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GRAND UNIFICATION

OF THE STANDARD MODEL WITH QUANTUM GRAVITY
BY

ASHAY DHARWADKER


 
INSTITUTE OF MATHEMATICS
H-501  PALAM VIHAR
DISTRICT  GURGAON
HARYANA  1 2 2 0 1 7
INDIA
ashay@dharwadker.org

Abstract
We show that the mathematical proof of the four colour theorem [1] directly implies the existence of the standard model, together with quantum gravity, in its physical interpretation. Conversely, the experimentally observable standard model and quantum gravity show that nature applies the mathematical proof of the four colour theorem, at the most fundamental level. We preserve all the established working theories of physics: Quantum Mechanics, Special and General Relativity, Quantum Electrodynamics (QED), the Electroweak model and Quantum Chromodynamics (QCD). We build upon these theories, unifying all of them with Einstein's law of gravity. Quantum gravity is a direct and unavoidable consequence of the theory. The main construction of the Steiner system in the proof of the four colour theorem already defines the gravitational fields of all the particles of the standard model. Our first goal is to construct all the particles constituting the classic standard model, in exact agreement with 't Hooft's table [8]. We are able to predict the exact mass of the Higgs particle and the CP violation and mixing angle of weak interactions. Our second goal is to construct the gauge groups and explicitly calculate the gauge coupling constants of the force fields. We show how the gauge groups are embedded in a sequence along the cosmological timeline in the grand unification. Finally, we calculate the mass ratios of the particles of the standard model. Thus, the mathematical proof of the four colour theorem shows that the grand unification of the standard model with quantum gravity is complete, and rules out the possibility of finding any other kinds of particles.
Acknowledgements
Recent collaborative work with Vladimir Khachatryan shows how to use the grand unified theory to calculate the values of the Cabibbo angle and CKM matrix and also predict the Higgs Boson Mass [arXiv:0912.5189] with precision. We are pleased to announce that The Grand Unification has been published by Amazon in 2011.
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Introduction

We show that the mathematical proof of the four colour theorem [1] directly implies the existence of the standard model, together with quantum gravity, in its physical interpretation. Conversely, the experimentally observable standard model and quantum gravity show that nature applies the mathematical proof of the four colour theorem, at the most fundamental level. We should emphasize that we preserve all the established working theories of physics: Planck's Quantum Mechanics, Einstein's Special and General Relativity, Maxwell's Electromagnetism, Feynman's Quantum Electrodynamics (QED), the Weinberg-Salam-Ward Electroweak model and Glashow-Iliopoulos-Maiani's Quantum Chromodynamics (QCD). We build upon these theories, unifying all of them with Einstein's law of gravity, exactly as dictated by the proof of the four colour theorem. There is no escaping gravity. The main construction of the Steiner system in the proof of the four colour theorem already defines the gravitational fields of all the particles of the standard model. We now present an overview of the paper.

In Section 1, we briefly review Special Relativity and Lorentz invariance. In Section 2, we sketch the derivation of the relativistic Schrödinger wave equation and define Schrödinger discs in the complex plane. A Schrödinger disc is defined in terms of the wave function that is a solution to the Schrödinger wave equation, and various copies of oriented Schrödinger discs will represent all the particles of the standard model. In Section 3, we present a short summary of the proof of the Four Colour Theorem and introduce the labeled t-Riemann surface. All the particles of the standard model will be obtained by arranging copies of oriented Schrödinger discs on the labeled t-Riemann surface, as dictated by the proof of the four colour theorem. 

In Section 4, we define the Particle Frame in terms of the labeled t-Riemann surface. Particle frames associated with space-time points constitute a vector bundle in mathematical terminology, and a section of the vector bundle i.e. a particle frame at a space-time point, is called a gauge in the physics terminology. Thus, physical symmetries associated with sets of particles defined on a particle frame correspond to gauge transformations. The particle frame provides the general mathematical framework from which all the particles of the standard model will be defined, together with their basic physical properties: spin, charge and mass. We first specify the evolution of the particle frame according to the cosmological timeline or equivalent energy scales. Our first goal is to specify all the particles of the standard model as it is presently observed. 

To achieve this goal, we work with the particle frame structure that corresponds to the present epoch in the cosmological timeline or equivalent energy scales. Each kind of particle in the standard model will be defined by selecting a particular Schrödinger disc or the intersection of a particular set of Schrödinger discs from a particle frame. In Section 4.1, we specify the Fermion Selection Rule; in Section 4.2, we specify the Boson Selection Rule; in Section 4.3, we specify the Higgs Selection Rule; in Section 4.4, we specify the Spin Rule; in Section 4.5, we specify the Electric Charge Rule; in Section 4.6, we specify the Electromagnetic, Weak, Strong and Gravitational Charge Rule; in Section 4.7, we specify the Mass Rule; in Section 4.8, we specify the Equivalence Rule; in Section 4.9, we specify the Antiparticle Rule; in Section 4.10, we specify the Helicity Rule; in Section 4.11, we specify the CP-Transformation Rule; and finally, in Section 4.12, we specify the Standard Model Completion Rule; Using these rules, we achieve our first goal of defining all the particles constituting the classic standard model, in exact agreement with 't Hooft's table [8], as follows. 

In Section 5, we explicitly define all the Fermions and their antiparticles, following all the above rules:
 

Fermions Particle Name Symbol Type Generation Spin Mass (MeV) Charge Nc
Section 5.1.1, 2 e-neutrino νe Lepton I 1/2 > 0 0 1
Section 5.1.3, 4 electron e Lepton I 1/2 0.510999 -1 1
Section 5.1.5, 6 μ-neutrino νμ Lepton II 1/2 > 0 0 1
Section 5.1.7, 8 muon μ Lepton II 1/2 105.6584 -1 1
Section 5.1.9, 10 τ-neutrino ντ Lepton III 1/2 > 0 0 1
Section 5.1.11, 12 tau τ Lepton III 1/2 1771 -1 1
Section 5.2.1, 2 up u Quark I 1/2 5 +2/3 3
Section 5.2.3, 4 down d Quark I 1/2 10 -1/3 3
Section 5.2.5, 6 charm c Quark II 1/2 1600 +2/3 3
Section 5.2.7, 8 strange s Quark II 1/2 180 -1/3 3
Section 5.2.9, 10 top (truth) t Quark III 1/2 180000 +2/3 3
Section 5.2.11, 12 bottom (beauty) b Quark III 1/2 4500 -1/3 3

In Section 6, we explicitly define all the Bosons and their antiparticles, following all the above rules:
 

Bosons Particle Name Symbol Associated Force Field Spin Mass (MeV) Charge Nc
Section 6.1.1, 2 photon γ electromagnetic force 1 0 0 1
Section 6.2.1, 2 vector boson Z0 Z0 neutral carrier of the weak force 1 91188 0 1
Section 6.3.1, 2 vector boson W+ W+ positive carrier of the weak force 1 80280 +1 1
Section 6.4.1, 2 vector boson W- W- negative carrier of the weak force 1 80280 -1 1
Section 6.5.1, 2 gluon As strong force 1 0 0 8
Section 6.6.1, 2 graviton g gravitational force 2 0 0 1
Section 6.7.1, 2 scalar boson Higgs H0 Higgs field 0 125874 0 1

Note that we are able to predict the value of the Higgs mass quite precisely. Also, the CP violation of the weak interactions is a natural consequence of our definitions. 

In Section 7, we define the Force Fields associated with the bosons in standard model. In Section 7.1, we define the Electromagnetic Force Field. In Section 7.1.1, we review Maxwell's Electromagnetic Field Equations. In Section 7.1.2, we show how the photon acts as the carrier of the electromagnetic force. In Section 7.1.3, we give an example of a typical electromagnetic interaction: electron-electron scattering. In Section 7.1.4, we define the electromagnetic gauge group
 

Ge = U(1)

We explicitly define the observable gauge photon and show how the electromagnetic gauge group acts on it by means of the electromagnetic gauge transformations. 

In Section 7.2, we define the Weak Force Field. In Section 7.2.1, we define the Yang-Mills Weak Field Equations. In Section 7.2.2, we show how the Z0 acts as the neutral carrier of the weak force. In Section 7.2.3, we give an example of a typical weak Z0 interaction: muonic neutrino-electron scattering. In Section 7.2.4, we show how the W+ acts as the positive carrier of the weak force. In Section 7.2.5, we give an example of a typical weak W+ interaction: transformation of a down quark into an up quark, responsible for radioactivity. In Section 7.2.6, we show how the W- acts as the negative carrier of the weak force. In Section 7.2.7, we give an example of a typical weak W- interaction: again, transformation of a down quark into an up quark, responsible for radioactivity. The W+ and W- are antiparticles of each other. In Section 7.2.8, we define the weak gauge group
 

Gw = SU(2)

We explicitly define the observable gauge vector bosons [Z0], [W+], [W-] and show how the weak gauge group acts on them by means of the weak gauge transformations. In Section 7.2.9, we show that the Weinberg angle θw = 30 degrees (this is a running value). The Weinberg angle θw is a parameter that gives a relationship between the W+, W- and Z0 masses, as well as the ratio of the weak Z0 mediated interaction, called its mixing. 

In Section 7.3, we define the Strong Force Field. In Section 7.3.1, we define the Yang-Mills Strong Field Equations. In Section 7.3.2, we show how the gluon acts as the carrier of the strong force. In Section 7.3.3, we give an example of a typical strong interaction: formation of a quark-antiquark pair, called a meson. In Section 7.3.4, we define the strong gauge group
 

Gs = SU(3)

We explicitly define the eight species of observable gauge gluons and show how the strong gauge group acts on them by means of the strong gauge transformations. 

In Section 7.4, we define the Gravitational Force Field. In Section 7.4.1, we review General Relativity and curved space-time. We define the curvature tensor, the Ricci tensor and formulate Einstein's law of gravitation. Comparison with Newton's law of gravitation in the special case of flat space-time shows that the components of the metric tensor must be viewed as potentials describing the gravitational field. In Section 7.4.2, we show how to embed the particle frame in curved space-time without self-intersections. Then each of the Schrödinger discs of the particle frame carry the curvature and Ricci tensors. Thus, in Section 7.4.3, we can define the Gravitational Field Equations precisely as given by Einstein's law of gravitation. In Section 7.4.4, we show how the graviton acts as the carrier of the gravitational force. In Section 7.4.5, we give an example of a possible gravitational interaction: neutrino oscillation. In Section 7.4.6, we define the gravitational gauge group
 

Gg= SU(5)

We explicitly define the twenty-four species of observable gauge gravitons and show how the gravitational gauge group acts on them by means of the gravitational gauge transformations. 

In Section 8, we achieve our second goal, the Grand Unification of all the forces: electromagnetic, weak, strong and gravitational. In Section 8.1, we follow the cosmological timeline into the past, from the present to the Big Bang (or equivalent energy scales), showing how the gauge groups are embedded in a sequence
 

U(1) → SU(2) → SU(3) → SU(5)

during the unification and also showing how the Schrödinger discs are identified on the particle frame during the unification. In Section 8.2, we explicitly calculate all the coupling constants using the topological structure of the particle frame. The coupling constants are in excellent agreement with experimentally observed values:
 

electromagnetic coupling constant αe = 1/137 7.29 × 10-3
weak coupling constant αw = 1/1373 3.89 × 10-7
strong coupling constant αs = 1
gravitational coupling constant αg (1/13724, 1/13716) (5.23 × 10-52, 6.49 × 10-35)

Finally, in Section 8.3, we explicitly calculate the mass ratios of the particles in the standard model, using the topological structure of the particle frame. The mass ratios are in excellent agreement with the known experimentally observed values. Selecting units such that the mass of the electron is 1:
 

up quark mass / down quark mass = 5 / 10 = 1/2
charm quark mass / strange quark mass = 1600 / 180  8
top quark mass / bottom quark mass = 180000 / 4500 = 40
tau mass / muon mass = 1771 / 105.658 16
electron mass / unit mass = 1 / 1 = 1
Z0 mass / top quark mass = 91188 / 180000 1/2
Average Mixed Z0, W±mass / tau mass = (91188+80280) / 2 / 1771 48
W± mass / Z0mass = 80280 / 91188 √3/2
Higgs mass / Sum of Z0, W+, W- masses = 125874 / (91188+80280+80280) = 1/2

We have now calculated all the parameters that define the standard model and its associated force fields, according to 't Hooft's specification [8]. Thus, the mathematical proof of the four colour theorem shows that the grand unification of the standard model with quantum gravity is complete, and rules out the possibility of finding any other kinds of particles.

 

1. Special Relativity

The space-time of physics is defined by four real coordinates: the three space coordinates X, Y, Z and the time coordinate T. The theory of special relativity [6][7] is concerned with inertial reference frames in which force-free particles do not experience any acceleration with respect to the coordinate system. Inertial reference frames are defined by the group of Lorentz transformations which are linear transformations of the space-time coordinates that leave the velocity of light, c, invariant. A Lorentz transformation transforms one inertial reference frame to another that is in uniform motion relative to the first. One of the main motivations for restricting the theory to inertial reference frames is that Maxwell's equations for electromagnetism remain unchanged if the space-time coordinates are subjected to Lorentz transformations. Thus, according to the theory of special relativity, light has a constant velocity of propagation, c. If a light signal in a vacuum starts from a space point (X, Y, Z) at the time T, it spreads as a spherical wave and reaches a neighboring space point (X+dX, Y+dY, Z+dZ) at the time T+dT. Measuring the distance traveled by the light signal, we must have  
(cdT)2 = (dX)2+(dY)2+(dZ)2
(1.1)
Figure 1.1. A light signal is represented by a sphere of radius cdT centered at (X, Y, Z)

The equation (1.1) may be rewritten as  

(cdT)2-(dX)2-(dY)2-(dZ)2 = 0
(1.2)
Equation (1.2) represents an objective relation between neighboring space-time points and it holds for all inertial reference frames provided the transformations of the coordinates are restricted to those of special relativity, i.e. Lorentz transformations. By considering the inertial reference frames of special relativity, it can also be shown that the Lorentz transformations are precisely the linear transformations that leave the more general quantity  
(dS)2 = (cdT)2-(dX)2-(dY)2-(dZ)2
(1.3)
invariant. Note, however, that the vanishing of (dS)2 in equation (1.3) does not imply that the two space-time points coincide; it means that the two space-time points can be connected by a light signal.
 

2. The Schrödinger Wave Equation

Several ingenious physical experiments performed during the early 1900's irrefutably demonstrated the dual nature of light waves (electromagnetic radiation) and particles of matter:  light waves behave like material particles and the particles of matter behave like light waves. Furthermore, physical quantities like energy always occur in discrete packets, called quanta. The theory of quantum mechanics [9] was developed to describe these phenomena, and one of its fundamental rules is the uncertainty principle: it is impossible to specify precisely and simultaneously both the position and the momentum of a physical particle. The main idea was to use concentrated bunches of waves (called wave packets) to describe all particles (localized particles of matter and quanta of electromagnetic radiation). A particle is described by a wave function  
Ψ(X, Y, Z, T)
(2.1)
that depends on the space-time coordinates X, Y, Z, T. The wave function Ψ must have three basic properties:
  • Ψ must be able to interfere with itself, so that it can account for the results of diffraction experiments;
  • Ψ must be large in magnitude where the particle is likely to be and small elsewhere, so that it describes the particle as a wave packet;
  • Ψ should describe the behavior of a single particle, not the statistical distribution of a number of particles, which accounts for the fact that a single particle always interferes with itself and never with other particles.
We sketch the derivation of the Schrödinger wave equation [9] whose solution is the required wave function Ψ. The basic physical properties of the wave packet/particle are denoted by the following symbols:
 
m = mass
E = energy
p = momentum
h = Planck's constant
h = h/2π
λ = wave length
ν = frequency
ω = 2πν = angular frequency
k = 2π/λ = propagation number

Let us first consider a free particle of completely undetermined position, moving with a velocity much smaller than c, with precisely known momentum and kinetic energy, in accordance with the uncertainty principle. The propagation vector for the wave packet/particle is written as  

k = (kX, kY, kZ), where |k| = k
and its momentum vector is written as  
p = (pX, pY, pZ), where |p| = p
The kinetic energy of the particle is given by the equation  
E = p2/(2m)
(2.2)
On the other hand, we have Planck's equation for the energy of the wave packet:  
E = hν = hω
(2.3)
We also have de Broglie's equation for the momentum of the wave packet:  
p = (hX, hY, hZ) = (hkX, hkY, hkZ) = hk
(2.4)
To obtain agreement with physical observations we select a wave function (2.1) of the form:  
Ψ(X, Y, Z, T) = ei(kXX+kYY+kZZT)
(2.5)
Computing partial derivatives  
∂Ψ/∂T = -iω ei(kXX+kYY+kZZT) = -iωΨ
(2.6)
 
∂Ψ/∂X = ikX ei(kXX+kYY+kZZT) = ikXΨ
 
∂Ψ/∂Y = ikY ei(kXX+kYY+kZZT) = ikYΨ
 
∂Ψ/∂Z = ikZ ei(kXX+kYY+kZZT) = ikZΨ
 
2Ψ/∂X2 = (ikX)2ei(kXX+kYY+kZZT) = -kX2Ψ
(2.7)
 
2Ψ/∂Y2 = (ikY)2ei(kXX+kYY+kZZT) = -kY2Ψ
(2.8)
 
2Ψ/∂Z2 = (ikZ)2ei(kXX+kYY+kZZT) = -kZ2Ψ
(2.9)
Introducing the Laplace operator, from (2.7), (2.8) and (2.9) we have  
2Ψ = 2Ψ/∂X2+∂2Ψ/∂Y2+∂2Ψ/∂Z2 = -k2Ψ
(2.10)
From (2.2), (2.4) and (2.10) we have  
EΨ = (p2/(2m))Ψ = (h2k2/(2m))Ψ = (-h2/(2m))(-k2Ψ) = (-h2/(2m))∇2Ψ
(2.11)
On the other hand, from (2.3) and (2.6) we have  
EΨ = (hω)Ψ = (ih)(-iωΨ) = (ih)∂Ψ/∂T
(2.12)
Equations (2.11) and (2.12) yield the Schrödinger wave equation  
(ih)∂Ψ/∂T = (-h2/(2m))∇2Ψ
(2.13)
whose solution is the wave function (2.5) representing the wave packet/particle. From (2.11) and (2.12), we see that the momentum and energy can be represented by differential operators acting on Ψ:  
p  →  (-ih)∇
(2.14)
 
E  →  (ih)∂/∂T
(2.15)
At the time when Schrödinger developed his non-relativistic wave equation (2.13), he also proposed an extension of it that meets the requirements of special relativity [9]. If the free wave packet/particle (as described above) moves with a velocity approaching c, then its motion may be described by the relativistic Schrödinger wave equation as follows. The equation (2.2) is replaced by the relativistic equation  
E2 = c2p2+m2c4
(2.16)
where now E includes the rest-mass energy mc2. Substituting (2.14) and (2.15) in (2.16) and operating on the wave function Ψ, we obtain the relativistic Schrödinger wave equation  
-h22Ψ/∂T2 = -h2c22Ψ+m2c4Ψ
(2.17)
whose solution is the again the original wave function (2.5) representing the relativistic wave packet/particle. Finally, if the particle is massless with m = 0 and moves with the velocity of light c, then the relativistic wave equation 2.17 reduces to  
-h22Ψ/∂T2 = -h2c22Ψ
(2.18)
The h2 in equation 2.18 cancels and we have  
2Ψ/∂T2 - c22Ψ = 0
(2.19)
Thus, the quantum-mechanical behavior of a particle is completely described by the wave function (2.5) which is a solution of the appropriate wave equation. At each point of space-time (X, Y, Z, T), the value of the wave function Ψ(X, Y, Z, T) = ei(kXX+kYY+kZZT) corresponds to a point on the boundary of a disc D centered at the origin in the complex plane C, as shown in figure 2.1 below. We shall call D a Schrödinger disc.
 
Figure 2.1. A particle is represented by a Schrödinger disc

By the the uncertainty principle, it is impossible to specify precisely and simultaneously both the position and the momentum of the particle. We have defined the Schrödinger disc D representing the particle at a precisely specified position (X, Y, Z, T) in space-time, with an uncertain momentum vector p. Instead, we may define the Schrödinger disc D representing a particle with a precisely specified momentum vector p and an uncertain position (X, Y, Z, T) in space-time. Then the Schrödinger disc D may be oriented in one of two possible ways: clockwise or anticlockwise, depending on whether we choose the normal vector to the complex plane according to the left-hand or the right-hand rule. We align this normal vector with the momentum vector p of the particle in the case that p is precisely specified. A left-handed orientation of a Schrödinger disc D will represent a particle of left-handed helicity and a right-handed orientation will represent a particle of right-handed helicity. Note that helicity is conserved for massless particles that always travel with the velocity of light c, but not for particles with positive mass: according to the theory of special relativity, the direction of the momentum vector p is reversed relative to any reference frame that moves faster than the particle. Thus, for a particle with positive mass, the helicity cannot be conserved with respect to all reference frames.

