The Wave Structure of Matter (WSM) and the Origin of the Natural Laws

Figure 1. The spherical waves of the electron or positron. An electron is a pair of waves, one converging, one diverging that form a continuous resonant structure in space. The wave amplitudes of the electron are exactly opposite to those of the positron. Rotation of an incoming wave is 720° at the center and transforms inwaves to outwaves. The resulting spin = ± h/4 . Superposition of an electron and a positron causes annihilation since all amplitudes are opposite. 
There are no orbits. For many (too many) years people imagined atoms as point electrons orbiting around a nucleus. This myth, obviously imitating our planetary system, was shown wrong by quantum theory more than sixty years ago. For example in the hydrogen atom, quantum theory predicts the electron presence as a symmetrical spherical cloud around the proton. Some physicists concluded that the point bits of matter were still there hidden inside, even though quantum theory contains no notion of point particles. The old myth dies hard!
The point particle hides the natural laws. The natural philosophers believed that there is a reality of nature which when we find it will tell us the origin of the natural laws and the connections between microphysics and cosmology. Richard Feynman wrote of this, in connection with the Conservation of Energy. He said, “Nature has a hidden accountant who keeps honest books of energy input and output but we don’t know how she does it.” Unfortunately, Feynman missed the WSM because he believed in the point particle. However his associate at CalTech Carver Meade did not and wrote a book (2000) Collective Electrodynamics, published by the MIT Press that describes the first application of the WSM to electromagnetism in microchips.
The great philosophers of cosmology, Clifford, Mach, Einstein, Dirac, Schroedinger, and others have pointed out that only a wave structure of matter (not ‘particles’) can conform to experimental data and fulfill the logic of reality and cosmology. Such a Quantum Wave Structure of Matter (WSM) has been found and is described here showing how it is the origin of the natural laws. Since the WSM provides a quantitative origin of the fundamental natural laws, it becomes the basis of the physical sciences built upon the natural laws.
William Kingdon Clifford^{1,2} was a superb brilliant
mathematician and astronomer! He worked nearly a halfcentury before
Einstein developed his general theory of relativity, and before even a
hint existed of quantum mechanics. He discussed (1876) our ability to
deduce the geometry of space at astronomical distances and in space too
small (i.e. atomsized) to be observed, stating:
I hold:
The fundamental element of space curvature in Clifford's mathematical model was the twist, which he planned to use to describe electromagnetic and atomic phenomena. But the complexity of the mathematics involved allowed only the best and brightest mathematicians to draw conclusions regarding the physics of space and time from Clifford's mathematics.
Unfortunately, Clifford's ideas were difficult to grasp in the 1870's. Einstein's version of curved space was not seen as radical in 1915 as Clifford's was in 1870. The rapid acceptance of relativity was made easier, by J.K.F. Gauss' measurements of space curvature from three mountaintops in Hanover, Germany in the 1820's. Gauss' measurements were interpreted as a measure of space curvature long before relativity theory was introduced.
The interest of physicists in Clifford's mathematical work received a large boost some years ago when John Archibald Wheeler developed some of Clifford's ideas in his book Geometrodynamics.
Like Einstein and Clifford he recognized that discrete ‘point particles’ were impossible. He wrote that the puzzle of matter will be found in the structure of space, not in pointlike bits of matter, and that the physical world is based upon geometry of space. During discussions with Neils Bohr concerning the probability interpretation of his own Schrödinger’s Equation, he firmly disagreed with Bohr and mainstream quantum mechanics and wrote^{3 }: "Let me say at the outset, that in this discourse, I am opposing not a few special statements of quantum mechanics held today (1950s), I am opposing as it were the whole of it. I am opposing its basic views that have been shaped 25 years ago, when Max Born put forward his probability interpretation, which was accepted by almost everybody. ... I don't like it, and I'm sorry I ever had anything to do with it." (Erwin Schrödinger, Life and Thought, Cambridge U. Press, 1989).
Schrödinger understood the requirements of particle structure when he wrote in 1937: "What we observe as material bodies and forces are nothing but shapes and variations in the structure of space. Particles are just schaumkommen" (appearances). He believed that quantum waves were real, not probability distributions with a particle hidden inside. He saw that abolishing the discrete point particle would remove the paradoxes of 'waveparticle duality' and the 'collapse of the wave function'
In Schrodinger’s thoughts, the mass and charge substances that were assumed, do not exist but are properties of a wave structure in space. In short, space waves were real, while mass and charge points are mere appearances of the wave structure.
