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Canadian Journal of Physics

Equation of motion for a charged bound particle in a zero-point field

Michele Battezzati

Published Jun 1, 1990• 7 citations
Publisher

Abstract

A method previously proposed by the same author, of solving the equations of motion in the presence of friction and an external stochastic force, is applied to the nonrelativistic Dirac equation for a charged bound pointlike particle in a black-body radiation field. It separates the particle velocity field, under the assumtion of stationarity, into a position-dependent component and a randomly fluctuating component depending mainly upon the time. This description of the possible stationary states of the bound particle in the random force field is taken as a starting point for establishing a diffusion equation in configuration space. Since we identify the first component with the drift velocity, we show that it must be the solution to a modified Hamilton–Jacobi equation by using a term that is the counterpart of the quantum modification to the same equation, which has as a first approximation exactly the same form. This term gives results proportional to the diffusion coefficient and, thereby, to the spectral density of the external random force. The diffusion equation that is obtained has complex coefficients and therefore it defines a probability density for the complex values of the variable coordinate. We propose an interpretation of this probability density, based on a specific example.

DOI
10.1139/p90-079

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•Jun 17, 2026•0 citations
A method previously proposed by the same author, of solving the equations of motion in the presence of friction and an external stochastic force, is applied to the nonrelativistic Dirac equation for a charged bound pointlike particle in a black-body radiation field. It separates the particle velocity field, under the assumtion of stationarity, into a position-dependent component and a randomly fluctuating component depending mainly upon the time. This description of the possible stationary states of the bound particle in the random force field is taken as a starting point for establishing a diffusion equation in configuration space. Since we identify the first component with the drift velocity, we show that it must be the solution to a modified Hamilton–Jacobi equation by using a term that is the counterpart of the quantum modification to the same equation, which has as a first approximation exactly the same form. This term gives results proportional to the diffusion coefficient and, thereby, to the spectral density of the external random force. The diffusion equation that is obtained has complex coefficients and therefore it defines a probability density for the complex values of the variable coordinate. We propose an interpretation of this probability density, based on a specific example.
The Lorentz-Dirac equation for a nonrelativistic particle in a blackbody radiation field is solved by a stationary trial solution formed by a slowly varying part depending on the coordinate, which is supposed to be separable, plus a rapidly fluctuating part with zero average and constant variance, depending essentially on time only. Then an equation is obtained for the slowly varying part, including the stochastic field parameters, and this dependence is evaluated explicitly for special cases. The diffusion equation in coordinate space is established for this system in a general stationary state. It is shown that the condition for the probability density P(chi,t) to be representable in the same form as a quantum-mechanical pure state, is a certain relation between the diffusion coefficient in any stationary state and the value of the mean squared fluctuation of the momentum.
The following calculations provide expressions for the drift velocity and the average diffusion coefficient, applying to diffusion in configuration space of a nonrelativistic charged particle interacting with the zero-point field of stochastic electrodynamics. The particle is assumed to evolve according to a widely popular reduced form of the Braffort–Marshall equation, which is free from runaway solutions, and supposed to be found in stationary conditions, with regard to the average values of physical quantities. The results show an interesting similarity with the equations of quantum mechanics.
We set up the Maxwell's equations and the corresponding classical wave equations for the electromagnetic waves which together with the generating source, a traveling oscillatory charge of zero rest mass, comprise a particle traveling in the force field of an usual conservative potential and an additional frictional force fff. At the de Broglie wavelength scale and in the classic-velocity limit, the total wave equation decomposes into a component equation describing the particle kinetic motion, which for f=0f=0f=0 identifies with the usual linear Schrödinger equation as previously. The fff-dependent probability density presents generally an observable diffusion current of a real diffusion constant; this and the particle's usual quantum diffusion current as a whole are under adiabatic condition conserved and obey the Fokker-Planck equation. The corresponding extra, fff-dependent term in the Hamiltonian operator identifies with that obtained by H.-D. Doebner and G.A. Goldin. The friction produces to the particle's wave amplitude a damping that can describe well the effect due to a radiation (de)polarization field, which is always by-produced by the particle's oscillatory charge in a (nonpolar) dielectric medium. The radiation depolarization field in a dielectric vacuum has two separate significances: it participates to exert on another particle an attractive, depolarization radiation force which resembles in overall respects Newton's universal gravity as we showed earlier, and it exerts on the particle itself an attractive, self depolarization radiation force whose time rate gives directly the frictional force fff.