The theory of quantum mechanics that we have briefly sketched so far describes only a single particle (often referred to as the first quantization). To describe interactions between pairs of particles we need to develop quantum field theory (often referred to as the second quantization) [9]. Here the one-particle Schrödinger wave equation is modified to its many-particle version and its solution Ψ is construed as having many copies that describe the associated fields of the particles. In particular, the copies of the wave function Ψ represent the processes of creation and destruction of particles and the interacting particles of the associated force fields. For example, if the particle is a photon, then we can interpret Ψ to be either the electric field or the magnetic field and equation 2.19 describes the propagation of electromagnetic waves in vacuum.

We shall build the standard model of particle physics from copies of oriented Schrödinger discs, arranged in a certain way as dictated by the mathematical proof of the four colour theorem.

 

3. The Four Colour Theorem

We shall now follow the main construction in the proof of the four colour theorem [1]. Let us briefly state the theorem and sketch the proof, referring to [1] for details. A map on the sphere or plane is a subdivision of the surface into finitely many regions. Two regions in a map are said to be adjacent if they share a whole segment of their boundaries in common. A proper colouring of the regions of a map is an assignment of a colour to each region such that no two adjacent regions receive the same colour.

3.1. The Four Colour Theorem. Given any map on the sphere or plane, it is always possible to properly colour the regions of the map using at most four colours 0, 1, 2, 3.

Sketch of the proof [1]: In section I on map colouring, we define maps on the sphere and their proper colouring. For purposes of proper colouring it is equivalent to consider maps on the plane and furthermore, only maps which have exactly three edges meeting at each vertex. Lemma 1 proves the six colour theorem using Euler's formula, showing that any map on the plane may be properly coloured by using at most six colours. We may then make the following basic definitions.

  • Define N to be the minimal number of colours required to properly colour any map from the class of all maps on the plane.
  • Based on the definition of N, select a specific map m(N) on the plane which requires no fewer than N colours to be properly coloured.
  • Based on the definition of the map m(N), select a proper colouring of the regions of the map m(N) using the N colours 0, 1, ..., N-1.
The whole proof works with the fixed number N, the fixed map m(N) and the fixed proper colouring of the regions of the map m(N). In section II we define Steiner systems and prove Tits' inequality and its consequence that if a Steiner system S(N+1, 2N, 6N) exists, then  N cannot exceed 4. Now the goal is to demonstrate the existence of such a Steiner system. In section III we define Eilenberg modules. The regions of the map m(N) are partitioned into disjoint, nonempty equivalence classes 0, 1, ..., N-1 according to the colour they receive. This set is given the structure of the cyclic group ZN = {0, 1, ..., N-1} under addition modulo N. We regard ZN as an Eilenberg module for the symmetric group S3 on three letters and consider the split extension ZN]S3 corresponding to the trivial representation of S3. By section IV on Hall matchings we are able to choose a common system of coset representatives for the left and right cosets of S3 in the full symmetric group on |ZN]S3| letters. For each such common representative and for each ordered pair of elements of S3, in section V on Riemann surfaces we establish a certain action of the two-element cyclic group on twelve copies of the partitioned map m(N) by using the twenty-fourth root function of the sheets of the complex plane. Using this action, section VI gives the details of the main construction. The 6N elements of ZN]S3 are regarded as the set of points and lemma 23 builds the blocks of 2N points with every set of N+1 points contained in a unique block. This constructs a Steiner system S(N+1, 2N, 6N) which implies by Tits' inequality that N cannot exceed 4, completing the proof. ☐

After the proof of the four colour theorem [1] is complete, a posteriori we know that N = 4. The four colours 0, 1, 2, 3 are represented by the following palette:  

0  → 
1  → 
2  → 
3  → 
We select the map m(4) on the surface of the sphere, with its proper colouring as shown below in figure 3.1. According to the colour each region receives, the regions are partitioned into four equivalence classes that form the cyclic group Z4 = {0, 1, 2, 3} under addition modulo 4.
 
Figure 3.1. The map on the sphere

By boring a small hole in the blue region 0, we may deform the surface of the sphere until it is flat, to obtain a copy of the map m(4) on the complex plane C. Let D denote a disc centered at the origin of the complex plane, with a fixed orientation. We may perform the deformation of the map in such a way that both the origin and the boundary of the disc D are contained entirely inside the blue region of the map. Thus, we obtain the map m(4) inside the disc D, with the origin inside the blue region, as shown in figure 3.2.
 

Figure 3.2. The map inside a disc 

Next, we cut the disc D along the positive real axis. This cut has an upper and a lower edge, as shown in figure 3.3.
 
 

Figure 3.3.1. The map inside a disc with a cut 

We now follow the construction of the t-Riemann surface in [1]. Consider the composition of the functions C  → C; z → t = z2 and C → C; t → w = t12. The composite is given by the assignment z → t = z2 → w = t12 = z24. Take twenty-four identical copies of the map m(4) on the disc with a cut, labeled k = 1, ..., 24.
 

Figure 3.3.2. Twenty-four copies of the map 

For k = 1, ..., 23 attach the lower edge of the cut of disc k with the upper edge of the cut of disc k+1. To complete the cycle, attach the lower edge of the cut of disc 24 with the upper edge of the cut of disc 1. This forms the w-Riemann surface. The point w = 0 connects all the discs and is called the branch point. There are twenty-four superposed copies of the map m(4) on the w-Riemann surface corresponding to the twenty-four sectors 

{z|(k-1)(2π/24) < arg z < k(2π/24)} (k = 1, ..., 24)
on the z-plane. These are divided into two sets. The first set consists of twelve superposed copies of the map m(4) corresponding to the sectors  
{z|(k-1)(2π/24) < arg z < k(2π/24)} (k = 1, ..., 12)
of the upper half of the z-plane which comprise the upper sheet of the t-Riemann surface. The second set consists of twelve superposed copies of the map m(4) corresponding to the sectors  
{z|(k-1)(2π/24) < arg z < k(2π/24)} (k = 13, ..., 24)
of the lower half of the z-plane which comprise the lower sheet of the t-Riemann surface. The t-Riemann surface is orientable, since every orientation of a disc is carried over to the disc next to it.
 
Figure 3.4. The t-Riemann surface

We shall now label the regions of the maps on the t-Riemann surface. Referring to [1] for details, let S3 = <σ, ρ> = {1, ρ, ρ2, σρ2, σρ, σ} denote the dihedral group of order 6, abstractly isomorphic to the symmetric group on 3 letters. Corresponding to the trivial representation of S3, we regard Z4 as its Eilenberg module and form the split extension Z4]S3 that is abstractly isomorphic to the direct product Z4×S3. Next, form the integral group algebras Z(Z4]S3) and ZS3. Again, corresponding to the trivial representation of ZS3, we regard Z(Z4]S3) as its Eilenberg module and form the split extension Z(Z4]S3)]ZS3 that is abstractly isomorphic to the direct product Z(Z4]S3ZS3. Let Sym(Z4]S3) denote the symmetric group of order 24! on |Z4]S3|=24 letters. Then S3 embeds in Sym(Z4]S3) via the Caley right regular representation R. Select a common system of representatives {φi | i = 1, 2, 3, ..., 24!/6} for the left and right cosets of the embedded subgroup S3 in the group Sym(Z4]S3). Fix a common coset representative φi and a pair of elements β, γ of S3. The regions of the maps on the t-Riemann surface are labeled by elements of the split extension Z(Z4]S3)]ZS3 according to the following scheme.
 

Labeling Scheme
The branch point at the center which connects all the sheets is labeled (0, β+γ).
The regions of the maps on the upper half of the upper sheet of the t-Riemann surface are labeled as follows (all labels have a positive sign):
 
+
1 2 3 4 5 6
(+(0, 1)φiR(β), β+γ) (+(0, ρ)φiR(β), β+γ) (+(0, ρ2iR(β), β+γ) (+(0, σρ2iR(β), β+γ) (+(0, σρ)φiR(β), β+γ) (+(0, σ)φiR(β), β+γ)
(+(1, 1)φiR(β), β+γ) (+(1, ρ)φiR(β), β+γ) (+(1, ρ2iR(β), β+γ) (+(1, σρ2iR(β), β+γ) (+(1, σρ)φiR(β), β+γ) (+(1, σ)φiR(β), β+γ)
(+(2, 1)φiR(β), β+γ) (+(2, ρ)φiR(β), β+γ) (+(2, ρ2iR(β), β+γ) (+(2, σρ2iR(β), β+γ) (+(2, σρ)φiR(β), β+γ) (+(2, σ)φiR(β), β+γ)
(+(3, 1)φiR(β), β+γ) (+(3, ρ)φiR(β), β+γ) (+(3, ρ2iR(β), β+γ) (+(3, σρ2iR(β), β+γ) (+(3, σρ)φiR(β), β+γ) (+(3, σ)φiR(β), β+γ)
The regions of the maps on the lower half of the upper sheet of the t-Riemann surface are labeled as follows (all labels have a negative sign):
 
-
7 8 9 10 11 12
(-(0, 1)R(γ)φi, β+γ) (-(0, ρ)R(γ)φi, β+γ) (-(0, ρ2)R(γ)φi, β+γ) (-(0, σρ2)R(γ)φi, β+γ) (-(0, σρ)R(γ)φi, β+γ) (-(0, σ)R(γ)φi, β+γ)
(-(1, 1)R(γ)φi, β+γ) (-(1, ρ)R(γ)φi, β+γ) (-(1, ρ2)R(γ)φi, β+γ) (-(1, σρ2)R(γ)φi, β+γ) (-(1, σρ)R(γ)φi, β+γ) (-(1, σ)R(γ)φi, β+γ)
(-(2, 1)R(γ)φi, β+γ) (-(2, ρ)R(γ)φi, β+γ) (-(2, ρ2)R(γ)φi, β+γ) (-(2, σρ2)R(γ)φi, β+γ) (-(2, σρ)R(γ)φi, β+γ) (-(2, σ)R(γ)φi, β+γ)
(-(3, 1)R(γ)φi, β+γ) (-(3, ρ)R(γ)φi, β+γ) (-(3, ρ2)R(γ)φi, β+γ) (-(3, σρ2)R(γ)φi, β+γ) (-(3, σρ)R(γ)φi, β+γ) (-(3, σ)R(γ)φi, β+γ)
The regions of the maps on the upper half of the lower sheet of the t-Riemann surface are labeled as follows (all labels have a negative sign):
 
-
13 14 15 16 17 18
(-(0, 1)R(β)φi, β+γ) (-(0, ρ)R(β)φi, β+γ) (-(0, ρ2)R(β)φi, β+γ) (-(0, σρ2)R(β)φi, β+γ) (-(0, σρ)R(β)φi, β+γ) (-(0, σ)R(β)φi, β+γ)
(-(1, 1)R(β)φi,β+γ) (-(1, ρ)R(β)φi, β+γ) (-(1, ρ2)R(β)φi,β+γ) (-(1, σρ2)R(β)φi, β+γ) (-(1, σρ)R(β)φi, β+γ) (-(1, σ)R(β)φi, β+γ)
(-(2, 1)R(β)φi, β+γ) (-(2, ρ)R(β)φi, β+γ) (-(2, ρ2)R(β)φi, β+γ) (-(2, σρ2)R(β)φi, β+γ) (-(2, σρ)R(β)φi, β+γ) (-(2, σ)R(β)φi, β+γ)
(-(3, 1)R(β)φi, β+γ) (-(3, ρ)R(β)φi, β+γ) (-(3, ρ2)R(β)φi, β+γ) (-(3, σρ2)R(β)φi, β+γ) (-(3, σρ)R(β)φi, β+γ) (-(3, σ)R(β)φi, β+γ)
The regions of the maps on the lower half of the lower sheet of the t-Riemann surface are labeled as follows (all labels have a positive sign):
 
+
19 20 21 22 23 24
(+(0, 1)φiR(γ), β+γ) (+(0, ρ)φiR(γ), β+γ) (+(0, ρ2iR(γ), β+γ) (+(0, σρ2iR(γ), β+γ) (+(0, σρ)φiR(γ), β+γ) (+(0, σ)φiR(γ), β+γ)
(+(1, 1)φiR(γ), β+γ) (+(1, ρ)φiR(γ), β+γ) (+(1, ρ2iR(γ), β+γ) (+(1, σρ2iR(γ), β+γ) (+(1, σρ)φiR(γ), β+γ) (+(1, σ)φiR(γ), β+γ)
(+(2, 1)φiR(γ),β+γ) (+(2, ρ)φiR(γ), β+γ) (+(2, ρ2iR(γ), β+γ) (+(2, σρ2iR(γ), β+γ) (+(2, σρ)φiR(γ), β+γ) (+(2, σ)φiR(γ), β+γ)
(+(3, 1)φiR(γ), β+γ) (+(3, ρ)φiR(γ), β+γ) (+(3, ρ2iR(γ), β+γ) (+(3, σρ2iR(γ), β+γ) (+(3, σρ)φiR(γ), β+γ) (+(3, σ)φiR(γ), β+γ)
Figure 3.5. Labeling scheme for the regions of the maps on the t-Riemann surface

The main construction in the proof [1] now defines the Steiner system S(5, 8, 24). The 24 points of the Steiner system are the elements of the underlying set Z4]S3

(0, 1)
(0, ρ)
(0, ρ2)
(0, σρ2)
(0, σρ)
(0, σ)
(1, 1)
(1, ρ)
(1, ρ2)
(1, σρ2)
(1, σρ)
(1, σ)
(2, 1)
(2, ρ)
(2, ρ2)
(2, σρ2)
(2, σρ)
(2, σ)
(3, 1)
(3, ρ)
(3, ρ2)
(3, σρ2)
(3, σρ)
(3, σ)
Figure 3.6. The points of the Steiner system S(5, 8, 24)

The rest of the main construction [1] builds the blocks of the Steiner system. Each block consists of 8 points such that any set of 5 points is contained in a unique block [2].

 

4. The Particle Frame

Section 4.1 The Fermion Selection Rule
Section 4.2 The Boson Selection Rule
Section 4.3 The Higgs Selection Rule
Section 4.4 The Spin Rule
Section 4.5 The Electric Charge Rule
Section 4.6 The Electromagnetic, Weak, Strong and Gravitational Charge Rule
Section 4.7 The Mass Rule
Section 4.8 The Equivalence Rule
Section 4.9 The Antiparticle Rule
Section 4.10 The Helicity Rule
Section 4.11 The CP-Transformation Rule
Section 4.12 The Standard Model Completion Rule
 
We specify the general mathematical framework from which all the particles of the standard model will be defined, together with their basic physical properties: spin, charge and mass. We call the labeled t-Riemann surface, constructed in section 3, a particle frame. Each kind of particle in the standard model will be defined by selecting a particular disc or the intersection of a particular set of discs from a particle frame. At a time, there can be only one particle on a particle frame and only the selected discs will be active. The selected discs are the Schrödinger discs that determine the quantum mechanical behavior of the particle at a space-time point (X, Y, Z, T), as described in section 2. Particle frames associated with space-time points constitute a vector bundle in mathematical terminology, and a section of the vector bundle i.e. a particle frame at a space-time point, is called a gauge in the physics terminology. Thus, physical symmetries associated with sets of particles defined on a particle frame correspond to gauge transformations. We shall explicitly construct all the gauge groups for the standard model, including gravity, in section 8. The labeling and topological structure of the t-Riemann surface according to the proof of the four color theorem provide us with the set of rules for determining the spin, charge and mass creating mechanism of a particle. Before we select any particular discs, a blank particle frame, which corresponds to a space-time point in vacuum, is shown below.
 
Figure 4.1. The particle frame

For the purposes of defining the particles of the standard model, it is convenient to draw the particle frame embedded in flat Euclidean three-dimensional space (X, Y, Z) and associate with the drawing an independent time dimension T (the measuring rods and clocks of special relativity). This makes it easy to see all parts of the particle frame and visualize how the spin, charge and mass rules work. However, in 7.4.2, we shall show how to explicitly embed the particle frame in curved four-dimensional space-time, as required by general relativity.

The particles of the standard model correspond to the present observable universe. We contend that the particle frame evolved to its present form by following the cosmological timeline cf. [Section 8]:

  • The Planck Epoch, when all the four forces are combined into one (Time T = 0 seconds to T = 10-43 seconds): The universe is created with the Big Bang at time T = 0. Let us suppose that immediately afterwards, all the energy of the universe is in the form of one single particle. The energy of the particle is in the form of its thermodynamic temperature and cannot be calculated from the theory. During this epoch the particle frame may be represented by the t-Riemann surface given by the composition of the functions C  → C; z → t = z2 and C → C; t → w = t2. The composite is given by the assignment z → t = z2 → w = t2 = z4. There is only one kind of force present in this early universe, the gravitational force. The carrier of this force, the graviton, may be represented on this version of the particle frame as it is in later epochs.
  • The Grand Unification Epoch, when the gravitational force separates from the strong-electroweak force (Time T = 10-43 seconds to T = 10-35 seconds): Many new particles and antiparticles are created: free quarks, antiquarks and photons in equilibrium with each other. The average temperature of the universe can now be calculated as 1032 K. During this epoch the particle frame may be represented by the t-Riemann surface given by the composition of the functions C  → C; z → t = z2 and C → C;t → w = t4. The composite is given by the assignment z → t = z2 → w = t4 = z8. There are two kinds of forces present in this universe, the gravitational force and strong-electroweak force. The carriers of these forces, the graviton and the photon, may be represented on this version of the particle frame as they are in later epochs.
  • The Inflationary Epoch, when the strong force separates from the electroweak force (Time T = 10-35 seconds to T = 10-12 seconds): The universe undergoes a rapid inflation. The average temperature of the universe can now be calculated as 1027 K. During this epoch the particle frame may be represented by the t-Riemann surface given by the composition of the functions C  → C; z → t = z2 and C → C; t → w = t6. The composite is given by the assignment z → t = z2 → w = t6 = z12. There are three kinds of forces present in this universe, the gravitational force, the strong force and the electroweak force. The carriers of these forces, the graviton, and two different types of photons, may be represented on this version of the particle frame as they are in later epochs.
  • The Later Epochs, when the the weak force separates from the electromagnetic force and all the four forces are distinct (Time T = 10-12 seconds upto the present): At time T = 10-12 seconds, the average temperature of the universe can be calculated to be 1015 K and the electroweak force separates into the electromagnetic force and the weak force. At this time the particle frame assumes its present form as shown in figure 4.1, represented by the t-Riemann surface given by the composition of the functions C  → C; z → t = z2 and C → C; t → w = t12. The composite is given by the assignment z → t = z2 → w = t12 = z24. All the four kinds of forces are now present in the universe, the gravitational force, the strong force, the weak force and the electromagnetic force. As the universe cools down to its present average temperature of 3 K, matter and antimatter as we know it emerges. Quarks are no longer free and become confined to give stable neutrons and protons. These form the atomic nuclei. Electrons and nuclei then combine to produce stable atoms corresponding to all the elements of the periodic table.
In the Sections 5 and 6 we shall define all the particles and antiparticles, including the carriers of the four forces, on this present version of the particle frame as shown in Figure 4.1. Before that, we must formulate the exact rules for the definition of the particles and antiparticles that make up the standard model, as it is presently observed. In Section 4.1, we specify the Fermion Selection Rule; in Section 4.2, we specify the Boson Selection Rule; in Section 4.3, we specify the Higgs Selection Rule; in Section 4.4, we specify the Spin Rule; in Section 4.5, we specify the Electric Charge Rule; in Section 4.6, we specify the Electromagnetic, Weak, Strong and Gravitational Charge Rule; in Section 4.7, we specify the Mass Rule; in Section 4.8, we specify the Equivalence Rule; in Section 4.9, we specify the Antiparticle Rule; in Section 4.10, we specify the Helicity Rule; in Section 4.11, we specify the CP-Transformation Rule; and finally, in Section 4.12, we specify the Standard Model Completion Rule.
 

4.1. The Fermion Selection Rule

Particles of the standard model that obey the Fermi-Dirac statistics [9] are called fermions. Hence, distinct particle frames with fermions defined on them cannot be superposed at a point in space-time because of the Pauli exclusion principle.  A fermion type particle will be selected from the particle frame as follows. First select a disc out of the 24 discs and then select a region of the map on the selected disc.
 
Figure 4.1.1. Fermion type particle

Referring to the labeling in figure 3.5, there are two types of fermions and each type of fermion comes in three generations. Type 1 fermions, called leptons, consist of a disc corresponding to a label 1 (generation I), ρ (generation II) or ρ2 (generation III) of the t-Riemann surface. Type 2 fermions, called quarks, consist of a disc corresponding to a label σ (generation I), σρ (generation II) or σρ2 (generation III) of the t-Riemann surface. Each generation I, II and III consists of one lepton doublet and one quark doublet. In section 5, we shall see how to use this rule to define all the fermions in the standard model.

 

4.2. The Boson Selection Rule

Particles of the standard model that obey the Bose-Einstein statistics [9] are called bosons. Hence, many distinct particle frames with bosons defined on them can be superposed at a point in space-time. A boson type particle will be selected from the particle frame as follows. First select a pair of fermion type particles (with the selected regions of the same colour respectively) such that the two discs have an intersecting boundary (a ray on the particle frame). Then select another pair of fermion type particles with selected regions of the same colour as before, but in such a way that the corresponding ray on the particle frame is distinct. Thus, a boson type particle is selected by choosing a pair of rays on the particle frame with a particular colour.
 
