Stochastic Electrodynamics

In this chapter I want to take a much closer look at one particular interpretation of quantum mechanics, Stochastic Electrodynamics. It is a remarkable theory that (in my opinion) ought to be much more widely studied than it is. It is a purely classical theory that nonetheless reproduces all the features of quantum mechanics. It produces quantitatively accurate results even for very subtle phenomena like the Lamb shift and the Casimir effect.

Let me be very clear: I am not claiming it is the “right” interpretation. We still do not know which theory is the true explanation for quantum mechanics. But if nothing else, it serves an existence proof of what classical physics can do. Whenever someone tells you about all the things that supposedly cannot be explained by any classical theory, you can point to Stochastic Electrodynamics as an example of one that does.

Stochastic Electrodynamics is built on classical physics as it existed just before the development of quantum mechanics: classical mechanics, classical electrodynamics, and special relativity. On top of this foundation it adds one extra assumption: that all of space is filled with radiation. There is no way to block it out or isolate yourself from it. It is everywhere, and it profoundly affects the behavior of all matter.

This radiation is referred to as the zero-point field, or ZPF for short. The idea of a zero-point field is familiar from quantum electrodynamics, but in that case it is a rather mysterious phenomenon. The mathematics tells us it must exist, but gives little insight into its nature or physical meaning. Every mode of the electromagnetic field is modeled as a harmonic oscillator. Since a harmonic oscillator has non-zero energy in its ground state, the field must somehow also contain energy in its ground state. In Stochastic Electrodynamics, the zero-point field is much less mysterious. It simply consists of ordinary electromagnetic radiation, propagating through space as described by Maxwell’s equations.

An obvious objection may have occurred to you. If all of space is filled with radiation, shouldn’t that fact be obvious to us? For example, if you put a photodetector in a dark room, why does it not detect anything? Before I can answer this question, I first need to discuss another question that worried physicists a great deal in the early 20th century: why are atoms stable?

They knew from experiment that atoms were composed of negatively charged electrons orbiting around a positively charged nucleus. If the electrons are orbiting, that means they are constantly undergoing acceleration. They knew from classical electrodynamics that an accelerating charge radiates. So why don’t the electrons radiate away their energy and spiral in toward the nucleus?

No completely satisfactory answer to this question was found. Eventually quantum mechanics came along and sidestepped the question by declaring that electrons do not really orbit in the classical sense. An electron in a stationary orbital corresponds to a time invariant charge distribution, so it does not radiate. Of course, this is despite the fact that many experiments still require us to treat electrons as point-like particles, not as smooth charge distributions. This contradiction simply has to be accepted as part of the mystery of wave-particle duality.

Stochastic Electrodynamics gives a very different answer to the question. It declares that electrons really do orbit in the classical sense, following well defined trajectories through space as they move around the nucleus. Furthermore, they really do radiate exactly as predicted by classical electrodynamics. But they also gain energy as they are pushed about by the zero-point field. They are constantly absorbing energy from the zero-point field, and they are constantly emitting energy back into it. These two effects immediately come into equilibrium, with the average rate of energy emitted equaling the average rate of energy absorbed, so that atoms are stable.

We can now understand why the presence of the zero-point field is not obvious to us. Normally we think of radiation as being very easy to detect. For example, if you place any object in its way, the object will absorb energy from it and heat up. But the zero-point field does not cause objects to heat up, because those objects are already in equilibrium with it.

Of course, in another sense the zero-point field is obvious, just as air is obvious to us or water is obvious to a fish. It profoundly affects the behavior of everything around us, but because we have never experienced its absence, we take those effects for granted. We just assume, “That’s the way matter behaves,” not realizing that it only behaves that way because of the presence of the zero-point field.

We also can now see why it is impossible to isolate an experiment from the zero-point field, for example by placing it inside a metal box. The box is also in equilibrium with the field, and therefore itself acts as a source of radiation.

In practice, Stochastic Electrodynamics turns out to be a very complicated theory. Everything that quantum mechanics assumes as a postulate, it must derive from first principles. If you are interested in the details of how quantum behavior arises from the zero-point field, I must refer you elsewhere. 1 To give some sense of how it works, here are high level descriptions of a few examples.

Tunneling is one of the easiest phenomena to understand. If a barrier is struck by a particle that does not have enough energy to cross it, there is nonetheless a small chance of it tunneling through and emerging on the other side. The explanation of this is simple. As the particle interacts with the zero-point field, its energy constantly fluctuates around the average value. Viewing the particle’s energy as a distribution rather than a fixed value, we see there is a non-zero chance of it having enough energy to cross the barrier.

