Historical research guide · 0795 · 2022

On the analogy between stochastic electrodynamics and nonrelativistic quantum electrodynamics

Emilio Santos

In one minute

Emilio Santos, at the University of Cantabria, sets two pictures of the vacuum side by side and shows where they are the same theory. Stochastic electrodynamics takes ordinary classical electrodynamics and adds one ingredient: a real random radiation filling all of space. Quantum electrodynamics keeps the field quantised. Santos rewrites the nonrelativistic quantum theory in the Weyl-Wigner phase-space language, where states become functions of position and momentum, and finds that the quantum vacuum’s phase-space distribution is exactly the Gaussian random field the stochastic theory assumes. Keep only the first order in Planck’s constant and the quantum evolution equation collapses into the classical one, with quantum starting conditions. That is why the classical random-field picture reproduces quantum answers so well for springs and oscillators — the harmonic ground state, the Casimir effect, atoms in cavities, the specific heats of solids — and why it drifts for anything strongly nonlinear. The vacuum as a real fluctuating field is the part that survives.

Connection

Chapter 2 treats the zero-point field as a real medium rather than an accounting device, and this paper shows exactly how far that reading can be pushed inside standard physics: the quantum vacuum and a real random background are the same object in phase space to first order in Planck’s constant, which is also the regime in which the Casimir effect lives.

From the source

If a random background radiation is to look the same to every inertial observer, its spectral energy density must be proportional to the cube of the frequency, and Planck’s constant enters only to set the scale of that one universal spectrum — which is the same as giving every plane-wave mode an average energy of half Planck’s constant times its angular frequency.Section 1, Eqs. 2 to 4
In the Weyl-Wigner phase-space representation the quantum vacuum of the radiation field is described by exactly the same Gaussian function of the mode amplitudes that stochastic electrodynamics assumes for its real random field; the mathematics is identical and the difference is one of reading, a probability distribution in one theory and the Wigner function of the vacuum state in the other.Section 2.3, Eqs. 22 and 23 compared with Eqs. 1 and 4
Keeping only the first order in Planck’s constant — dropping the terms of order Planck’s constant squared in the Moyal bracket — turns the quantum evolution of particles plus field into the classical Liouville equation, so the dynamics is Maxwell and Lorentz while the initial condition for the field stays the quantum zero-point state.Section 3.1, Eqs. 27 and 28
Santos’s Proposition 1 states the analogy exactly: formally, stochastic electrodynamics is that first-order theory, differing only in that the particles may start from any probability distribution in phase space rather than from a Wigner function.Section 3.1, Proposition 1
Where the particle Hamiltonian is at most quadratic in coordinates and momenta, nothing is approximated at all, because only the first term of the Moyal series contributes — which is why the classical random-field picture returns the quantum answers for the ground-state energy and the uncertainty relations, for the Casimir effect and atoms in cavities, and, with the thermal spectrum added, for the specific heats of solids, with corrections of the order of the fine-structure constant reproducing radiative effects such as the Lamb shift.Section 1, paragraphs on results from Eq. 5; Section 3.1, paragraph following Proposition 1
For Hamiltonians that are not quadratic the discarded terms are of the order of the square of Planck’s constant times frequency divided by the system energy, a ratio of order one in the microscopic domain, so the classical picture drifts there; Santos’s own reading is that stochastic electrodynamics is a semiclassical approximation to nonrelativistic quantum electrodynamics valid in a narrow domain, and that what it has earned is the reading of the vacuum fields as real stochastic fields.Section 3.1, final paragraph; Section 4, closing paragraphs

The way in

https://arxiv.org/abs/2212.03077

Historical source-access record. This guide links to the original; it does not reproduce the paper text.

Licence checked on the source itself. The arXiv posting 2212.03077, submitted 3 December 2022 from the Departamento de Física of the Universidad de Cantabria in Santander, is under the arXiv.org perpetual non-exclusive distribution licence, which does not grant redistribution, and no Creative Commons statement appears in the nineteen-page text — so this sheet carries the summary, the claims and the author’s own abstract and sends the reader to the source, which is free to read on arXiv. The arXiv record shows no journal reference. Claims are located against that posting, whose sections are Stochastic electrodynamics, Nonrelativistic quantum electrodynamics, The approximation leading to analogy with SED, and The problems of interpretation. Two companion sheets in this library carry the rest of this conversation: Boyer’s case for stochastic electrodynamics as the closest classical approximation to quantum theory at /library/stm-1011f1af4f, and Santos’s own earlier survey of stochastic electrodynamics and the interpretation of quantum theory at /library/stm-aed90bbe0c.

How to cite it

Emilio Santos (2022) On the analogy between stochastic electrodynamics and nonrelativistic quantum electrodynamics arXiv:2212.03077

Where it sits in the curriculum

What the vacuum is