A relativistic/quantum model of atomic structures from particles to the atom By
Charles A. Laster
Undergraduate Thesis is the title of this paper for several reasons. I am applying as an undergraduate, and an example of my independent study would be useful in evaluating my application. I consider my work at the undergraduate level and it is written at that level. As an undergraduate level work, there may be mistakes. I have made mistakes in earlier works trying to understand physics, and have learned from them, my ego will not be bruised if flaws are found by those more learned than myself. It is also a thesis in a sense, despite being an undergraduate level work, it shows my knowledge of the subject matter and my ability to conduct research independently, digest that information and apply it. Due to self study, lacking the guidance or explanations of a professor, I have had to try many methods to comprehend the subject matter. In so doing I feel I have tried some innovative approaches that might be useful to others. So while my work is not technically advanced and does not advance the field of physics to the degree that a normal thesis would, I hope I have made some small contribution to that goal. During my self study I have found it useful to put my thoughts down on paper in the form of a research paper. It has been a good learning tool, helping to clarify my thoughts and for latter evaluation. This is also excellent practice for writing real research papers. My work could be located on the web and in my book for evaluation, but that would be time consuming and not representative of my current level of understanding. Much of my work is updated and presented here, along with considerable new material. The focus of my research is in constructing educational models for introducing advanced physics concepts at the undergraduate and lower level. These models are the result of trying to comprehend such subjects myself and it is my hope they may help others as well. I found that research often asks more questions than it answers, and models, even simple ones, do not always give the answer you expect, leading you places you had not foreseen. I supposed this is something every researcher encounters at some point. I am prepared to do an oral defense without reference material of this paper if required. The point is not that my ideas are correct, I expect to have made mistakes, but to show that it is my work , rather than the ideas of others, and that I do comprehend the subject matter.
Introduction Quantum Spin Spin and the Magnetic Moment The Dirac Sea and its relation to the Vacuum and Source Field "Virtual" Pair-production with Vectors Dirac Equation or Electron Model of Choice Real Pair-production The Neutrino The Photon Spin 0 Particles Time S1 Circle Group Symmetry Conclusions Section 2 A "Double Solution" for 1/2 spin Particles.
Introduction Standing Waves Point Particle Component of a Standing Wave A Double Solution Interaction with the Vacuum and Source Field Conclusions Section 3 Atomic Units and Lattice Theories
Introduction The Fine-structure Constant Hartree Atomic Units Proposed Atomic Constants Lattice Theories The Electron Lattice The Atomic Lattice Conclusions
Introduction Quarks, Those Other 1/2 Spin Particles Fractional Elementary Charge of the Quarks Confinement MIT Bag Model Confinement Anti de Sitter Model Confinement A Relativistic/Quantum Model Confinement and a Unity Model for 1/2 Spin Particles Conclusions Section 5 The Nucleus
Introduction Liquid Drop Model of the Nucleus Shell Model of the Nucleus Combining Liquid Drop and Shell Models The Meson Cloud Conclusions Section 6 Section 7 References Suggestions for Farther Research Summary Conclusions
A relativistic/quantum model of atomic structures from particles to the atom. Abstract
A semi-classical approach incorporating relativistic consideration in quantum models is used to examine particle formation, symmetry in particles, electron models and confinement to arrive at a relativistic lattice model compatible with quantum theory. It offers a coherent model from basic particles to the atom upon which farther research can build. Maxwell concept of polarization of the vacuum is seen in the formation of ½ spin particles. This leads to a “double solution” for the electron that is compatible with Maxwell and relativity, but developed from a QED electron model, bridging these disciplines. Then the research is extended to cover the a new approach to quark confinement with aspects of the MIT bag and AdS models apparent. This method of describing confinement also can be applied to the nucleus, finishing the study of atomic structures.
Symmetry in Pair Production and The Importance of Quantum Spin in Particle Formation
An elementary vector examination of pair-production can be useful in studying the symmetry of particles and pair-production. In addition the importance of quantum spin is seen with vectors as well as some of the properties of particles are determined by the vector components of quantum spin. Such topics as quantum spin and spinors must be represented, by breaking them into their vector components for this examination. The role of the magnetic moment, and how the vacuum and source fields fit into the process of particle creation are also examined. The model used in this paper is as basic as possible and starts with few assumptions. The representation of the vacuum and source field given by this vector model can be easily understood by students and the "virtual" particles of the vacuum are clearly expressed. The importance of quantum spin vectors in symmetry become clearly evident as does its role in the polarization of the vacuum. The polarization of the vacuum proposed by Maxwell also plays a role in pair-production. An added bonus is that a number of electron models can be examined within the scope of the paper if desired. Thus this basic vector method can prove useful in examining specific electron models. This vector model of particle formation can be a useful teaching tool for undergraduates while providing serious researchers another tool in examining symmetry and particle formation. Page 1
Introduction The Dirac equation lies at the heart of physics and describes pair-production. Despite its long history, the interpretation is not straight forward and beyond the grasp of many who would like to know more about physics. Pair-production, and symmetry in particles is a subject of much interest and a number of papers have been written on the subject, it is no easy task to add to this body of knowledge. There have been some noteworthy attempts to simplify this popular topic for undergraduates. Feynman Diagrams for Beginners  examines the Dirac equation and was meant to be part of a QED introductory course. The similarity between Feynman diagrams and vectors makes them a concept that is easy to grasp. Complex four-vectors and the Dirac equation uses the vector math of Clifford algebra to examine the Dirac equation and offers some new perspectives. It also suggested that a purely vector approach to pair-production as a possibility that likewise might offer some new perspective on the subject. This elementary examination of pair-production and particles is written at and intended for undergraduate level, it does provide a unique view that highlights symmetry and the importance of quantum spin vectors. Quantum Spin One of the first problems Dirac had to overcome was the discovery of Quantum Spin by Wolfgang Pauli in 1924 and introduced what he called a "two-valued quantum degree of freedom". This quantum mechanical spin has not described by vectors as in classical manner, but by spinors. A spinor and a vector behave differently under coordinate rotations. If you rotate a 1/2 spin particle by only 360 degrees, it does not return to its starting position, but to the opposite quantum phase. To return the particle to its exact original state, two full rotations, or 720 degrees is needed. For a spherical object, one vector is on the y plane, equator, and the vector F is in the +z or −z direction. This is the classical spin, and as it is used as a reference, we will call primary spin. Quantum spin is represented by another vector with rotation in the x plane. Page 2
Blue = Primary classical spin Green = Quantum 1/2 spin
Put a dot on the equator, y axis, at the tip of the arrow as the starting point. in one 360 degree revolution it will return to its starting point. The spin on the x axis is moving at 1/2 the speed of the primary spin, so a dot at the North pole only makes it to the south pole in the same amount of time. To return the particle to its exact original state, both dots in the same spot, two full rotations, or 720 degrees is needed. However the dots we imagine on the surface do not travel a straight line due to the complex spin, but rather follow a curved or wavy path around the globe. To track this over time using vectors would be very time consuming. So quantum spin can be represented in a geometrical vector manner that is not efficient mathematically as matrices and spinors, but give an good visual representation that can be grasped intuitively. One paper that does this is Spin, Charge and the Fine-Structure Constant which develops a Feynman Diagram like vector notation for the spin and charge of particles . This approach of treating quantum spin and charge as a vector has worked well in the "Golden FaTe" string theory being developed by the "String Theory Development group. In it a string can be given a twist with a single vector, or a complex twist. This has allowed the developing string theory to predict the observed leptons . The Dirac equation also produced a negative energy solution when the EM field was taken into account. When Clifford algebra was applied to the Dirac equation, a second axis of rotation appears in the matrix when the EM field is taken into account as well . This second axis of rotation is hid with the normal matrix used by Dirac and most papers on pair-production. Page 3
Spin and the Magnetic Moment The magnetic moment can be defined by different methods. In older textbooks the example using a bar magnet. The magnetic moment is equal to the force between the poles, but reversed as if this force was keeping the poles apart and the real magnet was not there. So this force depends on two factors: one on the strength p of its poles, and on the vector l of the distance separating them. The moment is defined as μ=pxl It points from South to North pole, and obeys the right hand rule for the direction of the magnetic-field. Not a very eloquent description and today the "Current Loop" definition is commonly used. Place a closed loop carrying an electric current I and a vector area S. Its magnetic moment μ, is defined as: μ = I x S Here is a graph of the electron and positron's magnetic moment using the current loop definition. In addition to the above terms, the vector P is added show the primary spin of the charge in relation to current flow. Electron Magnetic Moment Right Hand Rule Positron Magnetic Moment Left Hand Rule
Both definitions are useful in this paper to avoid confusion. Dot and cross vector products do not have a radius defined and works best with the magnet definition. The current loop definition is used for particles as both will overlap in particle formation, and this choice makes the graphing much clearer. The Dirac Sea and its relation to the Vacuum and Source Field The "Dirac Sea" was how the existence of the positron could be explained. Today, Dirac's view of a sea of invisible particles that can pop into our reality sounds a little silly and simplistic but the underlying concepts are still used. Page 4
In condensed matter physics the Fermi sea, the virtual particles of quantum theory, the color of quarks in QCD and the Bogoliubov transformation of QFT can be considered functionally equivalent to concepts of the Dirac sea. Even the Golden FaTe string theory has the equivalent concept, as Mark Aaron Simpson said "Our model describes space as a vast continuous opportunity of singularities (particles)" . In this paper the vacuum will be modeled as a perfect liquid or gas depending on density. This approach is nothing new and has been used since the earliest days of the Big Bang theory till now and by the author in other work . Now Dirac started with a complete electron equation that described the complex quantum wave structure of the electron and worked backwards from it. In section 3.2, Plane wave solutions, of the paper Complex four-vectors and the Dirac equation, it shows that a plane wave solution would work, as a small section of any spherical wave can be considered a plane wave at distance from its origin . "Virtual" Pair-production with Vectors Two plane waves intersecting at the speed of light, with the vectors denoting direction opposed 90 degrees, is examined with the vector diagram on the side. Plane wave A is on the (x,z) plane and moving in the vector a direction. Plane wave B is on the (x,y) plane and moving in the vector b direction. Both also have a given intensity based on the Density of the vacuum energy.
