Historical research guide · 0799 · 2019
Stochastic Electrodynamics: Lessons from Regularizing the Harmonic Oscillator
Theodorus Maria Nieuwenhuizen
In one minute
Theo Nieuwenhuizen works inside Stochastic Electrodynamics — the programme that treats an electron as an ordinary classical particle immersed in a real classical electromagnetic vacuum whose fluctuations carry exactly the zero-point spectrum, half a quantum of energy in every mode, with Planck’s constant entering as a property of the field rather than of the particle. His test case is the simplest one available, a charged harmonic oscillator, and his purpose is to do it carefully. Two things, he shows, must be handled properly. The radiation-damping term has to be kept in its exact form rather than expanded, which is what prevents the classic run-away solutions; and the noise spectrum has to be cut off and regularised, otherwise very high-frequency modes pump the particle to absurd energies. Do both and every relevant integral is governed by resonance alone — and the oscillator’s average energy comes out at three halves of a quantum, precisely the ground-state energy quantum mechanics assigns it. He closes by asking whether the same discipline can rescue hydrogen.
Connection
Chapter 3 rests on the proposition that an atom holds its ground state in balance with the zero-point field — Puthoff’s 1987 result, which this paper cites and builds toward. Nieuwenhuizen states exactly what has to be done correctly for that balance to be a real physical result rather than an artefact of a divergent integral, and names hydrogen as the case still open. It is chapter 1’s evidence ladder applied from the inside, by a practitioner of the theory.
From the source
Stochastic Electrodynamics is a theory of classical particles embedded in a classical electromagnetic vacuum whose mode intensities are compatible with the quantum zero-point spectrum — each vacuum mode a plane wave carrying half a quantum of energy, with Planck’s constant setting the energy scale of the fluctuation spectrum and appearing as a new constant of nature.Section 1, Introduction, third paragraph
Once the noise is regularised and the damping treated exactly, the resonance around the oscillator frequency yields an average energy of three halves of a quantum — which is the ground state energy of the three-dimensional quantum oscillator, recovered from a classical particle in a classical fluctuating field.Section 3, following Equation (22); restated at Equation (28)
Using the exact shape of the Lorentz damping term prevents run-away effects, and the damping has to be treated non-perturbatively because it is what sets the width of the resonance window.Abstract; Section 5, lesson (2)
Without the regularisation the average kinetic energy can become very large, because energy is injected by high-frequency modes — a general aspect of Stochastic Electrodynamics, whether theoretical or numerical — so the noise must be renormalised to suppress the dominance of high frequencies.Section 5, lesson (1) and the paragraph following it; Equation (23)
The energy absorbed from the field quickly goes to its ensemble average, and the energy radiated contains effects from both the classical orbit and the stochastic field; although the latter is formally of higher order, it contributes to the leading behaviour because of the resonances, starting at zero and decaying to its ensemble average over a few damping periods.Section 5, lessons (3) and (4) and the following paragraph
Whether the same regularisation makes the hydrogen ground state in Stochastic Electrodynamics physically sound is the open question. Certain fluctuation modes there are secular, growing linearly in time; at the linear level they appear absorbable into the unperturbed orbit by shifting its angle in the orbital plane, and whether the harmonic-oscillator lessons then make the non-secular fluctuations well-behaved is yet to be investigated.Section 5, closing paragraph
The way in
https://doi.org/10.3390/atoms7020059How to cite it
Theodorus Maria Nieuwenhuizen (2019) Stochastic Electrodynamics: Lessons from Regularizing the Harmonic Oscillator doi:10.3390/atoms7020059