Figure 4.2.1. Boson type particle - as two pairs of discs
Figure 4.2.2. Boson type particle - as a pair of rays

In section 6, we shall see how to define all the bosons of the standard model using this rule. In particular, the two pairs of fermion type particles that define a boson are interpreted as creation and destruction operators during interactions in which the boson is exchanged.

 

4.3. The Higgs Selection Rule

There must be a unique particle called the Higgs particle in the standard model that attributes mass to all other particles. The Higgs particle is also a boson, but unlike the other bosons in the standard model, it is a scalar boson (it does not select a preferred direction in space like a vector boson). A Higgs type particle is selected as the intersection of all the 24 discs of the particle frame. This is the branch point of the t-Riemann surface and this selection of the Higgs type particle is unique. The origins of the upper and lower sheets of the t-Riemann surface form a Cooper pair and the Higgs particle undergoes Bose condensation, plunging to the lowest energy state possible.
 
Figure 4.3.1. Higgs type particle

For a description of the mass creating mechanism, see the mass rule 4.7 below.

 

4.4. The Spin Rule

The particle frame consists of four half-surfaces:
  • The upper half of the upper sheet
  • The lower half of the upper sheet
  • The upper half of the lower sheet
  • The lower half of the lower sheet
Given a particle as a selection S of the intersection of a set of discs or as a pair of rays, count the number n of half-surfaces of the particle frame that intersect with a whole segment of S. Define s = n/2 to be the spin of the particle. With this definition, the fermion type particle shown in figure 4.1.1 has n = 1 and spin s = 1/2; the boson type particle shown in figure 4.2.2 has n = 2 and spin s = 2/2 = 1; the Higgs type particle shown in figure 4.3.1 has n = 0 and spin s = 0/2 = 0. In sections 5 and 6, this rule is used to explicitly calculate the spin of each particle in the standard model.
 

4.5. The Electric Charge Rule

We first associate each colour with a unique absolute value of electric charge according to the following scheme:  
 →  0
 →  1/3
 →  2/3
 →  1
The particle frame is labeled according to the labeling scheme of the t-Riemann surface shown in figure 3.5. The signs of the labels according to figure 3.5 are as follows:
  • The upper half of the upper sheet has a + sign
  • The lower half of the upper sheet has a - sign
  • The upper half of the lower sheet has a - sign
  • The lower half of the lower sheet has a + sign
Given a particle as a selection S of the intersection of a set of discs or as a pair of rays, assign a signed electric charge to the particle according to this scheme. This is defined to be the electric charge of the particle. With this definition, the fermion type particle shown in figure 4.1.1 has electric charge -1; the boson type particle shown in figure 4.2.2 has electric charge 0; the Higgs type particle shown in figure 4.3.1 has electric charge 0. In sections 5 and 6, this rule is used to explicitly calculate the electric charge of each particle in the standard model.
 

4.6. The Electromagnetic, Weak, Strong and Gravitational Charge Rule

When we speak of charge without any further specification, we always mean the electric charge. However, in addition to an electric charge, a particle also has electromagnetic, weak, strong and gravitational charges according to the following scheme:
 
Type
Charge
Electromagnetic
1
Weak
ρ, ρ2
Strong
σ, σρ, σρ2
Gravitational 0, 1, 2, 3, σ

The electromagnetic and weak charges are neutral as far as the strong force is concerned, hence they are regarded as neutral strong charges. The number of different unsigned strong charges that a particular type of particle may have is denoted by Nc. The electromagnetic, weak, strong and gravitational charges are also given a sign (charge or anticharge), exactly as for the electric charge in 4.5.

  • Each lepton type fermion has exactly one neutral unsigned strong charge, so Nc = 1. Since the sign does not alter a neutral strong charge, there is only one neutral strong charge possible for each lepton type fermion.
  • Each quark type fermion may have one of three kinds of unsigned strong charges σ, σρ or σρ2, so Nc = 3. After assigning a sign according to the above scheme, the strong charge for a quark type fermion is one of ±σ, ±σρ or ±σρ2.
  • There is a special boson type particle in the standard model, called the photon, that is responsible for electromagnetic interactions. A photon type particle carries a neutral strong charge-anticharge pair with + and - signs respectively. Consider the unitary group U(1) consisting of 1×1 complex unitary matrices under matrix multiplication. Then, the single Lie group parameter corresponds to the neutral strong charge 1 and the generator of U(1) corresponds to the photon. So, for the photon, Nc = 1.
  • There are three special boson type particles in the standard model, called the Z0,W+and W-, that are responsible for weak interactions between leptons as well as between quarks. Each of these carries a neutral strong charge-anticharge, charge-charge or anticharge-anticharge  pair respectively.
    • The Z0 boson has Nc = 1
    • The W+ boson has Nc = 1
    • The W- boson has Nc = 1
    Consider the special unitary group SU(2) consisting of 2×2 complex unitary matrices of determinant 1, under matrix multiplication. Then, the 2 Lie group parameters correspond to the weak charges ρ, ρ2 and the 3 generators of SU(2) correspond to the three bosons Z0,W+and W-.
  • There is a special boson type particle in the standard model, called a gluon, that is responsible for strong interactions between quarks. A gluon type particle carries a superposition of strong charge-anticharge pair with + and - signs respectively. Consider the special unitary group SU(3) consisting of 3×3 complex unitary matrices of determinant 1, under matrix multiplication. Then, the 3 Lie group parameters correspond to the strong charges σ, σρ, σρ2 and the 8 generators of SU(3) correspond to the 8 distinct gluon species. So, for the gluon, Nc = 8.
  • There is a special boson type particle in the standard model, called a graviton, that is responsible for gravitational interactions between particles. A graviton type particle carries a strong charge-anticharge pair ±σ. So, for the graviton, Nc = 1. However, a graviton type particle also carries a superposition of electric charge-anticharge pairs ±0, ±1, ±2 and ±3. Thus, a graviton carries a superposition of all the gravitational charge-anticharge pairs ±0, ±1, ±2, ±3, ±σ. Consider the special unitary group SU(5) consisting of 5×5 complex unitary matrices of determinant 1, under matrix multiplication. Then, the 5 Lie group parameters correspond to the gravitational charges 0, 1, 2, 3, σ and the 24 generators of SU(5) correspond to the 24 distinct graviton species. Note that the labels for the gravitational charges correspond to the fixed points of the subgroup H of the group G in the proof of the four colour theorem [1].
Referring to the mass rule 4.7 below and figure 3.5, we shall use one of β or γ to specify the unsigned strong charge of a particle on the particle frame.
 

4.7. The Mass Rule

The rest mass of a particle in the standard model is usually determined from experimental observations. However, we shall show in 8.3 that the rest masses of all the particles in the standard model cannot be independent and most of the mass ratios must be fixed quite precisely due to the structure of the particle frame. We define the assignment of the rest mass of a particle of the standard model in the following way. Referring to section 3, recall that each element ψ of Sym(Z4]S3) is a permutation of the underlying set Z4]S3. Each permutation ψ may be thought of as representing the entropy or disorder of the set Z4]S3, and we relate ψ to a definite thermodynamic temperature. During the hot early universe, the energy of a particle is measured in the form its thermodynamic temperature. Let us suppose that at the time T = 10-12 seconds when the particle frame assumes its present form, each kind of particle S in the standard model is associated with a unique permutation ψS, which in turn is associated with a unique value of energy respecting the fixed mass ratios calculated in 8.3. We think of the permutation ψS as measuring the degree of disorder of the system defining the particle, so that ψS can be associated with a unique thermodynamic temperature or energy. We interpret the rest mass of the particle S as being created by the energy of the uniquely associated permutation ψS, as follows. Write ψS = R(δ)φj as the unique expression in terms of the common coset representatives and, by definition ψSμ = φjR(δ), as described in [1]. Then ψS and ψSμ act on any given region (m, α) by means of the ↑ and ↓ group actions, respectively [1]. Note that by lemma 17 [1], the two group actions ↑ and ↓ are equal, so (m, α)ψS = (m, α)ψSμ for any selected region. The given particle is represented on the particle frame corresponding to the t-Riemann surface with φi, β, γ chosen accordingly. This means that for any region of the type (m, α) in the selection S that represents the particle on the frame, we have the uniquely associated particle rest mass (m, α)ψS = (m, α)ψSμ. By the antiparticle rule 4.9 below, a particle and antiparticle will always have the same rest mass. We have special cases for the bosons:
  • For the massless bosons S, we assume that the uniquely associated permutation ψS = R(δ)φj has φj equal to the identity permutation.
  • For the massive bosons S, we assume that the uniquely associated permutation ψS = R(δ)φj has δ = 1.
    • Furthermore, for the massive Higgs boson, we assume that the permutation φj has the energy equal to half of the sum of the masses of all other bosons. Since the Higgs particle and antiparticle will be identified (as a Cooper pair), their combined energy would then be the sum of the masses of all other bosons. This must be the lowest energy state possible for the Higgs boson when it undergoes Bose condensation.
Thus, the inertial mass of each particle S in the standard model is associated with a unique permutation ψS, respecting the mass ratios calculated in 8.3. Then, by the Higgs-Kibble mechanism [8], the Higgs particles can undergo Bose condensation and assign this definite finite value to the rest mass of each particle in space-time. According to general relativity 7.4.1 and the embedding of the particle frame in curved space-time 7.4.2, we must also associate ψS with the curvature tensor R μντλ and the Ricci tensor R ντ carried by the Schrödinger discs of the particle frame. Also, with reference to figure 3.5, note that we only need to specify one of β or γ, not both, in the specification of rest mass (the other is used in the specification of unsigned strong charge).
 

4.8. The Equivalence Rule

Given two different selections S1 and S2 of particles, if the resulting particles have the same spin, charge and mass according to the above rules 4.4, 4.5 and 4.6, then we regard S1 and S2 as representing the same kind of particle in the standard model.
 

4.9. The Antiparticle Rule

To each particle there corresponds an antiparticle. Let the function π denote a rotation of the z-plane by π radians. Then π induces a rotation of the t-Riemann surface by 2π radians. Any point on the particle frame is transformed by π into the point superposed directly above or below it by a continuous rotation that winds exactly once around the branch point. Given a particle as a selection S of the intersection of a set of discs or as a pair of rays, the image of the particle under the function π is called its antiparticle. Note that by the above rules, an antiparticle is of the same type as the original particle, with identical spin and mass but the opposite charge. If a particle has no charge, then we cannot distinguish between the particle and its antiparticle. By the equivalence rule 4.7, a particle with no charge is equivalent to its antiparticle in the standard model. For example, as we shall see later, the photon is its own antiparticle. Fermions that are their own antiparticles are called Majorana fermions, for example the neutrinos are Majorana fermions.
 

4.10. The Helicity Rule

The helicity of the particle is defined by selecting one of two possible orientations (left-handed or right-handed) for all the active Schrödinger discs on the particle frame. This orientation carries over to neighboring discs and defines an orientation of the t-Riemann surface. Note that by section 2, the helicity is well-defined and conserved for massless particles but not for massive particles.
 

4.11. The CP Transformation Rule

Given a particle defined on the particle frame at a space-time point (X, Y, Z, T) with momentum vector p, we define the transformations C and P as follows:
  • P reverses the spatial coordinates to (-X, -Y, -Z). Note that this means that the direction of the momentum vector is also reversed to -p. Hence, the orientation of the particle frame is reversed and by the helicity rule 4.10, the helicity of the particle is reversed. Left-handed particles are transformed into right-handed particles of the same kind and vice-versa.
  • C transforms the particle into its antiparticle. In particular, the charge of the particle is reversed.
When both transformations are performed together, the particle is said to undergo a CP transformation. For example, a CP transformation of a left-handed electron gives a right-handed positron. The CP transformation is used in analyzing the symmetry of particle interactions. Since particle interactions are mediated by bosons and their associated force fields, we can characterize the symmetry of particle interactions by applying the CP transformation to the mediating bosons:
  • If a CP transformation applied to a boson in the standard model yields an equivalent boson, then we say that CP symmetry is preserved in interactions involving that boson. For example, CP symmetry is preserved by interactions involving the photon i.e. electromagnetic interactions.
  • If a CP transformation applied to a boson in the standard model does not yield an equivalent boson, then we say that CP symmetry is violated in interactions involving that boson. For example, CP symmetry is violated by interactions involving the W+ and W- vector bosons i.e. weak interactions.
Indeed, we shall see that CP symmetry is violated only by weak interactions. 
 

4.12. The Standard Model Completion Rule

If all the particle frames corresponding to all the particles in the universe were to be superimposed (hypothetically, of course), then the fermions and bosons should fit together perfectly according to the above rules, forming the complete standard model.
 
Figure 4.12.1. The perfect fitting of the fermions in the standard model

Each of the 24 discs of the particle frame represents the Schrödinger disc of a unique fermion in the standard model, respecting all the above rules. There cannot be any other fermions in the standard model.
 

Figure 4.12.2. The perfect fitting of the bosons in the standard model

Each of the 24 pairs of rays of the particle frame represents the four Schrödinger discs of a unique boson in the standard model, respecting all the above rules. There cannot be any other bosons in the standard model.

 

5. Fermions

 
Section 5.1 Leptons
Section 5.2 Quarks
 

5.1. Leptons

 
Leptons Particle Name Symbol Generation Spin Mass (MeV) Charge Nc
Section 5.1.1, 2 e-neutrino νe I 1/2 > 0 0 1
Section 5.1.3, 4 electron e I 1/2 0.510999 -1 1
Section 5.1.5, 6 μ-neutrino νμ II 1/2 > 0 0 1
Section 5.1.7, 8 muon μ II 1/2 105.6584 -1 1
Section 5.1.9, 10 τ-neutrino ντ III 1/2 > 0 0 1
Section 5.1.11, 12 tau τ III 1/2 1771 -1 1
 

5.1.1. The e-neutrino particle

Let ψe-neutrino be the unique permutation associated with the e-neutrino particle according to the mass rule 4.7. Select φi and β according to the unique expression ψe-neutrino = φiR(β) and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. We select disc 1 on the upper sheet of the particle frame and its blue region 0, that has the label (+(0, 1)φiR(β), β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the e-neutrino particle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the e-neutrino particle is a lepton of generation I. By the spin rule 4.4, the spin of the e-neutrino particle is 1/2. By the electric charge rule 4.5, the electric charge of the e-neutrino particle is 0 and by the strong charge rule 4.6, its strong charge is neutral with Nc= 1. The Super-Kamiokande experiment [10] demonstrated that the e-neutrino particle has a small positive rest mass (not precisely determined as yet), which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. This experiment also demonstrated that the e-neutrino particle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the e-neutrino particle and antiparticle 5.1.2 represent the same kind of particle in the standard model. Thus, the e-neutrino particle is a Majorana fermion. By the CP transformation rule 4.11, a right-handed e-neutrino particle is transformed into a left-handed e-neutrino particle and vice-versa.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
e-neutrino particle
νe
Lepton
I
1/2
> 0
0
1

 
Figure 5.1.1. The e-neutrino particle
 

5.1.2. The e-neutrino antiparticle

The e-neutrino antiparticle is equivalent to the e-neutrino particle, as shown by the following construction. Let ψe-neutrino be the unique permutation associated with the e-neutrino particle according to the mass rule 4.7. Select φi and β according to the unique expression ψe-neutrino = φiR(β) and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. Using the antiparticle rule 4.9, we select disc 13 on the lower sheet of the particle frame and its blue region 0, that has the label (-(0, 1)R(β)φi, β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the e-neutrino antiparticle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the e-neutrino antiparticle is a lepton of generation I. By the spin rule 4.4, the spin of the e-neutrino antiparticle is 1/2. By the electric charge rule 4.5, the electric charge of the e-neutrino antiparticle is 0 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. The Super-Kamiokande experiment [10] demonstrated that the e-neutrino antiparticle has a small positive rest mass (not precisely determined as yet), which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. This experiment also demonstrated that the e-neutrino antiparticle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the e-neutrino antiparticle and particle 5.1.1 represent the same kind of particle in the standard model. Thus, the e-neutrino antiparticle is a Majorana fermion. By the CP transformation rule 4.11, a right-handed e-neutrino antiparticle is transformed into a left-handed e-neutrino antiparticle and vice-versa.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
e-neutrino antiparticle
νe
Lepton
I
1/2
> 0
0
1

 
Figure 5.1.2. The e-neutrino antiparticle
 

5.1.3. The electron particle

Let ψelectron be the unique permutation associated with the electron particle according to the mass rule 4.7. Select φi and γ according to the unique expression ψelectron = R(γ)φi and β = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. We select disc 7 on the upper sheet of the particle frame and its red region 3, that has the label (-(3, 1)R(γ)φi, β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the electron particle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the electron particle is a lepton of generation I. By the spin rule 4.4, the spin of the electron particle is 1/2. By the electric charge rule 4.5, the electric charge of the electron particle is -1 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. From experimental observations the rest mass of the electron particle is 0.510999 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The electron particle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed electron particle is transformed into a left-handed electron antiparticle (positron) and a left-handed electron particle is transformed into a right-handed electron antiparticle (positron).
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
electron particle
e
Lepton
I
1/2
0.510999
-1
1

 
Figure 5.1.3. The electron particle
 

5.1.4. The electron antiparticle (positron)

Let ψelectron be the unique permutation associated with the electron particle according to the mass rule 4.7. Select φi and γ according to the unique expression ψelectron = R(γ)φi and β = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. Using the antiparticle rule 4.9, we select disc 19 on the lower sheet of the particle frame and its red region 3, that has the label (+(3, 1)φiR(γ), β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the electron antiparticle (aka positron) on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the electron antiparticle is a lepton of generation I. By the spin rule 4.4, the spin of the electron antiparticle is 1/2. By the electric charge rule 4.5, the electric charge of the electron antiparticle is +1 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. From experimental observations the rest mass of the electron antiparticle is 0.510999 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The electron antiparticle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed electron antiparticle is transformed into a left-handed electron particle and a left-handed electron antiparticle is transformed into a right-handed electron particle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
electron antiparticle (positron)
e
Lepton
I
1/2
0.510999
+1
1

 
Figure 5.1.4. The electron antiparticle (positron)
 

5.1.5. The μ-neutrino particle

Let ψμ-neutrino be the unique permutation associated with the μ-neutrino particle according to the mass rule 4.7. Select φi and β according to the unique expression ψμ-neutrino = φiR(β) and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. We select disc 2 on the upper sheet of the particle frame and its blue region 0, that has the label (+(0, ρ)φiR(β), β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the μ-neutrino particle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the μ-neutrino particle is a lepton of generation II. By the spin rule 4.4, the spin of the μ-neutrino particle is 1/2. By the electric charge rule 4.5, the electric charge of the μ-neutrino particle is 0 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. The Super-Kamiokande experiment [10] demonstrated that the μ-neutrino particle has a small positive rest mass (not precisely determined as yet), which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. This experiment also demonstrated that the μ-neutrino particle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the μ-neutrino particle and antiparticle 5.1.6 represent the same kind of particle in the standard model. Thus, the μ-neutrino particle is a Majorana fermion. By the CP transformation rule 4.11, a right-handed μ-neutrino particle is transformed into a left-handed μ-neutrino particle and vice-versa.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
μ-neutrino particle
νμ
Lepton
II
1/2
> 0
0
1

 
Figure 5.1.5. The μ-neutrino particle
 

5.1.6. The μ-neutrino antiparticle

The μ-neutrino antiparticle is equivalent to the μ-neutrino particle, as shown by the following construction. Let ψμ-neutrino be the unique permutation associated with the μ-neutrino particle according to the mass rule 4.7. Select φi and β according to the unique expression ψμ-neutrino = φiR(β) and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. Using the antiparticle rule 4.9, we select disc 14 on the lower sheet of the particle frame and its blue region 0, that has the label (-(0, ρ)R(β)φi, β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the μ-neutrino antiparticle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the μ-neutrino antiparticle is a lepton of generation II. By the spin rule 4.4, the spin of the μ-neutrino antiparticle is 1/2. By the electric charge rule 4.5, the electric charge of the μ-neutrino antiparticle is 0 and by the colour charge rule 4.6, its colour charge is neutral with Nc = 1. The Super-Kamiokande experiment [10] demonstrated that the μ-neutrino antiparticle has a small positive rest mass (not precisely determined as yet), which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. This experiment also demonstrated that the μ-neutrino antiparticle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the μ-neutrino antiparticle and particle 5.1.5 represent the same kind of particle in the standard model. Thus, the μ-neutrino antiparticle is a Majorana fermion. By the CP transformation rule 4.11, a right-handed μ-neutrino antiparticle is transformed into a left-handed μ-neutrino antiparticle and vice-versa.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
μ-neutrino antiparticle
νμ
Lepton
II
1/2
> 0
0
1

