STM-D-1087Paper1996Published and peer-reviewed

Quantum and classical statistics of the electromagnetic zero-point field

Michael Ibison · Bernhard Haisch

Abstract and summary · read the original at the source

In one page

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.

Why it matters hereChapter 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.

What it claims

01In 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
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02That 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
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03An 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
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04Driving 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
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05Confining 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
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06The 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
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Abstract

A classical electromagnetic zero-point field (ZPF) analogue of the vacuum of quantum field theory has formed the basis for theoretical investigations in the discipline known as random or stochastic electrodynamics (SED). In SED the statistical character of quantum measurements is imitated by the introduction of a stochastic classical background electromagnetic field. Random electromagnetic fluctuations are assumed to provide perturbations which can mimic certain quantum phenomena while retaining a purely classical basis, e.g. the Casimir force, the Van-der-Waals force, the Lamb shift, spontaneous emission, the RMS radius of a quantum-mechanical harmonic oscillator, and the radius of the Bohr atom. This classical ZPF is represented as a homogeneous, isotropic ensemble of plane electromagnetic waves whose amplitude is exactly equivalent to an excitation energy of hν/2 of the corresponding quantized harmonic oscillator, this being the state of zero excitation of such an oscillator. There is thus no randomness in the classical electric field amplitudes: Randomness is introduced entirely in the phases of the waves, which are normally distributed. Averaging over the random phases is assumed to be equivalent to taking the ground-state expectation values of the corresponding quantum operator. We demonstrate that this is not precisely correct by examining the statistics of the classical ZPF in contrast to that of the electromagnetic quantum vacuum. Starting with a general technique for the calculation of classical probability distributions for quantum state operators, we derive the distribution for the individual modes of the electric field amplitude in the ground-state as predicted by quantum field theory (QFT). We carry out the same calculation for the classical ZPF analogue, and show that the distributions are only in approximate agreement, diverging as the density of k states decreases. We then introduce an alternative classical ZPF with a different stochastic character, and demonstrate that it can exactly reproduce the statistics of the electromagnetic vacuum of QED. Incorporating this field into SED, it is shown that the full probability distribution for the amplitude of the ground-state of a quantum-mechanical harmonic oscillator can be derived within a classical framework. This should lead to the possibility of developing further successful correspondences between SED and QED.

Michael Ibison and Bernhard Haisch, "Quantum and classical statistics of the electromagnetic zero-point field", Physical Review A 54, 2737 (1996). Abstract reproduced with attribution. © 1996 The American Physical Society.

(Abstract only — no further text of the paper is reproduced here; see the rights note above. The authors’ own manuscript is free to read at arxiv.org/abs/quant-ph/9911057. On this site, the founding statement of the classical zero-point field programme is Marshall’s random electrodynamics at /library/stm-7faa238629 and Boyer’s 1975 formulation — the field this paper analyses — at /library/stm-8c24f62160; Puthoff’s treatment of quantum ground states as equilibrium particle–vacuum interaction states, whose oscillator equation is used in Section V.C, is at /library/stm-c7c1082f9b; Haisch, Rueda and Puthoff on inertia, gravitation and mass as zero-point-field effects is at /library/stm-7be6973b35; and Milonni’s book-length account of the quantum vacuum, the paper’s reference 2, is at /library/stm-d0a2779af6.)

The way in

https://doi.org/10.1103/physreva.54.2737Published 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 The unified picture