The cross product, denoted a x b, is a vector perpendicular to both a and b and is defined as a x b = ||a|| ||b|| sin (Ө) n Where Ө is the measure of the angle between a and b, and n is a unit vector perpendicular to both a and b, and which obeys the right-hand rule. There is also a solution for - n which obeys the lefthand rule. The two solutions for n are known a Pseudo-vectors. The distinction between vectors and pseudo-vectors is often ignored, but it becomes important in studying symmetry, which is needed in examination in the pair-production process. This relationship where vectors a x b = n, with right and left-handed solutions is the same relationship we see in the magnetic moment of the electron using the right-hand rule, and a negative solution for the positron using the left-hand rule. Premise: That a force, the vector cross product, obeying the right and left-hand rule, the same as the magnetic moment in particles and anti-particles, could induce a polarization of the vacuum energy (create a magnetic moment) and play a role in particle formation. Researchers have induced a magnetic moment in neutral atoms using lasers to induce a spin . The concept of polarization of the vacuum by a vector field that gives the density of a permanent or induced electric dipole moment is a critical part of the work by Maxwell on electromagnetism. On the following site is a video on YouTube that shows the Polarization of charged particles by right and left handed spin in a perfect fluid called a Hot Quark Soup, made at RHIC http://www.youtube.com/watch?v=kXy5EvYu3fw&feature=player_embedded
I placed the graphics of the electron and positron magnetic moments next to the vector diagram. Note that with the positron upside down, the vectors for the magnetic moment are aligned with the vector cross product, and that the flow of current for both magnetic moments are flowing in the same counter-clockwise direction. Next is a graph of this electrostatic magnetic moment and field lines.
The pseudo-vector force in effect create a pseudo-magnet. The pseudo-magnet sets up the electromagnetic field for the flow of current as it climbs in intensity from 0 to maximum as Maxwell suggested. When the current begins to flow, it causes the polarized vacuum energy to acquire a physical spin. The spin induced on the charge by current helps to contain the charge of a particle as does the Hall current towards the center. If there is sufficient energy for particle formation that is. Without the required energy for particle formation the forces from intersecting waves are just pseudo-particles. They have everything in place except the last ingredient, sufficient energy/mass. As the vacuum is filled with intersecting waves, there is a virtual sea of these pseudo-particles. They have no physical or quantum spin, no mass or charge to speak of. Particles of any quantum spin might arise from them depending on the energy density and the quantum spin vectors of the waves. Dirac Equation or Electron Model of Choice Even the Clifford algebra approach started with the complex quantum spin as part of the plane wave, but this paper did not set such complex starting conditions. It is best to start with as few assumptions as possible in any model, and complex spin could arise as part of the pair-production process. Page 7
As a result the pseudo-particles does not have quantum spin. But that could arise at several points in the formation of real particles. The flow of current in the plane wave may induce or modify the quantum spin as the sphere is formed from the plane wave. Or the spin may be induced at the time when the two south poles of the electric magnetic moment kicks them apart (as described in the next section). The force is inducing rotation from the pole, the right vector direction for quantum spin. It light of these variables, not setting starting conditions for quantum spin seemed the best approach. At this point one can pick the electron model of their choosing and try to see how it could form from this process of twisting a plane wave into a particle. The Dirac equation which assumes complex spin vectors on the plane wave as in the Clifford algebra approach to the Dirac equation  could be used to get the Dirac electron. Ring, torus, and other electron models could be used. For example the author  and the Golden FaTe  string theory both used a standing wave for particles that employed a "double solution" for particles. So this approach is compatible with various electron models. Real Pair-production If there is sufficient energy, real particles can arise from these pseudo-particles. When this happens each of the two particles will develop real spin, and generate their own real magnetic moment, which brings an added force to the picture.
I placed the magnetic moments to the side of the vector for clarity in the first diagram, but the second one is more accurate. When they develop their own electric magnetic moment, separate from the pseudo-magnet magnetic moment of the vector force, due to the symmetry of formation, the South poles of both are nearly touching. Page 8
This provides the final force needed to separate them and why when electron/positron pairs first appear, they are spiraling away from each other, despite their opposite charges. Everything revolves around the pseudo-magnet magnetic moment, which creates real particle magnetic moments. This model can be applied to other 1/2 spin particles like quarks, but what about the Neutrino? The Neutrino The neutrino was first predicted by Pauli in the process called beta decay, and that is an excellent example to use. In Beta decay a neutron turns into a proton and an electron and an antineutrino are produced.
A down quark with a -1/3 charge decays and up quark pair production occurs within a composite particle. The polarization for up quarks would be +2/3 and -2/3, however there is an additional -1/3 charge contributed from the decaying down quark available. The up quark with a +2/3 charge is stable and converts the neutron into a proton. However its partner has too much charge to form an anti-up quark. It either fails to form or immediately decays and pair-production occurs again. There is an excess of -1 charge and the available positive charge has just been used for the up quark. A positron can't form, but a neutral left-handed particle can.
Page 9 Neutrino's don't form under normal pair-production, but under special circumstances they can
form in the process of pair-production when one particle of the pair fails to form due to insufficient charge. The small observed mass of the neutrino comes from the non-polarized vacuum energy that was trapped during formation instead of charge polarized vacuum energy. Now the neutrino represented here is formed on the side of the vector cross product that follows the left-handed rule. All of the neutrino’s we have recorded have been left-handed. This electron/neutrino pair-production would also apply to the muon/neutrino and the tau/neutrino as well. Right-handed or anti-neutrino's are theoretically possible as well, but have not been observed. This may be due to the difficulties in distinguishing between left and right-handed neutrino's. Their high speed, low mass, and virtually undetectable magnetic moment, if it has one at all, has made this task impossible so far. There was even a theory that the photon was composed of a neutrino/anti-neutrino pair, known as the Neutrino Theory of Light. The theory held a lot of promise at one time as it satisfies Maxwell's equations. The lack of right-handed neutrino's and because no mass-less neutrino's have ever been found, the theory has been discredited. But that does bring the discussion to the next topic, the photon. The Photon Maxwell's equation describes the photons electromagnetic nature and its wave like form. Yet within its requirements, a number of models like the neutrino theory of light are possible. This vector force model can also be applied to the photon and remain within the requirements set by Maxwell. Note in the graph that two full amplitudes with reversed polarity that are within a single wavelength of the photon.
Now when 1/2 spin particles was examined, a "Double Solution" where two spherical standing waves was mentioned. One of the waves is the extended waveform normally applied to the electron, and the other wave reduced in amplitude, 1/137 of the extended wavelength and forming the point particle aspect due to mass and charge. . The photon has no mass and both waves have the same amplitude as in a normal standing wave. So a "Double Solution" standing wave where the amplitude of each wave is equal to λ will work if polarization can be accounted for. The photon is a spin 1 particle so both quantum spin and classical spin are the same magnitude. Page 10
QM states that quantum spin is an intrinsic property of particles. When the waveform collapses to a point particle all other vector forces cancel out, only the quantum spin vectors remain. If one of the two waves had its quantum spin in the opposite direction to the other wave, this changes the symmetry of the vector cross product, right and left handed sides are switched as well as polarity.
Unlike the example with plane waves where the pseudo-magnet was in a steady state, this pseudo-magnet is in existence only briefly. Maxwell showed that polarization of the vacuum happened over a finite amount of time. The separation of charge by polarization and its return to a neutral state is a simple harmonic motion. The force separates and/or creates and separates charge, and then returns to a state of equilibrium, and polarized again in the opposite direction.
Thus the EM wave produced by the pseudo-magnet takes the above form over time. This approach of a particle standing wave that accounts for quantum spin also solves one of the classical problems with the photon. Bohr had a problem in his early atomic model when the electron emitted or received a photon and its resultant change in orbit, energy conservation was not preserved, something was missing. Bohr and Kramer tried to correct this with BKS theory, but could not. Page 11
Bohr gave up on the problem and embraced the emerging field of quantum mechanics where this problem could be circumvented. Born and Jordan was also working on this classical problem in BKS theory. They found that energy conservation could be achieved during a quantum jump if a scalar waveform was emitted at the same time as the EM wave p 105 - 106 . Born and Jordan sent Bohr a letter to that effect, but Bohr had already given up on BKS and any hope of a classical understanding of the atom. Now the spherical standing wave needs to be mapped over the graph of the EM wave. The node of each spherical waves node matches up with every other node of the EM wave. The two spherical waves are out of phase, one at full amplitude when the other is at minimum, but traveling at the speed of light means two two waves no longer occupy the exact same region of space, but are offset by the movement of the photon.
This explains most of the properties of the photon. It gives a true 90 degree polarization to the direction of travel. Now examine polarization from a head on view. The pseudo-magnet caused by the cross product is a straight line force vector, but from this direction the poles could have a number of orientations. Generally we simplify this to having either a horizontal or vertical polarization.
