The quantum theory of light, the wave theory of the electron — both tell us that photons, electrons and other particles are somehow at the same time waves and particles. The two theories converge from different starting points: classically light consisted of waves, electricity of particles; now both kinds of entities partake of both forms of existence. Energy and matter converge in a grand synthesis; both theories are syntheses in themselves in a Hegelian sense (Newton’s light corpuscles and Huyghens’ light waves as thesis and antithesis for light, quantum theory as synthesis; electricity as continuous fluid and as discrete particles as another dialectical pair, with the wave theory of the electron as the synthesis), and then the two syntheses super-synthesize as a grand theory of “mattergy”. Didn’t Einstein unify the two and find the conversion factor (square of the velocity of light) in E = Mc squared?

But Einstein was never satisfied with Planck’s quantum theory or the Bohr theory of the atom, nor with Bohr’s notion of “complementarity”, according to which entities could somehow be both waves and particles. It seemed to Einstein that this violates a law of logic, the law of contradictions: a thing must be either this or that. Many scientists felt repelled by complementarity and other paradoxes of quantum and wave theories, like Einstein; but unlike Einstein, most became reconciled to the new theories, familiar with the new jargon (both verbal and mathematical), and even quite enamoured of the new philosophies, which some called “the Tao of physics”.

The story I wish to tell might satisfy Einstein. It might steer us away from complementarity and its contradictions and paradoxes, though not to classical mechanistic philosophies. It is more in line with modern notions of “chaos” and non-linearity. The story will be told imperfectly, because I don’t totally understand it. But to me it has the ring of truth, and my friend Robert Betchov, who told it to me, understands it, in fact he invented it. If only I could get him to write it down…

He told it to me when we were passengers in the back seat of a car, with our friends Ruth and Manja in the front, driving through the mountains of Norway. A high northern plain or plateau, barren and rock-strewn, wildly beautiful. I wanted to watch, but Robert kept interrupting, with his talk about non-linearity and turbulent flow. I wanted to ignore him and day-dream about Peer Gynt, but he captured my attention – this strange genius, old now and forgetful, but playing with concepts that gradually convinced me could be revolutionary in physics and philosophy. Yet I only half followed him, even when I concentrated. The mathematics I have now forgotten completely.

The Schrodinger equation for an electron or a quark apparently has a few linear terms and also a non-linear (high exponent) residue term. The latter has usually been ignored and the linear terms were the only ones considered; otherwise the problem would have been too difficult, in fact intractable. This has been the usual fate of non-linearities; even turbulent flow, a simple everyday occurrence at water taps or in mountain streams, has never been completely defined or described mathematically in fluid mechanics, which deals only with the case of laminar (linear) flow.

In the case of the electron or quark, the linear terms give a smooth attenuated (damped) sine wave, which has been the solution generally presented. The damped sine wave represents the probability of finding the “particle” at that spot if the appropriate experiment is performed. The non-linear residual term yields a sharp narrow dissonant peak, like a sudden loud noise interrupting a melody. We have tended to dismiss it as irrevelant.

Where in the wave packet is this dissonant peak? A hint may be gathered from another (unrelated) case of a non-linear residual, resonance in air columns enclosed in cylinders through which a piston is moving to generate pressure waves. Here the non-linear peak shuffles back and forth in the column, like the eternal outcast wandering the seven seas forever without finding a refuge.

Returning to the electron wave packet: perhaps the non-linearity peak oscillates extremely rapidly among the regular waves of probability, without becoming localized — somewhat as the double and single bonds in a benzene ring do. Yet the residence time of the peak is somewhat longer where the amplitude of the regular wave is higher.

Now a leap of faith: a young physicist at CERN near Geneva apparently said to Betchov when he expounded these theories, “Ah, there is the quark!” The electron or quark is unlocatable, because it oscillates much too rapidly; but it is more probable to be found where the regular wave amplitudes are greater. This is the link between the linear and non-linear terms of the Schrodinger equation; it is not really “decomposable” into linear and non-linear terms, they interact.

When we perform certain experiments, e.g. passing a stream of electrons through a slit, we get a wave pattern by diffraction or interference, as we would with light. But if we try to pass a single electron through two slits lined up behind each other, it will sometimes go through and sometimes be stopped. It goes through only if the peak (or particle) of non-linearity, in its super-rapid oscillation along the wave packet, happens to be at the exact position to go through the slits.

Sometimes the concept is put in this way: that the particle is brought into existence only when the experiment is carried out. This is counter-intuitive and Einstein did not like it. Probability is not good enough, he said. “God does not play dice with the universe.” Perhaps this would not matter if the question was only whether or not a single electron would go through two aligned slits; but in a famous “thought experiment”, the electron, if it goes through, could trigger an electric device that would electrocute a cat, that is, have large effects in the macro world. God would not leave this to chance, says Einstein.

Postulating the rapidly oscillating peak of non-linearity restores determinism to the sub-atomic world, and hence to the macro world, where cats and humans live and die. The probability waves are still there, but are subjective rather than objective. They reflect only our ignorance of the sub-micro phenomena in space and time (very small dimensions, very rapid oscillations). We don’t know where that peak (electron) is, but God knows all the time. His eye is not only on the sparrow, but on every quark and electron in the universe.

The barren high plain of Norway gives rise to brooding meditations. Did Einstein know God’s mind better than Planck or Bohr? Does anyone? But Betchov’s speculations are intriguing.