Around 1951 Dirac wrote a less-known paper compared to his famous Dirac equation paper. In that 1951 paper with title A new classical theory of electrons, Dirac wrote among other things:
"Recent works by Lamb, Schwinger, Feynman and others have been very successful in setting up rules for handling the infinities and subtracting them away, ... but the resulting theory is ugly and incomplete one."
With that statement it seems Dirac wants to say that the quantum electrodynamics theory is incomplete and unsatisfactory. I read elsewhere that Feynman also said that QED is like sweeping under the rug.
One problem related to this incompleteness of QED is incompatibility between quantum and classical picture of electron. Therefore Dirac's attempt to arrive at a new classical model of electrons should be appreciated.
What do you think? Do you know whether the classical finite-size models of electron have been proposed after Dirac? Has this problem been solved?
See: http://dieumsnh.qfb.umich.mx/archivoshistoricosMQ/ModernaHist/Dirac1951a.pdf
Classical charge distributions, that have a finite size, of course exist and can be described quantitatively. However such a description break down for the electron in a known way, since such a description is incompatible with the anomalous magnetic moment, on the one hand, and the weak interactions of the electron, on the other hand, to name but two, experimentally well studied properties. Experiments at LEP and the LHC have shown that quarks and lepton in general and the electron in particular are point-like up to the energies probed, I.e. O(TeV); and the classical approximation to the Lagrangian of the Standard Model is *many* I.e. more than 5 standard deviations, away from the results, that do agree with the loop corrections. Dirac's paper is seen as relevant in exploring how gravitational effects may be taken into account, where, of course, the notion of a point of finite mass breaks down.
It's not the fact that the electron is a point that leads to singularities in quantum field theory, but the fact that a quantum field theory describes infinitely many degrees of freedom. This is the infinity that must be defined-and renormalization theory achieves this in a way that's consistent with Lorentz invariance. incidentally, before Dirac, Heisenberg and Snyder (independently) had studied how the introduction of a minimal length might, indeed, remove the divergences in quantum field theory. These works are now recognized as precursors to the ideas of non-commutative geometry of spacetime, in the context, once more, of quantization of gravity.
A fixed length scale, (or energy scale, like a mass), incidentally, breaks Lorentz invariance, so care is required. And while a mass can define an ``infrared cutoff'', it can't regularize ``short-distance'', ``ultraviolet'' divergences.
All this is well known, within the context where quantum gravitational effects can be neglected, i.e. how spacetime reacts to quantum matter fluctuations. What's not known is how to describe the quantum fluctuations of spacetime, induced by matter fluctuations.
I think there is a feeling that the finite sized models of classical particles, in particular for the radiation reaction, are in a good place. See http://books.google.com/books?id=bZkaJZ5htiQC. The relevance to the quantum case is not clear. Ralfelski claims that QFT cannot adequately describe radiation reaction. I think the classical picture with finite size should be exported into QFT but in a manner much more sophisticated than the simple cut offs for regularization and that would require a truly initial value approach to QFT. It is not clear how path integrals or quantum fields would relate to this.
@Mohamed, Luiz, Stam, Clifford, and Remi: thank you for your comments. best wishes
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