There is one other factor that might affect polarization. So far this model of the photon has only looked at the interaction of the photon waves with the vacuum, but the source field may also affect polarization. As examined in the section on pair-production, source waves fill the vacuum. The same particle that emits a photon also emits a source wave with every oscillation that travels at the speed of light away from the particle as well. The photon can be thought of as "riding" a source wave from the particle that emitted it. Page 12
If the source wave from the particle has complex spin vectors, they would modify the orientation of the polarization. So the initial polarization of a photon would be affected by the particle which emitted it. Of course that polarization can change over the course of the photons journey. An electromagnetic wave like the photon can be created by a rotating dipole like the electron's magnetic moment. If we envision the quantum spin, which in the vector model had a polar orientation, as also spinning of the poles of the magnetic moment, a "virtual photon" would be created with every complete 720 degree spin. This vector model also shows that the point particle of an electron would have to be in the direct path of a photon to absorb it, and at the node of the photons wavelength as well. So while the larger photon waveform may encompass a number of electrons, only one can absorb it. That brings to a close a basic examination of the vector forces at work in the photon and can be applied to spin 1 particles. Next on the list, 0 spin particles. Spin 0 Particles The zero spin particles are perhaps the easiest to understand as they are the closest thing to a classical standing wave in particle physics. No mass, charge or quantum spin. Without complex spin, they do not create a cross vector product, so no polarization or EM forces either. It is simply a packet of wave energy going from one spot to another. An efficient means of transferring energy/force over distance. As such spin 0 particles can not be examined within the scope of this model. The theoretical Higgs Boson is a 0 spin particle. Super-symmetric theories predict more such particles like sleptons, sneutrino, squarks, axino, saxion, dilaton, majoron, and the radion. More theoretical particles of other quantum spins are possible, like the Spin 2 Graviton, and which like the other theoretical particles just mentioned, has not been observed. Page 13
Time Time has not been mentioned much so far for several reasons. Philosophically time for a particle does not begin for it till the particle is created, just as time did not exist for the universe till it was created in the Big Bang. A vector cross product is a 3-dimensional phenomenon, it is also possible in the 7th dimension in multidimensional theories. Time is often treated differently in multidimensional models, so it did not seem necessary to include time at the start, but only when needed, starting at the moment of particle formation when an electron equation can be derived with time included. A particle can only have a frequency once created based on its mass, and this frequency can be considered the internal clock of the particle. Prior work by the author  and the Golden FaTe string theory  showed a vector model would work with as good or better results than a completely time dependent model of particle formation in both string and quantum approaches. While time may not exist for the particle till it is created, it does exist for an independent observer observing the process of pair-production. The orientation of the plane waves in this paper were chosen to reduce the effects of time as much as possible. The point of intersection of the two waves would travel in a vector a + b with the waves, creating a stable system over time to examine. If the two waves meet head on at 180 degrees, the effects of time would be maximized and the pseudo-particle will exist only briefly. The intensity of a wave does not hit all at once. It goes from a value of 0 to maximum intensity and back to 0. Polarization takes a small amount of time, as Maxwell suggested. Due to the rapid expansion and contraction of the EM field during this time, simulating the movement of a standing wave, this type of wave interaction may be more conducive to pair-production. Note: Two simple scalar waves 180 degrees opposed cancel each other out. The plane waves must have complex quantum spin vectors in addition to the scalar direction to form a cross product . S1 Circle Group Symmetry Lie groups and symmetry are one of the subjects often found in research and of importance to physics. One of the basic Lie groups is the S1 Circle Group and serves as a good introduction to the subject for students. So lets use this basic symmetry group and look at charged 1/2 spin particles. At the most basic level we can view particle/anti-particle formation as having circular symmetry. Page 14
Starting at the top with 0 charge, travel distance = π clockwise for a value of -1 for the electron, or counter-clockwise to get +1 for the positron. The two halves are mirror images of each other, even if we take fractional charges like the quarks into account.
Split each half of the circle into three sections as above to represent fractional charge. Normally when viewing circle symmetry we normally travel only one direction, clockwise. It would be easy to take each side separately and expand it into a full circle. However it is ascetically pleasing to have the anti-particle traveling in the opposite direction from the norm, and allows both particle and antiparticles to be represented in a common group. This chart also represents the symmetry in pair-production. Lines of the same color link pairs found in pair production. The neutrino is paired with the electron in pair production, but does not rule out the possibility that a positron/neutrino pair-production could occur. It is a very simple model that should help students grasp the concept of symmetry and how it can be applied to a specific subject like charge. This model will be expanded upon latter in this paper. Page 15
Conclusions The discovery of quantum spin as provoked a lot of thought since its discovery. Spin is deeply connected to the study of symmetry in particles. This vector examination of 1/2 spin particles and the photon offers another view in that study of symmetry. Right and left-handedness are also aspects of symmetry in particles that a vector approach can help make its role in symmetry clearer. The symmetry of charge is likewise seen in this vector examination of particle creation. While the symmetry of charge in particle and anti-particle formation is evident, the positron is not an electron going backwards in time, as in the Dirac equation. That conclusion seems to be the result of Dirac working backwards from a time dependent equation. Otherwise all the aspects of particle formation are consistent with Dirac's work. The interesting aspect of this model is that the polarization of the vacuum proposed by Maxwell plays a role in particle formation and provides the connection to the vacuum as proposed in quantum theory. Mass: Most of the rest mass of a particle seems to come from charge and the energy to confine that charge and keep it from returning to an un-polarized ground state. The non-zero mass of the neutrino suggests that un-polarized vacuum energy can be trapped and contained during particle formation, but accounts for only a tiny fraction of a particles mass. In the standard model we could say that the particle traps some Higgs Bosons during particle formation, but the concept of the Higgs particle is not necessary. Wave energy can be canceled out at T = 0 as the wave collapses returning the energy to a ground state, the vacuum and seems a more elegant solution than inventing a particle to provide mass because we would still have to show how that particle particle acquires mass where as this solution will work for all particles with mass. Lastly this models shows no difference between 1/2 spin particle other than their mass, which is determined by the energy density of the environment they were formed in and the energy density at which they can obtain a stable state.
A "Double Solution" for 1/2 Spin Particles
A "double solution" for a standing wave is developed. While no electron model is perfect, the approach used solves many of the problems associated with other models. A standing wave models solve problems with point and shell models. A possible explanation for the electron coupling constant and renormalization in QED is found and the mechanism for solving the infinite mass problem as well. Page 16
Introduction There are a number of electron and quark models to choose from. All have their strength and weakness which restrict them. The electron model developed here is an attempt at a relativistic model of the electron, no easy task. Peter Milonni in The Quantum Vacuum stated "The relativistic theory of an extended charged particle has evidently not been developed. Such a theory would be complicated because the assumption of a rigid charge distribution as earlier is inconsistent with special relativity" (p.168 ) So the search for a better models goes on. This electron model starts with a standing wave, well known and used in physics, but first the very concept of what constitutes a standing wave is examined. A standing wave will always obey the uncertainty principle. In 1926, Schrödinger developed a mathematical atomic model with the electrons modeled as 3 dimensional standing waves. He realized it was mathematically impossible to obtain precise values for both the position and the momentum of the electron. This was the basis for the uncertainty principle as formulated by Heisenberg latter that same year. Point particle models like the Abraham-Lorentz electron has flaws like pre-acceleration. The discrepancy in time is equal to the time for a wave traveling at the speed of light, to cross the classical electron radius. This suggests an extended electron or some kind of structure. However ridged shell models like Lorentz's 1892 model are not compatible with special relativity because they can not follow the Fitzgerald-Lorentz contraction (p. 3-4 ) among other problems. A standing wave solves many of the problems of the models at the two extreme's of point and extended models and seems the most logical place to start, but standing waves themselves must be examined first. Standing Waves A standing wave can be defined by the wave equation which remains unchanged under rotation of the spatial coordinates. There are solutions that only depend on the radius from a point. This can be done for a 3-dimensional spherical wave. However such a waveform must meet the requirement that the positive and negative amplitudes sum to 0 as in this equation.
The quantity ru in the equation satisfies the 1-dimensional wave equation of the form
Now the values F and G are arbitrary functions which can each represent 1/2 of the spherical standing wave over time, the expanding and contracting phases. The Inward and Outward components of the wave cancel out over time, but not at any given point in time. Page 17
So in 1 and 3 dimensions a standing wave can be defined by the scalar distance from a point. However we can only do this in odd numbered dimensions. Most textbooks define a standing wave using two opposed waves on a string, it is a two dimensional description of a standing wave, and is used in even numbered dimensions where a spherical solution to the wave equation does not work. In string theory, strings were first defined as a number of 0 dimensional harmonic oscillators. The two opposed waves in this model sum to 0 at any given point in time as well as over time. So while both examples define what we call a standing wave, the two models are fundamentally different. A standing wave can also be defined in 3 dimensions using two opposed waves and such models have been called double solutions. Point Particle Component of a Standing Wave Many have pointed out problems with a pure point particle model, but one of the clearest examples is found in QED with the “observed coupling constant”. Feynman described it this way, it is “the amplitude for a real electron to emit or absorb a real photon.” P 129 . In QED, α is equal to the coupling constant, determining the strength of the interaction between electrons and photons. QED does not predict its value, so α must be determined experimentally. QED does provide a way to measure α directly using the Quantum Hall Effect or by the anomalous magnetic moment of the electron. If a ½ spin particle were truly dimensionless, the observed coupling constant would have a value of 0 in a graph of the wave function, which is the halfway point between the positive and negative amplitudes of the particle, known as the node when a standing wave is plotted over time. Instead it comes out just under this point. Why not just above or some other places if it is not on the node? It is as if the inward bound wave is disappearing into a finite sized object, 1/137 the size of the electron waveform, which absorbs and emits photons. Because a standing wave can be composed of two opposed waves, this finite point particle can be modeled as a standing wave. (1/2 spin particles Amplitude) X (1/137) = Size of Finite Point Particle. This gives a finite size that even in relation to the particles own standing wave, is so small it can effectively be considered a point particle. But even this finite size for the point particle is useful in a model of the electron. Relativistic Q.E.D. also supports the spatially extended charge and a mobile finite point model, as it gives a spread-out electron of length m-1which is its Compton wavelength (p.400-403 ), similar to the model developed here. Page 18
A Double Solution As a standing wave can be defined by a single vector value from a central point in 3dimensions, we can simplify the equation at the start by comparing the amplitude of the two wave components with a single vector value. One wave as a standing wave can be defined as:
AmpOut + AmpIn = 0
Now it is simple to include a second wave.
