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Journal of Mathematical Physics

Derivation of quantum mechanics from stochastic electrodynamics

L. de la Peña-Auerbach, Ana Marı́a Cetto

Published Aug 1, 1977• 70 citations
Publisher

Abstract

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.

DOI
10.1063/1.523448

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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.
A set of basic evolution equations for the mean values of dynamical variables is obtained from the Fokker–Planck equation applied to the general problem of a particle subject to a random force. The specific case of stochastic electrodynamics is then considered, in which the random force is due to the zero-point radiation field. Elsewhere it has been shown that when this system reaches a state of energy balance, it becomes controlled by an equation identical to Schrödinger’s, if the radiationless approximation is made. The Fokker–Planck equation was shown to lead to the Ehrenfest theorem under such an approximation. Here we show that when the radiative terms are not neglected, an extended form of the Ehrenfest equation is obtained, from which follow, among others, the correct formulas for the atomic lifetimes and the (nonrelativistic) Lamb shift.
We derive a classical Schrodinger type equation from the classical Liouville equation in phase space. The derivation is based on a Wigner type Fourier transform of the classical phase space probability distribution, which depends on an arbitrary constant ααα with dimension of action. In order to achieve this goal two requirements are necessary: 1) It is assumed that the classical probability amplitude Ψ(x,t)Ψ(x,t)Ψ(x,t) can be expanded in a complete set of functions Φn(x)Φ_n(x)Φn​(x) defined in the configuration space; 2) the classical phase space distribution W(x,p,t)W(x,p,t)W(x,p,t) obeys the Liouville equation and is a real function of the position, the momentum and the time. We show that the constant ααα appearing in the Fourier transform of the classical phase space distribution, and also in the classical Schrodinger type equation, has its origin in the spectral distribution of the vacuum zero-point radiation, and is identified with the Planck's constant ℏ\hbarℏ.
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.
A Schrödinger type equation for a mathematical probability amplitude ψ(x,t), is derived from the generalized phase space Liouville equation valid for the motion of a microscopic particle, with mass M, moving in a potential V(x). The particle phase space probability density is denoted W(x,p,t) and the entire system is immersed in the "vacuum" zero-point electromagnetic radiation . We show that the generalized Liouville equation is reduced to a non-quantized Liouville equation in the equilibrium limit where the small radiative corrections cancel each other approximately. Our derivation will be based on a simple Fourier transform of the non-quantized phase space probability distribution W(x,p,t). For convenience, we introduce in this Fourier transform an auxiliary constant α, with dimension of action, and an auxiliary coordinate denoted by y. We shall prove that α is equal to the Planck's constant present in the momentum operator of the Schrödinger equation. Moreover, we shall show that this momentum operator is deeply related with the ubiquitous zero-point electromagnetic radiation. It is also important to say that we do not assume that the mathematical amplitude ψ(x,t) is a de Broglie matter-wave, in other words, the wave-particle duality hypothesis is not used within our work. The implications of our study for the standard interpretation of the photoelectric effect is discussed by considering the main characteristics of the phenomenon. We also mention, briefly, the effects of the zero-point radiation in the tunnelling phenomenon and in the Compton's effect.
We study the behavior of the atomic matter immersed in a zero-point radiation field, which is taken as the source of the endless stochastic motion of electrons. The starting point is a statistical phase-space equation for a particle in interaction with the field. Under certain natural approximations, the system reaches a stationary condition; the corresponding reduced description obtained in configuration space is just the Schrodinger quantum mechanics. Quantum nonlocality is revisited from this point of view and is shown to appear as an artifact of the reduced description. The theory presented also serves to explain why usual quantum mechanics is not amenable to a phase-space formulation.