Quantum and classical statistics of the electromagnetic zero-point field

Michael Ibison and Bernhard Haisch
Phys. Rev. A 54, 2737 – Published 1 October 1996
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Abstract

A classical electromagnetic zero-point field (ZPF) analog 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. We carry out the same calculation for the classical ZPF analog, 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 quantum electrodynamics (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. © 1996 The American Physical Society.

  • Received 17 April 1996

DOI:https://doi.org/10.1103/PhysRevA.54.2737

©1996 American Physical Society

Authors & Affiliations

Michael Ibison

  • C-131 Engineering Quadrangle, Princeton University, Princeton, New Jersey 08544

Bernhard Haisch

  • Solar and Astrophysics Laboratory, Division 91-30, Building 252, Lockheed Martin, 3251 Hanover Street, Palo Alto, California 94304

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Issue

Vol. 54, Iss. 4 — October 1996

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