Machiavelli understood the human behavior of established science that rejected Schrodinger out of hand. He wrote 500 years ago [1513]: "There is nothing more difficult to plan, more doubtful of success, more dangerous to manage than the creation of a new system. The innovator has the enmity of all who profit by the preservation of the old system and only lukewarm defenders by those who would gain by the new system."
Law of Inertia. Ernst Mach and Bishop Berkeley had proposed about 1880, that the law of inertia depended on all the matter of the universe. This was known as Mach's Principle . It was the first recognition that a natural law depends on cosmology.
Ernst Mach’s9 observation in 1883 was the first hint of the mechanism of cosmological energy transfer. He noticed that the inertia of a body depended on the presence of the visible stars. He asserted: "Every local inertial frame is determined by the composite matter of the universe" and jokingly, "When the railway jerks, it is the fixed stars that throw you down.” His deduction arose from two different methods of measuring rotation. First, without looking at the sky one can measure the centrifugal force on a rotating mass m and use the inertia law F = ma to find circumferential speed and position, as in a gyroscope. The second method is to compare the object’s angular position with the fixed (distant) stars. Both methods give exactly the same result!
Mach’s Principle was criticized because it appeared to predict instantaneous actionatadistance across empty space. How can information travel from here to the stars and back again in an instant? The answer had to await the WSM where Nature’s energy exchange mechanism, formerly unknown, is now seen as the interaction of waves in an everpresent universal space medium. Space is not empty because it is a quantum wave medium created by oscillating waves from every particle in the universe (Principle II below). Inertia, charge, and other forces are mediated by the pervasive space medium. There is no need to travel across the universe.
Albert Einstein was greatly influenced by it when he deduced the General Theory of Relativity (GTR). Now, Mach's Principle, in a more exacting form, has become Principle II of the Wave Structure of Matter (below) from which all the laws derive.
The failure of the point particle concept was recognized by Einstein. He and Ernst Mach reasoned that space and thus matter cannot be discontinuous so that particles must be "spherically spatially extended in space." Einstein wrote, "…Hence the material particle has no place as a fundamental concept in a field theory.
Dirac's work was concerned with theoretical aspects of quantum mechanics. He received a Nobel Prize for his work (1923) predicting the spin of the electron and the existence of the positron. At the time he did not realize that he was describing a quantum wave electron because his work was purely mathematical and not related to experiments that came later. Dirac had developed much of the theory describing the quantum waves of the electron, but was never satisfied with its pointparticle character because the Coulomb electron required a mathematical correction termed "renormalization". In 1937, he wrote^{4}, "This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it turns out to be small — not neglecting it because it is infinitely large and you do not want it!" Like Schrödinger, he had confidence in what his mathematics was telling him and refused to go along with the speculations and approximations of mainstream science. Dirac seemed to foresee the WSM.
They^{5} (1945) modeled the electron as spherical inward and outward electromagnetic waves, seeking to explain radiation forces. Unhappily they failed because there are no spherical solutions of vector electromagnetic wave equations. Nevertheless, their work pioneered the concept that every particle sends quantum waves outward, and receives an inward response from the universe. In hindsight, if they had used scalar quantum waves instead, this paper would have appeared 55 years ago! So close and yet so far! "A hair’s breadth separates the false and true."
After 1945, research on wave structure stopped until 1985 when Milo Wolff ^{6,7}, using a scalar wave equation with spherical quantum wave solutions, found the Wave Structure of Matter described here. It successfully predicted the natural laws and all of the properties of the electron., Below, this paper reviews the WSM and provides a physical origin of spin that accords with quantum theory and the Dirac Equation.
For the electron, the structure is a pair of spherical outward and inward quantum waves, convergent to a center, existing in ordinary space and termed a space resonance . Every space resonance shares its binary wave structure pair with all other waves in our universe thus we exist in an interconnected binary universe.
The predictive power of the WSM is illustrated by describing in Part IV how the electron’s wave pair is the physical origin of the Natural Laws and the previously unknown quantum spin. These two waves are a Dirac spinor satisfying the theoretical Dirac Equation. You can easily see in your mind how spin occurs because the inward quantum wave continually undergoes a special spherical rotation at the center transforming it to the outward wave.  more on this later. Further predictive power will be seen from the role of the WSM in communication, life, and our connection with the binary universe.
Only three basic Principles of Nature describe the wave medium and enable quantitative calculation of the wave structure of matter and the origin of the natural laws. The medium of the waves, and the leading player in the new scenario, is space that supposed void of which we formerly knew little. The properties of particles, structured of waves, and the laws embedded in them, depend upon space.