 
Figure 5.1.6. The μ-neutrino antiparticle
 

5.1.7. The muon particle

Let ψmuon be the unique permutation associated with the muon particle according to the mass rule 4.7. Select φi and γ according to the unique expression ψmuon = R(γ)φi and β = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. We select disc 8 on the upper sheet of the particle frame and its red region 3, that has the label (-(3, ρ)R(γ)φi, β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the muon particle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the muon particle is a lepton of generation II. By the spin rule 4.4, the spin of the muon particle is 1/2. By the electric charge rule 4.5, the electric charge of the muon particle is -1 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. From experimental observations the rest mass of the muon particle is 105.6584 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The muon particle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed muon particle is transformed into a left-handed muon antiparticle and a left-handed muon particle is transformed into a right-handed muon antiparticle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
muon particle
μ
Lepton
II
1/2
105.6584
-1
1

 
Figure 5.1.7. The muon particle
 

5.1.8. The muon antiparticle

Let ψmuon be the unique permutation associated with the muon particle according to the mass rule 4.7. Select φi and γ according to the unique expression ψmuon = R(γ)φi and β = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. Using the antiparticle rule 4.9, we select disc 20 on the lower sheet of the particle frame and its red region 3, that has the label (+(3, ρ)φiR(γ), β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the muon antiparticle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the muon antiparticle is a lepton of generation II. By the spin rule 4.4, the spin of the muon antiparticle is 1/2. By the electric charge rule 4.5, the electric charge of the muon antiparticle is +1 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. From experimental observations the rest mass of the muon antiparticle is 105.6584 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The muon antiparticle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed muon antiparticle is transformed into a left-handed muon particle and a left-handed muon antiparticle is transformed into a right-handed muon particle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
muon antiparticle
μ
Lepton
II
1/2
105.6584
+1
1

 
Figure 5.1.8. The muon antiparticle
 

5.1.9. The τ-neutrino particle

Let ψτ-neutrino be the unique permutation associated with the τ-neutrino particle according to the mass rule 4.7. Select φi and β according to the unique expression ψτ-neutrino = φiR(β) and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. We select disc 3 on the upper sheet of the particle frame and its blue region 0, that has the label (+(0, ρ2iR(β), β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the τ-neutrino particle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the τ-neutrino particle is a lepton of generation III. By the spin rule 4.4, the spin of the τ-neutrino particle is 1/2. By the electric charge rule 4.5, the electric charge of the τ-neutrino particle is 0 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. The Super-Kamiokande experiment [10] demonstrated that the τ-neutrino particle has a small positive rest mass (not precisely determined as yet), which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. This experiment also demonstrated that the τ-neutrino particle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the τ-neutrino particle and antiparticle 5.1.10 represent the same kind of particle in the standard model. Thus, the τ-neutrino particle is a Majorana fermion. By the CP transformation rule 4.11, a right-handed τ-neutrino particle is transformed into a left-handed τ-neutrino particle and vice-versa.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
τ-neutrino particle
ντ
Lepton
III
1/2
> 0
0
1

 
Figure 5.1.9. The τ-neutrino particle
 

5.1.10. The τ-neutrino antiparticle

The τ-neutrino antiparticle is equivalent to the τ-neutrino particle, as shown by the following construction. Let ψτ-neutrino be the unique permutation associated with the τ-neutrino particle according to the mass rule 4.7. Select φi and β according to the unique expression ψτ-neutrino = φiR(β) and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. Using the antiparticle rule 4.9, we select disc 15 on the lower sheet of the particle frame and its blue region 0, that has the label (-(0, ρ2)R(β)φi, β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the τ-neutrino antiparticle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the τ-neutrino antiparticle is a lepton of generation III. By the spin rule 4.4, the spin of the τ-neutrino antiparticle is 1/2. By the electric charge rule 4.5, the electric charge of the τ-neutrino antiparticle is 0 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. The Super-Kamiokande experiment [10] demonstrated that the τ-neutrino antiparticle has a small positive rest mass (not precisely determined as yet), which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. This experiment also demonstrated that the τ-neutrino antiparticle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the τ-neutrino antiparticle and particle 5.1.9 represent the same kind of particle in the standard model. Thus, the τ-neutrino antiparticle is a Majorana fermion. By the CP transformation rule 4.11, a right-handed τ-neutrino antiparticle is transformed into a left-handed τ-neutrino antiparticle and vice-versa.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
τ-neutrino antiparticle
ντ
Lepton
III
1/2
> 0
0
1

 
Figure 5.1.10. The τ-neutrino antiparticle
 

5.1.11. The tau particle

Let ψtau be the unique permutation associated with the tau particle according to the mass rule 4.7. Select φi and γ according to the unique expression ψtau = R(γ)φi and β = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. We select disc 9 on the upper sheet of the particle frame and its red region 3, that has the label (-(3, ρ2)R(γ)φi, β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the tau particle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the tau particle is a lepton of generation III. By the spin rule 4.4, the spin of the tau particle is 1/2. By the electric charge rule 4.5, the electric charge of the tau particle is -1 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. From experimental observations the rest mass of the tau particle is 1771 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The tau particle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed tau particle is transformed into a left-handed tau antiparticle and a left-handed tau particle is transformed into a right-handed tau antiparticle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
tau particle
τ
Lepton
III
1/2
1771
-1
1

 
Figure 5.1.11. The tau particle
 

5.1.12. The tau antiparticle

Let ψtau be the unique permutation associated with the tau particle according to the mass rule 4.7. Select φi and γ according to the unique expression ψtau = R(γ)φi and β = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. Using the antiparticle rule 4.9, we select disc 21 on the lower sheet of the particle frame and its red region 3, that has the label (+(3, ρ2iR(γ), β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the tau antiparticle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the tau antiparticle is a lepton of generation III. By the spin rule 4.4, the spin of the tau antiparticle is 1/2. By the electric charge rule 4.5, the electric charge of the tau antiparticle is +1 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. From experimental observations the rest mass of the tau antiparticle is 1771 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The tau antiparticle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed tau antiparticle is transformed into a left-handed tau particle and a left-handed tau antiparticle is transformed into a right-handed tau particle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
tau antiparticle
τ
Lepton
III
1/2
1771
+1
1

 
Figure 5.1.12. The tau antiparticle
 

5.2. Quarks

 
Quarks Particle Name Symbol Generation Spin Mass (MeV) Charge Nc
Section 5.2.1, 2 up u I 1/2 5 +2/3 3
Section 5.2.3, 4 down d I 1/2 10 -1/3 3
Section 5.2.5, 6 charm c II 1/2 1600 +2/3 3
Section 5.2.7, 8 strange s II 1/2 180 -1/3 3
Section 5.2.9, 10 top (truth) t III 1/2 180000 +2/3 3
Section 5.2.11, 12 bottom (beauty) b III 1/2 4500 -1/3 3
 

5.2.1. The up quark particle

Let ψup-quark be the unique permutation associated with the up quark particle according to the mass rule 4.7. Select φi and β according to the unique expression ψup-quark = φiR(β) and γ = σ, σρ or σρ2. Then the particle frame corresponds to the t-Riemann surface with these choices of φi, β, γ. We select disc 6 on the upper sheet of the particle frame and its green region 2, that has the label (+(2, σ)φiR(β), β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the up quark particle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the up quark particle is a quark of generation I. By the spin rule 4.4, the spin of the up quark particle is 1/2. By the electric charge rule 4.5, the electric charge of the up quark particle is +2/3. By the strong charge rule 4.6, the up quark particle has one of three possible strong charges +σ, +σρ or +σρ2, so Nc = 3. Due to quark confinement, no free quarks can be observed. From indirect experimental observations the effective rest mass of the up quark particle is 5 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The up quark particle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed up quark particle is transformed into a left-handed up quark antiparticle and a left-handed up quark particle is transformed into a right-handed up quark antiparticle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
up quark particle
u
Quark
I
1/2
5
+2/3
3

 
Figure 5.2.1. The up quark particle
 

5.2.2. The up quark antiparticle

Let ψup-quark be the unique permutation associated with the up quark particle according to the mass rule 4.7. Select φi and β according to the unique expression ψup-quark = φiR(β) and γ = σ, σρ or σρ2. Then the particle frame corresponds to the t-Riemann surface with these choices of φi, β, γ. Using the antiparticle rule 4.9, we select disc 18 on the lower sheet of the particle frame and its green region 2, that has the label (-(2, σ)R(β)φi,β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the up quark antiparticle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the up quark antiparticle is a quark of generation I. By the spin rule 4.4, the spin of the up quark antiparticle is 1/2. By the electric charge rule 4.5, the electric charge of the up quark antiparticle is -2/3. By the strong charge rule 4.6, the up quark antiparticle has one of three possible strong charges -σ, -σρ or -σρ2, so Nc = 3. Due to quark confinement, no free quarks can be observed. From indirect experimental observations the effective rest mass of the up quark antiparticle is 5 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The up quark antiparticle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed up quark antiparticle is transformed into a left-handed up quark particle and a left-handed up quark antiparticle is transformed into a right-handed up quark particle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
up quark antiparticle
u
Quark
I
1/2
5
-2/3
3

 
Figure 5.2.2. The up quark antiparticle
 

5.2.3. The down quark particle

Let ψdown-quark be the unique permutation associated with the down quark particle according to the mass rule 4.7. Select φi and γ according to the unique expression ψdown-quark = R(γ)φi and β = σ, σρ or σρ2. Then the particle frame corresponds to the t-Riemann surface with these choices of φi, β, γ. We select disc 12 on the upper sheet of the particle frame and its yellow region 1, that has the label (-(1, σ)R(γ)φi, β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the down quark particle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the down quark particle is a quark of generation I. By the spin rule 4.4, the spin of the down quark particle is 1/2. By the electric charge rule 4.5, the electric charge of the down quark particle is -1/3. By the strong charge rule 4.6, the down quark particle has one of three possible strong charges -σ, -σρ or -σρ2, so Nc = 3. Due to quark confinement, no free quarks can be observed. From indirect experimental observations the effective rest mass of the down quark particle is 10 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The down quark particle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed down quark particle is transformed into a left-handed down quark antiparticle and a left-handed down quark particle is transformed into a right-handed down quark antiparticle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
down quark particle
d
Quark
I
1/2
10
-1/3
3

 
Figure 5.2.3. The down quark particle
 

5.2.4. The down quark antiparticle

Let ψdown-quark be the unique permutation associated with the down quark particle according to the mass rule 4.7. Select φi and γ according to the unique expression ψdown-quark = R(γ)φi and β = σ, σρ or σρ2. Then the particle frame corresponds to the t-Riemann surface with these choices of φi, β, γ. Using the antiparticle rule 4.9, we select disc 24 on the lower sheet of the particle frame and its yellow region 1, that has the label (+(1, σ)φiR(γ), β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the down quark antiparticle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the down quark antiparticle is a quark of generation I. By the spin rule 4.4, the spin of the down quark antiparticle is 1/2. By the electric charge rule 4.5, the electric charge of the down quark antiparticle is +1/3. By the strong charge rule 4.6, the down quark antiparticle has one of three possible strong charges +σ, +σρ or +σρ2, so Nc = 3. Due to quark confinement, no free quarks can be observed. From indirect experimental observations the effective rest mass of the down quark antiparticle is 10 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The down quark antiparticle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed down quark antiparticle is transformed into a left-handed down quark particle and a left-handed down quark antiparticle is transformed into a right-handed down quark particle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
down quark antiparticle
d
Quark
I
1/2
10
+1/3
3

 
Figure 5.2.4. The down quark antiparticle
 

5.2.5. The charm quark particle

Let ψcharm-quark be the unique permutation associated with the charm quark particle according to the mass rule 4.7. Select φi and β according to the unique expression ψcharm-quark = φiR(β) and γ = σ, σρ or σρ2. Then the particle frame corresponds to the t-Riemann surface with these choices of φi, β, γ. We select disc 5 on the upper sheet of the particle frame and its green region 2, that has the label (+(2, σρ)φiR(β), β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the charm quark particle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the charm quark particle is a quark of generation II. By the spin rule 4.4, the spin of the charm quark particle is 1/2. By the electric charge rule 4.5, the electric charge of the charm quark particle is +2/3. By the strong charge rule 4.6, the charm quark particle has one of three possible strong charges +σ, +σρ or +σρ2, so Nc = 3. Due to quark confinement, no free quarks can be observed. From indirect experimental observations the effective rest mass of the charm quark particle is 1600 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The charm quark particle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed charm quark particle is transformed into a left-handed charm quark antiparticle and a left-handed charm quark particle is transformed into a right-handed charm quark antiparticle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
charm quark particle
c
Quark
II
1/2
1600
+2/3
3

 
Figure 5.2.5. The charm quark particle
 

5.2.6. The charm quark antiparticle

Let ψcharm-quark be the unique permutation associated with the charm quark particle according to the mass rule 4.7. Select φi and β according to the unique expression ψcharm-quark = φiR(β) and γ = σ, σρ or σρ2. Then the particle frame corresponds to the t-Riemann surface with these choices of φi, β, γ. Using the antiparticle rule 4.9, we select disc 17 on the lower sheet of the particle frame and its green region 2, that has the label (-(2, σρ)R(β)φi, β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the charm quark antiparticle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the charm quark antiparticle is a quark of generation II. By the spin rule 4.4, the spin of the charm quark antiparticle is 1/2. By the electric charge rule 4.5, the electric charge of the charm quark antiparticle is -2/3. By the strong charge rule 4.6, the charm quark antiparticle has one of three possible strong charges -σ, -σρ or -σρ2, so Nc = 3. Due to quark confinement, no free quarks can be observed. From indirect experimental observations the effective rest mass of the charm quark antiparticle is 1600 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The charm quark antiparticle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed charm quark antiparticle is transformed into a left-handed charm quark particle and a left-handed charm quark antiparticle is transformed into a right-handed charm quark particle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
charm quark antiparticle
c
Quark
II
1/2
1600
-2/3
3

 
Figure 5.2.6. The charm quark antiparticle
 

5.2.7. The strange quark particle

Let ψstrange-quark be the unique permutation associated with the strange quark particle according to the mass rule 4.7. Select φi and γ according to the unique expression ψstrange-quark = R(γ)φi and β = σ, σρ or σρ2. Then the particle frame corresponds to the t-Riemann surface with these choices of φi, β, γ. We select disc 11 on the upper sheet of the particle frame and its yellow region 1, that has the label (-(1, σρ)R(γ)φi, β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the strange quark particle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the strange quark particle is a quark of generation II. By the spin rule 4.4, the spin of the strange quark particle is 1/2. By the electric charge rule 4.5, the electric charge of the strange quark particle is -1/3. By the strong charge rule 4.6, the strange quark particle has one of three possible strong charges -σ, -σρ or -σρ2, so Nc = 3. Due to quark confinement, no free quarks can be observed. From indirect experimental observations the effective rest mass of the strange quark particle is 180 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The strange quark particle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed strange quark particle is transformed into a left-handed strange quark antiparticle and a left-handed strange quark particle is transformed into a right-handed strange quark antiparticle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
strange quark particle
s
Quark
II
1/2
180
-1/3
3

 
Figure 5.2.7. The strange quark particle
 

5.2.8. The strange quark antiparticle

Let ψstrange-quark be the unique permutation associated with the strange quark particle according to the mass rule 4.7. Select φi and γ according to the unique expression ψstrange-quark = R(γ)φi and β = σ, σρ or σρ2. Then the particle frame corresponds to the t-Riemann surface with these choices of φi, β, γ. Using the antiparticle rule 4.9, we select disc 23 on the lower sheet of the particle frame and its yellow region 1, that has the label (+(1, σρ)φiR(γ), β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the strange quark antiparticle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the strange quark antiparticle is a quark of generation II. By the spin rule 4.4, the spin of the strange quark antiparticle is 1/2. By the electric charge rule 4.5, the electric charge of the strange quark antiparticle is +1/3. By the strong charge rule 4.6, the strange quark antiparticle has one of three possible strong charges +σ, +σρ or +σρ2, so Nc = 3. Due to quark confinement, no free quarks can be observed. From indirect experimental observations the effective rest mass of the strange quark antiparticle is 180 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The strange quark antiparticle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed strange quark antiparticle is transformed into a left-handed strange quark particle and a left-handed strange quark antiparticle is transformed into a right-handed strange quark particle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
strange quark antiparticle
s
Quark
II
1/2
180
+1/3
3

 
Figure 5.2.8. The strange quark antiparticle
 

5.2.9. The top (truth) quark particle

Let ψtop-quark be the unique permutation associated with the top quark particle according to the mass rule 4.7. Select φi and β according to the unique expression ψtop-quark = φiR(β) and γ = σ, σρ or σρ2. Then the particle frame corresponds to the t-Riemann surface with these choices of φi, β, γ. We select disc 4 on the upper sheet of the particle frame and its green region 2, that has the label (+(2, σρ2iR(β), β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the top (aka truth) quark particle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the top quark particle is a quark of generation III. By the spin rule 4.4, the spin of the top quark particle is 1/2. By the electric charge rule 4.5, the electric charge of the top quark particle is +2/3. By the strong charge rule 4.6, the top quark particle has one of three possible strong charges +σ, +σρ or +σρ2, so Nc = 3. Due to quark confinement, no free quarks can be observed. From indirect experimental observations the effective rest mass of the top quark particle is 180000 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The top quark particle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed top quark particle is transformed into a left-handed top quark antiparticle and a left-handed top quark particle is transformed into a right-handed top quark antiparticle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
top (truth) quark particle
t
Quark
III
1/2
180000
+2/3
3

 
Figure 5.2.9. The top (truth) quark particle
 

5.2.10. The top (truth) quark antiparticle

Let ψtop-quark be the unique permutation associated with the top quark particle according to the mass rule 4.7. Select φi and β according to the unique expression ψtop-quark = φiR(β) and γ = σ, σρ or σρ2. Then the particle frame corresponds to the t-Riemann surface with these choices of φi, β, γ. Using the antiparticle rule 4.9, we select disc 16 on the lower sheet of the particle frame and its green region 2, that has the label (-(2, σρ2)R(β)φi, β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the top (aka truth) quark antiparticle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the top quark antiparticle is a quark of generation III. By the spin rule 4.4, the spin of the top quark antiparticle is 1/2. By the electric charge rule 4.5, the electric charge of the top quark antiparticle is -2/3. By the strong charge rule 4.6, the top quark antiparticle has one of three possible strong charges -σ, -σρ or -σρ2, so Nc = 3. Due to quark confinement, no free quarks can be observed. From indirect experimental observations the effective rest mass of the top quark antiparticle is 180000 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The top quark antiparticle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed top quark antiparticle is transformed into a left-handed top quark particle and a left-handed top quark antiparticle is transformed into a right-handed top quark particle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
top (truth) quark antiparticle
t
Quark
III
1/2
180000
-2/3
3

 
Figure 5.2.10. The top (truth) quark antiparticle
 

5.2.11. The bottom (beauty) quark particle

Let ψbottom-quark be the unique permutation associated with the bottom quark particle according to the mass rule 4.7. Select φi and γ according to the unique expression ψbottom-quark = R(γ)φi and β = σ, σρ or σρ2. Then the particle frame corresponds to the t-Riemann surface with these choices of φi, β, γ. We select disc 10 on the upper sheet of the particle frame and its yellow region 1, that has the label (-(1, σρ2)R(γ)φi, β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the bottom (aka beauty) quark particle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the bottom quark particle is a quark of generation III. By the spin rule 4.4, the spin of the bottom quark particle is 1/2. By the electric charge rule 4.5, the electric charge of the bottom quark particle is -1/3. By the strong charge rule 4.6, the bottom quark particle has one of three possible strong charges -σ, -σρ or -σρ2, so Nc = 3. Due to quark confinement, no free quarks can be observed. From indirect experimental observations the effective rest mass of the bottom quark particle is 4500 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The bottom quark particle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed bottom quark particle is transformed into a left-handed bottom quark antiparticle and a left-handed bottom quark particle is transformed into a right-handed bottom quark antiparticle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
bottom (beauty) quark particle
b
Quark
III
1/2
4500
-1/3
3

 
Figure 5.2.11. The bottom (beauty) quark particle
 

5.2.12. The bottom (beauty) quark antiparticle

Let ψbottom-quark be the unique permutation associated with the bottom quark particle according to the mass rule 4.7. Select φi and γ according to the unique expression ψbottom-quark = R(γ)φi and β = σ, σρ or σρ2. Then the particle frame corresponds to the t-Riemann surface with these choices of φi, β, γ. Using the antiparticle rule 4.9, we select disc 22 on the lower sheet of the particle frame and its yellow region 1, that has the label (+(1, σρ2iR(γ), β+γ), according to figures 3.4 and 3.5. This represents the Schrödinger disc of the bottom (aka beauty) quark antiparticle on the particle frame, according to figures 2.1 and 4.1. By the fermion selection rule 4.1, the bottom quark antiparticle is a quark of generation III. By the spin rule 4.4, the spin of the bottom quark antiparticle is 1/2. By the electric charge rule 4.5, the electric charge of the bottom quark antiparticle is +1/3. By the strong charge rule 4.6, the bottom quark antiparticle has one of three possible strong charges +σ, +σρ or +σρ2, so Nc = 3. Due to quark confinement, no free quarks can be observed. From indirect experimental observations the effective rest mass of the bottom quark antiparticle is 4500 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The bottom quark antiparticle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. By the CP transformation rule 4.11, a right-handed bottom quark antiparticle is transformed into a left-handed bottom quark particle and a left-handed bottom quark antiparticle is transformed into a right-handed bottom quark particle.
 
