Engelman Review

Review by Dr. Engelman

November 11, 1996

“The unquestioning acceptance of the Copenhagen interpretation of quantum theory has, in the last 40 years or so, held back progress on the development of alternative theories. … Blind acceptance of the orthodox position cannot produce the challenges needed to push the theory eventually to its breaking point. And break it will, probably in a way no one can predict to produce a theory no one can imagine.”

Jim Baggott, 1992 [1]

The grand unified theory of Randell L. Mills: a natural unification of quantum mechanics and relativity ? This could well evolve as being true. At any rate, Mills proposes such a basic approach to quantum theory that it deserves considerably more attention from the general scientific community than it has received so far. The new theory appears to be a realization of Einstein’s vision and a fitting closure of the “Quantum Century” that started in 1900 with Max Planck’s work on black-body radiation and his subsequent postulate of energy quanta.

It was Einstein’s lifelong dream to unify the quantum world with his theory of (special and general) relativity [2]. Even though he was one of the three eminent fathers of quantum mechanics – besides Planck and Bohr – Einstein had serious doubts about the uncertainties that were a basic feature of its theoretical framework. In his response to Born’s interpretation of the wave function as a probability-field (“ghost-field”) he made the now famous statement: “I am at all events convinced that He does not play dice.” [1,2]

In addition, of course, quantum mechanics is fundamentally inconsistent with relativity. The somewhat forced unification in Dirac’s approach was hardly satisfying to that great genius of an Einstein: “I incline to belief that physicists will not be permanently satisfied with … an indirect description of Reality, even if the [quantum] theory can be fitted successfully to the General Relativity postulates.” [2] Einstein’s dream for a unified field theory envisioned a “programme which may suitably be called Maxwell’s … .” As his biographer Abraham Pais put it, his vision called for “start[ing] with a classical field theory, a unified field theory, and demand[ing] of that theory that the quantum rules should emerge as constraints imposed by that theory itself.” [2]

Randell L. Mills proposes such a unified field theory. He outlines a quantum theory for the atomistic world that is fully consistent with “Maxwell’s programme”: It is founded solely upon the classical laws of physics in the framework of Einstein’s relativity with an additional Lorentz-invariant scalar wave equation for de-Broglie matter waves. This additional wave equation is completely compatible with Maxwell’s vector wave equation of electromagnetism. The key for quantization of the steady-state is the well known physical law that a steady-state of moving charge or matter, with or without acceleration, must not radiate either electromagnetic or gravitational waves. This postulate was originally derived from Maxwell’s Equations in 1986 by Hermann Haus for a moving charge [3] and was generalized by Mills as follows: the steady-state eigenfunction of charge/matter has to be free of Fourier components synchronous with waves traveling at the speed of light. The condition is equivalent to the violation of phase-matching for the exchange of energy in coupled mode theory.

Retrospectively, the non-radiation postulate is the only quantization condition that seems to make perfect sense. Applied to the central force field of a hydrogen atom, Mills derives eigenfunction solutions that correspond to concentric spherical shells (called “orbitspheres”) with radii that are integer multiples n of the Bohr radius ao. These eigenfunctions can be naturally interpreted as two-dimensional charge/mass density functions of the electron confined to a spherical surface. Charge/mass points on the orbitsphere move along great circles with a fixed magnitude of linear velocity in a strictly coordinated motion to each other (the orbitsphere is not a rigid spinning globe).

All electromagnetic field energy is trapped inside the orbitsphere as in a resonant cavity with perfectly conducting walls, except for a static magnetic field produced by the surface currents of the orbitsphere. In the excited states n > 1 this trapping is meta-stable. The well-known quantized energy states of the hydrogen atom are predicted by Mills’ solutions.

As a corollary, Mills derives the properties of the electron spin and the Bohr magneton in agreement with the Stern-Gerlach experiment. These properties arise out of a constant “spin-term” in the angular function required in the solutions to satisfy the condition of negative definite charge (or positive definite mass) everywhere on the orbitsphere for any set of quantum numbers. Mills assigns to this spin-term the quantum number s = 1/2. Thus, the spin is a natural by-product of the theory, whereas in the traditional quantum mechanics of Schrödinger and Heisenberg it had to be introduced artificially.

In the ground state (n = 1, l = 0) Mills derives a homogeneous charge/mass distribution on the orbitsphere surface, in the excited states (n > 1, l > 0), on the other hand, the charge distribution becomes non-uniform and generates, together with the central charge of the nucleus, multipoles. Transition probabilities would follow from classical multipole radiation theory. At ionization the orbitsphere would expand to infinity, thus becoming the wavefront of a quasi-plane de-Broglie wave traveling away from the central nucleus, and once “free” from the nucleus, the electron orbitsphere would collapse into a spinning disk in order to conserve angular momentum.

