What was Feynman's "much better way of presenting the electrodynamics" -- which did **not** appear in the Feynman lectures?
I spent a long time researching this question for Carver Mead (mentioned by Art Brown) in 2008, because we were both curious what Feynman meant. Carver thought Feynman's "better way of presenting electrodynamics" would be something along the lines of his own "Collective Electrodynamics," but that turned out to be only partly true, as I discovered in four pages of Feynman's notes, written during the year he was teaching the FLP lectures on electrodynamics, which briefly explains his new program. [These notes can be found in The Caltech Archives: Box 62, Folder 8 of The Feynman Papers, "Working Notes And Calculations: Alternate Way to Handle Electrodynamics, 13 Dec 1963."] I asked Matt Sands if he knew anything about it, and he told me that in about the middle of the 2nd year of the FLP lectures, Feynman started to complain that he was disappointed that he had been unable to be more original. He explained that he thought he had now found the "right way to do it" -- unfortunately too late. He said that he would start with the vector and scalar potentials, then everything would be much simpler and more transparent. The notes are much more detailed than that. Unfortunately I don't have the right to publish them myself (without asking Caltech's permission)... but there is a plan to digitize the Feynman Papers and put them online - funding is being sought for that now.
Mike Gottlieb: Editor, The Feynman Lectures on Physics & Co-author, Feynman's Tips on Physics
P.S. As mentioned in my comment below, the notes have been posted. They can now be found here.
I am not sure, but maybe this is about Feynman's derivation of the Maxwell equations outlined in Dyson's article http://signallake.com/innovation/DysonMaxwell041989.pdf (Am. J. Phys. 58(3), March 1990, p. 209). However, my impression was that derivation is deficient.
Opening with an aside:
Interestingly, one of Feynman's students, Carver Mead, of VLSI fame, expressed similar dissatisfaction with these EM lectures and actually wrote a monograph, "Collective Electrodynamics", which attempts to reformulate the discipline using the potentials, not the fields, as the primary entities, and quantum systems (superconducting loop, coherent quantum resonator) as the canonical examples.
It's not a difficult read. I'm not qualified to pass judgment on its success, but I do know I wouldn't want this approach to be my first course in EM.
Anyway, all that is only tangential to your question. I believe the interview you quoted is from 1966. Much later, in 1983, Feynman gave a series of public lectures on his theory of Quantum Electrodynamics (QED), which were subsequently published as QED, The Strange Theory of Light and Matter.
The bulk of this book describes the probability amplitudes of interactions of photons and electrons, and their applications in various settings ("calculating the sum of all the little arrows"). Near the end of Chapter 3, there is a schematic argument which may refer to the "1966 approach":
There are circumstances, for example, where the amplitude to emit a photon by a source is independent of whether another photon has been emitted. This can happen ... when a very large number of electrons are all moving the same way, such as up and down in the antenna of a broadcasting station or going around in the coils of an electromagnet. Under such circumstances a large number of photons are emitted, all of exactly the same kind. The amplitude of an electron to absorb a photon in such an environment is independent of whether it or any other electron has absorbed other photons before. Therefore its entire behavior can be given by just this amplitude for an electron to absorb a photon, which is a number - called a "field" - that depends only on the electron's position in space and time.... When we take polarization into account, there are more components to the field. (There are four components - corresponding to the amplitude to absorb each of the different kinds of polarization (X, Y, Z, T) the photon might be in - technically called the vector and scalar electromagnetic potentials.
In other words, Feynman is claiming to derive classical EM as a particular limit of QED. Of course that should be possible; the impressive thing here is that, if this approach is indeed what he was referring to in 1966, Feynman felt he could explain it to a "very intelligent person" (or maybe sufficiently intelligent?).
I should emphasize that "QED" contains no more detail on this topic than what I quoted above. It's not going to satisfy someone looking for a detailed exposition.
And maybe his "1966 approach" was something completely different than that of "QED". Feynman was nothing if not creative.