Name
Symbol
Type
Generation
Spin
Mass (MeV)
Charge
Nc
bottom (beauty) quark antiparticle
b
Quark
III
1/2
4500
+1/3
3

 
Figure 5.2.12. The bottom (beauty) quark antiparticle
 

6. Bosons

 
Bosons Particle Name Symbol Associated Force Field Spin Mass (MeV) Charge Nc
Section 6.1.1, 2 photon γ electromagnetic force 1 0 0 1
Section 6.2.1, 2 vector boson Z0 Z0 neutral carrier of the weak force 1 91188 0 1
Section 6.3.1, 2 vector boson W+ W+ positive carrier of the weak force 1 80280 +1 1
Section 6.4.1, 2 vector boson W- W- negative carrier of the weak force 1 80280 -1 1
Section 6.5.1, 2 gluon As strong force 1 0 0 8
Section 6.6.1, 2 graviton g gravitational force 2 0 0 1
Section 6.7.1, 2 scalar boson Higgs H0 Higgs field 0 125874 0 1
 

6.1.1. The photon particle

The photon is the carrier of the electromagnetic force. The unique permutation ψphoton = R(β)φi associated with the massless photon particle must have φi equal to the identity permutation according to the mass rule 4.7. We select β = 1 and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. For the first ray we select discs 1, 2 on the upper sheet of the particle frame and their blue regions 0, that have the labels (+(0, 1)φiR(β), β+γ), (+(0, ρ)φiR(β), β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 7, 8 on the upper sheet of the particle frame and their blue regions 0, that have the labels (-(0, 1)R(γ)φi, β+γ), (-(0, ρ)R(γ)φi, β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs corresponding to the electromagnetic field and the pair of rays represent the photon particle on the particle frame, according to figures 2.1 and 4.1. By the boson selection rule 4.2, the photon particle is a boson. By the spin rule 4.4, the spin of the photon particle is 1. By the electric charge rule 4.5, the electric charge of the photon particle is 0 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. From experimental observations the photon particle is massless and moves with the velocity of light, so by the mass rule 4.7, the Higgs-Kibble mechanism must assign it zero mass. The photon particle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the photon particle and antiparticle 6.1.2 represent the same kind of particle in the standard model. By the CP transformation rule 4.11, a right-handed photon is transformed into a left-handed photon and vice-versa. Thus, CP symmetry is preserved in particle interactions involving the photon, i.e. electromagnetic interactions.
 
Name
Symbol
Associated Force Field
Spin
Mass (MeV)
Charge
Nc
photon particle
γ
electromagnetic force
1
0
0
1

 
Figure 6.1.1. The photon particle
 

6.1.2. The photon antiparticle

The photon antiparticle is equivalent to the photon particle, as shown by the following construction. The unique permutation ψphoton = R(β)φi associated with the massless photon particle must have φi equal to the identity permutation according to the mass rule 4.7. We select β = 1 and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. By the antiparticle rule 4.9, for the first ray we select discs 13, 14 on the lower sheet of the particle frame and their blue regions 0, that have the labels (-(0, 1)R(β)φi, β+γ), (-(0, ρ)R(β)φi, β+γ) respectively, according to figures 3.4 and 3.5. Again by the antiparticle rule 4.9, for the second ray we select discs 19, 20 on the lower sheet of the particle frame and their blue regions 0, that have the labels (+(0, 1)φiR(γ), β+γ), (+(0, ρ)φiR(γ), β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs corresponding to the electromagnetic field and the pair of rays represent the photon antiparticle on the particle frame, according to figures 2.1 and 4.1. By the boson selection rule 4.2, the photon antiparticle is a boson. By the spin rule 4.4, the spin of the photon antiparticle is 1. By the electric charge rule 4.5, the electric charge of the photon antiparticle is 0 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. From experimental observations the photon antiparticle is massless and moves with the velocity of light, so by the mass rule 4.7, the Higgs-Kibble mechanism must assign it zero mass. The photon antiparticle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the photon particle 6.1.1 and antiparticle represent the same kind of particle in the standard model. By the CP transformation rule 4.11, a right-handed photon antiparticle is transformed into a left-handed photon antiparticle and vice-versa. Thus, CP symmetry is preserved in particle interactions involving the photon, i.e. electromagnetic interactions.
 
Name
Symbol
Associated Force Field
Spin
Mass (MeV)
Charge
Nc
photon antiparticle
γ
electromagnetic force
1
0
0
1

 
Figure 6.1.2. The photon antiparticle
 

6.2.1. Vector boson Z0 particle

The vector boson Z0 particle is the neutral carrier of the weak force. The unique permutation ψZ0 = R(β)φi associated with the massive Z0 particle must have β = 1 according to the mass rule 4.7 and we select γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. For the first ray we select discs 3, 4 on the upper sheet of the particle frame and their blue regions 0, that have the labels (+(0, ρ2iR(β), β+γ), (+(0, σρ2iR(β), β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 9, 10 on the upper sheet of the particle frame and their blue regions 0, that have the labels (-(0, ρ2)R(γ)φi, β+γ), (-(0, σρ2)R(γ)φi, β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs corresponding to the neutral component of the weak field and the pair of rays represent the Z0 particle on the particle frame, according to figures 2.1 and 4.1. By the boson selection rule 4.2, the Z0 particle is a boson. By the spin rule 4.4, the spin of the Z0 particle is 1. By the electric charge rule 4.5, the electric charge of the Z0 particle is 0 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. From experimental observations the rest mass of the Z0 particle is 91188 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The Z0 particle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the Z0 particle and antiparticle 6.2.2 represent the same kind of particle in the standard model. By the CP transformation rule 4.11, a right-handed Z0 particle is transformed into a left-handed Z0 particle and vice-versa. Thus, CP symmetry is preserved in particle interactions involving the Z0 particle, i.e. neutral weak interactions.
 
Name
Symbol
Associated Force Field
Spin
Mass (MeV)
Charge
Nc
vector boson Z0 particle
Z0
neutral carrier of the weak force
1
91188
0
1

 
Figure 6.2.1. The vector boson Z0 particle
 

6.2.2. Vector boson Z0 antiparticle

The Z0 antiparticle is equivalent to the Z0 particle, as shown by the following construction. The unique permutation ψZ0 = R(β)φi associated with the massive Z0 particle must have β = 1 according to the mass rule 4.7 and we select γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. By the antiparticle rule 4.9, for the first ray we select discs 15, 16 on the lower sheet of the particle frame and their blue regions 0, that have the labels (-(0, ρ2)R(β)φi, β+γ), (-(0, σρ2)R(β)φi, β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 21, 22 on the lower sheet of the particle frame and their blue regions 0, that have the labels (+(0, ρ2iR(γ), β+γ), (+(0, σρ2iR(γ), β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs corresponding to the neutral component of the weak field and the pair of rays represent the Z0 antiparticle on the particle frame, according to figures 2.1 and 4.1. By the boson selection rule 4.2, the Z0 antiparticle is a boson. By the spin rule 4.4, the spin of the Z0 antiparticle is 1. By the electric charge rule 4.5, the electric charge of the Z0 antiparticle is 0 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. From experimental observations the rest mass of the Z0 antiparticle is 91188 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The Z0 antiparticle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the Z0 particle 6.1.1 and antiparticle represent the same kind of particle in the standard model. By the CP transformation rule 4.11, a right-handed Z0 antiparticle is transformed into a left-handed Z0 antiparticle and vice-versa. Thus, CP symmetry is preserved in particle interactions involving the Z0 particle, i.e. neutral weak interactions.
 
Name
Symbol
Associated Force Field
Spin
Mass (MeV)
Charge
Nc
vector boson Z0 antiparticle
Z0
neutral carrier of the weak force
1
91188
0
1

 
Figure 6.2.2. The vector boson Z0 antiparticle
 

6.3.1. Vector boson W+ particle

The vector boson W+ particle is the positive carrier of the weak force. The unique permutation ψW+ = R(β)φi associated with the massive W+ particle must have β = 1 according to the mass rule 4.7 and we select γ = 1.Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. For the first ray we select discs 2, 3 on the upper sheet of the particle frame and their red regions 3, that have the labels (+(3, ρ)φiR(β), β+γ), (+(3, ρ2iR(β), β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 20, 21 on the lower sheet of the particle frame and their red regions 3, that have the labels (+(3, ρ)φiR(γ), β+γ), (+(3, ρ2iR(γ), β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs corresponding to the positive component of the weak field and the pair of rays represent the W+ particle on the particle frame, according to figures 2.1 and 4.1. By the boson selection rule 4.2, the W+ particle is a boson. By the spin rule 4.4, the spin of the W+ particle is 1. By the electric charge rule 4.5, the electric charge of the W+ particle is +1 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. From experimental observations the rest mass of the W+ particle is 80280 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The W+ particle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the W+ particle and the W- antiparticle 6.4.2 represent the same kind of particle in the standard model. By the CP transformation rule 4.11, a right-handed W+ particle is transformed into a left-handed W- particle and a left-handed W+ particle is transformed into a right-handed W- particle. Since the W+ and W- particles are not equivalent, CP symmetry is violated in particle interactions involving the W+ particle, i.e. positive weak interactions.
 
Name
Symbol
Associated Force Field
Spin
Mass (MeV)
Charge
Nc
vector boson W+ particle
W+
positive carrier of the weak force
1
80280
+1
1

 
 
Figure 6.3.1. The vector boson W+ particle
 

6.3.2. Vector boson W+ antiparticle

The W+ antiparticle is equivalent to the W- particle, as shown by the following construction. The unique permutation ψW+ = R(β)φi associated with the massive W+ particle must have β = 1 according to the mass rule 4.7 and we select γ = 1.Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ.  By the antiparticle rule 4.9, for the first ray we select discs 14, 15 on the lower sheet of the particle frame and their red regions 3, that have the labels (-(3, ρ)R(β)φi, β+γ), (-(3, ρ2)R(β)φi, β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 8, 9 on the upper sheet of the particle frame and their red regions 3, that have the labels (-(3, ρ)R(γ)φi, β+γ), (-(3, ρ2)R(γ)φi, β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs corresponding to the negative component of the weak field and the pair of rays represent the W+ antiparticle on the particle frame, according to figures 2.1 and 4.1. By the boson selection rule 4.2, the W+ antiparticle is a boson. By the spin rule 4.4, the spin of the W+ antiparticle is 1. By the electric charge rule 4.5, the electric charge of the W+ antiparticle is -1 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. From experimental observations the rest mass of the W+ antiparticle is 80280 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The W+ antiparticle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the W+ antiparticle and the W- particle 6.4.1 represent the same kind of particle in the standard model. By the CP transformation rule 4.11, a right-handed W+ antiparticle is transformed into a left-handed W- antiparticle and a left-handed W+ antiparticle is transformed into a right-handed W- antiparticle. Since the W+ and W- antiparticles are not equivalent, CP symmetry is violated in particle interactions involving the W+ antiparticle, i.e. negative weak interactions.
 
Name
Symbol
Associated Force Field
Spin
Mass (MeV)
Charge
Nc
vector boson W+ antiparticle
W+
negative carrier of the weak force
1
80280
-1
1

 
Figure 6.3.2. The vector boson W+ antiparticle
 

6.4.1. Vector boson W- particle

The vector boson W- particle is the negative carrier of the weak force. The unique permutation ψW- = R(β)φi associated with the massive W- particle must have β = 1 according to the mass rule 4.7 and we select γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. For the first ray we select discs 16, 17 on the lower sheet of the particle frame and their red regions 3, that have the labels (-(3, σρ2)R(β)φi, β+γ), (-(3, σρ)R(β)φi, β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 10, 11 on the upper sheet of the particle frame and their red regions 3, that have the labels (-(3, σρ2)R(γ)φi, β+γ), (-(3, σρ)R(γ)φi, β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs corresponding to the negative component of the weak field and the pair of rays represent the W- particle on the particle frame, according to figures 2.1 and 4.1. By the boson selection rule 4.2, the W- particle is a boson. By the spin rule 4.4, the spin of the W- particle is 1. By the electric charge rule 4.5, the electric charge of the W- particle is -1 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. From experimental observations the rest mass of the W- particle is 80280 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The W- particle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the W- particle and the W+ antiparticle 6.3.2 represent the same kind of particle in the standard model. By the CP transformation rule 4.11, a right-handed W- particle is transformed into a left-handed W+ particle and a left-handed W- particle is transformed into a right-handed W+ particle. Since the W- and W+ particles are not equivalent, CP symmetry is violated in particle interactions involving the W- particle, i.e. negative weak interactions
 
Name
Symbol
Associated Force Field
Spin
Mass (MeV)
Charge
Nc
vector boson W- particle
W-
negative carrier of the weak force
1
80280
-1
1

 
Figure 6.4.1. The vector boson W- particle
 

6.4.2. Vector boson W- antiparticle

The W- antiparticle is equivalent to the W+ particle, as shown by the following construction. The unique permutation ψW- = R(β)φi associated with the massive W- particle must have β = 1 according to the mass rule 4.7 and we select γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ.  By the antiparticle rule 4.9, for the first ray we select discs 4, 5 on the upper sheet of the particle frame and their red regions 3, that have the labels (+(3, σρ2iR(β), β+γ), (+(3, σρ)φiR(β), β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 22, 23 on the lower sheet of the particle frame and their red regions 3, that have the labels (+(3, σρ2iR(γ), β+γ), (+(3, σρ)φiR(γ), β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs corresponding to the positive component of the weak field and the pair of rays represent the W- antiparticle on the particle frame, according to figures 2.1 and 4.1. By the boson selection rule 4.2, the W- antiparticle is a boson. By the spin rule 4.4, the spin of the W- antiparticle is 1. By the electric charge rule 4.5, the electric charge of the W- antiparticle is +1 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. From experimental observations the rest mass of the W- antiparticle is 80280 MeV, which can be attributed by the mass rule 4.7 to the Higgs-Kibble mechanism. The W- antiparticle can be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the W- antiparticle and the W+ particle 6.3.1 represent the same kind of particle in the standard model. By the CP transformation rule 4.11, a right-handed W- antiparticle is transformed into a left-handed W+ antiparticle and a left-handed W- antiparticle is transformed into a right-handed W+ antiparticle. Since the W- and W+ antiparticles are not equivalent, CP symmetry is violated in particle interactions involving the W- antiparticle, i.e. positive weak interactions.
 
Name
Symbol
Associated Force Field
Spin
Mass (MeV)
Charge
Nc
vector boson W- antiparticle
W-
positive carrier of the weak force
1
80280
+1
1

 
Figure 6.4.2. The vector boson W- antiparticle
 

6.5.1. The gluon particle

The gluon is the carrier of the strong force. The unique permutation ψgluon = R(β)φi associated with the massless gluon particle must have φi equal to the identity permutation according to the mass rule 4.7. We select β = σ, σρ or σρ2 and γ = σ, σρ or σρ2. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. For the first ray we select discs 5, 6 on the upper sheet of the particle frame and their blue regions 0, that have the labels (+(0, σρ)φiR(β), β+γ), (+(0, σ)φiR(β), β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 11, 12 on the upper sheet of the particle frame and their blue regions 0, that have the labels (-(0, σρ)R(γ)φi, β+γ), (-(0, σ)R(γ)φi, β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs corresponding to the strong field and the pair of rays represent the gluon particle on the particle frame, according to figures 2.1 and 4.1. By the boson selection rule 4.2, the gluon particle is a boson. By the spin rule 4.4, the spin of the gluon particle is 1. By the electric charge rule 4.5, the electric charge of the gluon particle is 0. By the strong charge rule 4.6, the gluon particle carries 8 possible strong charge/anticharge pairs as superpositions of particle frames (there are 8 gluon species), so Nc = 8. Due to quark confinement, no free gluons can be observed. From indirect experimental observations the gluon particle is massless and moves with the velocity of light, so by the mass rule 4.7, the Higgs-Kibble mechanism must assign it effective zero mass. The gluon particle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the gluon particle of a species and the antiparticle 6.5.2 of the corresponding species represent the same kind of particle in the standard model. By the CP transformation rule 4.11, a right-handed gluon is transformed into a left-handed gluon and vice-versa. Thus, CP symmetry is preserved in particle interactions involving the gluon, i.e. strong interactions.
 
Name
Symbol
Associated Force Field
Spin
Mass (MeV)
Charge
Nc
gluon particle
As
strong force
1
0
0
8

 
Figure 6.5.1. The gluon particle
 

6.5.2. The gluon antiparticle

The gluon antiparticle is equivalent to the gluon particle, as shown by the following construction. The unique permutation ψgluon = R(β)φi associated with the massless gluon particle must have φi equal to the identity permutation according to the mass rule 4.7. We select β = σ, σρ or σρ2 and γ = σ, σρ or σρ2. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. By the antiparticle rule 4.9, for the first ray we select discs 17, 18 on the lower sheet of the particle frame and their blue regions 0, that have the labels (-(0, σρ)R(β)φi, β+γ), (-(0, σ)R(β)φi, β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 23, 24 on the lower sheet of the particle frame and their blue regions 0, that have the labels (+(0, σρ)φiR(γ), β+γ), (+(0, σ)φiR(γ), β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs corresponding to the strong field and the pair of rays represent the gluon antiparticle on the particle frame, according to figures 2.1 and 4.1. By the boson selection rule 4.2, the gluon antiparticle is a boson. By the spin rule 4.4, the spin of the gluon antiparticle is 1. By the electric charge rule 4.5, the electric charge of the gluon antiparticle is 0. By the strong charge rule 4.6, the gluon antiparticle carries 8 possible strong anticharge/charge pairs as superpositions of particle frames (there are 8 antigluon species), so Nc = 8. Due to quark confinement, no free antigluons can be observed. From indirect experimental observations the gluon antiparticle is massless and moves with the velocity of light, so by the mass rule 4.7, the Higgs-Kibble mechanism must assign it effective zero mass. The gluon antiparticle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the gluon antiparticle of a species and the particle 6.5.1 of the corresponding species represent the same kind of particle in the standard model. By the CP transformation rule 4.11, a right-handed gluon antiparticle is transformed into a left-handed gluon antiparticle and vice-versa. Thus, CP symmetry is preserved in particle interactions involving the gluon, i.e. strong interactions.
 
Name
Symbol
Associated Force Field
Spin
Mass (MeV)
Charge
Nc
gluon antiparticle
As
strong force
1
0
0
8

 
Figure 6.5.2. The gluon antiparticle
 

6.6.1. The graviton particle

The graviton is the carrier of the gravitational force. The unique permutation ψgraviton = R(β)φi associated with the massless graviton particle must have φi equal to the identity permutation according to the mass rule 4.7. We select β = 1 and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. Let m = 0, 1, 2, or 3, corresponding to the gravitational charge rule 4.6. For the first ray we select discs 1, 12 on the upper sheet of the particle frame and their regions m, that have the labels (+(m, 1)φiR(β), β+γ), (-(m, σ)R(γ)φi, β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 6, 7 on the upper sheet of the particle frame and their regions m, that have the labels (+(m, σ)φiR(β), β+γ), (-(m, 1)R(γ)φi, β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs corresponding to the gravitational field and the pair of rays represent the graviton particle on the particle frame, according to figures 2.1 and 4.1. By the boson selection rule 4.2, the graviton particle is a boson. By the spin rule 4.4, the spin of the graviton particle is 2. By the electric charge rule 4.5, the electric charge of the graviton particle is 0 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. By the gravitational charge rule 4.6, the graviton particle carries 24 possible gravitational charge/anticharge pairs as superpositions of particle frames (there are 24 graviton species). The graviton particle has not been observed yet, but it is an inevitable consequence of the gravitational force and quantum mechanics [8]. It is believed [8] that the graviton particle is massless and moves with the velocity of light, so by the mass rule 4.7, the Higgs-Kibble mechanism must assign it zero mass. The graviton particle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the graviton particle and antiparticle 6.6.2 represent the same kind of particle in the standard model. By the CP transformation rule 4.11, a right-handed graviton is transformed into a left-handed graviton and vice-versa. Thus, CP symmetry is preserved in particle interactions involving the graviton, i.e. gravitational interactions.
 
