Stochastic Electrodynamics: Lessons from Regularizing the Harmonic Oscillator

Stochastic Electrodynamics: Lessons from Regularizing the Harmonic Oscillator

Theodorus Maria Nieuwenhuizen — Institute for Theoretical Physics, 1098 XH Amsterdam, The Netherlands

Atoms 2019, 7, 59. Received 29 April 2019; accepted 9 June 2019; published 10 June 2019.

Abstract

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.

Keywords: stochastic electrodynamics; harmonic oscillator; harmonic oscillator ground state

1. Introduction

Quantum mechanics (QM) is a statistical theory, which does no less and no more than provide the Born probabilities for the outcomes of experiments. Though many interpretations have been put forward and various ontologies have been sought within the quantum theory, a deep analysis of the dynamics of a quantum measurement in a rich enough, but solvable, model has strengthened the case that QM provides no more than statistics. It is even capable of being connected to individual measurements, thus addressing the celebrated “measurement problem”. However, in this philosophy, QM does not provide a framework to describe individual systems in detail.

In practice, in each single experiment, something occurs inside the apparatus which triggers the specific outcome for its pointer. It is our thesis that we do not have a theory for such events. The theory can not be QM, which provides only the statistics of outcomes; it is not classical electrodynamics, nor any other known theory. Nature produces individual events “every day, the whole day”, and we are still lacking a theory for these. The challenge to find this theory is as large as it was to find special relativity and quantum mechanics in the early 1900s.

The theory of Stochastic Electrodynamics (SED) is a bold and attractive attempt to describe nature at a deeper level than by quantum theory, aiming to capture individual events. As such, it is an attempt to provide a sub-quantum mechanics. SED is a theory for classical particles embedded in a classical electromagnetic vacuum, with the intensity of its modes being compatible with the quantum zero-point spectrum. In particular, each vacuum mode is a plane wave with energy one half of Planck’s reduced constant times the angular frequency, where Planck’s constant sets the energy scale of the fluctuation spectrum and appears as a new constant of nature. The SED theory has been well-developed; the reader is referred to the books in the reference list. Before addressing deeper questions, one, first, has to demonstrate that SED explains various properties of QM at the statistical level. This has been shown, on general grounds, within a certain flow of arguments. Hence, specific cases are required to support the derivation and for gaining deeper insights. Progress has been made for harmonic oscillators, where a proper ground state emerges. Several phenomena related to oscillators have been explained as well, such as van der Waals forces, the Casimir effect, and the Unruh effect. Additionally, the leading logarithm of the Lamb shift between the hydrogen 1s and 2p states has emerged in the harmonic oscillator and the hydrogen problems; however, the finite part is not in agreement for the harmonic oscillator problem, and it has not been worked out for the hydrogen problem.

In a recent publication, we considered the hydrogen ground state in SED. To lay the basis for a possible reformulation of that problem, we revisit, here, the harmonic oscillator problem, paying special attention to the prevention of peculiarities related to damping and ultraviolet divergences. We outline the problem in Section 2 and consider the average dynamical properties in Section 3. In Section 4, we consider the average progression from a general initial condition. We conclude with a summary.

2. The basis of SED

The harmonic oscillator problem for an electron in SED is set by a stochastic differential equation in which the mass times the acceleration equals the restoring force of the oscillator, minus the charge times the stochastic electric field, plus the Lorentz radiation-reaction term — the mass times a characteristic time multiplied by the third time derivative of position. That characteristic time is two thirds of the squared charge divided by the mass times the speed of light cubed, equivalently two thirds of the fine-structure constant times Planck’s reduced constant divided by the electron rest energy, and equals 6.266 times ten to the minus twenty-four seconds. In Gaussian units the fine-structure constant is approximately one over one hundred and thirty-seven.

We study this problem on both space and time scales, inspired by the Kepler problem for the hydrogen atom. Hence, we consider a typical frequency and a typical deviation such that the oscillator energy at that scale equals one quantum at that frequency. This leads us to define an adimensional time, distance, and adimensional charge, and, dropping the adimensional index, an adimensional equation of motion in which the acceleration equals minus the squared scaled frequency times the position, minus the adimensional charge times the field, plus the squared adimensional charge times the third derivative of position. Though the adimensional charge and the scaled frequency are dimensionless, the combination that replaces the frequency times the radiation-reaction time makes the squared adimensional charge appear to have the dimension of time.

(The remainder of Section 2 and the derivations of Sections 3 and 4 are omitted for length; they carry the frequency representation of the field and the particle position, the exact damping kernel with its exponential cut-off, the regularization of the stochastic force, and the evaluation of the average energy and the average rate of energy change. The complete text is at the source.)