The proof of the WSM is that all the natural laws can be obtained mathematically from the three basic principles describing the wave space medium. The natural laws are simply obtained from the Wave Structure of Matter and match experimental measurements. In contrast, conventional physics required dozens of assumptions plus many more arbitrary constants to explain the operation of the laws. Even then some laws, like spin, were a puzzle with no origin known.
The wavestructured particle, Figure 1, is termed a space resonance (SR). The medium of the waves, is space. The properties of space resonances and the laws that they produce are derived from properties of space. Thus, this single entity, space, described by three principles, underlies everything .
Principle I is a Wave Equation. This Principle describes how quantum waves are formed and travel in the medium of space. The wave amplitudes are scalar numbers. If the medium is uniform, typical nearly everywhere, only spherical waves occur. If observed in relative motion, Doppler modulation and elliptical waves appear. If the medium is locally dense, as in the central region of a proton, waves circulate like sound waves in a drum or a sphere. The heavier particles, Hadrons, are protons whose waves circulate in more complex modes. Principle I is:
Quantum matter waves exist in space and are solutions of a scalar wave equation.
A wave equation is actually a statement of the conservation of energy in space:
Potential Energy + Kinetic Energy = a constant.
An increase of one energy requires a decrease of the other. It is written as a differential equation:
Where is a scalar amplitude, is the velocity of light, and is the time. Its only linear solutions shown in Figure 1, are a pair of spherical in/out waves that form the simple structure of the electron or positron. The waves decrease in intensity with increasing radius, like the forces of charge and gravity.
There are only two combinations of the two in/out waves. These form electrons and positrons, with opposite phase and spin rotation. Thus matter is constituted of two binary elements  like computer hardware. Although the variety of molecules and materials populating the universe is enormous, the basic building bricks are just two. Is there a profound meaning to this? Is the Universe binary?
Solutions of the Wave Equation (Principle I). The wave equation is written (following Wolff ^{6,7,8}) in spherical coordinates because cosmological space has spherical symmetry. Uniform density of the medium (space) is assumed which yields a constant speed c of the waves and 'light'. These only two solutions describe the charge waves of common charged particles including the electron, positron, proton, and antiproton. They are:
INwave amplitude:  (a) 
OUTwave amplitude:  (b) 
= wave amplitude, k = mc/h = wave number, w = 2f, r = radius from wave center, and energy = E = hf = mc^{2}.
At the center, the inwave continuously rotates to transform it to the outwave. Superposition of the two amplitudes produces a standing wave that can occur in two ways depending on rotation, CW or CCW. One way is the electron, the other the positron with opposite spin. To see this write the rotation operators as R^{ccw} or R^{cw} . Then the two amplitudes are:
The electron = E() = { INwave + OUTwave} R^{ccw}  (c) 
The positron = E(+) = {+ INwave  OUTwave} R^{cw}  (b) 
You can experiment with particle inversions (switching electrons to positrons) by changing the (+ or ) signs in the amplitude equations. To perform a Time inversion, change t to –t, which converts the positron into an electron. To perform a mirror inversion (Parity), imagine that the waves are viewed in a mirror. You will see that a positron is a mirror image of the electron. To change a particle to an antiparticle (Charge inversion), switch the inwaves and the out waves, and the spin direction. Successive C, P, and T inversions returns to the initial state which is a proof of the empiricaltheoretical CPT rule, now seen to be a property of the wave structure of matter. If you add the electron amplitude to the positron amplitude, the resulting amplitude is zero or annihilation as is well known experimentally.
Principle II  Space Density Principle (SDP). This principle determines the density of the quantum wave medium  space. It is fundamental because the properties of waves depend the density of their medium. As is true for all oscillators in Nature the properties of the waves, such as speed and amplitude, derive from properties of their medium. Density of the space medium is given below. Since the natural laws depend on the waves, we deduce that the natural laws in turn depend on the medium. Thus the medium of space is the wellspring of everything.
Principle II is:
At each point in space, waves from all particles in the universe combine their intensities to form the wave medium of space.
The space medium density ~ 
In other words, at every point in space, the frequency f or the mass m of a particle depends on the sum of squares of all wave amplitudes _{n} from the N particles inside the “Hubble universe”, whose distance decreases inversely with their range r_{n} squared. The "Hubble Universe" is of radius R = c/H, where H is the Hubble constant.
This principle is a quantitative version of Mach’s Principle because the space medium is the inertial frame of the law F = ma. When mass or charge is accelerated, energy exchange takes place between it and the surrounding space medium. This principle also appears in General Relativity in a much more complex math form.