In this work we analyze the relation between the stochastic representation of nonrelativistic quantum mechanics and the Newtonian stochastic mechanics. In order to understand how the nodes of a wave function are accounted for in stochastic quantum mechanics, we solve the Fokker-Planck equation associated with the quantum-mechanical density ${\ensuremath{\rho}}_{0}$=x2{x}^{2}x2 and current j0{j}_{0}j0​=0, and find that the stochastically perturbed particle does not cross the nodal surface x=0 of the density ${\ensuremath{\rho}}_{0}$. Further we consider the classical motion of a particle subject to a stochastic perturbation, in the absence of conventional external forces and friction, and moreover we consider the classical stochastic motion of a harmonically bound particle in the presence of a friction force proportional and opposite to the velocity of the particle, to show that although the quantum-mechanical equations for the probability density and current can be put in a form resembling Newton's second law, the actual significance of those equations is not related to Newtonian stochastic mechanics. From the fact that, in the stochastic representation of quantum mechanics, the particles have continuous trajectories, we infer that the interactions would possess, in this theory, a global character. We finally show that the property of reversibility in stochastic quantum mechanics depends on the existence of suitably chosen compensating drifts in the mean forward velocity or acceleration of the particle under consideration.
For the Langevin model of the dynamics of a Brownian particle with perturbations orthogonal to its current velocity, in a regime when the particle velocity modulus becomes constant, an equation for the characteristic function ψ(t,λ)=M[exp⁡(λ,x(t))/V=v(0)]ψ(t,λ)=M\left[\exp (λ,x(t))/V={\rm v}(0)\right]ψ(t,λ)=M[exp(λ,x(t))/V=v(0)] of the position x(t)x(t)x(t) of the Brownian particle. The obtained results confirm the conclusion that the model of the dynamics of a Brownian particle, which constructed on the basis of an unconventional physical interpretation of the Langevin equations, i. e. stochastic equations with orthogonal influences, leads to the interpretation of an ensemble of Brownian particles as a system with wave properties. These results are consistent with the previously obtained conclusions that, with a certain agreement of the coefficients in the original stochastic equation, for small random influences and friction, the Langevin equations lead to a description of the probability density of the position of a particle based on wave equations. For large random influences and friction, the probability density is a solution to the diffusion equation, with a diffusion coefficient that is lower than in the classical diffusion model.
This paper extends the stochastic electrodynamic framework of Papers 1, 2, and 3 [1–3] to relativistic quantum mechanics, deriving the Dirac equation from the relativistic Coulomb field. The non-relativistic Brownian motion of Paper 1 is generalised to a four-dimensional stochastic process in Minkowski spacetime, parameterised by proper time τ . The relativistic diffusion coefficient isD4D = ℏ/m—exactly twice the non-relativistic value ℏ/2m—arising from the additional contribution of the temporal component of the four-dimensional Wiener process. The covariant zero-point field spectral tensor Sμν(ω) = (ℏω^3/6π^2ϵ0c^3)gμν is derived from the relativistic Coulomb field through the Lorentz-covariant Einstein–Hopf detailed balance and the ergodic theorem, whose applicability to the SED electron-vacuum system is proved in Paper 3 [3]. Relativistic stochastic optimal control gives the Hamilton–Jacobi–Bellman (HJB) equation in four-dimensional configuration space, which yields the Klein–Gordon equation for spin-0 particles through the relativistic Itô correction (ℏ/2m)□. Linearisation of the relativistic Lagrangian −mc√uμuμ via Dirac matrices γμ satisfying {γμ, γν} = 2gμν converts the second-order HJB equation into a first-order equation—the Dirac equation: (iℏγμ∂μ−eγμAμ/c−mc)ψ = 0. The central new result is that spin- 1/2 is not introduced by the linearisation: it is already present in the geometry of the four-dimensional stochastic path. The mass-shell constraint uμuμ = −c^2 forces the relativistic Brownian path to be helical, with radius ℏ/mc and angular frequency 2mc^2/ℏ (Zitterbewegung). The angular momentum of this helix is ℏ/2 (spin- 1/2 ) and the magnetic moment gives g = 2—both following from the Coulomb field without additional assumptions. The linearisation then finds the unique first-order wave equation consistent with this pre-existing helical spin structure. The four-component Dirac spinor encodes four degrees of freedom already determined by the stochastic path: two spin states (helix handedness) and two time directions (particle/antiparticle).