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It is commonly known that Feynman's path integral was inspired by Dirac's observation that the kernel is proportional to $\exp{i\hbar S}$. It was Feynman, however, who had the idea of expressing the kernel as the integral of such an expression over all possible paths.

Is it known what led Feynman to try out this idea of an integral over all paths? He provides motivation from a probabilistic understanding of quantum mechanics in "Non-relativistic Quantum Mechanics" and Quantum Mechanics and Path Integrals, but is this motivation what actually led him to try out the idea of the path integral and discover that up to suitable normalization, it recovered the correct expression for the kernel and the Schrodinger equation?

Moreover, why was Dirac's observation of the proportionality of the final answer to $\exp{i\hbar S}$ insufficient as the basis for an alternative formulation of quantum mechanics.

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    $\begingroup$ Feynman was certainly inspired by the variational principles of classical mechanics. $\endgroup$ – Alexandre Eremenko Apr 13 at 23:03
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It was Dirac's paper The Lagrangian in Quantum Mechanics. He gave more than just a remark about $\exp{i\hbar S}$, he described the general structure of the path integral expression for the transition amplitude, see Dirac's remark that inspired Feynman when formulating path integral on Physics SE. But he did not come up with the suggestive intuition of summing over paths, or developed a computational procedure that could be followed, or related it to the semi-classical approximation. Fortunately, Dirac was pretty explicit as to what motivated him:

"Quantum mechanics was built up on a foundation of analogy with the Hamiltonian theory of classical mechanics... it would seem desirable to take up the question of what corresponds in the quantum theory to the Lagrangian method of the classical theory. A little consideration shows, however, that one cannot expect to be able to take over the classical Lagrangian equations in any very direct way. These equations involve partial derivatives of the Lagrangian with respect to the coordinates and velocities and no meaning can be given to such derivatives in quantum mechanics... We must therefore seek our quantum Lagrangian theory in an indirect way. We must try to take over the ideas of the classical Lagrangian theory, not the equations of the classical Lagrangian theory."

He then relates traditional amplitudes to the integral from the least action principle. Feynman uses similar motivation in QED: The Strange Theory of Light and Matter, for example, where he introduces the path integrals starting from the various paths a photon could take, and using the Fermat's variational principle.

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