Variation of Space Density. Because there are a large number of particles, N ~ 10^{80}, in the Hubble universe, the medium is nearly constant everywhere and we observe a nearly constant speed of light. But near a large astronomical body like the Sun the larger space density produces a measurable curvature of the paths of the IN and OUT waves and thus of light and the motion of matter. We observe these as the effect of gravity described by Newton and calculated by the space curvature of Einstein's general relativity.
Origin of ‘charge’. The density of space changes drastically near the center of a space resonance, for example an electron. Space becomes dense near the resonance centers due to their own large wave amplitude. Dense space at the central region propagates waves nonlinearly, which allows energy transfer or coupling between two resonances. We observe this process and call it ‘charge.’ But there is no ‘charge substance’ involved. It is a property of the wave structure at the center.
Equation of the Cosmos. Can this charge mechanism be tested? Yes. Let’s calculate using Principle II. If a resonance’s selfwaves can be dominate in its space, then at some radius, ro from the center, selfwave density must equal the total integrated density of waves from the other N particles in the Universe. This is
Selfwave intensity at r_{o} = Space Intensity at r_{o}
where N = number of particles in Hubble Universe, V is the volume of that Universe and R is its radius. Evaluating this equality^{6,7,8} yields
Equation of the Cosmos 
The best astronomical measurements, R = 10^{26} meters, N = 10 ^{80} particles, yield r_{o} = 6 x 10^{15} meters. This should be near the classical radius, e^{2}/mc^{2 } of an electron, which is 2.8 x 10^{15} meters. The test is satisfied.
This is called the Equation of the Cosmos a relation between the 'size' r_{o} of the electron and the size R of the Hubble Universe. Astonishingly, it describes how all the N particles of the Hubble Universe create the space medium and the 'charge' and 'mass' of each electron as a property of space.
Principle III  Minimum Amplitude Principle (MAP) This third principle can be obtained from Principle II, but because it is a powerful law of the universe and very useful because it determines how interactions take place and how wave structures will move, I write it separately as a third Principle:The total amplitude of all particle waves in space everywhere always seeks a minimum.
This principle is the disciplinarian of the universe. That is, energy transfers take place and wavecenters move in order to minimize total wave amplitude. Amplitudes are additive, so if two opposite resonances move together, the motion will minimize total amplitude. This explains the origin of empirical rules like, "Like charges repel and unlike charges attract" because those rules minimize total amplitude. The origins of many rules are now understood. For example, MAP produces the Heisenberg Exclusion Principle, which prevents two identical resonances (two fermions) from occupying the same state. This is not allowed because total amplitude would be a maximum, not a minimum. The operation of MAP is seen in ordinary situations like the water of a lake, which levels itself, and in the flow of heat that always moves from a hot source to a cold sink.
Origin of Special Relativity (SRT) and the De Broglie Wavelength of QM Theory. Before the WSM, there had been no known physical reason for the theoretical mass increase of special relativity. Likewise there were no physical reasons for quantum theory or spin. Were these apparently separate laws connected or not? Indeed, many theorists proclaimed that these phenomena were irreconcilable! Few had thought about a connection because most physicists were satisfied with the separate and reasonably accurate theoretical results using a discrete point particle model. The WSM changes this.
The Natural laws that involve relative motion can be found by calculating the Doppler effect between two Space Resonances (say electrons) in relative motion. To do this write the equation of a SR, as seen by an observer with relative velocity b = v/c, as shown in Wolff^{8}. Then, insert relativistic Doppler factors, g = [1v^{2}/c^{2 }]^{ 1/2}. The amplitudes received by the observer are:
Received amplitude 
This equation is an exponential oscillator modulated by a sine factor. The origin of both the de Broglie wavelength (QM) and the relativistic energy and momentum are seen as follows:
In the exponential factor:
Wavelength = h/mvg = de Broglie
wavelength with relativistic
momentum.
Frequency = kgc / 2
= gmc^{2} / h = mass frequency with
relativistic energy.
And in the sine factor:
Wavelength = h/mcg = Compton wavelength with
relativistic momentum.
Frequency = b gmc2/h = b x (mass frequency) =
relativistic momentum frequency.
Origin of electron properties. Thus the space resonance physically displays all properties of an electron, viz: QM, SRT, forces, annihilation, conversion to a positron, and CPT relations between Charge, Parity (mirror image) and Time; all of which were formerly empirical or theoretical properties. These properties depend on the spherical wave structure and ultimately on the wave medium  space.
