Historical research guide · 1087 · 1996

Quantum and classical statistics of the electromagnetic zero-point field

Michael Ibison · Bernhard Haisch

In one minute

Michael Ibison, then at the Institute for Advanced Studies at Austin, and Bernhard Haisch of the Lockheed Martin Solar and Astrophysics Laboratory take the classical model of the vacuum used in stochastic electrodynamics and test it against quantum field theory at the level of statistics rather than averages. In stochastic electrodynamics the vacuum is a real classical field: an ensemble of plane waves whose amplitude is fixed at exactly half a quantum, with all the randomness carried in their phases. That picture already reproduces the Casimir force, the van der Waals force, the Lamb shift, spontaneous emission and the radius of the Bohr atom. Ibison and Haisch show that its mode-by-mode statistics match the quantum vacuum only approximately, and drift apart as the density of modes falls. Then they repair it: let each mode's amplitude fluctuate as well as its phase, and the classical field reproduces the QED vacuum exactly. Feed that field to a classical electron oscillator and the full quantum ground-state distribution comes out, and with it the Bohr radius.

Connection

Chapter 2 rests on the claim that the vacuum is a real, structured field rather than a bookkeeping device, and stochastic electrodynamics is the programme that takes that most literally. This paper is what a healthy version of that programme looks like: two of its own practitioners find the exact point where the classical model parts company with quantum electrodynamics, and then close the gap. For chapter 13 it is the standard to hold every classical-vacuum model to — reproduce the distribution, not only the average.

From the source

In stochastic electrodynamics the classical zero-point field is represented as a homogeneous, isotropic ensemble of plane electromagnetic waves whose amplitude is exactly equivalent to an excitation energy of one half of Planck's constant times the frequency, with no randomness in the amplitudes at all — the randomness is introduced entirely in the phases of the waves, which are normally distributed.Abstract; Section IV, Boyer's classical stochastic zero-point field, Equations 37 and 46
That assumption is not precisely correct: the individual modes of the Boyer field are not normally distributed, and reproduce the normal distribution of the full field only by virtue of the mean value theorem, in the limit of an infinite density of wavevector states — so the agreement with quantum field theory diverges as the density of those states decreases.Section IV, the conclusion following Equation 46; Section VI, Discussion, paragraph 2
An alternative classical zero-point field, in which each mode carries a stochastic amplitude built from two independent zero-mean unit-variance normal deviates — equivalently, each mode's intensity is exponentially distributed — makes every mode match the quantum field theory distribution, and is therefore the correct classical analogue of the QFT zero-point field.Section V.A, Equations 47 to 49; Section V.B, Equations 50 and 51
Driving a classical electron oscillator with that modified field, using Puthoff's equation of motion with radiation damping, yields a Gaussian distribution for the oscillator coordinate that agrees with the distribution quantum mechanics predicts for the ground state of the non-relativistic harmonic oscillator — the full probability distribution, not only the root-mean-square amplitude the Boyer field recovers.Section V.C, Equations 52 to 62
Confining that distribution to two dimensions, and treating the ground-state Bohr atom as a pair of one-dimensional oscillators in quadrature, gives the mean square radius required by quantum theory — the Bohr radius, derived inside a classical framework.Section V.D, Equations 63 and 64
The authors' own reading of what the exercise shows is that to reproduce quantum field theory statistics the classical field must borrow the appropriate distributions from quantum field theory; the next step they name is a classical field whose stochastic character imitates the elevated and mixed states of the quantized field, not only the ground state.Section VI, Discussion, final paragraph

The way in

https://doi.org/10.1103/physreva.54.2737

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

Published in Physical Review A, volume 54, page 2737, 1996, © 1996 The American Physical Society. The only licence on the Crossref record is the APS default licence and there is no Creative Commons statement, so this page reproduces the abstract only. The abstract is given as the authors print it on their own manuscript, which is free to read at arxiv.org/abs/quant-ph/9911057 and carries the line ‘Published in Phys. Rev. A, 54, 2737 (1996)’; the version indexed in Crossref carries LaTeX markup artefacts around the half-quantum energy and the copyright line, which are rendered plainly here and nothing else is changed. The claims below are read from that manuscript and the locators name its own sections and numbered equations. Michael Ibison was at the Institute for Advanced Studies at Austin; Bernhard Haisch was at the Solar and Astrophysics Laboratory, Lockheed Martin, Palo Alto.

How to cite it

Michael Ibison, Bernhard Haisch (1996) Quantum and classical statistics of the electromagnetic zero-point field doi:10.1103/physreva.54.2737

Where it sits in the curriculum

What the vacuum is