Name
Symbol
Associated Force Field
Spin
Mass (MeV)
Charge
Nc
graviton particle
g
gravitational force
2
0
0
1

 
Figure 6.6.1. The graviton particle
 

6.6.2. The graviton antiparticle

The graviton antiparticle is equivalent to the graviton particle, as shown by the following construction. The unique permutation ψgraviton = R(β)φi associated with the massless graviton particle must have φi equal to the identity permutation according to the mass rule 4.7. We select β = 1 and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. Let m = 0, 1, 2, or 3, corresponding to the gravitational charge rule 4.6. By the antiparticle rule 4.9, for the first ray we select discs 13, 24 on the lower sheet of the particle frame and their regions m, that have the labels (-(m, 1)R(β)φi, β+γ), (+(m, σ)φiR(γ), β+γ) respectively, according to figures 3.4 and 3.5. Again by the antiparticle rule 4.9, for the second ray we select discs 18, 19 on the lower sheet of the particle frame and their regions m, that have the labels (-(m, σ)R(β)φi, β+γ), (+(m, 1)φiR(γ), β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs corresponding to the gravitational field and the pair of rays represent the graviton antiparticle on the particle frame, according to figures 2.1 and 4.1. By the boson selection rule 4.2, the graviton antiparticle is a boson. By the spin rule 4.4, the spin of the graviton antiparticle is 2. By the electric charge rule 4.5, the electric charge of the graviton antiparticle is 0 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. By the gravitational charge rule 4.6, the graviton antiparticle carries 24 possible gravitational charge/anticharge pairs as superpositions of particle frames (there are 24 antigraviton species).The graviton antiparticle has not been observed yet, but it is an inevitable consequence of the gravitational force and quantum mechanics [8]. It is believed [8] that the graviton antiparticle is massless and moves with the velocity of light, so by the mass rule 4.7, the Higgs-Kibble mechanism must assign it zero mass. The graviton antiparticle can theoretically be observed with both right-handed and left-handed helicities, in agreement with the helicity rule 4.10. Note that by the equivalence rule 4.8, the graviton particle 6.6.1 and antiparticle represent the same kind of particle in the standard model. By the CP transformation rule 4.11, a right-handed graviton antiparticle is transformed into a left-handed graviton antiparticle and vice-versa. Thus, CP symmetry is preserved in particle interactions involving the graviton, i.e. gravitational interactions.
 
Name
Symbol
Associated Force Field
Spin
Mass (MeV)
Charge
Nc
graviton antiparticle
g
gravitational force
2
0
0
1

 
Figure 6.6.2. The graviton antiparticle
 

6.7.1. The Higgs particle

The Higgs particle attributes mass to all the particles in the standard model, including itself. The unique permutation ψHiggs = R(β)φi associated with the massive Higgs particle must have β = 1 according to the mass rule 4.7. Furthermore, by the mass rule 4.7, the permutation φj has the energy equal to half of the sum of the masses of all other bosons. Since the Higgs particle and antiparticle will be identified (as a Cooper pair), their combined energy would then be the sum of the masses of all other bosons. This must be the lowest energy state possible for the Higgs boson when it undergoes Bose condensation. We select γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. By the Higgs selection rule 4.3, the Higgs particle is given as the intersection of all the 24 discs of the t-Riemann surface. We may regard the Higgs particle as the intersection of the discs 1 ,..., 12 of the upper sheet (the origin of the upper sheet) and the Higgs antiparticle as the intersection of the discs 13, ..., 24 of the lower sheet (the origin of the lower sheet). However, the particle and antiparticle 6.7.2 are identified as the branch point (0, β+γ) of the t-Riemann surface. These 24 discs together represent the Schrödinger discs corresponding to the Higgs field and the branch point represents the Higgs particle on the particle frame, according to figures 2.1 and 4.1. By the Higgs selection rule 4.2, the Higgs particle is a scalar boson. By the spin rule 4.4, the spin of the Higgs particle is 0. By the electric charge rule 4.5, the electric charge of the Higgs particle is 0 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. The Higgs particle has not been observed yet, but it is an inevitable consequence of the Higgs-Kibble mechanism [8]. By the mass rule 4.7, we can predict the mass of the Higgs particle (in MeV) as

(91188 + 80280 + 80280)/2 = 125874


Name
Symbol
Associated Force Field
Spin
Mass (MeV)
Charge
Nc
Higgs particle
H0
Higgs field
0
125874
0
1

 
Figure 6.7.1. The Higgs particle
 

6.7.2. The Higgs antiparticle

The Higgs antiparticle is equivalent to the Higgs particle (they are actually identified on the particle frame), as shown by the following construction. The Higgs particle attributes mass to all the particles in the standard model, including itself. The unique permutation ψHiggs = R(β)φi associated with the massive Higgs particle must have β = 1 according to the mass rule 4.7. Furthermore, by the mass rule 4.7, the permutation φj has the energy equal to half of the sum of the masses of all other bosons. Since the Higgs particle and antiparticle will be identified (as a Cooper pair), their combined energy would then be the sum of the masses of all other bosons. This must be the lowest energy state possible for the Higgs boson when it undergoes Bose condensation. We select γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. By the Higgs selection rule 4.3 and the antiparticle rule 4.9, the Higgs antiparticle is given as the intersection of all the 24 discs of the t-Riemann surface. We may regard the Higgs particle as the intersection of the discs 1 ,..., 12 of the upper sheet (the origin of the upper sheet) and the Higgs antiparticle as the intersection of the discs 13, ..., 24 of the lower sheet (the origin of the lower sheet). However, the particle 6.7.1 and antiparticle are identified as the branch point (0, β+γ) of the t-Riemann surface. These 24 discs together represent the Schrödinger discs corresponding to the Higgs field and the branch point represents the Higgs antiparticle on the particle frame, according to figures 2.1 and 4.1. By the Higgs selection rule 4.2, the Higgs antiparticle is a scalar boson. By the spin rule 4.4, the spin of the Higgs antiparticle is 0. By the electric charge rule 4.5, the electric charge of the Higgs antiparticle is 0 and by the strong charge rule 4.6, its strong charge is neutral with Nc = 1. The Higgs antiparticle has not been observed yet, but it is an inevitable consequence of the Higgs-Kibble mechanism [8]. By the mass rule 4.7, we can predict the mass of the Higgs antiparticle (in MeV) as

(91188 + 80280 + 80280)/2 = 125874


Name
Symbol
Associated Force Field
Spin
Mass (MeV)
Charge
Nc
Higgs antiparticle
H0
Higgs field
0
125874
0
1

 
Figure 6.7.2. The Higgs antiparticle
 

7. Force Fields

 
Section 7.1 The Electromagnetic Force Field
Section 7.2 The Weak Force Field
Section 7.3 The Strong Force Field
Section 7.4 The Gravitational Force Field
 

7.1. The Electromagnetic Force Field

In Section 7.1.1, we review Maxwell's Electromagnetic Field Equations. In Section 7.1.2, we show how the photon acts as the carrier of the electromagnetic force. In Section 7.1.3, we give an example of a typical electromagnetic interaction: electron-electron scattering. In Section 7.1.4, we define the electromagnetic gauge group
 
Ge = U(1)

We explicitly define the observable gauge photon and show how the electromagnetic gauge group acts on it by means of the electromagnetic gauge transformations. 

 

7.1.1. Maxwell's Electromagnetic Field Equations

Maxwell's equations for electromagnetism remain unchanged if the space-time coordinates are subjected to Lorentz transformations given by equation 1.3. To demonstrate Lorentz invariance, we must put Maxwell's equations into the four-dimensional form required by special relativity. We use Einstein's notation for tensors [5], with superscripts for components of contravariant four-vectors, subscripts for components of covariant four-vectors and subscript commas for partial derivatives. We choose units of distance and time such that the velocity of light c = 1. Maxwell's equations are usually written as:  
E = (-1/c)(∂A/∂T) - grad Φ
(7.1.1.1)
 
H = curl A
(7.1.1.2)
 
(1/c)(∂H/∂T) = - curl E
(7.1.1.3)
 
div H = 0
(7.1.1.4)
 
(1/c)(∂E/∂T) = curl H - 4πJ
(7.1.1.5)
 
div E = 4πρ
(7.1.1.6)
where E = (E 1, E 2, E 3) denotes the electric field, H = (H 1, H 2, H 3) denotes the magnetic field, Φ denotes the electric potential, A = (A 1, A 2, A 3) denotes the magnetic potential, ρ = J 0 denotes the charge density, J = (J 1, J 2, J 3) denotes the current density and c denotes the velocity of light. Put T = X 0 for the time coordinate, X = X 1, Y = X 2, Z = X 3 for the space coordinates, Φ = K 0 for the electric potential and A 1 = K 1, A 2 = K 2, A 3 = K 3for the magnetic potential. Then the contravariant four-vector (X 0, X 1, X 2, X 3) represents space-time, the contravariant four-vector (K 0, K 1, K 2, K 3) represents the electromagnetic potential and the contravariant four-vector (J 0, J 1, J 2, J 3) represents the charge-current density. Define the contravariant and covariant four-vectors  
(F 0, F 1, F 2, F 3) and (F 0, F 1, F 2, F 3)
(7.1.1.7)
respectively, where F 0 = F 0, F 1 = - F 1, F 2 = - F 2, F 3 = - F 3, by means of the tensor equations  
Fμν = FμFν = Kμ, ν - Kν, μ = (∂Kμ / ∂X ν) - (∂Kν / ∂X μ)
(7.1.1.8)
By Maxwell's first equation 7.1.1.1,  
E 1 = (-1/c)(∂A 1/∂T) - (∂Φ/∂X) = (-1/c)(∂K 1/∂X 0) - (∂K 0/∂X 1)
= (1/c)(∂K 1/∂X 0) - (∂K 0/∂X1) = F10 = F1F0 = - F 1F 0 = - F 10
 
E 2 = (-1/c)(∂A 2/∂T) - (∂Φ/∂Y) = (-1/c)(∂K 2/∂X 0) - (∂K 0/∂X 2)
= (1/c)(∂K 2/∂X 0) - (∂K 0/∂X 2) = F20 = F2F0 = - F 2F 0 = - F 20
 
E 3 = (-1/c)(∂A 3/∂T) - (∂Φ/∂Z) = (-1/c)(∂K 3/∂X 0) - (∂K 0/∂X 3)
= (1/c)(∂K 3/∂X 0) - (∂K 0/∂X 3) = F30 = F3F0 = - F 3F 0 = - F 30
Thus, the three components of the electric field are given by  
E 1 = F10 = - F 10
(7.1.1.9)
 
E 2 = F20 = - F 20
(7.1.1.10)
 
E 3 = F30 = - F 30
(7.1.1.11)
By Maxwell's second equation 7.1.1.2,  
H 1 = (∂A 3/∂Y) - (∂A 2/∂Z) = (∂K 3/∂X 2) - (∂K 2/∂X 3)
= - (∂K 3/∂X 2) + (∂K 2/∂X 3) = F23 = F2F3 = (- F 2)(- F 3) = F 2F 3  = F 23
 
H 2 = (∂A 1/∂Z) - (∂A 3/∂X) = (∂K 1/∂X 3) - (∂K 3/∂X 1)
= - (∂K 1/∂X 3) + (∂K 3/∂X 1) = F31 = F3F1 = (- F 3)(- F 1) = F 3F 1  = F 31
 
H 3 = (∂A 2/∂X) - (∂A 1/∂Y) = (∂K 2/∂X 1) - (∂K 1/∂X2)
= - (∂K 2/∂X 1) + (∂K 1/∂X 2) = F12 = F1F2 = (- F 1)(- F 2) = F 1F 2  = F 12
Thus, the three components of the magnetic field are given by  
H 1 = F23 = F 23
(7.1.1.12)
 
H 2 = F31 = F 31
(7.1.1.13)
 
H 3 = F12 = F 12
(7.1.1.14)
By definition,  
Fμν, τ = (Kμ, ν - Kν, μ), τ = ((∂Kμ / ∂X ν) - (∂Kν / ∂X μ)), τ
= ∂((∂Kμ / ∂X ν) - (∂Kν / ∂X μ)) / ∂X τ = (∂2Kμ / ∂X νX τ) - (∂2Kν / ∂X μX τ)
Hence,  
Fμν, τ = (∂2Kμ / ∂X νX τ) - (∂2Kν / ∂X μX τ)
Fντ, μ = (∂2Kν / ∂X τX μ) - (∂2Kτ / ∂X νX μ)
Fτμ, ν = (∂2Kτ/ ∂X μX ν) - (∂2Kμ / ∂X τX ν)
Adding the three terms on the left-hand sides of the three equations, the terms on the right-hand sides cancel in pairs and we obtain  
Fμν, τ + Fντ, μ + Fτμ, ν = 0
(7.1.1.15)
From equation 7.1.1.15 we can derive Maxwell's third equation 7.1.1.3:  
(1/c)(∂H 1/∂T) = F23, 0 = - F02, 3 - F30, 2 = F20, 3 - F30, 2 = - ((∂E 3/∂Y) - (∂E 2/∂Z))
(1/c)(∂H 2/∂T) = F31, 0 = - F03, 1 - F10, 3 = F30, 1 - F10, 3 = - ((∂E 1/∂Z) - (∂E 3/∂X))
(1/c)(∂H 3/∂T) = F12, 0 = - F01, 2 - F20, 1 = F10, 2 - F20, 1 = - ((∂E 2/∂X) - (∂E 1/∂Y))
(7.1.1.16)
From equation 7.1.1.15 we can also derive Maxwell's fourth equation 7.1.1.4:  
(∂H 1/∂X) + (∂H 2/∂Y) + (∂H 3/∂Z) = F23, 1 + F31, 2 + F12, 3 = 0
(7.1.1.17)
By Maxwell's fifth equation 7.1.1.5,  
(1/c)(∂E 1/∂T) = F10, 0 = F12, 2 - F31, 3 - 4πJ 1 = (∂H 3/∂Y) - (∂H 2/∂Z) - 4πJ 1
(1/c)(∂E 2/∂T) = F20, 0 = F23, 3 - F12, 1 - 4πJ 2 = (∂H 1/∂Z) - (∂H 3/∂X) - 4πJ 2
(1/c)(∂E 3/∂T) = F30, 0 = F31, 1 - F23, 2 - 4πJ 3 = (∂H 2/∂X) - (∂H 1/∂Y) - 4πJ 3
(7.1.1.18)
Finally, by Maxwell's sixth equation 7.1.1.6,  
(∂E 1/∂X) + (∂E 2/∂Y) + (∂E 3/∂Z) = F10, 1 + F20, 2 + F30, 3 = J 0 = 4πρ
(7.1.1.19)
This demonstrates the Lorentz invariance of Maxwell's equations. We can express the correspondence between the antisymmetric tensor Fμν and the components of the electromagnetic field as follows:
  
 F 00
 F 01
 F 02
 F 03
F 10
F 11
F 12
F 13
F 20
F 21
F 22
F 23
F 30
F 31
F 32
F 33
 → 
0
- E 1
- E 2
- E 3
  E 1
0
H 3
- H 2
  E 2
- H 3
0
H 1
  E 3
H 2
- H 1
0
(7.1.1.20)
 

7.1.2. The photon as the carrier of the electromagnetic force

We now show how the photon 6.1.1 can be regarded as the carrier of the electromagnetic force. The unique permutation ψphoton = R(β)φi associated with the massless photon particle must have φi equal to the identity permutation according to the mass rule 4.7. We select β = 1 and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. For the first ray we select discs 1, 2 on the upper sheet of the particle frame and their blue regions 0, that have the labels (+(0, 1)φiR(β), β+γ), (+(0, ρ)φiR(β), β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 7, 8 on the upper sheet of the particle frame and their blue regions 0, that have the labels (-(0, 1)R(γ)φi, β+γ), (-(0, ρ)R(γ)φi, β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs of the photon 6.1.1 and the electromagnetic field via the following correspondence:
  
F 0  →  (+(0, 1)φiR(β), β+γ)
F 1  →  (+(0, ρ)φiR(β), β+γ)
F 2  →  (-(0, 1)R(γ)φi, β+γ)
F 3  →  (-(0, ρ)R(γ)φi, β+γ)
(7.1.2)

 
Figure 7.1.2. The photon and the electromagnetic field
 

7.1.3. Electromagnetic Interactions

A typical electromagnetic interaction is shown in the following Feynman diagram for like-charge repulsion: a pair of electrons experience a repulsive force and a virtual photon (the carrier of the electromagnetic force) is exchanged.
 
Figure 7.1.3.1. Feynman diagram for a typical electromagnetic interaction: "like charges repel"

We can visualize this electromagnetic interaction as shown below in figure 7.1.3.2: (a) the two electrons approach each other; (b) two of the Schrödinger discs of the electromagnetic field destroy the two old electrons (the corresponding wave functions are now interpreted as destruction operators); (c) the other two Schrödinger discs of the electromagnetic field create two new electrons (the corresponding wave functions are now interpreted as creation operators); (d) together, the four Schrödinger discs of the electromagnetic field constitute the virtual photon that is exchanged while transmitting the electromagnetic force; (e) the two electrons are repelled.
 
 

Figure 7.1.3.2. A typical electromagnetic interaction: "like charges repel"
 

7.1.4. The Electromagnetic Gauge Group

The electromagnetic field described by the tensor Fμν is a special case of a Yang-Mills gauge field, corresponding to the photon. The electromagnetic Yang-Mills gauge field is specified by its gauge group Ge and a constant of interaction αe, called its coupling constant. The coupling constant will be calculated explicitly in 8.2. We shall now construct the gauge group Ge for the electromagnetic field. Consider the unitary group U(1) consisting of 1×1 complex unitary matrices under matrix multiplication. The unitary group U(1) is generated (as a Lie group) by a single 1×1 matrix, whose only entry can be assumed to be the real number 1 (for the unitary group with n parameters, any set of n2 linearly independent n×n Hermitian matrices is a set of generators). We first define two copies of U(1), called U(1)e and U(1)w, as follows:
  • U(1)e = U(1) is the gauge group for the electromagnetic force after it separated from the weak force in the cosmological timeline. We label the row and the column of the generator of U(1)e as follows:
Γ1
+ 1 
- 1
1
Figure 7.1.4.1. U(1)e generator
  • U(1)w = U(1) is the gauge group for the electromagnetic force before it separated from the weak force in the cosmological timeline. We label the row and the column of the generator of U(1)w as follows:
Γ1
+ ρ 
- ρ
1
Figure 7.1.4.2. U(1)w generator
  • Let U(1)e×U(1)w = {(a, b) | aU(1)e , bU(1)w} denote the group direct product and define the electromagnetic gauge group
 
Ge = (U(1)e×U(1)w)diagonal = {(a, a) | aU(1)e , aU(1)w} = U(1)
(7.1.4.1)

 
Γ1
+ 1
+ ρ
- 1
- ρ
1
Figure 7.1.4.3. The electromagnetic gauge group Ge= U(1) and its generator

The row and column labels
 

± 1
± ρ

in figure 7.1.4.3 specify the Schrödinger discs of the photon and the electromagnetic force as defined by equations 7.1.2 and figure 7.1.2. The first component of the row and column labels consists of electromagnetic charge; the second component specifies how the electromagnetic gauge group will embed in the weak gauge group 7.2.9. The observable photon is defined as Γ1. The observable photon is regarded as a superposition of the electromagnetic charge - electromagnetic anticharge labels of its row and column multiplied by a complex number (just 1, in this case) viewed on the z-plane. Since t = z2, a rotation by an angle θ around the origin of the z-plane corresponds to a rotation by an angle 2θ around the branch point of the t-Riemann surface. In this case, multiplication by 1 on the z-plane corresponds to a trivial rotation of the t-Riemann surface by 0 degrees around the branch point. This means that for the observable photon, the rays defining the photon (and its equivalent antiparticle) are permuted trivially amongst themselves (and not any other rays) on the particle frame. Thus, the observable photon corresponds to (trivial) superpositions of particle frames for the photon.
 

Figure 7.1.4.4. The observable photon

The electromagnetic gauge group acts (trivially) on the observable photon by conjugation, viewed on the z-plane:
 

Γ1-1 Γ1Γ1 = 1

Again, since t = z2, a rotation by an angle θ around the origin of the z-plane corresponds to a rotation by an angle 2θ around the branch point of the t-Riemann surface and multiplication by 1 on the z-plane corresponds to a trivial rotation of the t-Riemann surface by 0 degrees around the branch point. This means that for any electromagnetic gauge transformation, the rays defining the photon (and its equivalent antiparticle) are permuted amongst themselves (and not any other rays) on the particle frame. The electromagnetic gauge transformations will always transform superpositions of photons to other superpositions of photons.

 

7.2. The Weak Force Field

In Section 7.2.1, we define the Yang-Mills Weak Field Equations. In Section 7.2.2, we show how the Z0 acts as the neutral carrier of the weak force. In Section 7.2.3, we give an example of a typical weak Z0 interaction: muonic neutrino-electron scattering. In Section 7.2.4, we show how the W+ acts as the positive carrier of the weak force. In Section 7.2.5, we give an example of a typical weak W+ interaction: transformation of a down quark into an up quark, responsible for radioactivity. In Section 7.2.6, we show how the W- acts as the negative carrier of the weak force. In Section 7.2.7, we give an example of a typical weak W- interaction: again, transformation of a down quark into an up quark, responsible for radioactivity. The W+ and W- are antiparticles of each other. In Section 7.2.8, we define the weak gauge group
 
Gw = SU(2)

We explicitly define the observable gauge vector bosons [Z0], [W+], [W-] and show how the weak gauge group acts on them by means of the weak gauge transformations. In Section 7.2.9, we show that the Weinberg angle θw = 30 degrees (this is a running value). The Weinberg angle θw is a parameter that gives a relationship between the W+, W- and Z0 masses, as well as the ratio of the weak Z0 mediated interaction, called its mixing. 

