STM-D-0783Paper2012Published and peer-reviewed

Stochastic electrodynamics and the interpretation of quantum theory

Emilio Santos

Abstract and summary · read the original at the source

In one page

Emilio Santos takes one idea as far as it will go: that the quantum vacuum is a real random electromagnetic field filling space, and that quantum behaviour is what charged particles do when they are shaken by it. That programme is called stochastic electrodynamics, and Santos defines it strictly — ordinary classical electrodynamics, plus the zero-point field, with Planck’s constant entering nowhere except as the scale of that field. He then works through the standard textbook systems one by one. The oscillator comes out right: the quantum ground state becomes a particle in ceaseless irregular motion, balanced between absorbing energy from the field and radiating it back, with the Heisenberg relations following from that motion rather than being assumed. Entanglement becomes a correlation carried between two places by the vacuum. Specific heats of solids come out right. He is equally clear about where the strict theory disagrees — rotators, the hydrogen atom, atomic spectra — and names exactly what a fuller theory would have to add.

Why it matters hereChapter 2 says the vacuum is a real, structured medium, and this is the most careful audit anyone has written of what follows if you take that literally: which quantum results fall straight out of it, which do not yet, and what is missing. Chapter 3’s picture of matter held in balance with the field is Santos’s oscillator result, and chapter 13 gets his most useful proposal — that the quantum commutator is the signature of a stochastic process whose spectrum is odd in the frequency.

What it claims

01Requiring the background radiation field to be Lorentz invariant fixes its spectrum completely: the energy per unit volume per unit frequency interval is proportional to the cube of the frequency, which corresponds to an average energy of half a Planck quantum per normal mode. Planck’s constant enters the theory only as the parameter fixing the scale of that universal random radiation.Sect. 1.2, Eq. (1)
Settled physics
02Stochastic electrodynamics as Santos defines it — one interaction only, non-relativistic energies, and Planck’s constant appearing exclusively in the vacuum field — is an approximation to quantum electrodynamics at the lowest non-trivial order in Planck’s constant, with purely classical electrodynamics as the zeroth order. The equation of motion is the classical one plus a radiation damping term and the vacuum field force, known as the Braffort-Marshall equation, and the whole idea traces back to Walter Nernst, who extended Planck’s 1912 zero-point fluctuations to the field itself and suggested they might explain the stability of atoms and the chemical bond.Sect. 1.1 and 1.2; Sect. 2.1, Eqs. (2) and (3)
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03The quantum ground state of a particle in a potential well corresponds to a stationary state of a particle performing a highly irregular motion driven by the vacuum field, in dynamical equilibrium between absorption and emission. The mean square position and mean square momentum agree with the quantum predictions and satisfy the Heisenberg uncertainty relations, which are here interpreted as a consequence of that unavoidable random motion — and radiative corrections such as the Lamb shift appear naturally, as the interaction between the charged particle and the real vacuum field.Sect. 2.6, Lessons for a realistic interpretation of quantum theory
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04Commutation rules get a stochastic reading. Santos defines a two-time commutator for a stationary stochastic process as the sine Fourier transform of its spectrum, the counterpart of the self-correlation function which is the cosine transform, and shows that for the stochastic-electrodynamics oscillator this closely resembles the quantum commutator in the Heisenberg picture. His proposal is that non-commuting mathematical objects succeed in quantum mechanics because the underlying stochastic processes have spectra that are odd under a change of sign of the frequency.Sect. 3.5, Eqs. (47) and (48); restated in Sect. 7
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05Two oscillators coupled at zero Kelvin give an intuitive picture of entanglement as a correlation between fluctuations in two different places, mediated by the vacuum field, and the same model reproduces the London theory of van der Waals forces. At finite temperature the same calculation returns a mean mode energy of half a Planck quantum times the hyperbolic cotangent of the quantum divided by twice the thermal energy, which is the Debye result for the specific heat of solids — an argument, Santos says, that the energies of the normal modes are continuous though random, and that phonons need not be particles.Sect. 4.1 and 4.2, Eq. (54); Sect. 4.5, Eq. (63)
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06Santos is precise about where the strict theory and quantum mechanics part company, and all of it is in the non-linear systems: the planar rigid rotator, the emission and absorption spectrum of a particle in a well, and the hydrogen atom, whose early analytic treatment predicted spontaneous ionisation — though numerical solutions since 2003 do give stationary distributions fairly close to the quantum position distribution. He names the sharpest open puzzle as the dispersion-free zero angular momentum of a ground state, and proposes reading it as the sum of two highly correlated random angular momenta, the material system’s and the vacuum field’s, exchanged continuously so the total is conserved at zero. The missing ingredients he names for a fuller theory are the back action of the particles on the field, and the other vacuum fields including metric fluctuations of spacetime.Sect. 6.3, 6.4 and 6.5; Sect. 7, the summary of clues
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Read it · abstract

Abstract

I propose that quantum mechanics is a stochastic theory and quantum phenomena derive from the existence of real vacuum stochastic fields filling space. I revisit stochastic electrodynamics (SED), a theory that studies classical systems of electrically charged particles immersed in an electromagnetic (zeropoint) radiation field with spectral density proportional to the cube of the frequency, Planck's constant appearing as the parameter fixing the scale. Asides from briefly reviewing known results, I make a detailed comparison between SED and quantum mechanics. Both theories make the same predictions when the stochastic equations of motion are of first order in Planck constant, but not in general. I propose that SED provides a clue for a realistic interpretation of quantum theory.

(Abstract only — see the rights note above. The full text is free to read at arXiv:1205.0916. Santos’s later and shorter statement of the same programme, published open access, is at /library/stm-06eb6dfba7. Timothy Boyer’s survey of the field is at /library/stm-2d2d42369d and his case for stochastic electrodynamics as the closest classical approximation to quantum theory — the paper Santos argues with in section 1.3 — is at /library/stm-1011f1af4f. The numerical hydrogen work Santos points to in section 6.4 is at /library/stm-fca91e355a.)

The way in

https://arxiv.org/abs/1205.0916Posted to arXiv on 4 May 2012 and revised through version 3, dated 11 April 2020, under the arXiv.org perpetual non-exclusive distribution licence version 1.0 — checked on the arXiv abstract page 2026-09-08, where the rights link resolves to arxiv.org/licenses/nonexclusive-distrib/1.0/. That is not a Creative Commons licence and does not grant redistribution, so this page carries the summary, the claims and the author’s own abstract, and sends the reader to the source. The claims below are read against the version 3 manuscript, a book-length chapter of about 20,000 words whose section and equation numbers are cited in the locators. Santos wrote from the Departamento de Física, Universidad de Cantabria, Santander.

How to cite it

Emilio Santos (2012) Stochastic electrodynamics and the interpretation of quantum theory. arXiv:1205.0916

Where it sits in the curriculum

What the vacuum is Inertia and gravity from the vacuum The unified picture