The two combinations contain all experimental electronpositron properties. Charge depends on whether there is a + or  amplitude of the IN wave at the center. If a resonance is superimposed upon an anti resonance, they annihilate. The amplitude at the center is finite as observed, not infinite as in the Coulomb rule. They obey Feynman’s Rule: "A positron is an Electron going Backward in Time."
The properties of quantum mechanics (QM) and special relativity (SRT) are the result of the motion of one space resonance relative to another, which produces a Doppler shift in both the IN and OUT waves. All parameters of QM and SRT for a moving particle; that is, the de Broglie wavelength of QM and the relativistic mass and momentum changes, appear as algebraic factors in the Dopplershifted waves exactly as experimentally measured.
Our present knowledge of science is based on the natural laws that describe the behavior of particles. These laws are the rules for calculating electricity, gravity, relativity, quantum mechanics, and conservation of energy and momentum. The origins have been unknown. Now, the origin of the natural laws is found to be a quantitative result of the WSM.
The origins of the laws had been sought for centuries. Finding all of them at the same time is a philosopher’s dream come true. This is of great importance to science because the natural laws and quantum spin determine the structure of the Atomic Table, which dictates the varied forms of matter: metals, crystals, semiconductors, and the molecules of life. The deep understanding of basic physics that is revealed opens a door to broad fields of applied technology such as integrated circuits, photonics, and commercial energy. It reveals a universe of real quantum wave structures that we live in but seldom see.
Finding these origins was very simple. The ancient notion of a point particle was replaced with a spherical wave structure, which had already been predicted by Clifford, Einstein and Irwin Schroedinger, a century ago. It was not complex or difficult but it was different because it disagreed with human emotional experience of point particles that most people were satisfied with. Emotions play a powerful, often hidden role, in the motivation for research.
We don't easily see the space medium because our survival as an animal species depends mostly on our ability to fight with other animals seeking food, and to compete for mates that produce children, not closely related to the quantum space medium. In our selffocused human perspective few of us are even aware of the wave medium in which we exist. For survival, it doesn't matter what space is, or whether we can observe it  it exists nevertheless. This situation is much like the life of a fish who cannot comprehend the existence of water because he is too deeply immersed in it. Like the fish, traditional science has tended to comprehend the universe in terms of our own local experiences. It was assumed that matter particles are like tiny grains of sand  as proposed by the Greek philosopher Democritus many centuries ago. As microscopy improved, models were created subdividing grains into smaller ‘grains’  ad infinitum. But only a few people like Einstein saw that grains were impossible or ever asked how the grains communicate forces  unexplained by the old science. Formerly, forces were accepted as faith in Nature.
Our quantum wave universe is not as helpful to survival of our personal genes as recognizing apples we can eat and avoiding tigers who want to eat us. We need to recognize those quickly. But it was not necessary that nature equip us to observe quantum waves quickly. Lacking personal experience of simple quantum waves, people chose to imagine that the electron is a "particle," like a bullet. Laboratory evidence does not support this humanoriented idea. Accordingly, belief must change from discrete particles to wave structure. But emotional rejection can occur if the new truth conflicts with established belief. Max Planck once said, "New scientific truth does not triumph by convincing its opponents, but because the opponents die and a new generation grows up unopposed to the new idea."
Origin of Energy Transfer, and the ActionataDistance Paradox. A new facet of the WSM is the mechanism of energy exchange. Experience tells us that communication or acquisition of knowledge of any kind occurs only with an energy transfer. Storage of information, whether in a computer disk or in our brain, always requires an energy transfer. Energy is required to move a needle, to magnetize a tape, to stimulate a neuron. There are no exceptions. This rule of nature is embedded in biology and our instruments. Finding the energy transfer mechanism between particles is essential to understanding the natural laws. It is found naturally in Principle II of the WSM.
A major deficiency of the classical force laws is that they have no theoretical or physical mechanism for energy transfer. The formulas contain only constants, "mass" and "charge,"  no mechanism. This was an inherent defect of the static point particle model. Einstein, Wheeler and Feynman^{5} knew this, recognizing that there must exist a continual dynamic means for forces to transfer energy and sought it in electromagnetic waves. Unfortunately there are no spherical solutions of the vector em wave equation. Hence the mechanism had to await the scalar waves of the WSM. We will see that wave communication is the means.