The Pauli exclusion principle also has a straightforward explanation. Remember that I told you atoms are larger than would be expected based only on Coulomb repulsion between the electrons? That analysis ignored the effect of radiation, because in the conventional picture, an electron in a stationary orbital does not radiate. But in Stochastic Electrodynamics it does, and that radiation must be taken into account. If two electrons occupy the same orbital, they resonantly couple to the same modes of the electromagnetic field. This leads to the extra repulsion between them.

Now I need to discuss a phenomenon that is considered rather esoteric in quantum mechanics, but takes on much greater significance in Stochastic Electrodynamics. In 1930, Schrödinger analyzed the solutions to the Dirac equation for a free electron. 2 He concluded that it undergoes constant oscillations around its average position, rotating at the speed of light with a frequency equal to twice the Compton frequency, \(\omega = 2\omega_C = 2mc^2/\hbar\). He named this motion “zitterbewegung” (“shaking motion” in German).

Unfortunately, there was no obvious way to test this prediction experimentally. It attracted occasional interest over the years, but largely was neglected. Most physicists viewed it as either an obscure phenomenon with little practical importance, or else as a mathematical artifact with no physical reality.

In 1990, David Hestenes revived interest in it with his “zitterbewegung interpretation of quantum mechanics”. 3 He began by observing that the angular momentum of the oscillatory motion exactly equals the electron’s spin; that the magnetic moment induced by the rotating charge equals the electron’s magnetic moment; and that the energy of the motion equals the electron’s rest mass. This led him to propose that an electron is an intrinsically massless particle that truly is in constant motion at the speed of light. Far from being an obscure phenomenon, this is the origin of its spin, magnetic moment, and effective mass.

Stochastic Electrodynamics allows us to take this interpretation further by also providing a physical origin for the oscillatory motion: it is created by the interaction with the zero-point field. More specifically, it is created by the circularly polarized modes of the field whose frequencies equal the de Broglie frequency of the electron.

But the significance goes even deeper. If the amplitudes of those modes are spatially varying, this leads to a net force on the electron’s average position. The spatially varying field modes effectively steer the particle, influencing the path it follows.

Does this sound familiar? It should! It is essentially pilot wave theory. The particle is a classical object following a well defined classical trajectory, while a wave directs its motion. But we no longer need to postulate the existence of the pilot wave or arbitrarily assume a form for its interaction with the particle. The pilot wave has a simple physical interpretation: it consists of those modes of the electromagnetic field whose frequencies equal the de Broglie frequency of the electron. The interaction between the zero-point field and the zitterbewegung is the orgin of the phenomena we conventionally describe as “wave-particle duality”.

I must emphasize again that we do not know whether Stochastic Electrodynamics is correct or not. For all its successes, it still contains many unsolved problems. There is a great deal of work left to be done.

Still, it is interesting to speculate on how the history of physics might have been different if Stochastic Electrodynamics had been developed in the early 20th century. Quantum mechanics was created to explain phenomena that, it was believed at the time, could not be explained by classical physics. If they had known those phenomena could, in fact, be explained by classical physics, would anyone have interpreted quantum mechanics as a fundamental theory? Or would it have been seen from the start as just an approximation, a set of empirical rules describing the effects of the complex interactions between matter and radiation?

Generations of theorists have struggled to unite general relativity and quantum mechanics, two deeply incompatible theories that describe the world in radically different ways. Stochastic Electrodynamics, being a classical theory, does not share those difficulties. If quantum mechanics were not viewed as a fundamental theory, would anyone have even thought to search for a “quantum” theory of gravity? Could all those years of effort have been avoided?

Perhaps Stochastic Electrodynamics would eventually have been abandoned. Perhaps phenomena would have been identified that truly could not be explained by any classical theory. Perhaps that will still happen. But it has not happened yet.

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1

de la Peña, L., Cetto, A. M., Hernández, A. V. “The Emerging Quantum: The Physics Behind Quantum Mechanics.” Springer International Publishing (2015).

2

Schrödinger, E. “Über die kräftefreie Bewegung in der relativistischen Quantenmechanik.” Sitzungsberichte der Preussischen Akademie der Wissenschaften: Physikalisch-Mathematische Klasse 24: 418-428 (1930).

3

Hestenes, D. “The Zitterbewegung Interpretation of Quantum Mechanics.” Foundations of Physics 20: 1213–1232 (1990).