3. The harmonic oscillator in SED and its steady state

The harmonic oscillator problem in SED has been studied by many leaders in the field, and it is also discussed at length in the standard book. This is typically done by taking, in frequency integrals, the contributions from resonances and not bothering much about high- or low-frequency peculiarities. It is our purpose to clarify where regularizations are needed and which form they should have, in order to derive these physically relevant results in a proper fashion.

The damping term, if taken from the naive adimensional equation of motion, would be proportional to the cube of the frequency; fortunately, it has been derived from first principles in the standard book, and in our notation that exact result is a truncated convolution of the velocity with a damping kernel, in which we assume an exponential cut-off in frequency similar to the one in the stochastic spectrum. Because the structure is a truncated convolution, it still leads to a product in Fourier space.

There is a resonance around the oscillator frequency, yielding a mean squared velocity of three halves of the scaled frequency and hence an average energy of three halves of the scaled frequency, which is the ground state energy of the three-dimensional quantum oscillator. However, the large-frequency limit is only suppressed by the exponential. Evaluating its contribution to leading order adds a further term to the mean squared velocity, proportional to the squared adimensional charge divided by the cut-off time. That large extra term comes from energy injection by high-frequency modes and is removed by the regularization.

After the regularization, the average energy agrees, to leading order, with that of the ground state of the quantum mechanical oscillator: three halves of a quantum at the oscillator frequency.

4. Average progression of a specific orbit

Following specific orbits in time reveals the structure of the dynamics. Due to the stochastic force, this can only be done numerically. The idea to look at the average progression of a collection of orbits, starting at some initial time at a general initial position and speed, was put forward by de la Peña and by Puthoff for the hydrogen problem. Our numerical results motivated us to revisit this average progression of a set of orbits.

The resulting expression for the average rate of energy change exhibits that orbits with initial energy above three halves of a quantum have the tendency to lose energy, and orbits with initial energy below it have the tendency to gain energy, again demonstrating the stability of the ground state. The adimensional characteristic timescale is one over twice the damping rate, which reads one over twice the squared frequency times the radiation-reaction time in the physical units of the original equation of motion. The radiated power decays, in an algebraic and damped oscillatory fashion, to its ensemble average.

For the hydrogen problem, we have found numerical and analytical support for the thesis that SED does not produce a proper stationary state, but instead leads to self-ionization. For analyzing such cases, the best one can do is to evaluate the average rate of energy change of sets of specific orbits.

5. Summary: lessons from the harmonic oscillator analysis

The regularized equation of motion can be viewed as the motion of the particle in a stochastic potential — the ordinary harmonic potential evaluated not at the particle position but at the position shifted by the adimensional charge times a regularized stochastic displacement. For the hydrogen problem, this approach was considered in the literature, and that formulation was used as a check for our numerics.

The important lessons we learned for this harmonic oscillator problem were:

1. The noise must be renormalized to suppress the dominance of high frequencies;
2. the damping has to be treated non-perturbatively to set the width of the resonance window;
3. the energy absorbed from the field quickly goes to its ensemble average; and
4. the energy radiation contains effects from both the classical orbit and the stochastic field.

Though the latter is formally of fourth order in the adimensional charge, it contributes to the leading second-order behavior due to the resonances. It starts out at zero and decays to its ensemble average over a few damping periods.

Notice, however, that if the regularization is not adopted, the average kinetic energy can become very large. Such excessive behavior arises from energy injection by high-frequency modes, a general aspect of SED, be it theoretical or numerical.

Noise renormalization is also important for the energy radiation rate. If one calculates the fluctuating part of the radiated power by using the unregularized equation of motion, one observes a large term proportional to the mean squared field, scaling as the inverse square of the fine-structure constant times the fourth power of the nuclear charge, which does not disappear when one solves the same effect from the same equation in an equivalent way; however, when employing the renormalized equation of motion it yields, at most — actually, not even — a logarithmic divergency.

We hope that these insights will improve the understanding of other properties of SED; in particular, for the hydrogen ground state. In that problem, it has been established that certain fluctuation modes are secular, that is, growing linearly in time. Clearly, this leads, formally, to corrections which relatively quickly exceed the leading order effects. It appears that, at the linear level, these secular terms can be absorbed in the unperturbed orbit by taking its angle in the plane of the orbit at a slightly modified value. Working with this expression corresponds to taking the effect to all orders. The non-secular fluctuations are bounded, as in the harmonic oscillator problem, and will likely lose their correlation with the shifted angle quite quickly. It is yet to be investigated whether the above lessons make the non-secular fluctuations well-behaved and, ideally, provide a regularization that makes the hydrogen ground state problem in SED physically sound.

Funding and acknowledgments

This research received no external funding. This work is inspired by the workshop on stochastic electrodynamics, SED2018 in Boston, 18 to 20 July 2018. It is a pleasure to thank the organizers Herman Batelaan, Ana-Maria Cetto, and Daniel Cole for the invitation and for creating a stimulating atmosphere. The author declares no conflict of interest.