AmpOut F + AmpIn F = 0 AmpOut E + AmpIn E = 0
Where : F is the finite point particle. E is the extended standing wave. This however is not enough the connection between the two waves is not shown. How are they connected to form standing wave? At this point a double solution for this model can take two forms. It can be modeled as two separate spherical waves, one contained inside the other, sharing the same energy/mass. If the two standing waves are sharing the same mass/energy, than one will be at full amplitude at the same time the other is at minimum. When the extended wave is at full amplitude, the point particle will be at 0 amplitude, a true point particle. When the extended wave collapses into the particle, the finite point standing wave is approaching full amplitude. The two standing waves undergo a 1/2 quantum spin, turning themselves inside out in 3-dimensions. ( AmpIn E - AmpOut F ) + ( AmpOut E - AmpIn F ) = 0 The other way a double solution can be modeled is with a visual representation of quantum 1/2 spin in 2-dimensions. If we grasp the ends of a circle, O, and give one side a 180 turn, we create a single wave with two spherical components, ∞, connected by a 180 degree, an inherent1/2 spin. This general equation can be modified to serve as a description of 1/2 spin particles in several disciplines of physics and mathematical methods as diverse as String theory, QM, QED, and Clifford algebra, yet it still simple enough for use as an educational model. Page 19
A double solution for 1/2 spin particles more accurately reflects the wave/particle duality than some other models. In most models of the electron, the wave is dominant 99.993% of the time, with the particle aspect appearing only momentarily. The pint particle aspect still collapses to a pure point, but has a finite size most of the time, just like the wave. Interaction with the Vacuum and Source Field To be compatible with modern theory, the electron's interaction with the vacuum and source field should be explained. Before the concept of the vacuum and source field was introduced, electron models had a host of problems like the lack of persistence. Lack of persistence in early electron models, the probability wave of QM was first put forth as a way to work around this problem. In modern electron models, like QED, 1/2 the mass of a particle comes from the source field, as in some older models, and 1/2 the mass, known as the bare mass, comes from the vacuum. Even old electron models have been improved by adding bare mass as Arthur Yaghian did with the AbrahamLorentz electron, solving its problems with persistence . While it is evident that the vacuum must be part of the electron, the method for modeling the vacuum must be chosen. To be compatible with relativity, modeling the vacuum as a perfect fluid or gas, depending on density, with the density being equal to the cosmological constant in free space, though it may vary locally. Compared to the vacuum, the source field is easy to model. That one electron receives a de Broglie wave wave from another electron was one of the experimental proofs of emerging quantum theory. Even those with only a classical background in physics can envision a particle emitting a wave and another receiving it. Now that the vacuum and source field has been defined within the scope of the 1/2 spin particle model, how it interacts with them must be examined. As a standing wave can be defined by a single scalar distance from a central point, the scalar values cancel each other out at T=0, when the extended wave collapses to a point. The inward bound wave has been coupled with incoming source waves, and therefore has a vector value due to wave intensity. So as the wave collapses to a point this energy will cancel each other out, returning that energy to a ground state, the vacuum. Of course there is no reason to suppose that the incoming wave intensity is the same in all directions, but is a simplified example. As the standing wave begins to expand, it must first travel through the smaller standing wave forming the point particle. Wave intensity is determined by the density of the medium it moves through. The vacuum in quantum theory acts much like a medium does in classical physics (p. 2 ). This process forms an energy/mass control mechanism for the particle. Wave energy is converted to the vacuum ground state and the energy density of the vacuum determines the outgoing wave intensity. Page 20
Likewise all source waves from other particles will have varying intensity. The similarity with Larmor radiation is apparent. The Larmor formula calculates the total power radiated by a nonrelativistic particle as it accelerates. This model of the electron is relativistic, and the radiation of energy with the source waves can then account for Larmor radiation in this electron model. The same process can also be used for modeling absorption and emission of a photon. "In a process in which a photon is annihilated (absorbed). we can think of the photon as making a transition into a vacuum state. Similarly, when a photon is created (emitted), it is occasionally useful to imagine that the photon has made a transition out of the vacuum state" (p. 48 ). The standing wave that creates the finite point particle component of the double solution would have a magnetic moment as well. A rotating dipole generating an EM wave, along with the vector cross product model for photons , provides a mechanism for photon creation within the particle model developed here. Because there are two waves composing the standing wave, this model solves a classical problem. When Bohr first calculated the electron magnetic dipole moment, the classic value, ≈ 1, was only half the measured value. Today constant of 2 is used as suggested by the Dirac equation. Two waves would double the classic value bringing it in line with the quantum value of 2 in use today. The value of 2 in the Dirac equation may be suggestive of a double solution. One final note concerning the Hall current. This model shows that the hall current will help contain the charge, for both waves, keeping it within the combined waveform. The Hall current in quarks will be examined latter as well. Conclusions The two possible versions of a double solution for 1/2 spin particles presented here are not the only models one can consider. De Brogolie's Theory of the Double solution was perhaps the first and most famous, and T. S. Natarajan , one of the more recent versions, and there are others as well. That said the double solution for 1/2 spin particles presented here appears robust enough to warrant farther study. Page 21
Atomic Units and Lattice Theories
The Hartree atomic unit system is examined ans some new constants proposed. These constants are then helpful in building a lattice theory based on atomic units that is as background independent as nature will allow. Introduction The Hartree atomic unit system has proved useful in the field of atomic physics. It can also prove useful in education by simplifying equations and quantities, while allowing the student to see relationships within the mathematics not clearly visible in other number systems. One example is how prominent the fine-structure constant is within the atomic units. A study of the fine-structure constant is used to define some boundary condition that can then be used to build an atomic lattice for modeling the atom. While examining the atomic unit system, some other constants are found that are useful and are proposed as additional constants to the Hartree atomic unit system. The Fine-structure Constant 1 / α = 137.035 999 044, or α = 0.007 297 352 5714 The fine-structure constant is one of those mysteries that have captivate physicists and anyone who loves a good mystery. The number keeps popping up in unexpected places, like the quantum Hall effect. Most people get introduced to the fine-structure constant by the relationship between the Bohr radius, the Compton wavelength and the classical electron radius making it an excellent starting point. The Bohr radius for the electron orbit is the largest of these related distances. The Compton wavelength is 137 times smaller than that, and the classical radius of the electron is 137 times smaller than the Compton wavelength. In effect, when physics defines some boundaries, the fine-structure constant shows up. so a logical approach would be to examine the types of boundaries defined by it. The elementary charge of the electron is considered a point in quantum theory. Classically the electron had a radius as if the charge was a number of smaller charges distributed in or on a sphere. That would take a large number of small charges that have never been observed. All evidence seems to point to a single point charge. Still this is interesting if we think about it. The classical radius is the distance at which a point charge behaves as a classically extended charge. At this boundary it behaves in a more classical fashion. Page 22
The Compton wavelength represents the wavelength of a particle if all of its mass was converted to energy, like a photon. Just as we consider this the boundary for a photon, we can consider this the boundary or wavelength for the EM field of a particle. Before continuing to the Bohr radius, the use of atomic Hartree units proves useful as it also has a deep connection with the fine-structure constant.
Hartree Atomic Units Hartree Atomic Units is based on four fundamental values reduced to unity, that is having a value of one. The rest mass of the electron is the base unit of mass. The elementary charge of the electron is the basic unit of charge. Angular momentum is defined by the reduced Planck constant, and the Electric constant is equal to the Coulomb force constant. Of course dimensionless physical constants like the fine-structure constant retain their value in any system of units used. When the speed of light is calculated in Hartree atomic units, the speed of light is 1 / α ≈ 137. Now back to the Bohr radius. In Hartree atomic units the equation for the Bohr radius is:
With the fine-structure constant being present in the equation explicitly and by it's relationship to the speed of light, it comes as no surprise that it is the next scale up where the fine-structure constant is seen. The Bohr radius is the basic unit of length in the Hartree atomic unit system. As this is the basic unit of length in the system, we can use it to define the magnetic moment of two elementary charges separated at this distance. The force of the magnetic moment depends on two factors: one on the strength p of its poles, and on the vector l of the distance separating them, μ = p x l. As both p and l have a unitary value of 1 in this system, the energy between them is equal to 1 Hartree. This is also how Coulomb's law can be interpreted within the Hartree atomic unit system. Now we can consider half of the force arising from each of the two equal charges we arrive at a value of 1/2 Hartree, which is equal to the Rydberg unit of energy relating to the atomic spectra. The Rydberg unit of energy corresponds to the energy of a photon whose wave number is equal to the Rydberg constant which is the inverse wavelength of any photon emitted by the hydrogen atom, or the photon energy needed to ionize the atom. Thus atomic units allow us to more directly see the relationship between charge, orbit, and the energy of photons emitted and effects of photons absorbed in the atom. This system of units was specifically designed for working at the atomic scale. Page 23
The Hartree atomic system hints at a dimensional value for the fine-structure constant. The speed of light is 1 / α ≈ 137 suggesting a minimum value for α defining length. The classical radius of the electron is α squared in Hartree units. The fine-structure is considered a dimensionless constant even in atomic units, yet here it is defining a known radius! So a length value for the fine-structure could be calculated, but does that mean anything? If we adjust the scale of the units and set a different particle, say the up quark as having elementary charge and mass of 1, it also would have a radius α squared. All this at best suggests that the Compton radius of a particle is related to α and suggests there is a fundamental unit of length, but we already know that, it is called Planck's length. Proposed Atomic Constants One of the constants defined by Hartree units is the mass of the proton, defined as the mass of the proton divided by the mass of the electron which is of course the fundamental mass. The mass of the proton then is ≈ 1836. We could then define the mass of the neutron as ≈ 1839. The CODATA value for the neutronelectron mass ratio is 1838 but values up to 1840 are cited. The mass of the electron is ≈ 40% of the mass difference between the neutron and proton. If the assigned value was 1838, the electron would be 50% of the mass difference. At a value of 1839 the electron would be 30% of the mass difference. So while not exact a value of ≈ 1839 in atomic units approximates observation. This gives us a usable mass value for the neutron and proton. Now let us examine the importance of the classical size of these composite particles. For many years the size of the proton was estimated to be ≈ 1/3 the size of the electron. Recent measurements suggests it is slightly smaller that that. The approximation of 1/3 was used successfully in predicting the mass of the Up and Down quarks (p. 149-151 ), but the application of Hartree atomic units suggests a ratio of π, pi would be more appropriate. Angular momentum in this system is defined by the reduced Planck constant which is Planck length divided by 2π. Thus the ratio of 1/π is less arbitrary than the value 1/3 and gives a more accurate prediction. At a ratio 1/π instead of 1/3 the size of the proton will be slightly less than the classical approximation and closer to observations. So we can now define hadron radius as the electron radius divided by π. As frequency is inversely proportional to mass, if the electron was reduced in size it would become more massive, so what would the mass of an electron be if it were the size of the nucleons. The electron mass of 0.511 MeV x π = 1.6 MeV which is close to the observed value of ≈ 1.9 MeV for the Up quark. Page 24
The Down quark mass is also ≈ Up quark mass x π. 1.6 MeV x π = 5.0 MeV, which is close to the observed value of ≈ 4.6 MeV. The problem is that we can not measure the mass of the Up and Down quarks accurately enough to confirm or deny the relationship with pi. The symmetry with the Compton radius and that this symmetry fits well with the π S1 Circle Group Symmetry discussed in the section on pair-production lends credence for the view, and it gives a value to test, even if it does turn out to be an approximation.
