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arXiv:2212.03077 · The European Physical Journal Plus

On the analogy between stochastic electrodynamics and nonrelativistic quantum electrodynamics

Emilio Santos

Published Dec 2, 2022• 10 citations
arXivHTMLPDF

Abstract

Abstract I expose nonrelativistic quantum electrodynamics in the Weyl–Wigner representation. Hence, I prove that an approximation to first order in Planck constant has a formal analogy with stochastic electrodynamics (SED), that is classical electrodynamics of charged particles immersed in a random radiation filling space. The analogy elucidates why SED agrees with quantum theory for particle Hamiltonians quadratic in coordinates and momenta, but fails otherwise.

DOI
10.1140/epjp/s13360-022-03500-1

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•May 4, 2012•1 citation
Abstract I expose nonrelativistic quantum electrodynamics in the Weyl–Wigner representation. Hence, I prove that an approximation to first order in Planck constant has a formal analogy with stochastic electrodynamics (SED), that is classical electrodynamics of charged particles immersed in a random radiation filling space. The analogy elucidates why SED agrees with quantum theory for particle Hamiltonians quadratic in coordinates and momenta, but fails otherwise.
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.
During the last decades there has been a relatively extensive attempt to develop the theory of stochastic electrodynamics (SED) with a view to establishing it as the foundation for quantum mechanics. The theory had several important successes, but failed when applied to the study of particles subject to nonlinear forces. An analysis of the failure showed that its reasons are not to be ascribed to the principles of SED, but to the methods used to construct the theory, particularly the use of a Fokker-Planck approximation and perturbation theory. A new, non perturbative approach has been developed, called linear stochastic electrodynamics (LSED), of which a clean form is presented here. After introducing the fundamentals of SED, we discuss in detail the principles on which LSED is constructed. We pay attention to the fundamental issue of the mechanism that leads to the quantum behaviour of field and matter, and demonstrate that indeed LSED is a natural way to the quantum formalism by demanding its solutions to comply with a limited number of principles, each one with a clear physical meaning. As a further application of the principles of LSED we derive also the Planck distribution. In a final section we revisit some of the most tantalizing quandaries of quantum mechanics from the point of view offered by the present theory, and show that it offers a clear physical answer to them.
From the equation of motion for a radiating charged particle embedded in the zero-point radiation field we construct a stochastic Liouville equation which serves to derive, by a smoothing process, a Fokker–Planck-type equation with infinite memory. We show that an exact alternative form of this phase-space equation is the Schrödinger equation in configuration space, with radiative corrections. In the asymptotic, radiationless limit (when the radiative corrections became negligible), the phase-space density reduces to Wigner’s distribution, thus confirming Weyl’s rule of correspondence. We briefly discuss several other implications of stochastic electrodynamics which are relevant for quantum theory in general.
We apply the theory of stochastic electrodynamics to the study of the (nonrelativistic) harmonic oscillator by using the Fokker–Planck method. It is demonstrated that the equilibrium distribution in phase space is exactly equal to that given by quantum statistical mechanics, i.e., the corresponding Wigner distribution, and that analysis of this distribution by means of a decomposition in terms of canonical densities leads automatically to the usual description of quantum mechanics in terms of excited states. All fundamental equations of quantum mechanics are recovered as aprproximations to zero order in the radiation terms; the first-order terms lead to the radiative corrections predicted by quantum electrodynamics, namely, the decay of states and the Lamb shift of the energy levels. The necessary differences between both treatments of the oscillator and their implications are briefly discussed.
Arguments are provided for the plausibility that quantum mechanics is a stochastic theory and that many quantum phenomena derive from the existence of a real noise consisting of vacuum fluctuations of the fields existing in nature. I revisit stochastic electrodynamics, a theory that studies classical systems of electrically charged particles immersed in a real electromagnetic zeropoint 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 introduce a calculational quantumlike method resting upon the commutator of stochastic processes. An intuitive explanation is offered for some typical quantum features, like the uncertainty principle, the entanglement and the fact that quantum noise looks as not dissipative.