 

7.2.1. Yang-Mills Weak Field Equations

Corresponding to the three vector bosons Z0, W+, W- we have three electromagnetic type fields defined by the tensors Fμν(0), Fμν(+), Fμν(-) respectively. Thus, we have three covariant 4-vectors  
(F 0(w), F 1(w), F 2(w), F 3(w)) where w = 0, +, -
(7.2.1.1)
We also have three electric and three magnetic fields corresponding to the antisymmetric tensors Fμν(0), Fμν(+), Fμν(-) given by the following correspondence:
  
 F 00(w)
 F 01(w)
 F 02(w)
 F 03(w)
F 10(w)
F 11(w)
F 12(w)
F 13(w)
F 20(w)
F 21(w)
F 22(w)
F 23(w)
F 30(w)
F 31(w)
F 32(w)
F 33(w)
 → 
0
- E 1(w)
- E 2(w)
- E 3(w)
  E 1(w)
0
H 3(w)
- H 2(w)
  E 2(w)
- H 3(w)
0
H 1(w)
  E 3(w)
H 2(w)
- H 1(w)
0
where w = 0, +, -
(7.2.1.2)
 

7.2.2. The vector boson Z0 as the neutral carrier of the weak force

We now show how the vector boson Z0 particle 6.2.1 can be regarded as the neutral carrier of the weak force. The unique permutation ψZ0 = R(β)φi associated with the massive Z0 particle must have β = 1 according to the mass rule 4.7 and we select γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. For the first ray we select discs 3, 4 on the upper sheet of the particle frame and their blue regions 0, that have the labels (+(0, ρ2iR(β), β+γ), (+(0, σρ2iR(β), β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 9, 10 on the upper sheet of the particle frame and their blue regions 0, that have the labels (-(0, ρ2)R(γ)φi, β+γ), (-(0, σρ2)R(γ)φi, β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs of the vector boson Z0 particle and the neutral component of the weak field via the following correspondence:  
F 0(0)  →  (+(0, ρ2iR(β), β+γ)
F 1(0)  →  (+(0, σρ2iR(β), β+γ)
F 2(0)  →  (-(0, ρ2)R(γ)φi, β+γ)
F 3(0)  →  (-(0, σρ2)R(γ)φi, β+γ)
(7.2.2)
Figure 7.2.2. The Z0 and the neutral component of the weak field
 

7.2.3. Weak Z0 Interactions

A typical weak Z0 interaction is shown in the following Feynman diagram for a muonic neutrino - electron collision: a μ-neutrino and an electron collide (elastically) and a virtual Z0 (the neutral carrier of the weak force) is exchanged.
 
Figure 7.2.3.1. Feynman diagram for a typical weak Z0 interaction: a muonic neutrino - electron collision

We can visualize this weak Z0 interaction as shown below in figure 7.2.3.2: (a) a μ-neutrino and an electron approach each other; (b) two of the Schrödinger discs of the weak Z0 field destroy the old μ-neutrino and the old electron (the corresponding wave functions are now interpreted as destruction operators); (c) the other two Schrödinger discs of the weak Z0 field create a new μ-neutrino and a new electron (the corresponding wave functions are now interpreted as creation operators); (d) together, the four Schrödinger discs of the weak Z0 field constitute the virtual Z0that is exchanged while transmitting the neutral weak force; (e) the μ-neutrino and electron have collided (elastically).
 
 

Figure 7.2.3.2. A typical weak Z0 interaction: a muonic neutrino - electron collision
 

7.2.4. The vector boson W+ as the positive carrier of the weak force

We now show how the vector boson W+ particle 6.3.1 can be regarded as the positive carrier of the weak force. The unique permutation ψW+ = R(β)φi associated with the massive W+ particle must have β = 1 according to the mass rule 4.7 and we select γ = 1.Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. For the first ray we select discs 2, 3 on the upper sheet of the particle frame and their red regions 3, that have the labels (+(3, ρ)φiR(β), β+γ), (+(3, ρ2iR(β), β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 20, 21 on the lower sheet of the particle frame and their red regions 3, that have the labels (+(3, ρ)φiR(γ), β+γ), (+(3, ρ2iR(γ), β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs of the vector boson W+ particle and the positive component of the weak field via the following correspondence:
  
F 0(+)  →  (+(3, ρ)φiR(β), β+γ)
F 1(+)  →  (+(3, ρ2iR(β), β+γ)
F 2(+)  →  (+(3, ρ)φiR(γ), β+γ)
F 3(+)  →  (+(3, ρ2iR(γ), β+γ)
(7.2.4)
Figure 7.2.4. The W+ and the positive component of the weak field
 

7.2.5. Weak W+ Interactions

When a neutron interacts with a neutrino, a W+ can be exchanged, transforming the neutron into a proton and producing an electron. A down quark in the neutron changes into an up quark due to an intermediate interaction with a virtual W+, transforming the neutron into a proton. Although quarks are not directly observed due to confinement, we may still represent this interaction by the following Feynman diagram.
 
Figure 7.2.5.1. Feynman diagram for a typical weak W+ interaction: transformation of a down quark into an up quark

We can visualize this weak W+ interaction as shown below in figure 7.2.5.2: (a) a down quark and a neutrino approach each other; (b) two of the Schrödinger discs of the weak W+ field destroy the old down quark and the old neutrino (the corresponding wave functions are now interpreted as destruction operators); (c) the other two Schrödinger discs of the weak W+ field create a new up quark and a new electron (the corresponding wave functions are now interpreted as creation operators); (d) together, the four Schrödinger discs of the weak W+ field constitute the virtual W+ that is exchanged while transmitting the positive weak force; (e) an up quark and an electron have been created.
 
 

Figure 7.2.5.2. A typical weak W+ interaction: transformation of a down quark into an up quark
 

7.2.6. The vector boson W- as the negative carrier of the weak force

We now show how the vector boson W- particle 6.4.1 can be regarded as the negative carrier of the weak force. The unique permutation ψW- = R(β)φi associated with the massive W- particle must have β = 1 according to the mass rule 4.7 and we select γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. For the first ray we select discs 16, 17 on the lower sheet of the particle frame and their red regions 3, that have the labels (-(3, σρ2)R(β)φi, β+γ), (-(3, σρ)R(β)φi, β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 10, 11 on the upper sheet of the particle frame and their red regions 3, that have the labels (-(3, σρ2)R(γ)φi, β+γ), (-(3, σρ)R(γ)φi, β+γ) respectively, according to figures 3.4 and 3.5. These together represent the Schrödinger discs of the vector boson W- particle and the negative component of the weak field via the following correspondence:
  
F 0(-)  →  (-(3, σρ2)R(β)φi, β+γ)
F 1(-)  →  (-(3, σρ)R(β)φi, β+γ)
F 2(-)  →  (-(3, σρ2)R(γ)φi, β+γ)
F 3(-)  →  (-(3, σρ)R(γ)φi, β+γ)
(7.2.6)
Figure 7.2.6. The W- and the negative component of the weak field
 

7.2.7. Weak W- Interactions

When a neutron interacts with a neutrino, a W- can be exchanged, transforming the neutron into a proton and producing an electron. A down quark in the neutron changes into an up quark due to an intermediate interaction with a virtual W-, transforming the neutron into a proton. Although quarks are not directly observed due to confinement, we may still represent this interaction by the following Feynman diagram.
 
Figure 7.2.7.1. Feynman diagram for a typical weak W- interaction: transformation of a down quark into an up quark

We can visualize this weak W- interaction as shown below in figure 7.2.7.2: (a) a down quark and a neutrino approach each other; (b) two of the Schrödinger discs of the weak W- field destroy the old down quark and the old neutrino (the corresponding wave functions are now interpreted as destruction operators); (c) the other two Schrödinger discs of the weak W- field create a new up quark and a new electron (the corresponding wave functions are now interpreted as creation operators); (d) together, the four Schrödinger discs of the weak W- field constitute the virtual W- that is exchanged while transmitting the negative weak force; (e) an up quark and an electron have been created.
 
 

Figure 7.2.7.2. A typical weak W- interaction: transformation of a down quark into an up quark
 

7.2.8. The Weak Gauge Group

The weak Yang-Mills field 7.2.1 consists of three electromagnetic type fields defined by the tensors Fμν(0), Fμν(+), Fμν(-) corresponding to the three vector bosons Z0, W+, W- respectively. The weak Yang-Mills gauge field is specified by its gauge group Gw and a constant of interaction αw, called its coupling constant. The coupling constant will be calculated explicitly in 8.2. We shall now construct the gauge group Gw for the weak field. Consider the special unitary group SU(2) consisting of 2×2 complex unitary matrices of determinant 1, under matrix multiplication. The special unitary group SU(2) is generated (as a Lie group) by the three 2×2 unitary matrices of determinant 1 (called Pauli generators). We first define two copies of SU(2), called SU(2)w and SU(2)s, as follows:
  • SU(2)w = SU(2) is the gauge group for the weak force after it separated from the strong force in the cosmological timeline. We label the row and the column of each generator of SU(2)w as follows:
Γ1
+ ρ 
+ ρ2
- ρ 
0
1
- ρ2
1
0
Γ2
+ ρ
+ ρ2
- ρ 
0
- i
- ρ2
i
0
Γ3
+ ρ 
+ ρ2
- ρ 
1
0
- ρ2
0
- 1
Figure 7.2.8.1. SU(2)w generators
  • SU(2)s = SU(2) is the gauge group for the weak force before it separated from the strong force in the cosmological timeline. We label the row and the column of each generator of SU(2)s as follows:
Γ1
+ σρ 
+ σρ2
- σρ 
0
1
- σρ2
1
0
Γ2
+ σρ 
+ σρ2
- σρ 
0
- i
- σρ2
i
0
Γ3
+ σρ 
+ σρ2
- σρ 
1
0
- σρ2
0
- 1
Figure 7.2.8.2. SU(2)s generators
  • Let SU(2)w×SU(2)s = {(a, b) | aSU(2)w , bSU(2)s} denote the group direct product and define the weak gauge group
 
Gw = (SU(2)w×SU(2)s)diagonal = {(a, a) | aSU(2)w , aSU(2)s} = SU(2)
(7.2.8.1)
Γ1
+ ρ
+ σρ
+ ρ2
+ σρ2
- ρ
- σρ
0
1
- ρ2
- σρ2
1
0
Γ2
+ ρ
+ σρ
+ ρ2
+ σρ2
- ρ
- σρ
0
- i
- ρ2
- σρ2
i
0
Γ3
+ ρ
+ σρ
+ ρ2
+ σρ2
- ρ
- σρ
1
0
- ρ2
- σρ2
0
- 1
Figure 7.2.8.3. The weak gauge group Gw = SU(2) and its generators

The row and column labels
 

± ρ
± σρ
,
± ρ2
± σρ2

in figure 7.2.8.3 specify the Schrödinger discs of the three vector bosons Z0, W+, W- and the weak force as defined by equations 7.2.2, 7.2.4 and 7.2.6, respectively. The first component of the row and column labels in figure 7.2.8.3 consists of weak charge; the second component specifies how the weak gauge group will embed in the strong gauge group 7.3.4. The three observable vector bosons are defined as
 

[Z0] = Γ1
[W+] = Γ2
[W-] = Γ3

Each observable vector boson [Z0], [W+] or [W-] is regarded as a superposition of the weak charge - weak anticharge labels of its row and column multiplied by a complex number viewed on the z-plane. Since t = z2, a rotation by an angle θ around the origin of the z-plane corresponds to a rotation by an angle 2θ around the branch point of the t-Riemann surface. In particular, notice that multiplication by ±i, ±1 on the z-plane correspond to rotations of the t-Riemann surface by ±180, ±360 degrees around the branch point, respectively. This means that for the three observable vector bosons [Z0], [W+], [W-], the rays defining the three vector bosons Z0, W+, W- are permuted amongst themselves (and not any other rays) on the particle frame. Thus, the three observable vector bosons [Z0], [W+], [W-] correspond to superpositions of particle frames for the three vector bosons Z0, W+, W-.
 

Figure 7.2.8.4. The observable vector bosons Z0, W+, W-

The weak gauge group acts on the three observable vector bosons [Z0], [W+], [W-] by conjugation, viewed on the z-plane:
 

Γi-1 ΓjΓi for i, j = 1, 2, 3

Again, since t = z2, a rotation by an angle θ around the origin of the z-plane corresponds to a rotation by an angle 2θ around the branch point of the t-Riemann surface and multiplication by ±i, ±1 on the z-plane correspond to rotations of the t-Riemann surface by ±180, ±360 degrees around the branch point, respectively. This means that for any weak gauge transformation, the rays defining the three vector bosons Z0, W+, W- are permuted amongst themselves (and not any other rays) on the particle frame. The weak gauge transformations will always transform superpositions of the three vector bosons Z0, W+, W- to other superpositions of the three vector bosons Z0, W+, W-.

 

7.2.9. The Weinberg Angle

The Weinberg angle θw is a parameter that gives a relationship between the W+, W- and Z0 masses, as well as the ratio of the weak Z0 mediated interaction, called its mixing. Indeed, from figures 7.2.2, 7.2.4, 7.2.6 and 7.2.8.4, it is apparent that the components of the weak Z0 field mix with the components of the weak W+, W- fields and the angle subtended by the mixing Schrödinger discs on the particle frame is exactly π/6 radians or 30 degrees. Hence, we predict that θw = 30 degrees. This is in good agreement with the SLAC experiment [12] which estimates sin2w) = 0.2397, i.e. θw =  29.3137 degrees (this is a running value, depending on the momentum at which it is measured, with a significance of 6 standard deviations).
 
Figure 7.2.9. The Z0 subtends an angle θw = 30 degrees with the W+ and the W- components of the weak field
 

7.3. The Strong Force Field

In Section 7.3.1, we define the Yang-Mills Strong Field Equations. In Section 7.3.2, we show how the gluon acts as the carrier of the strong force. In Section 7.3.3, we give an example of a typical strong interaction: formation of a quark-antiquark pair, called a meson. In Section 7.3.4, we define the strong gauge group
 
Gs = SU(3)

We explicitly define the eight species of observable gauge gluons and show how the strong gauge group acts on them by means of the strong gauge transformations.

 

7.3.1. Yang-Mills Strong Field Equations

Corresponding to the eight gluon species described in 6.5.1, we have eight electromagnetic type fields defined by the eight tensors Fμν(s), s = 1, ..., 8. Thus, we have eight covariant 4-vectors  
(F 0(s), F 1(s), F 2(s), F 3(s)) where s = 1, ..., 8
(7.3.1.1)
We also have eight electric and eight magnetic fields corresponding to the antisymmetric tensors Fμν(s) given by the following correspondence:
  
 F 00(s)
 F 01(s)
 F 02(s)
 F 03(s)
F 10(s)
F 11(s)
F 12(s)
F 13(s)
F 20(s)
F 21(s)
F 22(s)
F 23(s)
F 30(s)
F 31(s)
F 32(s)
F 33(s)
 → 
0
- E 1(s)
- E 2(s)
- E 3(s)
  E 1(s)
0
H 3(s)
- H 2(s)
  E 2(s)
- H 3(s)
0
H 1(s)
  E 3(s)
H 2(s)
- H 1(s)
0
where s = 1, ..., 8
(7.3.1.2)
 

7.3.2. The gluon as the carrier of the strong force

We now show how the gluon 6.5.1 can be regarded as the carrier of the strong force. The unique permutation ψgluon = R(β)φi associated with the massless gluon particle must have φi equal to the identity permutation according to the mass rule 4.7. We select β = σ, σρ or σρ2 and γ = σ, σρ or σρ2. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. For the first ray we select discs 5, 6 on the upper sheet of the particle frame and their blue regions 0, that have the labels (+(0, σρ)φiR(β), β+γ), (+(0, σ)φiR(β), β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 11, 12 on the upper sheet of the particle frame and their blue regions 0, that have the labels (-(0, σρ)R(γ)φi, β+γ), (-(0, σ)R(γ)φi, β+γ) respectively, according to figures 3.4 and 3.5. By the strong charge rule 4.6, the gluon particle carries eight possible strong charge/anticharge pairs as superpositions of particle frames (there are eight gluon species indexed by s = 1, ..., 8). These together represent the Schrödinger discs of the gluon 6.5.1 and the strong field via the following correspondence, for s = 1, ..., 8:  
F 0(s)  →  (+(0, σρ)φiR(β), β+γ)
F 1(s)  →  (+(0, σ)φiR(β), β+γ)
F 2(s)  →  (-(0, σρ)R(γ)φi, β+γ)
F 3(s)  →  (-(0, σ)R(γ)φi, β+γ)
(7.3.2)
Figure 7.3.2. The gluon and the strong field. Each of the eight species of gluon also carries a strong charge-anticharge pair.
 

7.3.3. Strong Interactions

No free quarks are observed because of the phenomenon of quark confinement. Quarks are bound together by the strong force forming neutrons, protons and mesons. If one were to try and isolate a quark at a distance greater than the proton volume, the energy required would be greater than the energy required to form a quark-antiquark pair (a meson) and the lower energy process is favoured in nature. The strong force obeys the law of asymptotic freedom: at distances comparable to the proton volume the strong force effectively vanishes, so that quarks are essentially free to move about within this confined volume. A typical strong interaction is shown in the following Feynman diagram for the formation of an up quark - strange antiquark pair (a meson), called the K+ kaon. An up quark and a strange antiquark experience an attractive force (which is asymptotically free) and a virtual gluon (the carrier of the strong force) is exchanged. Suppose that the incoming up quark has a strong charge σ and the incoming strange antiquark has a strong charge σρ. The exchanged virtual gluon As has a pair of strong charges σ - σρ and causes the strong charge of the outgoing up quark to change to σρ and the strong charge of the outgoing strange antiquark to change to σ.
 
Figure 7.3.3.1. Feynman diagram for a typical strong interaction: formation of a quark - antiquark pair

We can visualize this strong interaction as shown below in figure 7.3.3.2: (a) the up quark (with strong charge σ) and the strange antiquark (with strong charge σρ) within the confinement volume ; (b) two of the Schrödinger discs of the strong field destroy the up quark and the strange antiquark (the corresponding wave functions are now interpreted as destruction operators); (c) the other two Schrödinger discs of the strong field create a new up quark (with strong charge σρ) and a new strange antiquark (with strong charge σ) (the corresponding wave functions are now interpreted as creation operators); (d) together, the four Schrödinger discs of the strong field constitute the virtual gluon that is exchanged while transmitting the strong force; (e) the up quark (with strong charge σρ) and the strange antiquark (with strong charge σ) stay within the confinement volume, forming a meson.
 
 

Figure 7.3.3.2. A typical strong interaction: formation of a quark - antiquark pair
 

7.3.4. The Strong Gauge Group

The strong Yang-Mills field 7.3.2 consists of eight electromagnetic type fields defined by the tensors Fμν(s) for s = 1, ..., 8, corresponding to the eight gluon species. The strong Yang-Mills gauge field is specified by its gauge group Gs and a constant of interaction αs, called its coupling constant. The coupling constant will be explicitly calculated in 8.2. We shall now construct the gauge group Gs for the strong field. Consider the special unitary group SU(3) consisting of 3×3 complex unitary matrices of determinant 1, under matrix multiplication. The special unitary group SU(3) is generated (as a Lie group) by the eight 3×3 unitary matrices of determinant 1 (called Gell-Mann generators). We first define two copies of SU(3), called SU(3)s and SU(3)g, as follows:
  • SU(3)s = SU(3) is the gauge group for the strong force after it separated from the gravitational force in the cosmological timeline. We label the row and the column of each generator of SU(3)s as follows:
Γ1
+ σρ 
+ σρ2
+ σ 
- σρ 
0
1
0
- σρ2
1
0
0
- σ 
0
0
0
Γ2
+ σρ 
+ σρ2
+ σ 
- σρ 
0
- i
0
- σρ2
i
0
0
- σ 
0
0
0
Γ3
+ σρ 
+ σρ2
+ σ 
- σρ 
1
0
0
- σρ2
0
- 1
0
- σ 
0
0
0
Γ4
+ σρ 
+ σρ2
+ σ 
- σρ 
0
0
1
- σρ2
0
0
0
- σ 
1
0
0
Γ5
+ σρ 
+ σρ2
+ σ 
- σρ
0
0
- i
- σρ2
0
0
0
- σ
i
0
0
Γ6
+ σρ 
+ σρ2
+ σ 
- σρ 
0
0
0
- σρ2
0
0
1
- σ
0
1
0
Γ7
+ σρ 
+ σρ2
+ σ 
- σρ 
0
0
0
- σρ2
0
0
- i
- σ
0
i
0
Γ8
+ σρ 
+ σρ2
+ σ 
- σρ 
1/√3
0
0
- σρ2
0
1/√3
0
- σ
0
0
-2/√3
Figure 7.3.4.1. SU(3)s generators
  • SU(3)g = SU(3) is the gauge group for the strong force before it separated from the gravitational force in the cosmological timeline. We label the row and the column of each generator of SU(3)g as follows:
Γ1
+ 1
+ 2
+ σ
- 1
0
1
0
- 2
1
0
0
- σ
0
0
0
Γ2
+ 1
+ 2
+ σ
- 1
0
- i
0
- 2
i
0
0
- σ
0
0
0
Γ3
+ 1
+ 2
+ σ
- 1
1
0
0
- 2
0
- 1
0
- σ
0
0
0
Γ4
+ 1
+ 2
+ σ
- 1
0
0
1
- 2
0
0
0
- σ
1
0
0
Γ5
+ 1
+ 2
+ σ
- 1
0
0
- i
- 2
0
0
0
- σ
i
0
0
Γ6
+ 1
+ 2
+ σ
- 1
0
0
0
- 2
0
0
1
- σ
0
1
0
Γ7
+ 1
+ 2
+ σ
- 1
0
0
0
- 2
0
0
- i
- σ
0
i
0
Γ8
+ 1
+ 2
+ σ
- 1
1/√3
0
0
- 2
0
1/√3
0
- σ
0
0
-2/√3
Figure 7.3.4.2. SU(3)g generators
  • Let SU(3)s×SU(3)g = {(a, b) | aSU(3)s , bSU(3)g} denote the group direct product and define the strong gauge group
 
Gs = (SU(3)s×SU(3)g)diagonal = {(a, a) | aSU(3)s , aSU(3)g} = SU(3)
(7.3.4.1)

 
Γ1
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
0
1
0
- σρ2
- 2
1
0
0
- σ
- σ
0
0
0
Γ2
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
0
- i
0
- σρ2
- 2
i
0
0
- σ
- σ
0
0
0
Γ3
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
1
0
0
- σρ2
- 2
0
- 1
0
- σ
- σ
0
0
0
Γ4
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
0
0
1
- σρ2
- 2
0
0
0
- σ
- σ
1
0
0
Γ5
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
0
0
- i
- σρ2
- 2
0
0
0
- σ
- σ
i
0
0
Γ6
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
0
0
0
- σρ2
- 2
0
0
1
- σ
- σ
0
1
0
Γ7
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
0
0
0
- σρ2
- 2
0
0
- i
- σ
- σ
0
i
0
Γ8
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
1/√3
0
0
- σρ2
- 2
0
1/√3
0
- σ
- σ
0
0
-2/√3
Figure 7.3.4.3. The strong gauge group Gs=SU(3) generators

The row and column labels
 

± σρ
± 1
,
± σρ2
± 2
,
± σ 
± σ

in figure 7.3.4.3 specify the Schrödinger discs of the gluon and the strong force as defined by equations 7.3.2 and figure 7.3.2. The first component of the row and column labels in figure 7.3.4.3 consists of strong charges; the second component specifies how the strong gauge group will embed in the gravitational gauge group 7.4.6. The eight observable gluon species are defined as
 

[A(s)] = Γs for s = 1, ..., 8

Each observable gluon [A(s)] is regarded as a superposition of the strong charge - strong anticharge labels of its row and column multiplied by a complex number viewed on the z-plane. Since t = z2, a rotation by an angle θ around the origin of the z-plane corresponds to a rotation by an angle 2θ around the branch point of the t-Riemann surface. In particular, note that multiplication by ±i, ±1 on the z-plane correspond to rotations of the t-Riemann surface by ±180, ±360 degrees around the branch point, respectively. This means that for the eight observable gluons [A(s)], for s = 1, ..., 8, the rays defining the eight gluons A(s), for s = 1, ..., 8, are permuted amongst themselves (and not any other rays) on the particle frame. Thus, the eight observable gluons [A(s)], for s = 1, ..., 8, correspond to superpositions of particle frames of the eight gluons A(s), for s = 1, ..., 8.
 

