Dear Luis de la Pena, Ana Maria Cetto, and Andrea Valdes Hernandez,
For many physicist the idea that a classical equation of motion plus a ZPE effect can explain many phenomena traditionally associated to QED sound repugnant, but it has an easy explanation.
The best way to understand the success of SED (and variations such as LSED) is using the Wigner & Moyal formulation of quantum mechanics. As you report, a quantum field is a collection of quantum harmonic oscillators. The question here is that those oscillators belong to a very special kind of quantum systems.
When we work the Wigner & Moyal equivalent of the Schrödinger equation we find that it is equal to the classical equation of motion plus quantum corrections involving third and higher-order corrections in the potential V. The potential of the harmonic oscillator is quadratic and therefore the exact quantum equation of motion is classical! (The other special system whose quantum equation of motion is classical is the free particle).
Of course, the quantum harmonic oscillator is not a classical system. The corresponding quantum properties are encoded in its quantum state (the Wigner function). For instance, the ground state Wigner function predicts a non-zero energy (the ZPE) in agreement with Heisenberg uncertainty. This explains why taking a classical equation of motion and adding a ZPE correction works so well for SED. With some modifications (some of which are already incorporated in LSED) we can mimic a lot of QED but without the inherent conceptual and computational complexity of this latter. By this same reason the Wigner & Moyal formulation is very popular in applications in quantum optics!
Of course, this of above is specific to the harmonic oscillator. More complex quantum systems cannot be described with a classical equation of motion plus a ZPE. We need a complete quantum treatment. Moreover, even at the level of classical dynamics, the Abraham-Lorentz equation (2) in your essay is only an approximation to more general classical equations of motion. For the derivation of (2) we must ignore Poincaré resonances, approximate the memory kernel by a delta-like kernel --which means that the equation (2) is valid only for times larger than a characteristic scale-- and ignore the long-time behaviour of the correlations --which means that the equation (2) is valid in scales before the long-tail effects develop--...
The very important lesson from SED is that quantum mechanics does not need to reply on operators, wavefunctions, Hilbert spaces, and all that, but that can be obtained from adding 'something' to classical mechanics.
That 'something' that we added to classical physics is a stochastic correction to classical momentum --see below link--. This correction term depends on the shape of the state. The resulting quantum equation of motion contains the Schrödinger equation and even the Von Neumann equation as special cases. All the paradoxes and puzzles of quantum mechanics are gone in the spirit of your essay.
In the work Positive Definite Phase Space Quantum Mechanics, finished yesterday, I offer details on the Liouvillian formulation of quantum mechanics, which I mentioned in my essay.
Regards
