On the electron Zitterbewegung structure—The Abraham-Lorentz model revisited

Ph. Blanchard, Ph. Combe, M. Sirugue +1 more

William D. Thacker

Jürgen Bosse

By solving the non-relativistic Abraham–Lorentz (AL) equation, I demonstrate that the AL equation of motion is not suited for treating the Lorentz atom, because a steady-state solution does not exist. The AL equation serves as a tool, however, for deducing the appropriate parameters Ω and Γ to be used with the equation of forced oscillations in modelling the Lorentz atom. The electric polarisability, which many authors “derived” from the AL equation in recent years, is shown to violate Kramers–Kronig relations rendering obsolete the extracted photon-absorption rate, for example. Fortunately, errors turn out to be small quantitatively, as long as the light frequency ω is neither too close to nor too far from the resonance frequency Ω. The polarisability and absorption cross section are derived for the Lorentz atom by purely classical reasoning and are shown to agree with the quantum mechanical calculations of the same quantities. In particular, oscillator parameters Ω and Γ deduced by treating the atom as a quantum oscillator are found to be equivalent to those derived from the classical AL equation. The instructive comparison provides a deep insight into understanding the great success of Lorentz’s model that was suggested long before the advent of quantum theory.

Jürgen Bosse

Abstract By solving the non-relativistic Abraham–Lorentz (AL) equation, I demonstrate that the AL equation of motion is not suited for treating the Lorentz atom, because a steady-state solution does not exist. The AL equation serves as a tool, however, for deducing the appropriate parameters Ω and Γ to be used with the equation of forced oscillations in modelling the Lorentz atom. The electric polarisability, which many authors “derived” from the AL equation in recent years, is shown to violate Kramers–Kronig relations rendering obsolete the extracted photon-absorption rate, for example. Fortunately, errors turn out to be small quantitatively, as long as the light frequencyωis neither too close to nor too far from the resonance frequency Ω. The polarisability and absorption cross section are derived for the Lorentz atom by purely classical reasoning and are shown to agree with the quantum mechanical calculations of the same quantities. In particular, oscillator parameters Ω and Γ deduced by treating the atom as a quantum oscillator are found to be equivalent to those derived from the classical AL equation. The instructive comparison provides a deep insight into understanding the great success of Lorentz’s model that was suggested long before the advent of quantum theory.

Michael K.‐H. Kiessling

We discuss a class of quantum Abraham models in which the N‐particle spinor wave function of N electrons solves a Pauli respectively Schrödinger equation, featuring regularized classical electromagnetic potentials which solve the semi‐relativistic Maxwell‐Lorentz equations for regularized point charges, which move according to some de Broglie‐Bohm law of quantum motion. Thus there is a feedback loop from the actual particle motions to the wave function. The electrons have a bare charge and positive bare mass different from their empirical charge and mass due to renormalization by the self‐fields. In the classical limit the various models reduce to the Hamilton‐Jacobi version of corresponding Abraham models of classical electron theory.

Alexei A. Deriglazov