Origin of the Conservation of Energy. The transfer mechanism between combinations of resonances is a result of the dense (nonlinear) space at resonance centers, which permits coupling or exchanges of the same wave frequency. When the waves of a potential source and a potential receiver, pass through both centers in an allowed transition, MAP minimizes the total of both amplitudes. In the source, the frequency (energy) of a wave state shifts downward. In the receiver there is an equal shift upward. Only wave states (oscillators) with equal frequencies 'tuned' to each other can couple and shift frequency. Accordingly, the frequency (energy) changes must be equal and opposite. This is exactly the content of the Conservation of Energy law. The origin of this universal law is reduced to the matching of waves of two particles  not too different from tuning up an orchestra matched to the 'A' played by the first violin!
The Origin of the IN Waves and the Response of the Universe. At first thought, it is puzzling where the IN waves come from. This puzzle is our own fault  a result of looking at the waves of only one particle, and ignoring the waves of all other particles in space – over simplification! To find reality, we must deal with the real wavefilled universe. When we study this question^{13} we find a rational origin of the inward waves: Two hundred years ago Christian Huygens, a Dutch mathematician, found that if a surface containing many separate wave sources was examined at a distance, the combined wavelets appeared as a single wave front having the shape of the surface. This wave front is termed a 'Huygens Combination' of the separate wavelets (Figure 2). This mechanism is the origin the inwaves, as follows: When an outgoing wave encounters other particles, their outwaves are joined with part of the initial out wave. Then outwaves from all other particles can form a Huygens Combination wave front that is the inwave of the initial particle. These waves arrive in phase at the initial center. This occurs throughout the universe so that every particle depends on all others to create its in wave.
We see it is wrong to imagine each particle as one pair of in and out waves, because one pair cannot exist alone. We have to think of each particle as inextricably joined with other matter of the universe. Although particle centers are widely separated, all particles together are one unified structure. Thus, we are part of a unified universe and the universe is part of us.
Figure 2. Formation of inwaves. The outwave of every particle interacts with other matter in the universe. The response to the outgoing wave is Huygens wavelets from other matter that converge back to the center of the initial outwave. When the wavelets approach the center, their combined amplitude becomes larger, forming the IN wave. Thus every particle depends on all other particles for its existence. We are part of the universe and the universe is part of us. 
The origin of the electron's spin. As an example of the depth of understanding and universality of the Wave Structure of Matter, I will describe the origin of the spin of the electron. The physical nature and cause of electron spin has been sought for 75 years ever since a theoretical theory of spin had been developed by Nobel laureate Paul Dirac^{4} in 1926. He theoretically predicted the positron , found in cosmic rays five years later by C. D. Anderson.
Dirac’s Theory. Dirac studied the connection between the conservation of relativistic energy given by
(1D) 
and Schroedinger’s quantum theory. E is energy and p is momentum of a particle of mass m. Schroedinger’s procedure was to use an energy relation for a particle, like (1D) and change it to a wave equation. His usual way was to change the terms for E and p into two wave equation operators by substituting,
and  (2D) 
Where is the amplitude of the Schroedinger wave function sought. Then the solutions should describe the amplitude of waves of the particle. No one knew why this worked but the results for the hydrogen atom are amazingly accurate so it is trusted.
Unfortunately for Dirac, Eqn (1D) uses squared terms whereas Eqns (2D) cannot be squared! He had a new idea, 'Try replacing Eqn (1D) with a matrix equation:'
(3D) 
Where the new factors [Identity], [alpha], and [beta] are matrix operators. This avoided squares of E and p but placed severe restrictions on the new operators and their solutions.
Dirac saw that Eqn (1D) is similar to squaring the sides of a triangle. He deduced that in Eqn (3D) the equivalent of squaring had to become part of the matrix algebra. He found that solutions existed if E and p had fixed values. This theoretical matrix algebra produced correct values of the electron's energy ±mc^{2}, and spin h/4 angular momentum units, but gave no hint of the physical structure of the electron. But Equation (3D) became famous as the Dirac Equation.
Dirac also noticed that only two functions were needed in the electron's solution. So Dirac simplified the algebra by introducing number pairs, termed spinors, and 2x2 matrices called spin operators creating a twonumber algebra instead of a fouralgebra or our common onenumber algebra. His spinors correspond to the INOUTwaves , Eqns(a,b,c,d) above. A detailed readerfriendly review of Dirac’s Equation is given by Eisele^{10}.
Origin of the Mechanism of Spin. Spin occurs when the inwave arrives at the center and rotates continuously in order to transform into the outwave. To accomplish rotation there are strict boundary conditions on the amplitudes and polarity of the IN and OUT waves. Rotation cannot be allowed to twist up space without limit. The spherical wave amplitudes must continually and smoothly change while changing direction from in to out. At the center, the inwave amplitude must be equal and opposite (one half cycle different) to the outwave amplitude.