The simple π S1 Circle Group Symmetry is clearly seen stating with the electron mass, then the mass or the Up quark and the mass of the Down quark brings us full circle. The neutrino, not have charge, does not fit this model of mass progression.
Now we look at the symmetry of charge chart we made earlier, and if we travel 1 pi from any particle we arrive at another whose mass is a ratio of π between them, except for the neutrino/electron of course. Earlier it was noted that this graph reflects the model for pair-production given in this paper by particles connected by the same color lines. Stable quark pairs are found opposed to each other 180 degrees, on different color lines. Page 25
Let's review the proposed constants added for atomic units based on the above relationships. Neutron to Electron Mass Ratio ≈ 1839 Hadron Radius ≈ Electron Radius / π Up Quark mass ≈ Electron mass x π Down Quark mass ≈ Up Quark mass x π The points brought up in this section on atomic units is directly related to the construction of an atomic lattice theory. Lattice Theories The term lattice theory in this paper refers to those theories or methods that uses a lattice or grid as a background reference. Because they have an arbitrary reference frame many feel they are not in accord with relativity which is background independent, and the frame of reference depends on the observer. It was noted that in atomic units that any particle when normalized to a value of 1 like the electron, has a Compton radius of α squared in Hartree units. This suggests something that should be obvious, the frame of reference for any particle is itself. Metaphorically every particle "sees" itself as the center of the universe, its mass and charge the rule-stick. Every particle has its own clock as well. In QM, f = mc2/h, so whenever we have a change in mass, there will be a change in the frequency of the particle and the size of the standing wave itself. This frequency is the internal clock of a particle. While an outside observer sees a change in frequency/time, for the particle, time always moves at a steady beat. The particle always "sees" the speed of light as being 137. Because the scale is always normalized to a value of 1 for a particle, if mass or frequency changes for an outside observer like us, it always remains the same from the view of the particle. How can a lattice theory that is in accord with relativity be created if every particle has its own frame of reference? If a relativistic electron is complicated, then a relativistic lattice model of the atom would be extremely complex. Many have tried to resolve this and other problems with lattice theories, including an attempt that was outlined by Heisenberg in 1930 , known as Heisenberg's lattice world or Heisenberg's unfinished theory of 1930. At this point in time Heisenberg and others were bothered by the infinite energy of the electron point particle. Heisenberg found a solution to the problem by using a lattice theory based on the radius of the electron and the differential equations replaced with difference equations. One of the things that bothered him about his lattice theory was that it implied that there existed a fundamental unit of length, which at the time he felt was not in agreement with relativity and length contraction and may also have suggested a fundamental unit of time. Page 26
Heisenberg apparently wrote other physicists about his idea as the Soviet physicists Demitri Iwanenko and Victor Ambarzumian, also in 1930. attempted a similar lattice theory and cited Heisenberg's work on a lattice theory as proof for their work. Some of the more interesting aspects of his lattice world was that it was a "unity" theory, meaning that the electron was the fundamental particle. Heisenberg even suggested that the mass of the electron could be dispensed with in calculations as it was not needed. Mass depended on the energy density it was found in. Also the speed of light was a function of frequency, meaning a smaller more massive particles standing wave is smaller due to a reduction in the speed of light. At the time it was assumed that the proton of the nucleus was the anti-particle of the electron. His lattice theory also predicted a neutral particle in the nucleus he called a "heavy photon" as the photon was the only known neutral particle at the time. When the positron was discovered, that was the end of his lattice theory, but he had already started backing away from it, as his friend Bohr and others had strong objections regarding the theory which was never completed or published. Still many of the problems faced by Heisenberg in his lattice theory are with us today, and must be considered in any lattice theory. The Electron Lattice Heisenberg treated the radius of the electron as a non-reducible unit of length which caused problems with the Lorentz contraction. In Hartree atomic units the electron radius is equal to the finestructure constant squared, and that this is true for any particle whose charge and mass are normalized to a value of one. This suggests the same thing Heisenberg saw, that the radius of a particle is its fundamental unit of length. Heisenberg was looking for what we would call today as Planck's length, an its use can resolve this problem. The fine-structure constant is a running constant, so the α squared radius of a particle can be considered as being a multiple of Planck's length, the exact number being dependent on the current radius of a particle. In this manner a particle can undergo a Lorentz contraction as in relativity, but only in quantum steps which are a multiple of Planck's length. Now the field beyond the classical radius of a particle must be considered. This is a good opportunity to incorporate the source field into a lattice model. The fine-structure constant of a particle can also be considered the coupling constant of the particle with the source field. Radian measurement is based on an arc that is equal to the radius. The atomic radius of a particle is α squared in this lattice model. When the standing wave of an electron couples with source waves, the electron only interact with a small section of the source wave, defined by the arc of the intersection. In this lattice model that arc is defined by α. In short, fine structure constant is a measure of an angle in radians for a particle in atomic units. Page 27
With the relativistic model of the electron as a standing wave, the distance between each source wave is determined by the radius of the particle. A lattice built on the particle radius would then reflect the expansion of the source wave over time. Such a lattice would also represent lines of equal-potential for the electrostatic force surrounding the particle, and be useful for the EM field as well. This type of lattice works for an isolated particle, but problems arise when more particles are considered. In the atom, electrons in in different orbital shells have a different energy/mass. This contracts our particle model so they have a slightly different radius and therefore lattice structure as well. How do you determine the distance between two particles when each has its own frame of reference for distance? The simplest approach is to establish an arbitrary frame of reference for this. This approach is neither ascetically pleasing nor representative of reality. Luckily nature solves this problem in the atom for us. The Atomic Lattice Let's examine how this would work in the Hydrogen atom. The proton is a composite particle ≈ 3.14 times smaller than the electron orbiting it. We can then apply the lattice model to the proton forming the nucleus. This lattice can logically be extended much farther than an individual electron due to the mass of the proton. The lattice of the proton then forms the background reference frame for the orbiting electron. Now in our relativistic electron model, excess energy is emitted with the source wave as a model for radiation. The electron is gaining and losing energy all the time as it interacts with the vacuum within it and the external source field. When a Bohr orbit is placed on this lattice, it matches the grid structure. The energy received from the source waves of the proton is consistent over the course of an orbit. Likewise the energy received by the proton from the electron is consistent over the course of an orbit. We can consider a Bohr orbit as an equal-potential line, with the energy gained and lost by the electron and proton to be the same and sum to 0. In a stable elliptical orbit, the energy lost and gained over the course of one orbit would still sum to 0. Bohr assumed that an electron in a Bohr orbit neither gained nor lost energy. While this assumption works, it is not compatible with classical physics, or the concept of the source field inherent to quantum theory. In this lattice model Bohr's assumption must be modified though the end result is the same. In a stable orbital for an electron, the gain and loss of energy sums to 0 over the period of one orbital. Page 28
Just as in QM, only the probability of the electron's position can be determined. What this means for a lattice theory is that the probable energy density of source waves can be calculated, and due to spherical symmetry, though tedious can be graphed, which is one of the advantages of a lattice theory. Another advantage is that a position can be arbitrarily chosen and momentum excluded for a static classroom example. Such an approach is also useful it examining the accuracy of the lattice model. Such a static "snapshot" does not reflect the complexities involved in such a relativistic lattice model. The complexity of the model increases when larger atoms are considered. As mentioned before, the unit of length for the lattice will be different for electrons in different orbitals. How to treat the lattice of the nucleus in larger atoms also has to be addressed. If the lattice for the atom be kept at the radius of the proton, then each Hadron would have its own lattice overlapping the other Hadrons. Despite this problem, it would be more accurate than treating the whole nucleus as a composite particle. Conclusions The mystery of the fine-structure constant is far from solved, if anything its complexity grows some more in this paper. The fine-structure pervades the Hartree atomic unit system, and has lead to some interesting observations like its role in boundary conditions and the fundamental length of a particle, its radius. The proposed constants to be added to the Hartree units was useful in expanding on the symmetry found in 1/2 spin particles. It also gives a good prediction of the up and down quark masses based on the radius and mass of the electron. These proposed additions would also be useful in the examination of the nucleus. The foundation for a relativistic lattice theory compatible with quantum theory is established. While it is intuitively easy to grasp the basic hydrogen atom, the model gains complexity rapidly with larger atoms.
