Dirac's original argument behind this statement.
 A: Dirac is indeed the forefather of the path integral approach to quantum mechanics. His reasoning was described in the 1933 paper "Lagrangian in/of quantum mechanics". See the full text:

http://www.ifi.unicamp.br/~cabrera/teaching/aula%2015%202010s1.pdf

Conceptually, he had the whole thing. He realized that the Poisson brackets have a counterpart in quantum mechanics, the commutators, and because the Lagrangian seems so natural and effective in classical physics, it should have a counterpart in quantum mechanics, too. He indeed derived the possibility to compute the transition amplitude as a sum over histories with the Feynmanian exponential inserted in.
He had formally derived this insight by inserting lots of completeness relations, as integrals over the position $Q$, inside the transition amplitude.
Dirac really didn't get too far in calculating this sum over histories, verifying that it's the right thing, dealing with subtleties that arise in the picture, and he didn't make it any useful. Those things were done by Feynman but all the formal features of the path-integral approach were already known to Dirac.