These conditions are a known^{12} property of 3D space called spherical rotation Figure 3, in which space rotates continually around a point and returns to its initial state after two turns. In spherical rotation there is no fixed axis like cylindrical rotation of a wheel. Spherical symmetry is preserved because the center of rotation is a point. One direction of rotation produces the electron, the other the positron. This is why every charged particle has an antiparticle.
BattyPratt & Racey^{11} (1980) analyzed spherical rotation and showed that an exponential oscillator, e^{iwt}, was a Dirac spinor. Wolff^{8} realized in 1989 that the exponential inout oscillator waves of the WSM were the real physical spinors satisfying the theoretical Dirac Equation. It is significant that only 3D space^{12} has this remarkable property. If this real property of 3D space did not exist, particles and matter could not exist! Life and the Universe as we know it could not exist.
Calculate spin using the WSM. Think of how the INwave becomes an OUT wave. Then calculate the rotation of an inwave of frequency = mc^{2}/h and wavelength = h/mc, that changes to an OUT wave. Use a rotation rate two turns each cycle. You will get an angular momentum of ±h/4. You obtain Dirac's result simply.
The results differ interestingly from the Dirac Equation. The spin is the same but energy from the Dirac Equation is ± mc^{2} while the WSM says both the electron and positron have positive energy +mc^{2}. Dirac was forced to interpret the puzzling negative energy as an unseen "sea of negative energy particles". This strange concept has never been observed and has been abandoned but the reason was unknown until the WSM.
Figure 3. A model of spherical rotation. Spherical rotation is modeled by a ball held by threads inside a cubical frame. The threads represent the coordinates of the space and the rotating ball represents the space at the center of the converging and diverging quantum waves. [ Click to see a YouTube video demonstrating this.] 
Interpretation of the Dirac negative energy. Look at the electron/positron wave algebra in Eqns. (a,b,c,d) above. In the electron wave write the sign of the product of energy and time as negative, that is: Et/h = wt. But this is the same as exchanging the inwave with the outwave, which changes the electron into a real positron! We see Dirac’s numerical result was right but the interpretation was wrong. He had no wave structure theory to guide him. He assumed the electron was a discrete particle instead of a wave structure! This mistaken assumption has plagued physics for a century. Note that the Dirac Equation only describes rotations at the electron center, not the entire WSM. The work of BattyPratt^{11} (pp 453455) analyzed the rotation of local exponential waves, found that these spinors and their rotation produced the Dirac Equation.
Examples of the Dirac matrices and spinors: Dirac matrices perform geometric operations. Spinors are written as a pair of 0 or 1 elements in a column. An element may be an exponential with unit magnitude. Matrices (operators) are written as four elements in two columns. Matrices operate on (multiply) spinors to produce new spinors. The rule of matrix operation is to multiply the row elements of a spin operator by the column elements of the initial spinor and add them. Each result is the new element of the final spinor.
For example, inverting a spin state is produced by the inversion matrix operating on a spinor,
You can also reverse the spin axis, by simply reversing time ( t > t). This is equivalent to exchanging the outgoing spherical wave of an electron with the incoming wave. In spinor notation, the change would be,
In these examples, note that inversion and spin reversal are not the same. Our human view of cylindrical rotation differs from quantum rotation. The difference is characteristic of the quantum wave electron and is important to understand particle structure.
Another illustrative example is the Identity operator
Spin also connects Quantum Theory and Relativity. Before the WSM, there had been no known physical reasons for special relativity, quantum theory or spin. The Doppler connection of QM and relativity were shown in Part III above. Dirac's work also connected spin with relativity and QM as follows:
The increase of mass (energy or frequency) m = m_{o}[1 – v^{2}/c^{2}]^{1/2} of a space resonance was seen due to the Doppler increase of frequency seen by a moving observer. This equation can be squared to get the energy equation used by Dirac:
(1) 
Then the union of spin, mass increase, and quantum theory occurs in the spherical rotation at the wave center, and is part of the WSM in general.
The Wave Structure of Matter is the true physical reality of the universe. The required proof is that the experimental evidence, which empirically began the natural laws, must necessarily agree with the laws predicted by the WSM. It does. This is enough.
There is further confidence because empirical evidence agrees better with the simple WSM rules and explains many phenomena not explained with conventional rules. For example, an infinity of charge potential at r=0 in Coulomb's law is not found experimentally, in agreement with the WSM. There are more examples: Conventional physics has no explanation for the energy exchange mechanism of forces, or the Pauli Principle, or inertia, or gravity, or spin, or charge attraction and repulsion. These are now direct conclusions from the WSM.