The MIT Bag model and the Anti de Sitter model are examined and adapted to construct a relativistic/quantum model for quark confinement. Only the foundation of such a model is established, but seems sufficient to warrant a deeper analysis of the basic concept. The basic model remains in harmony with current models. It can be applied to the nucleus as well to provide confinement for the meson cloud. At the end the possibility of a unitary model for 1/2 spin particles is explored. While this section departs from standard theory, the subject is worth exploring and not required for the basic model. Introduction Quarks are the key to understanding the nucleus, but they are elusive particle to study. To study them we break them out of their region of confinement, which alters them and what we observe as well. It is the proverbial catch-22 situation. There are models that give good predictions, like QCD. While it is accurate, the QCD models concept of color is abstract, does not fit well with other models, and requires quarks to emit a large number of particles that other 1/2 spin particles do not. The large number of particles in turn causes problems with the value for the vacuum density. The success of QCD does point out that abstract or not, it is an invaluable tool. The MIT Bag model of confinement was one of the first to incorporate QCD and is the best place to start an examination of quark confinement. Quarks, Those Other 1/2 Spin Particles Heisenberg's unfinished theory of 1930 was called a unity theory because he thought that the proton might be the electron at a different stable mass. As we can see for electrons in orbit, electrons in different orbitals have different masses, but are stable at that energy level. This of course implies that electron could be stable at a number of energy levels so long as the energy/mass lost and gained sum to 0 as in the orbitals. In 1930 we simply did not know enough about the atom to attempt a unity model like Heisenberg's, when the positron was discovered and he realized the proton could not be the electron. Such an approach however is ascetically pleasing and if it had worked, could resolve some of the problems facing quantum theory today. Quarks had not yet been discovered. Today modern theories like string theory are still trying to make a unity model for particles a viable solution. There is nothing in this paper that would rule out such a possibility. The research herein can be used in support of a unity model for 1/2 spin particles and has been in one string theory model . Page 30
Looking at the vector cross product model for particle formation, the only difference between 1/2 spin particles is the energy density they form in. If the energy density surrounding the newly formed particles were stable, like within a composite particle, they could find or form at a mass that was in equilibrium with that energy density. A case for quarks being electrons at a different energy density from a classical standpoint. A wave when entering a region of different density will cause a change in the amplitude of the wave and therefore the mass as well. The symmetry in the masses and radius of the electron, up and down quarks found in atomic units can support this view. So there are some evidence that quarks are electrons at a different mass, the most compelling comes from looking at electron capture and emission from the nucleus. We already looked at Beta decay where a neutron turns into a proton, and an electron and an anti-neutrino are produced. Likewise the nucleus can capture an electron. Despite the observation that electrons and positrons can come out of the nucleus, and electrons can enter the nucleus, no electron has ever been found within the nucleus and no quark ever been found outside the nucleus. Any model that reduces the number of known particles also helps to solve on of the problems with quantum theory, the cosmological constant problem. Quantum field theory predicts a large energy density for the vacuum, the discrepancy between theory and observation is 120 orders of magnitude . If quarks were electrons at a different energy level, that would mean the proposed gluons were just photons. This would reduce the the discrepancy between theory and observation enormously. As fermions and bosons cancel each other out, if the fundamental particles were fermion/boson pairs, the value for the vacuum density would be 0. The research and educational models developed so far can be used in the standard model and are generic enough to be used in varying disciplines of physics. However the thought of a unity model and the number of theoretical problems such an approach could resolve, it is a concept worth exploring. This paper will keep close to standard models, but when the opportunity arises to explore a unity model for 1/2 spin particles, it will be examined. The greatest obstacle to a unity model where quarks are electron is the fractional charge of the quarks. Fractional Elementary Charge of the Quarks In such theories as QCD, this problem is resolved by the assumption that the proton and neutron can only have a total elementary charge of 1 or 0, as this is what we see in observation, the fractional quark charges sum to 1 or 0.
A clue is found in the boundaries defined by the fine-structure constant examined earlier. The classical radius is the distance at which a point charge behaves as a classically extended charge. That fits with observation, but a good question would be why does the electron have the specific charge it has? Quarks have a fractional elementary charge, but no other magnitudes of charge are found in particles. The answer to the question of charge will continue unanswered for now, but it also raises another question, is there a limit to the total charge in an area of space that can be defined as a boundary condition? If we looked at two classically extended electrons, they would be touching at this distance from each other. If they were any closer, the charges would have to be combined into a single composite particle. Now with a standing wave model presented here, the distance between the electron's can be considerably less than in a purely extended shell model, and the distance varies over time, but the two electron's still can not enter the others standing wave with out having to combine the particles charges into a single composite particle. There is some justification then in this model for the assumption that there is a maximum amount of charge within a composite particle. As such this model can make a functionally equivalent assumption; A 1/2 spin particle, or an elementary composite particle formed of 1/2 spin particles; the total charge can not exceed an elementary charge value of 1, 0, or -1. This assumption also in effect includes the Pauli exclusion principle within it as like charges would then exceed the elementary value if they occupied the same quantum state. Within the scope of this paper it is also necessary to define composite particles and types. The most basic type of composite particle of formed of elementary 1/2 spin particles. This will be refereed to as an Elementary Composite particle. As this paper progresses, concepts like confinement and boundary conditions found in them can also be applied to the nucleus, and as such it can be treated like a Complex Composite particle formed of elementary composite particles. As we examine quark confinement, the Hall current that confines the charge within the particle may be responsible for the fractional charge of quarks. The Hall effect and the Fractional Hall effect are well known in both classical and quantum physics models. To this researchers knowledge, the Hall current has never been factored into any particle model. Confinement MIT Bag Model In most modern models of the nucleus, the quarks are confined within the proton or neutron in agreement with observation. In the MIT Bag model, quarks are considered to move freely within the confines of the bag. The bag defines the area of confinement, which is viewed as a bubble of gas within a perfect fluid. The pressure of the gas, which comes from the vacuum state of the quarks, determines the size of the bag. In addition the quarks are considered mass-less within the bag and having mass outside the bag. The total color change must sum to a neutral color in any interactions, limiting the quark types to observed values. Also no quark currents can travel through the bag, which is considered a perfect color dielectric. Page 32
A relativistic quantum model of the atom constructed in this paper has a lot in common with the MIT Bag model, and the simple MIT Bag model is a good place to start an examination of quark confinement. One vital topic must be explored before we continue, and that is determining what is a vacuum effect and what is a source field effect within the scope of our model. Peter Milonni also commented on this topic"..., as in the case of the Lamb shift, the interpretation of the Casimir force in terms of the vacuum field is largely a matter of taste: underling this interpretation is a particular and arbitrary choice of ordering of field operators...and in particular a normal ordering allows us to attribute the Casimir force entirely to the source fields...Why has it taken so long to recognize that the Casmir effect and other vacuum field effects have equivalent derivations in terms of source fields?..."( p.250-251 ). The Casimir effect can be described as a source field wave effect in a relativistic model and remain compatible with quantum theory, even on a large scale. The Casimir force can be used to explain the attractive force between two ships at sea in high waves . In that example wave force is treated as radiation pressure. Radiation pressure is a universal concept that holds for all kinds of waves. Thus Van der Waals forces can be modeled as the sum of all wave activity. As in the MIT Bag model, pressure can define the boundary of a proton or neutron within the scope of this paper. Modeling the source field as radiation pressure will help make a relativistic quantum atomic model possible. Does not matter if the pressure is a vacuum or source field effect, the basics of the MIT model can still be applied. In the lattice model for the atom discussed, the main contributor of external radiation pressure is from the electron's in orbitals around it, and were at the point where energy gained and lost to the nucleus and other electrons sum to 0. Thus the pressure from the waves balance out to help define the region of confinement. In this paper the vacuum is treated as a perfect gas/fluid/jelly depending on energy density. While the MIT Bag model is a sphere of gas within a liquid, within the scope of this model it would be a liquid sphere within a gas, a gas that in free space would be equal to the cosmological constant and within a neutron star or other dense object, more like a jelly. This seems a more consistent approach with other fields of physics. By applying the basic MIT model of quark confinement to our atomic model, we can begin to explore greater complexities of confinement by looking at the Anti de Sitter model. Confinement Anti de Sitter Model In the AdS model for quark confinement, the proton and neutron are described as closed universes. The quarks are cut off within a confined section of space, but they are not freely roaming, harmonics determines orbitals where the quarks are likely to be found. Page 33
At first it does not seem such a model could be useful in a relativistic quantum model. Its role in the structure of the protons and neutrons becomes clear when we use it in conjunction with the MIT model, from a relativistic perspective. One of the important aspect of particle models not discussed in basic texts it the problem with persistence in particles. Problems with persistence in electron models plagued classical physics. The problem is largely ignored in quantum theories. It is one of those topics that only surfaces in importance briefly every decade or so. The AdS model can be used to explain not only the orbitals for the quarks, but also shows the role of persistence in quarks. It is clear that in the MIT model, the bag defines the boundary between two regions of different energy densities. Now we must look at a classical wave mechanics effect to connect the MIT and AdS models. Classically when a wave moves from a region of one density into another, their will be both reflection and transmission of waves at the boundary. In terms of particle persistence, the quarks are largely dependent on their own reflected wave energy for their existence. Modeling it as a closed sphere simplifies this reflected wave energy. As in the AdS model the harmonics of the reflected wave energy determines probability of a quark being found within an orbital. Harmonics of 1/3 and 2/3 would, for example, tend to favor stable standing waves at these orbitals.
