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There are a number of questions on this site that explain the many wave-like behaviors of photons by making reference to wave-particle duality.

However, I have just finished reading Feynman's book QED, where he seems to go to great effort to explain things without resorting to this wave-particle duality. He explains the double-slit interference pattern entirely as a matter of the interaction of probability amplitudes. If anything, he seems to consider wave-particle duality to be an obsolete concept that physicists used to rely on to explain photon's behavior, before we "knew better".

I know that Feynman's book is > 20 years old by now, but even the recent new introduction doesn't try to explain how things have evolved since then. Am I interpreting Feynman's explanation correctly, and if so, is this still considered the "best" way of explaining the wave-like behavior of particles?

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Physics is all about constructing approximate mathematical models to describe the real world. For example Newton's laws are a mathematical model, and they do a pretty good job of describing the motion of the planets round the Sun. Pretty good, but not perfect, because Newton's laws fail to fully describe the motion of Mercury. To fix this we use a more accurate, but also far more complicated, mathematical model called General Relativity. But for most purposes, e.g. sending spaceships to Mars, we don't need the accuracy of General Relativity and Newton's laws will do, which is just as well as GR is extremely hard to do calculations with.

The point of this rambling is that physicists continually face decisions as to what mathematical models to use, and they'll normally choose the simplest one that works. Quantum Electrodynamics is the most accurate theory we have for describing the interactions of light, but it's also very hard and in many cases we don't need to go that far. If you're asked to calculate the diffraction pattern from a Young's slits experiment you can do it to better than experimental accuracy using the wave model: you don't need to use QED.

Feynmann is quite correct that the wave/particle duality is a naive idea, and one now superceded by quantum field theory. But just because an idea is naive doesn't mean it isn't useful. There isn't a best way of calculating the behaviour of physical systems just like there isn't a best car. You could use QED to calculate what happens in the Young's slits experiment, but you wouldn't, any more than you'd use a Ferrari to go to the supermarket.

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that makes sense; though in QED Feymnan actually does use QED to calculate the results of Young's experiments, but I guess because it's a famous experiment he kinda had to? – Mike Edenfield Nov 12 '12 at 15:24
Well Feynmann invented the path integral method, so he had a vested interest in using it :-) – John Rennie Nov 12 '12 at 16:40

Effectively the wave-particle duality is an obsolete concept that had its origin in a technological limitation. Everything around us can be explained using the quantum theory of particles [*] An extensive and up to date discussion of the wave-particle duality misconception is found in the Klein site and in the literature cited therein.

[*] Feynman uses the old formulation in terms of probability amplitudes and this is a bit outdated for some applications. Several applications in modern quantum optics use a recent formalism that avoids the amplitudes and uses pseudo-probability distributions for the particles (photons).

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