The principle of least action features prominently in Richard P. Feynman's (1918--88) alternative formulation of quantum mechanics. It is prominent in his PhD thesis (1942) and in his article "Space-Time Approach to Non-Relativistic Quantum Mechanics" (1948). As a complement to existing discussions (e.g. by Silvan Schweber 1994), I will argue that Feynman's use of the action function was characteristic of his way to solve problems. Also, I will trace the role of the action function through Feynman's subsequent work which led to his proposal of a divergence-free theory in his "Space-Time Approach to Quantum Electrodynamics" (1949).
In the framework of his PhD thesis Feynman used the action function to extend the domain of application of the traditional way of constructing quantum versions of classical theories. I see this as an instance of Feynman's characteristic heuristics of using alternative formulations for precise purposes. By divorcing the action function from its exclusive definition as the intergral of a Lagrangian, Feynman was able to extend a quantization procedure, partially borrowed from Paul Dirac, to cases in which no classical Hamiltonian or Lagrangian function was available. Such cases were of interest to Feynman because, together with his supervisor John Archibald Wheeler, he had developed a divergence-free classical electrodynamics which awaited quantization.
In Feynman's subsequent work, leading to his proposal of a divergence-free quantum electrodynamics, the action seems to be less important. However, the Green's functions, which Feynman then used, played almost exactly the same role as the action did in his earlier work. The slight change in focus, or maybe even only in terminology, was probably due to Feynman's war-related work where he often dealt with diffusion processes as noted by Peter Galison (1998). The more significant change in Feynman's theoretical proposals was that he abandoned the search to justify the action or Green's function by a microscopic model following Gregory Breit (1928), Erwin Schrödinger (1930), and Dirac (1933, 1935). Instead, he made those functions the basic elements of his theory.