Mills’ orbitspheres, the electron eigenfunctions of the atom, emerge as complete charge/matter equivalents of standing electromagnetic waves in a resonant cavity. The compatibility of the respective wave equations allows a harmonic self-consistent description of electron (charge/mass) and electromagnetic-field (energy) distribution in the atom. It would bring back determinism to quantum theory, a heroic task that Schrödinger set out to accomplish with his wave mechanics but, to his own dismay [1], tragically failed to do.

If, then, the charge/mass density functions of Mills were the correct solutions, the “real thing” that Einstein’s “inner voice” predicted [1,2], what are Schrödinger’s wave functions? In order to find some answer to this question one has to realize that in the case of time harmonic motion the steady-state Schrödinger equation is identical to the steady-state charge/matter wave equation in Mills’ theory, specified for the non-relativistic limit. The connection is provided by the de-Broglie relation combined with conservation of energy. What is vastly different, of course, is the boundary condition! The Haus criterion in Mills’ theory, which was outlined above, leads to non-radiating eigenfunctions. This situation is equivalent to a perfectly closed lossless resonant cavity. Schrödinger’s boundary condition, on the other hand, requires that the wave function vanishes at infinity and is well behaved anywhere else. As demonstrated by Mills, the resulting eigenfunctions have Fourier transforms with components traveling at the speed of light and, thus, should involve radiation. Schrödinger’s eigenfunctions can be considered the normal modes of a spherical resonator of infinite extent.

In such a context, how could Schrödinger’s solution describe a steady-state that has some physical meaning? To this reviewer it is quite conceivable that such a state, for each set of quantum numbers, can be characterized by the superposition of two dynamic states: one would consist of a continually contracting orbitsphere emitting “virtual” photons, whereas the other one would constitute the reverse, the orbitsphere continually expanding and thereby re-absorbing these photons, where the principle quantum number n refers to the “home” orbit and the angular quantum numbers (s,l,m) determine the angular charge/mass distribution on the contracting and expanding orbitsphere surface. No net emission or absorption of photons takes place. Such a quasi-dynamic state could, perhaps, best be compared with the superposition of a lasing resonant cavity emitting a light beam to infinity (there vanishing just like a spherical wave) and its time-reversed counterpart, i.e. a lossy cavity absorbing the opposite light beam as it travels from infinity into the cavity. This would result, in effect, in a leaky resonant cavity with lossless feedback from infinity.

Obviously, the described hybrid quasi-dynamic state could not be a real state. Rather it should be viewed as a virtual state. As such, it is expected to provide some statistical information about the possible dynamic behavior which the real steady-state of a Mills charge/mass eigenfunction may be subjected to. In the hydrogen atom the statistics would refer to all possible expansion and contraction events starting from a particular orbitsphere with a given set of quantum numbers: the orbitsphere expansion/contraction events are an endless “Monte Carlo game” forced onto the Schrödinger eigenfunction by perfect feedback from infinity! Thus, in a quasi-dynamic sense, one could consider a Mills orbitsphere of a given set of quantum numbers as being statistically “projected” onto a Schrödinger wave function of the same set of quantum numbers. To make this projection complete the latter needs to be generalized by adding the spin-term in the angular function which Schrödinger did not consider.

Such a view point would lead to the conclusion that the statistical interpretation of the Schrödinger wave function remains compatible with the unified field theory of Mills. However, the statistics became purely classical, they were totally equivalent, e.g., to the statistics of thermodynamics: statistics of, in effect, an infinite number of real individual events that proceed in a completely deterministic way without the intervention of a measurement apparatus. Hence, Heisenberg’s uncertainty principle would loose all its mystique, in the context of the Mills theory it just became the charge/mass-density-function equivalent of the classical relation between the decay time and bandwidth of a damped harmonic oscillator and its spatial twin for propagating waves!

The unified theory of Mills provides a simple, exceptionally pleasing, resolution of the conceptual problems with the traditional quantum mechanics of Schrödinger and Heisenberg. In fact, this resolution is amazingly close to Einstein’s vision [2]: Quantum mechanics is revealed as incomplete but remains a valid branch of statistical physics. It is highly accurate when dealing with a large number of quantum events, but utterly fails in the description of individual “quantum jumps”. Here, the unified theory of Mills re-establishes determinism, as is demonstrated by Mills with the example of electron scattering from a He atom: Schrödinger ‘s approach provides accurate results only for relatively large scattering angles for which the statistics are expected to be good. Mills’s deterministic approach, however, provides an accurate solution for the full angular range. Hence, traditional quantum mechanics – a better term in the framework of Mills’ theory would be statistical quantum mechanics – is related to the quantum laws of the Mills unified theory – Mills calls it classical quantum mechanics – in a way that is somewhat reminiscent of the relation between Newton’s mechanics and its generalization in special relativity. May, at last, Einstein’s spirit rest in peace?