Aspects of both the MIT and the AdS models can be used in a relativistic/quantum model for quark confinement. Concepts like radiation pressure, harmonics and wave effects are universal. Confinement A Relativistic/Quantum Model Both the MIT and the AdS models can be considered a model for a stable composite particle. In both models the particles are self contained, nothing moves through the boundary. The models could be equally valid for a stable particle if the sum of energy through the boundary summed to 0. So far only the boundary in terms of total charge, energy density and wave reflection effects. Certain orbitals are given, but little else. While that is sufficient for a basic structure, factors like electrostatic forces must be taken into account if concepts like color change are not used. While such concepts as quark color has proved very useful, it is an abstract concept that is hard to fit into a relativistic model. So in this model the quarks do not exchange charge, so quarks must be tracked much like the electrons in QM with a quantum number assigned to them. Page 34
In the proton there are two positive Up quarks and a negative Down quark. The Up quarks are going to be attracted to the one Down, but repelled by each other. From an electrostatic approach the most stable formation would place the heavier Down quark between the two Up quarks. As in the MIT model, all the electrostatic field lines are contained within the composite particle. Quarks are 1/2 spin particles and like the electron they can emit an EM particle like the photon. The gluon's in QCD are like the photon, but control things like the concept of color. The paper is not constructing a theory as in-depth as QCD, so it is best to start with just the basic photon as an EM particle, and then see it any other particles are required as this basic work is examined and expanded upon. In the Neutron the same electrostatic forces are at work, only the situation is reversed. The positive Up quark will tend to be between the two Down quarks. Electrostatic forces suggest a more linear arrangement of quarks rather than the triangular arrangement in the QCD model. This more linear arrangement is supported by the study of standing waves within a spherical cavity. In section 4, figure 4 of the paper by Daniel Russell, the modes for standing waves within a spherical cavity are represented. The only example with 3 modes within the cavity is interesting as they form a linear arrangement as in the electrostatic model. Lobes with opposite phase are designated with + and - signs. In that example the - lobe is between the two + lobes . In the simple hydrogen atom, the proton as the nucleus exchanges energy with the source field, and both transmission and reflection of source field wave energy occurs at the boundary. A particle is never truly isolated and interacts with the rest of the universe. When we add a neutron to the proton of the hydrogen atom, some aspects of the basic model does change. Electrons in orbit around the nucleus are affected by the electrostatic forces of the nucleus, so there is no reason to assume the electrostatic forces of the quarks are confined to the Proton or Neutron. When additional nucleons are added to the model, the linear arrangement of quarks is farther reinforced. The order of quark charges in the proton and neutron compliment each other. The complimentary arrangement is easy to see, Proton + - + Neutron - + - such as the quark charges tend to line up in the nucleus like magnets would + - + - + - but unlike magnets, the length between charges is not fixed.
Like electrons, quarks can emit/absorb photons in a quantum jump from one orbital to another and they are not restricted to a spherical orbital. So while any graph is an oversimplification, it does show how the electrostatic forces can affect the orbitals of quarks within the confinement of the composite particle. From a view based on harmonics as in the AdS model, the orbital surrounding the center is the most stable for either 1/3 or 2/3 standing waves. The 1/3 standing wave would be equally stable in either of the other two outer orbitals or an elliptical orbital. In addition to the central orbital, the 2/3 standing wave would be most stable in the blue outer orbital or an elliptical one. Another force dealing with confinement that must be accounted for in this model that has not been considered in other models is the Hall current.. The model for 1/2 spin particles used in this paper has the Hall current working to confine the charge/polarized vacuum energy. The two similarly charged particles avoid each other within the region of confinement, and the hall currents do not overlap. However the Hall currents of oppositely charged particles will overlap within the region of confinement. Quarks have also been frequently modeled as couples harmonic oscillators. The finite point particle of the quark, being a small standing wave, is a harmonic oscillator. In addition to the likelihood of being in an orbital based on their standing wave, the finite point particle is continually receiving and transmitting wave energy. Coupled harmonic oscillators is one way to model the wave energy between quarks. The model for confinement used in this paper can be extended to include larger nuclei. A nucleus larger than a single photon can be modeled as a complex composite particle and a boundary condition established for it. The nucleus is already considered to have a radius. By doing this we can define a region of confinement for the meson cloud as well. This extends the model to where it can be applied to larger atoms. Before moving on to complex nuclei the effect of confinement on a unity model for 1/2 spin particles needs to be addressed. Confinement and a Unity Model for 1/2 Spin Particles Earlier it was noted that if an electron were reduced in size to that of a proton, it would have the mass of the Up quark, that being π times the mass of the electron. The Down quark is π times the mass of the Up quark. Given the harmonics seen in the AdS model and the similarity to the model developed here, it seems plausible that the quarks are sub-harmonics of the electron standing wave due to the nature of confinement. The limit on total charge within the region of confinement causes some charge of the particles to be canceled out. Within the scope of this paper that means it becomes un-polarized vacuum energy having been returned to the ground state, increasing the vacuum density within the composite particle. Thus some of the mass of the nucleon comes from this canceled charge. From the study of the formation of the neutron would suggest the contribution to the mass would be small. Page 36
In addition to the quarks being modeled as sub-harmonics of the electron, there is another approach. In classic wave mechanics, when a wave moves into a region of different density, the amplitude of the wave is changed. The process of barrier penetration can be done from a classic or quantum approach, so electron capture by the nucleus can be done in a relativistic/quantum approach. The high energy density of the nucleus would reduce the amplitude of the electron, with a corresponding increase in mass to become stable within the denser region of space. This provides a good explanation why electrons are never found within the nucleus, and quarks never seen outside it, even though electron are known to be captured by the nucleus, and electron/positrons escape. While such an approach answers some questions, it asks more, there would be many obstacles in trying to develop a unity model for all 1/2 spin particles. Still the overall benefits of a unity model makes it worth exploring. Conclusions When attempting to treat quantum models in a relativistic fashion, what is a vacuum effect and what is a source field effect must be defined at the start. Doing so allows quantum models of confinement like the MIT and AdS to have relevance in a relativistic model of confinement. Only the basic structure is developed, and a complete model would be an extensive undertaking in its own right. Because it has much in common with quantum models it will be easier to flesh out the common aspects before moving on to newer, unexplored material.
A lattice theory for the nucleus is explored which is compatible with the electron and atomic lattice already constructed. This completes a lattice model of the atom that so far seems to be compatible with both quantum theory and relativity. Introduction No model of the atom can be considered complete without examining atoms with a large number of composite particles in them. Like modeling the quarks, that is easier said than done, the limits of observation means we must look for models that matches what we do know about the nucleus. There are over 50 models of the nucleus put forth. While not as many as there are electron models, it underscores the vast range of possible models that fit with observation to a significant amount. The two leading models of the nucleus are the Liquid drop and the Shell models. Both have aspects that are incorporated into a relativistic/quantum model of the nucleus. Liquid Drop Model of the Nucleus The liquid drop model of the nucleus as in some respects the nucleus does act as a drop of liquid. The density of the liquid compared to free space also limits the movement of composite particles within the nucleus. The model developed here has much in common with the liquid drop model. Energy density is based on an ideal gas/liquid and as a result composite particles are like a denser drop of liquid within a liquid nucleus as given in the model for confinement. This”Liquid Bag” approach also ties in well with the model for the source field and there will be a Casmir force between nucleons in addition the the electrostatic forces from the quarks. A great deal of wave energy will be concentrated in the central region of the nucleus and may prove important in the forming of new particles and composite particles within larger nuclei. It would also be possible to model the protons/neutrons of the nucleus as coupled harmonic oscillators as an addition to the basic liquid bag concept. The Shell Model of the nucleus can be modeled as a group of 3-dimensional harmonic oscillators. Page 38
Shell Model of the Nucleus Observations have noticed that there are certain “Magic Numbers” of nucleons that are more tightly bound then the next larger number. These magic number of protons or neutrons are; 2, 8, 20, 28, 50, 82 and 126. There is some evidence that 16 and 40 may also be magic or semi-magic numbers. There are also “Double Magic Number atoms where the protons and neutrons are both at the same magic number. The Pauli exclusion principle is important to the shell model as it describes the shells in terms of energy levels like the electrons in the atom while preventing overlapping of the composite particles. How the Pauli exclusion principle applies to this model has already been covered. When the nucleus is modeled as a group of 3-dimensional harmonic oscillators the magic numbers that are predicted are 2, 8, 20, 40, 70, 112. The first 3 numbers agree with observation, but it diverges after that point. The forth number predicted, 40, has been suggested as being a semi-magic number. The value of 16 does not appear at all in the harmonic model used. The harmonic oscillator potential grows infinity as r goes to infinity. But such a model does not account for the inward pressure from confinement. The shell model also saw that the basic harmonic model was not realistic and an approach like the Woods Saxon potential would be more suitable. By doing so the average radius is a little larger, and larger shells have a total energy less than in a simple harmonic model. Combining Liquid Drop and Shell Models Another approach predicting magic and semi-magic numbers in shell models is by using an idealized filling order where energy levels do nor overlap. This is perhaps the best way to combine the magic numbers of the shell model with the utility of the liquid drop model. Pressure and confinement in the model developed here would pack the nucleons into the smallest and most efficient use of space, without overlap, as this accounts for Pauli exclusion principle as applied to the liquid drop model and ours as well. The first magic number is 2, so a double magic nucleus would have 2 protons and 2 neutrons. This atom is familiar to most people being Helium. The structure of the nucleons is the same as in the standard model seen below.
And here in a probability distribution and relative size.
As in the liquid drop model, the vibrations of the nucleons is one means of interaction binding the nucleus together. The pressure on the bag of confinement for the nucleus is also a factor. An additional complication in the model developed here is the electrostatic interactions of quarks between composite particles. In the section on confinement it was discussed how this interaction would affect the probability of quark orbitals within the particle. First graphic represents the ground state of a proton. The two Up quarks are both in the lowest orbital where the highest probability of stability is. The quarks are close together and all the electrostatic forces are contained as in other models.