The philosophical conclusions are fascinating, particularly concerning the connectedness of the universe because of the origin of the inwaves and inertia (Mach’s Principle and Principle II): Everything we are and observe here on Earth, matterlawslife, necessarily depends on the existence of all the matter in the universe. We must conclude, if the stars and galaxies were not in the heavens, we could not exist! Thus, we are part of the universe, and the universe is part of us.
But the practical value of the WSM theory is the insight it provides. It allows scientists to deeply analyze quantum wave structures, the cosmos, and the natural laws. In the R&D laboratory, the new insight should advance biology and electronic applications, especially IC and memory devices because their tiny transistor elements use quantum effects to control the flow of currents.
The new knowledge should improve communication and the efficiency of energy transmission. For example, conduction of electric energy along a wire is a quantum energy transfer process between SRs. Knowing this, energy losses might be reduced, lowering cost and increasing transmission distances.
Speculations of the future. Is it possible the nature has made use of the fact that each particle is connected with the waves of other particles? Researchers often make a new discovery, only to realize later that the discovery already existed in nature; "Nature did it first!" is a common occurrence. We can speculate that this interconnectedness could affect human thinking, physiological, psychological, or genetic processes ^{15}. That is, quantum energy transfers, as a requirement of the MAP, could create hidden communication links between parts of our bodies or with neighboring matter, like computer programs.
What is the mechanism of these matter wave connections? Each atom of our body structure arranges its waves to form minimum amplitudes that satisfy MAP. In the process, each of its neighbors makes a contribution to its inwave structure. If one atom is disturbed, others must be affected. This is an elementary information transfer. Since every life cell (each particle) contains the holographic image of waves from other parts (other particles) of the body, each cell is a memory device. So the apparatus of computers appears to be there but whether or not nature has arranged our body cells into a physiological computer we do not know. Indeed, the mechanisms of ordinary sensory communication in the nerves of the body, and in the brain are still poorly known. The new understanding of energy transfer may clarify them but we won't know much about their quantum communication without more lab research.
Interested readers should begin the study of the quantum universe in references 7 and 8, at SpaceAndMotion.com and at the "Quantum Science Corner" web site: QuantumMatter.com. A future article will describe the origin of the Hubble universe and the red shift.
[1]  Clifford, William, "On the Space Theory of Matter", Cambridge Philosophical Society, 2, pp. 157158 (1870). 
[2]  Clifford, William, Mathematical Papers, Editor: Robert Tucker, (Chelsea Publishing, N.Y. 1968) (1882). 
[3]  Moore, Walter, Schroedinger  Life and Thought, (Cambridge U. Press, England, p. 327, 1989) 
[4]  Dirac, Paul, "Quantum Electrodynamics", Nature, London, 174, p. 321. (1937) 
[5]  Wheeler, J. A. and Feynman, R., "Interaction with the Absorber....", Rev. Mod. Phys., 17, p.157, (1945) 
[6]  Wolff, Milo, "Microphysics, Fundamental Laws and Cosmology", Invited paper at 1st Sakharov Conf. Phys, Moscow, May 2131, pp. 11311150. Proceedings by Nova Scientific Publ., NY, (1991) 
[7]  Wolff, Milo, "Fundamental Laws, Microphysics and Cosmology", Physics Essays, 6, pp. 181203, (1993) 
[8]  Wolff, Milo Exploring the Unknown Universe, ISBN 0962778702, (Technotran Press, Manhattan Beach, CA, 1990). 
[9]  Mach, Ernst (1883  in German), English edition: The Science of Mechanics, Open Court, London (1960) 
[10]  Eisele, John A., Modern Quantum Mechanics with Elementary Particle Physics, (Wiley, NY, 1960). 
[11]  BattyPratt, E. and Racey, T., "Geometric Model for Fundamental Particles", Int. J. Theor. Phys.19, pp. 437475 (1980). 
[12]  Misner, C.W., Thorne, K. and Wheeler, J.A. Gravitation, p. 1149, (W.H. Freeman Co. San Francisco, 1973). 
[13]  Wolff, Milo, "Exploring the Universe", Temple University Frontier Perspectives, 6, No 2, pp. 4456., (1997). 
[14]  Wolff, Milo, "Beyond the Point Particle – A Wave Structure for the Electron", Galilean Electrodynamics, SeptOct, pp. 8391, (1995). 
[15]  Wolff, Milo, "Matter Waves and Human Consciousness", The Noetic Journal, 4, No 2, pp. 6775, (1999). 