A free particle under the action of a zero-point field and a small perturbation

The main scope of this article is to give an exposition, which is both simplified and satisfactory of the results that were obtained in BATTEZZATI M., Nuovo Cimento B, 108 (1993) 559, which was concerned with the same problem, namely, the steady-state solutions for the probability density distribution of a unidimensional system under the action of a zero-point field and a small static perturbation. These solutions belong to a diffusion equation, whose parameters are to be evaluated by the procedures which were described in full details in previous articles by the same author. A semi-Eulerian representation is used for the velocity field, whose deterministic part, which has been proved to satisfy a Hamilton-Jacobi-Riccati equation, represents the drift velocity of the system. The full equations of motion in Lagrangian representation are here solved in the presence of a zero-point field without external potential, so as to obtain a representation of the response functions satisfying the prerequisites which are necessary in order to evaluate the diffusion coefficient by the present method. The further developments considering a nonvanishing potential are therefore obtained through a limiting procedure, exploiting the typical stability theorems of classical mechanics. Consequently, it is shown that, in the presence of a static perturbation, a resonance phenomenon is mainly responsible for diffusion. Nonresonant terms reproduce fairly well the Lamb shifts obtained from quantum electrodynamics.