As neutrons are added, the probability of the quarks being found in the higher energy orbital within the proton and neutron increases. In most models of the nucleus, only the total charge of a composite particle is taken into account. If the electrostatic force of the quarks are allowed to extend through, then the quarks also contribute to the binding energy. In some arrangements of nucleons, especially when the number of protons and neutrons are equal and at a magic number, the distribution of electrostatic forces are balanced. The first such double magic nucleus is Helium. Now lets add the quarks to the standard picture of the Helium nucleus. Note in the following graphic the quarks tend to fall into an orbital as well.
The increase in binding energy is the same for all composite particles and the electrostatic forces are balanced, as much as possible. Such representations are at best an ideal state the nucleus seldom achieves but briefly, but can be seen more as an ideal state it is trying to maintain in a complex and chaotic environment. Page 41
The neutrons of the nucleus play an important part in the stability of the nucleus. Thus the Double magic numbers are of special interest. If the above illustration had only one neutron, the system would not be as stable and the quarks would have a higher probability of being found in various orbits. A shell is an accurate description as spherical arrangements of nucleons distribute the forces equally for a more stable arrangement. However a shell that is completely full may not be the most stable arrangement. The double magic shell above is very stable. However in 3 dimensions, if we envision the nucleus as roughly spherical, there is room for a nucleon on the z axis above and below the 4 shown. In this first shell, there is room for 6 but is the most stable at 4 nucleons. Instead of a pure packing problem in 3-D, magic numbers seem to be a product of stable configurations. The lowest possible shell for a nucleus would be a single proton. The shell just described does nor however have room for a single nucleon at the center. It overlaps and excludes the first shell level. Geometrically we could construct a shell that surrounded a space the size of a single proton, and it would hold 12 nucleons. It would also overlap the shell just described. The next shell that would fit around the shell described would hold 16 nucleons. By assigning a color to a shell, it will be easy to see which shells could coexist in a nucleus. Only shells of the same color can be in the same nucleus, so as not to violate the exclusion principle. Shell 4, with room for 16 also happens to be the next double magic number of 8 each. Here are the shells so far. Shell Number Max # Shell 1 Shell 2 Shell 3 Shell 4 1 nucleon 6 nucleons 12 nucleons 16 nucleons Notes Most stable at 4 nucleons Slightly more stable at 10 Most stable at 16 nucleons and possibly 12
Only shells of the same color can be in the same nucleus, so as not to violate the exclusion principle. Shell 4, with 16 also happens to be the next double magic number of 8 protons and neutrons each. This shell when full is stable, but would leave a rather large empty region in the center. If 2 nucleons from each of the top and bottom Z axis were removed and used to make shell 2, the result would still be a stable structure. That does give the model two possible configurations for this double magic number to investigate. From this cursory examination it would seem that the shell and liquid drop models can be combined in a relativistic/quantum model. However the magic numbers appear to be a more complex subject that just adding up shells to math the observed magic numbers. It would probably need computer simulations to arrive at the stable configurations for magic numbers in this model. Page 42
Without the use of computer graphics for a geometrical examination of larger shells and possible stable configurations for magic numbers, this cursory examination will have to stop here for now, but at least a possible path of exploration has been found. This brings this part of the nucleus examination to a close, leaving only the meson cloud left to finish up this perspective on the nucleus. The Meson Cloud Within this model the meson cloud is easy to explain with the liquid drop model as a basis and confinement as discussed in this paper. The nucleus has a boundary that defines a region of confinement for the nucleus just as the protons and neutrons have within it. This is the boundary defining the meson cloud as well. The nucleons as dense liquid drops suspended in a less dense liquid which is the meson cloud region. It exists between the nucleons and the spaces where there are incomplete shell allotments. As ½ spin particles can move through the bag of confinement through the process of barrier penetration, and considerable wave activity, both reflected and transmitted wave, this is a highly chaotic and unstable region for ½ spin and composite particles like the mesons. Even though the nucleus model is not complete, it would appear that there is a small region at the center not used in larger nucleus for a nucleon, which provides a space where energy would be concentrated for particle and composite particle formation. Mesons like protons and neutrons, would also have a region of confinement defining them as well, with all of the composite particles suspended in the less dense region of the meson cloud. In larger nuclei the electrostatic forces from the quarks of the mesons may also play an important role in the stability of atoms. In very large atoms the ratio of neutrons to protons changes with an excess of neutrons. As the electrostatic forces balance when the number of neutrons and protons are the same, these large nuclei would be less stable. The mesons may be standing in for protons in reducing this imbalance. Conclusions A relativistic/quantum model of the nucleus does seem a plausible avenue of research. While it is primarily a liquid drop model, aspects of energy shell levels and magic numbers can be incorporated. The main drawback is complexity. While the model can be grasped on an intuitive level, the number of variables and their interactions make it a very complex model mathematically. The model developed is consistent with the work in this paper and comparable with current models as well, though many details need worked out, being only the bare bones of a model is presented here. Page 43
Suggestions for Father Research
This thesis covers a considerable amount of ground. This is due to being an undergraduate thesis, it lacks the depth of detail a PhD thesis would have, and is meant to show a broad range of knowledge rather than advanced technical skill in a specific area. Thus the possibilities for research due to this paper considerable. First the material presented here need to be examined independently, as an undergraduate paper, it is not above errors. Then if the material presented is confirmed through peer review, then whole avenues of research present itself. In section one a geometrical vector examination of particles was used. To my knowledge this is the first time spinors were broken down into their vector components for this. Like Feynman Diagrams, expanding the vector examination in detail would be time consuming and a paper in its own right. However calculations could be performed on just the material given, as this paper has little mathematical material in it, and number of papers could just focus on this aspect of the material. The topic of symmetry in particles was the focus of the section, but still did not cover the topic in depth. Only basic S 1 circle symmetry was examined, so there is considerable room for the topic of symmetry to be examined in depth. In that vector examination, all types of particles were addressed as too how they fit into this model. The photon will probably attract the most attention. For the most part the other particles match well with current models, but to fit the photon into this model created a more unique view of the photon. Section one ties directly into the next section on ½ spin particles. Every new model of the electron, and there has been a number, often generates a number of papers confirming, refuting, or expanding on the model. The two double solutions presented for ½ spin particles will not doubt elicit more research as well. Of particular interest is that quarks and electrons obey the same rules, suggesting a unity model for ½ spin particles. This concept alone should spark some research. Lattice theories have long attracted the interest of researchers. It has been assumed that a relativistic approach to a atomic lattice theory was not possible. While complex, it does seem to be a possibility, but much work and research is still needed to see if it is indeed a viable model. The next section on confinement offers a number of research opportunities, much like the introduction of the Bag and AdS models did. There have been attempts to combine the MIT and the AdS models. This is not such an attempt, it has aspects of both these models, but has fundamental differences, such as transmission of energy through the boundary of the composite particle. Page 44
As the MIT bag model does not forbid energy through the bag, the work in this paper can be used directly in that model. This might be one of the first avenues of research that could be spurred by the work presented in this paper. The model for the nucleus is also defined by its confinement. This ties in well with the liquid drop model. Only the basics of incorporating shells and magic numbers were touched on, and new research could help combine the best of both these two models, completely separate from the nuclear model presented here. In closing a number of novel ideas are presented that may prove useful outside the scope of this paper. Add to that the work in peer review and building upon the models presented here, there is room for considerable farther research.
As this paper covered a broad range, each section had its own part for conclusions. Rather than repeat them here, this section will focus on the paper as a whole much like the abstract at the start of the thesis. Overall the paper is not technically advanced and lacks rigorous mathematical support. This is a major shortcoming. Even compared to other thesis’s, the paper is long and covers a number of topics rather than a narrower focus as would be common. For being written at the 2 year undergraduate level, the paper takes on topics normally reserved for graduate coursework. This complicates peer review due to its lack of mathematical support that would be present in a paper written at a higher level. The paper does meet its primary objectives. The research project was initially to design semiclassical models to introduce more advanced physics concepts at the undergraduate level. The splitting of spinors into their vector components allowed the use of vectors, a simpler physics and math concept than the matrices of the Dirac equation, to introduce pair production at the undergraduate level. The introduction of Maxwell equations can also use vectors to show the polarization of the vacuum at the undergraduate level. Likewise the model of the atom and nucleus, as well as advanced concepts like quark confinement are explainable at the undergraduate level. The paper was meant to update and incorporate as much of the independent research previously covered into a coherent model. Research however can lead to expected avenues of study, and this thesis is no exception. Previous models were discarded as new information was incorporated or a different approach used. In physics, many things are interconnected, and that is seen in this thesis as one section ties in and leads to the next section. Also more questions arise, perhaps more question are asked than answers given in this paper. Still much of this should be expected at the undergraduate level as new knowledge is absorbed and connections seen. Page 45
The paper does try to push past the limits of an undergraduate work. Developing an electron model is not normally within the realm of most undergraduate work. A “double solution” for an electron was compatible with the work done with vectors and polarization of the vacuum. The previous work on a double solution electron model has been used in 2 string theory models. Hopefully it will withstand farther peer review. The work on ½ spin particles included quarks, which lead to the work on confinement, as did some of the research in the section on lattice theories. The use of atomic units proved useful in a lattice model, and helps to introduce advanced concepts to the undergraduate student. A lattice model also highlight the problems relativity poses at the atomic level. There is however much work needed to see if the lattice model is workable as only a basic framework is established. The section on quark confinement moves past just an undergraduate explanation of leading models to exploring a more complex model for confinement. While the MIT and AdS models work well for an isolated, stable composite particle, a model for the gain or loss of energy is established which gives a mechanism for radioactive decay and electron capture by the nucleus as well as the exchange of energy from photons between the nucleus and electrons in orbit. The interaction between particles, composite particles and the nucleus is much more dynamic. The section on the nucleus does not extend current models as much as some other sections. Some of the problems with combining the liquid drop and shell models from a geometrical approach are found. The possible solution that magic numbers are specific configuration is interesting as it is directly related to the binding energy of the model outlined. This section also ties up all the loose ends from other sections to give a complete view of atomic structures from a semi-classical approach bridging quantum and relativistic concepts. Page 46
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