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Initially Wheeler and Feynman postulated that, the electromagnetic field is just a set of bookkeeping variables required in a Hamiltonian description. This is very neat because makes the point of divergent vacuum energy a moot point (i.e: an example of asking the wrong question)

However, a few years later (1951), Feynman wrote to Wheeler that this approach would not be able to explain vacuum polarization.

Anyone knows what was the argument for saying so? I don't see how allowing both processes with entry and exit particles and processes that begin in pair-creation and end in pair-annihilation makes the existence of a field a requirement.

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As phrased it seems like a question about the history of physics, not really about physics. Our understanding of all the issues involved (vacuum polarization, quantum fields, divergences) is entirely different now, and those early pioneering efforts to grapple with difficult subjects and gain partial understanding are now replaced with a much more complete and comprehensive picture. – user566 Jan 9 '11 at 22:31
@Moshe: True but I guess we still can understand some physics by getting the reason the approach cannot be used. I even don't know that theory :) So, nice question lurscher. – Robert Filter Jan 10 '11 at 0:16
up vote 5 down vote accepted

the original strategy of Feynman and wheeler was really about the desire to get rid of all self-interactions. In the modern language, it would eliminate most loop diagrams.

In particular, consider an electron propagator, in the modern language. One may attach a photon propagator on it. That modifies the electron's self-energy, and this is the kind of a term that the Wheeler-Feynman program wanted to eradicate completely. However, if you add another complexity to the photon propagator - namely an electron-positron loop in the middle - then it is a nontrivial contribution, especially because the vacuum polarization loop may be attached to different parts of the diagram as well.

Their very idea would be that it is impossible for the same electron propagator to have two photon end points attached - that would be connected with one another. That would throw the baby out with the bath water. At any rate, no complete theory of their picture exists (or is mathematically possible) and their dreams and partial hints have only been a motivation for them to get the really important insights.

Best wishes Lubos

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In CED one can obtain an exact solution of equations and exclude the electromagnetic field from the "mechanical" equations. In QED it was impossible and the field remained in some approximation in the perturbative calculations. Perturbative modifications of this field were called "vacuum polarization" effects. They thought to be "real physical effects"; that is why R. Feynman was of that opinion.

I would draw the following analogy: let us consider an atom-atomic scattering in the first Born approximations. It is described with unperturbed atomic wave functions and the Coulomb potentials acting between charges. The exact solutions differs from such a picture: atoms are polarized while scattering, for example, and the cross section is slightly different. It is like using "modified" Coulomb potentials or even more complicated description if one refers to the first Born approximation picture.

In fact, the Coulomb forces remain themselves in the exact equation and in the exact solution. It is simply the exact solution differs from the approximate one and nothing else. No true "vacuum polarization" takes place in our atomic problem. Similarly, I think, it is in QED: the exact solution is different from any perturbative one and some corrections are called "vacuum polarization".

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