Einstein’s theory of relativity modified Newton’s law but yielded more: it predicted the equivalence of matter and energy! What are the exciting new predictions of the grand unified theory of Mills? This theory predicts the existence of so-called “shrunken” atomic states, substates below the ground state. These substates are non-radiating electron orbitspheres at the simple fractions n = 1/2, 1/3, 1/4, … of the Bohr radius ao (the “subharmonics” of the atom!). The existence of these substates is consistent with the above speculation for a new statistical interpretation of Schrödinger’s wave functions, since these wave functions remain finite below the Bohr radius dropping to zero only at the nuclear center.

The ground state is completely stable, so the substates are generally inaccessible. According to Mills’ hypothesis, however, the atomic substates can be accessed by interaction with the proper partner atom(s) or ion(s) in a resonant energy exchange. For hydrogen, Mills calculates this critical energy to be just twice the hydrogen ionization energy from the ground state (2 x 13.6 eV). Once this energy quantum is transferred from the hydrogen ground state orbitsphere to the interacting partner atom(s) or ion(s) by exciting it to a higher orbitsphere level, the hydrogen orbitsphere becomes unstable and collapses to its next lower stable non-radiating substate with additional release of energy. Thus, to activate such a Coulomb field collapse the hydrogen atom has to absorb – as Mills calls it – an “energy hole” of 27.2 eV. According to Mills, absorption of multiples of energy holes is also allowed for this activation, and the size of the energy hole remains the same for activating further collapse from any of the substates. Considerable shrinkage should, hence, be possible in a catalytic process with the release of considerable amount of energy.

Mills also predicts that atomic Coulomb field collapse can proceed to such a degree that, with fusible atomic nuclei, e.g. deuterons, fusion can set in: Mills predicts the possibility of cold fusion or, in his terminology, Coulombic annihilation fusion (CAF)! Fleischmann and Pons [4] appear to be vindicated.

But the postulated Coulomb field collapse itself is predicted by Mills to lead to the release of large amounts of energy and by itself could explain the observed excess heat in electrolytic cell experiments. The process, thus, deserves earnest attention as a potential future energy source. Mills has some convincing experimental evidence for both catalytic exothermic formation of shrunken substates of hydrogen (so-called “hydrinos”), as well as for catalytic cold fusion. Only the future can tell, if these catalytic processes can be made efficient enough to be viable for useful energy production. As an encouraging sign, Mills has designed a hydrogen gas energy cell based on his shrinkage reaction that provides far superior performance than any of the original liquid electrolytic cells.

A most exciting feature of the Mills theory is, however, that it promises to be a true grand unified theory: Mills applies the orbitsphere concept not only to single and multiple electron atoms and ions, to the hydrogen molecule and the chemical bond, but also to pair production and positronium, and to the weak and strong nuclear forces. Mills proposes that just three basic concepts, i.e., electromagnetism, gravity, and mass/energy, suffice to describe all known phenomena from the dimensions of the atomic nucleus to those of the cosmos.

Mills’ new “classical” quantum mechanics is of beautiful conceptual simplicity and fully deterministic without the uncertainties, quantum jumps and probability functions of traditional quantum mechanics, without “spooky action at a distance” [1]. This should be a pure joy for every searching scientist! It appears that the scientific community has taken little notice of this new theory. Considering its revolutionary nature and seemingly far-out conclusions this is perhaps not too surprising. On the other hand, critical dialogue is necessary for any new and unconventional thinking in order to mature and reach a high degree of rigor and precision in its formulation. In view of the fact that recently receptiveness for alternate views on quantum theory has increased, as e.g. the renewed interest in the deterministic interpretation of Bohm attests [5], it is hoped that the theory of Randell L. Mills will find its deserved resonance.

Let me close, as I started this review, with a quote from Jim Baggott [1]: “Science is a democratic activity. It is rare for a new theory to be adopted by the scientific community overnight. Rather scientists need a good deal of persuading before they will invest belief in a new theory … . This process of persuasion must be backed up by hard experimental evidence, preferably from new experiments designed to test the predictions of the new theory. Only when a large cross-section of the scientific community believes in the new theory is it accepted as ‘true’.”

I am indebted to Professor Anthony Bell for originally bringing the 1992 edition of Mills’ book to my attention and to Professor Lawrence Ruby and Mr. Thomas Stolper for helpful advice.

Reinhart Engelmann – Professor of Electrical Engineering
Oregon Graduate Institute of Science and Technology,
Portland, OR 97291-1000


[1] Jim Baggott, The Meaning of Quantum Theory, Oxford University Press, 1992

[2] Abraham Pais, ‘Subtle is the Lord…’ The Science and the Life of Albert Einstein, Oxford University Press, 1982

[3] H.A Haus, “On the radiation from point charges,” Am. J. Phys. 54 (12), 1126 (December 1986)

[4] M. Fleischmann and S. Pons, “Electrochemically induced nuclear fusion of deuterium,” J. Electroanal. Chem. 261, 301 (1989)

[5] David Z. Albert, “Bohm’s Alternative to Quantum Mechanics,” Scientific American, May 1994, p.58