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Frontiers in Physics

Stochastic Electrodynamics: Renormalized Noise in the Hydrogen Ground-State Problem

Theo M. Nieuwenhuizen

Published Oct 23, 2020• 3 citations
Publisher

Abstract

The hydrogen ground state problem is a touchstone for the theory of Stochastic Electrodynamics. Recently, we have shown numerically and theoretically that the H-atom self-ionizes after a characteristic time. In another approach we reconsidered the harmonic oscillator and renormalized the stochastic force in order to suppress high-frequency tails so that all frequency integrals are dominated by the physical resonances. In the present work we consider the regularization of the noise in the hydrogen ground state problem. Several renormalization schemes are considered. Some are well behaved, while in other ones, the high frequency renormalization induces pathologies at low frequencies. In no situation we find a way to escape from the previously signaled self-ionization.

DOI
10.3389/fphy.2020.00335

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The hydrogen ground state problem is a touchstone for the theory of Stochastic Electrodynamics. Recently, we have shown numerically and theoretically that the H-atom self-ionizes after a characteristic time. In another approach we reconsidered the harmonic oscillator and renormalized the stochastic force in order to suppress high-frequency tails so that all frequency integrals are dominated by the physical resonances. In the present work we consider the regularization of the noise in the hydrogen ground state problem. Several renormalization schemes are considered. Some are well behaved, while in other ones, the high frequency renormalization induces pathologies at low frequencies. In no situation we find a way to escape from the previously signaled self-ionization.
In this paper, the harmonic oscillator problem in Stochastic Electrodynamics is revisited. Using the exact shape of the Lorentz damping term prevents run-away effects. After introducing a cut-off in the stochastic power spectrum and regularizing the stochastic force, all relevant integrals are dominated by resonance effects only and results are derived that stem from those in the quantum ground state. For an orbit with specific position and momentum at an initial time, the average energy and the average rate of energy change are evaluated, which stem with each other. Resonance effects are highlighted along the way. An outlook on the hydrogen ground state problem is provided.
Stochastic electrodynamics is a classical theory which assumes that the physical vacuum consists of classical stochastic fields with average energy 1 2 ℏ ω in each mode, i.e., the zero-point Planck spectrum. While this classical theory explains many quantum phenomena related to harmonic oscillator problems, hard results on nonlinear systems are still lacking. In this work the hydrogen ground state is studied by numerically solving the Abraham–Lorentz equation in the dipole approximation. First the stochastic Gaussian field is represented by a sum over Gaussian frequency components, next the dynamics is solved numerically using OpenCL. The approach improves on work by Cole and Zou 2003 by treating the full 3 d problem and reaching longer simulation times. The results are compared with a conjecture for the ground state phase space density. Though short time results suggest a trend towards confirmation, in all attempted modellings the atom ionises at longer times.
In a recent paper [arXiv:1502.06856] the authors studied numerically the hydrogen ground state in stochastic electrodynamics (SED) within the the non-relativistic approximation. In quantum theory the leading non-relativistic corrections to the ground state energy dominate the Lamb shift related to the photon cloud that should cause the quantum-like behaviour of SED. The present work takes these corrections into account in the numerical modelling. It is found that they have little effect; the self-ionisation that occurs without them remains present. It is speculated that the point-charge approximation for the electron is the cause of the failure.
De la Peña 1980 and Puthoff 1987 show that circular orbits in the hydrogen problem of Stochastic Electrodynamics connect to a stable situation, where the electron neither collapses onto the nucleus nor gets expelled from the atom. Although the Cole-Zou 2003 simulations support the stability, our recent numerics always lead to self-ionisation. Here the de la Peña-Puthoff argument is extended to elliptic orbits. For very eccentric orbits with energy close to zero and angular momentum below some not-small value, there is on the average a net gain in energy for each revolution, which explains the self-ionisation. Next, an 1 / r 2 potential is added, which could stem from a dipolar deformation of the nuclear charge by the electron at its moving position. This shape retains the analytical solvability. When it is enough repulsive, the ground state of this modified hydrogen problem is predicted to be stable. The same conclusions hold for positronium.
Stochastic electrodynamics (SED) is a theory of nature that seeks to derive key aspects of quantum mechanics and quantum electrodynamics from a purely classical physics perspective. While SED has been successful in many areas, it has largely failed to stably simulate the 3D classical hydrogen atom. The computational methodology presented here overcomes these foundational challenges. We develop a pseudo-wave formulation that integrates the solid angle in k-space utilizing symmetry, resulting in a significantly reduced memory footprint and lower computational overhead for generating the zero-point radiation spectrum. This optimization is paired with a logarithmic frequency grid, a sliding window tracking the instantaneous Keplerian frequency, and a Tukey smoothing function to eliminate discrete force discontinuities. Integrated via an adaptive RKF45 solver across highly dense grids of up to 100-million bins, these techniques demonstrate the asymptotic convergence of ensemble stability. We show that increasing the grid resolution in the high-frequency domain systematically mitigates artificial electron decay, yielding a simulated probability density function that closely aligns with the quantum mechanical ground state P(r).
Abstract. Quantum mechanics is considered to arise from an underlying classical structure (“hidden variable theory”, “subquantum mechanics”), where quantum fluctuations follow from a physical noise mechanism. The stability of the hydrogen ground state can then arise from a balance between Lorentz damping and energy absorption from the noise. Since the damping is weak, the ground state phase space density should predominantly be a function of the conserved quantities, energy and angular momentum. A candidate for this phase space density is constructed for ground state of the relativistic hydrogen problem of a spinless particle. The first excited states and their spherical harmonics are also considered in this framework. The analytic expression of the ground state energy can be reproduced, provided averages of certain products are replaced by products of averages. This analysis puts forward that quantum mechanics may arise from an underlying classical level as a slow variable theory, where each new quantum operator relates to a new, well separated time interval.