Figure 7.3.4.4. The observable gluons

The strong gauge group acts by conjugation on the eight observable gluons [A(s)], for s = 1, ..., 8, viewed on the z-plane:
 

Γi-1 ΓsΓi for i, s = 1, ..., 8

Again, since t = z2, a rotation by an angle θ around the origin of the z-plane corresponds to a rotation by an angle 2θ around the branch point of the t-Riemann surface and multiplication by ±i, ±1 on the z-plane correspond to rotations of the t-Riemann surface by ±180, ±360 degrees around the branch point, respectively. This means that for any strong gauge transformation, the rays defining the eight gluons A(s), for s = 1, ..., 8, are permuted amongst themselves (and not any other rays) on the particle frame. The strong gauge transformations will always transform superpositions of the eight gluons A(s), for s = 1, ..., 8, to other superpositions of the eight gluons A(s), for s = 1, ..., 8.

 

7.4. The Gravitational Force Field

In Section 7.4.1, we review General Relativity and curved space-time. We define the curvature tensor, the Ricci tensor and formulate Einstein's law of gravitation. Comparison with Newton's law of gravitation in the special case of flat space-time shows that the components of the metric tensor must be viewed as potentials describing the gravitational field. In Section 7.4.2, we show how to embed the particle frame in curved space-time without self-intersections. Then each of the Schrödinger discs of the particle frame carry the curvature and Ricci tensors. Thus, in Section 7.4.3, we can define the Gravitational Field Equations precisely as given by Einstein's law of gravitation. In Section 7.4.4, we show how the graviton acts as the carrier of the gravitational force. In Section 7.4.5, we give an example of a possible gravitational interaction: neutrino oscillation. In Section 7.4.6, we define the gravitational gauge group
 
Gg= SU(5)

We explicitly define the twenty-four species of observable gauge gravitons and show how the gravitational gauge group acts on them by means of the gravitational gauge transformations. 

 

7.4.1. General Relativity

To account for the gravitational force, Einstein [7] assumed that physical space-time forms a curved Riemann space and thereby laid the foundation for his theory of gravitation. Generalizing equation 1.3 of special relativity to general relativity, the invariant distance dS between a point (with space-time coordinates written as a contravariant four-vector) X μ and a neighbouring point X μ + dX μ is given by  
dS 2 = gμν dX μdX ν
(7.4.1.1)
 where the gμν are given as functions of the space-time coordinates and define the metric. The Christoffel symbols of the first kind are defined as  
Γμντ = (gμν, τ+ gμτ, ν- gντ, μ)/2
(7.4.1.2)
where the subscript commas denote partial derivatives in Einstein's notation. The Christoffel symbols of the second kind are defined as  
Γμντ = gμλ Γλντ
(7.4.1.3)
To a contravariant four-vector A μ at a point X μ we associate a contravariant four-vector A μ + dA μ at the infinitesimally close point X μ + dX μ by the bilinear expression
  
dA μ = - Γμντ A νdX τ
(7.4.1.4)
The infinitesimal displacement field Γ is not a tensor. However, it implies the existence of a tensor called the curvature tensor. This is obtained by displacing a contravariant four-vector A μ according to equation 7.4.1.4 along the circumference of an infinitesimal two-dimensional surface element and computing its change in one circuit. The curvature tensor is given by
  
R μντλ = Γμνλ, τ - Γμντ, λ+ Γκνλ Γμκτ - Γκντ Γμκλ
(7.4.1.5)
The curvature tensor satisfies the Bianci identities :
  
R μντλ = - R μνλτ
(7.4.1.6)
 
R μντλ + R μτλν + R μλντ = 0
(7.4.1.7)
 
R μντλ = - R νμτλ
(7.4.1.8)
 
R μντλ = R τλμν = R λτνμ
(7.4.1.9)
 
R κμντ + R κμτλ:ν + R κμλν:τ = 0
(7.4.1.10)
where the subscript colons denote covariant derivatives in Einstein's notation. As a consequence of the Bianci identities 7.4.1.6 - 7.4.1.10, only 20 of the 256 components of the curvature tensor R μντλ are independent. Space-time is flat if and only if the curvature tensor R μντλ= 0.
  • For flat space-time, we may choose a rectilinear system of coordinates and the gμν are constant.
  • For curved space-time, we need a curvilinear system of coordinates and the gμν are not constant.
Let us contract two of the suffixes in the curvature tensor R μντλ. If we take two with respect to which the curvature tensor is antisymmetric, we get zero. If we take any other two, the value of the curvature tensor is preserved, apart from the sign, because of the Bianci identities 7.4.1.6, 7.4.1.8 and 7.4.1.9. We contract the first and the last suffixes in the curvature tensor and put R μντμ = R ντ. This is called the Ricci tensor. By definition, the Ricci tensor is symmetric. From 7.4.1.5, the Ricci tensor is explicitly given by  
R ντ = Γμνμ, τ - Γμντ, μ+ Γκνμ Γμκτ - Γκντ Γμκμ
(7.4.1.11)
We can now state Einstein's law of gravitation in empty space:  
R ντ = 0
(7.4.1.12)
In an approximately flat and static region of space-time, we can show that g00 = 1 + 2V, where V = -m/r is the Newtonian gravitational potential due to a particle of mass m at the origin. Thus, Newton's law of gravitation can be obtained from Einstein's law of gravitation, provided the gravitational field strength is low (the space-time curvature is small) and approximately static (the velocities of the particles under consideration are small compared to the velocity of light). This gives us a valuable insight into the physical meaning of the gμν:
  • The gμν are interpreted as potentials describing the gravitational field
Newtonian mechanics and Einstein's special relativity were both based on the concept of an inertial reference frame. The principle of equivalence in general relativity does away with the concept of an inertial reference frame, replacing it instead with the concept of the curvature tensor (this observation is attributed to Levi-Civita). According to general relativity, we must replace the concept of inertial mass by the corresponding curvature tensor.
 

7.4.2. Embedding the particle frame in space-time

So far, we have drawn the particle frame (figure 4.1) embedded in flat Euclidean three-dimensional space (X, Y, Z) and associated with the drawing an independent time dimension T. Each of the discs in the complex plane have been drawn as discs embedded in flat Euclidean three-dimensional space (X, Y, Z). Such an embedding of the Riemann surface (figure 3.4) in flat Euclidean three-dimensional space has self-intersections. We shall show how to embed the particle frame in curved four-dimensional space-time without self-intersections. Let us first review the construction of the particle frame as discs embedded in flat Euclidean three-dimensional space. Consider the composition of the functions C  → C; z → t = z2 and C → C; t → w = t12. The composite is given by the assignment z → t = z2 → w = t12 = z24. Take twenty-four identical copies of the map m(4) on the disc with a cut, labeled k = 1, ..., 24.
 
Figure 7.4.2.1. Twenty-four copies of the map in Euclidean three-dimensional space 

For k = 1, ..., 23 attach the lower edge of the cut of disc k with the upper edge of the cut of disc k+1. To complete the cycle, attach the lower edge of the cut of disc 24 with the upper edge of the cut of disc 1. This forms the w-Riemann surface embedded in Euclidean three-dimensional space (X, Y, Z).
 

Figure 7.4.2.2. The w-Riemann surface in Euclidean space 

The point w = 0 connects all the discs and is called the branch point. There are twenty-four superposed copies of the map m(4) on the w-Riemann surface corresponding to the twenty-four sectors  

{z|(k-1)(2π/24) < arg z < k(2π/24)} (k = 1, ..., 24)
on the z-plane. As we can see in figure 7.4.2.2, the w-Riemann surface embedded in flat Euclidean three-dimensional space has many self-intersections. We shall now explicitly specify an embedding of the w-Riemann surface in curved four-dimensional space-time: consider the ray arg z = T on the w-Riemann surface, then as T goes continuously from 0 to 2π, the w-Riemann surface is embedded continuously in curved four-dimensional space-time (X, Y, Z, T). This embedding has no self-intersections.
 
Figure 7.4.2.3. The w-Riemann surface is embedded in space-time by the ray arg z = T,  0  ≤ T  ≤  2π

Note that the ray shown in figure 7.4.2.3 will go once around each of the 24 Schrödinger discs of the particle frame represented by the w-Riemann surface, as T goes continuously from 0 to 2π. This embedding will also carry the metric gμνof space-time, since the space-time distance for any two points on the w-Riemann surface is given via the embedding. If we take an infinitesimal contravariant four-vector on the ray arg z = 0 and compute its change in one circuit given by arg z = T, as T goes continuously from 0 to 2π, we obtain precisely the curvature tensor given by equation 7.4.1.5. Hence, each of the 24 Schrödinger discs of the particle frame represented by the w-Riemann surface carry the curvature tensor R μντλ and the Ricci tensor R ντ.
 

Figure 7.4.2.4. Each Schrödinger disc carries the curvature tensor

Referring to the mass rule 4.7, the inertial mass of each particle S in the standard model is associated with a unique permutation ψS. According to general relativity 7.4.1, we must also associate ψS with the curvature tensor R μντλ and the Ricci tensor R ντ carried by the Schrödinger discs of the particle frame.

 

7.4.3. The Gravitational Field Equations

In analogy with the definition of the electromagnetic field by Maxwell's equations 7.1.1, we can now define the gravitational field as follows. The contravariant four-vector (X 0, X 1, X 2, X 3) represents space-time and Kμν = gμν represents the gravitational potential. Since the gμν are symmetric, only 10 out of the 16 are independent, hence the gravitational potential K may be represented by a ten-vector (instead of a four-vector as in the case of the electromagnetic potential). Define the contravariant and covariant four-vectors  
(F 0, F 1, F 2, F 3) and (F 0, F 1, F 2, F 3)
(7.4.3.1)
respectively, where F 0 = F 0, F 1 = - F 1, F 2 = - F 2, F 3 = - F 3, by means of the tensor equations  
Fμν = FμFν = R μν
(7.4.3.2)
Note that by equations 7.4.1.2, 7.4.1.3, 7.4.1.5 and 7.4.1.11, the Ricci tensor R μν can be expressed purely in terms of the gμν and its partial derivatives, hence the tensor Fμν is expressed purely in terms of the gravitational potential Kμν and its partial derivatives. The gravitational field equations are given by Einstein's law of gravitation as  
Fμν = 0
(7.4.3.3)
Note that by equations 7.4.3.2, the tensor Fμν is expressed purely in terms of the gravitational potential Kμν and its partial derivatives, just as in the case of the electromagnetic field 7.1.1. However, the field equations for gravitation 7.4.3.3 are considerably more difficult to solve because they are not linear and also contain partial derivatives of the second order.

Corresponding to the twenty-four graviton species described in 6.6.1, we have twenty-four gravitational type fields defined by the twenty-four tensors Fμν(g), g = 1, ..., 24. Thus, we have twenty-four covariant 4-vectors  

(F 0(g), F 1(g), F 2(g), F 3(g)) where g = 1, ..., 24
(7.4.3.4)
The twenty-four symmetric tensors Fμν(g) are related to the curvature and twenty-four copies of the Ricci tensor by the following correspondence:
  
 F 00(g)
 F 01(g)
 F 02(g)
 F 03(g)
F 10(g)
F 11(g)
F 12(g)
F 13(g)
F 20(g)
F 21(g)
F 22(g)
F 23(g)
F 30(g)
F 31(g)
F 32(g)
F 33(g)
 → 
R 00(g)
R 01(g)
R 02(g)
R 03(g)
  R 01(g)
R 11(g)
R 12(g)
R 13(g)
  R 02(g)
R 12(g)
R 22(g)
R 23(g)
  R 03(g)
R 13(g)
R 23(g)
R 33(g)
where g = 1, ..., 24
(7.4.3.5)
 

7.4.4. The graviton as the carrier of the gravitational force

We now show how the graviton 6.6.1 can be regarded as the carrier of the gravitational force. The unique permutation ψgraviton = R(β)φi associated with the massless graviton particle must have φi equal to the identity permutation according to the mass rule 4.7. We select β = 1 and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φi, β, γ. Let m = 0, 1, 2, or 3, corresponding to the gravitational charge rule 4.6. For the first ray we select discs 1, 12 on the upper sheet of the particle frame and their regions m, that have the labels (+(m, 1)φiR(β), β+γ), (-(m, σ)R(γ)φi, β+γ) respectively, according to figures 3.4 and 3.5. For the second ray we select discs 6, 7 on the upper sheet of the particle frame and their regions m, that have the labels (+(m, σ)φiR(β), β+γ), (-(m, 1)R(γ)φi, β+γ) respectively, according to figures 3.4 and 3.5. By the gravitational charge rule 4.6, the graviton particle carries 24 possible gravitational charge/anticharge pairs as superpositions of particle frames (there are 24 graviton species). These together represent the Schrödinger discs of the graviton 6.6.1 and the gravitational field via the following correspondence, for g = 1, ..., 24:  
F 0(g)  →  (+(m, 1)φiR(β), β+γ)
F 1(g)  →  (-(m, σ)R(γ)φi, β+γ)
F 2(g)  →  (+(m, σ)φiR(β), β+γ)
F 3(g)  →  (-(m, 1)R(γ)φi, β+γ)
(7.4.4)
Figure 7.4.4. The graviton and the gravitational field. Each of the twenty-four species of graviton also carries a gravitational charge-anticharge pair.
 

7.4.5. Gravitational Interactions

The gravitational force is extremely weak compared to the other forces. In most cases, it would be very difficult to detect a gravitational interaction in an experiment, where it would be masked by an electromagnetic, weak or strong interaction. However, there is a possibility that such an interaction has already been observed, albeit indirectly, in the Super-Kamiokande experiment [10], where evidence for neutrino oscillations was found. A possible gravitational interaction is shown in the following Feynman diagram that may explain neutrino oscillations. A μ-neutrino is converted into a τ-neutrino, and vice versa, by the exchange of a virtual graviton. Suppose that the incoming μ-neutrino has a gravitational charge + 0 and the incoming τ-neutrino has a gravitational charge - 0. The exchanged virtual graviton g has a pair of gravitational charges 0 - 0 and causes the gravitational charge of the outgoing τ-neutrino to change to - 0 and the gravitational charge of the outgoing μ-neutrino to change to + 0.
 
Figure 7.4.5.1. Feynman diagram for a possible gravitational interaction: neutrino oscillation

We can visualize this gravitational interaction as shown below in figure 7.4.5.2: (a) the μ-neutrino (with gravitational charge + 0 ) and the τ-neutrino (with gravitational charge - 0); (b) two of the Schrödinger discs of the gravitational field destroy the μ-neutrino and the τ-neutrino (the corresponding wave functions are now interpreted as destruction operators); (c) the other two Schrödinger discs of the gravitational field create a new τ-neutrino (with gravitational charge - 0) and a new μ-neutrino (with gravitational charge + 0 ) (the corresponding wave functions are now interpreted as creation operators); (d) together, the four Schrödinger discs of the gravitational field constitute the virtual graviton that is exchanged while transmitting the gravitational force; (e) the μ-neutrino (with gravitational charge + 0 ) and the τ-neutrino (with gravitational charge - 0) have oscillated.
 
 

Figure 7.4.5.2. A possible gravitational interaction: neutrino oscillation
 

7.4.6. The Gravitational Gauge Group

The gravitational field 7.4.4 consists of twenty-four gravitational type fields defined by the tensors Fμν(g) for g = 1, ..., 24, corresponding to the twenty-four graviton species. The gravitational gauge field is specified by its gauge group Gg and a constant of interaction αg, called its coupling constant. The coupling constant will be calculated explicitly in 8.2. We shall now construct the gauge group Gg for the gravitational field. Consider the special unitary group SU(5) consisting of 5×5 complex unitary matrices of determinant 1, under matrix multiplication. The special unitary group SU(5) is generated (as a Lie group) by the twenty-four 5×5 unitary matrices of determinant 1. We first define two copies of SU(5), called SU(5)g and SU(5)e, as follows:
  • SU(5)g = SU(5) is the gauge group for the gravitational force after the end of the Planck epoch, upto the present, in the cosmological timeline. We label the row and the column of each generator of SU(5)g as follows:
Γ1
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
1
0
0
0
- 2
1
0
0
0
0
- σ
0
0
0
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ2
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
- i
0
0
0
- 2
i
0
0
0
0
- σ
0
0
0
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ3
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
1
0
0
0
0
- 2
0
- 1
0
0
0
- σ
0
0
0
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ4
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
1
0
0
- 2
0
0
0
0
0
- σ
1
0
0
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ5
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
- i
0
0
- 2
0
0
0
0
0
- σ
i
0
0
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ6
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
1
0
0
- σ
0
1
0
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ7
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
- i
0
0
- σ
0
i
0
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ8
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
1/√3
0
0
0
0
- 2
0
1/√3
0
0
0
- σ
0
0
-2/√3
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ9
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
1
0
- 2
0
0
0
0
0
- σ
0
0
0
0
0
- 0
1
0
0
0
0
- 3
0
0
0
0
0
Γ10
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
- i
0
- 2
0
0
0
0
0
- σ
0
0
0
0
0
- 0
i
0
0
0
0
- 3
0
0
0
0
0
Γ11
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
0
1
- 2
0
0
0
0
0
- σ
0
0
0
0
0
- 0
0
0
0
0
0
- 3
1
0
0
0
0
Γ12
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
0
- i
- 2
0
0
0
0
0
- σ
0
0
0
0
0
- 0
0
0
0
0
0
- 3
i
0
0
0
0
Γ13
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
1
0
- σ
0
0
0
0
0
- 0
0
1
0
0
0
- 3
0
0
0
0
0
Γ14
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
- i
0
- σ
0
0
0
0
0
- 0
0
i
0
0
0
- 3
0
0
0
0
0
Γ15
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
1
- σ
0
0
0
0
0
- 0
0
0
0
0
0
- 3
0
1
0
0
0
Γ16
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
- i
- σ
0
0
0
0
0
- 0
0
0
0
0
0
- 3
0
i
0
0
0
Γ17
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
0
- σ
0
0
0
1
0
- 0
0
0
1
0
0
- 3
0
0
0
0
0
Γ18
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
0
- σ
0
0
0
- i
0
- 0
0
0
i
0
0
- 3
0
0
0
0
0
Γ19
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
0
- σ
0
0
0
0
1
- 0
0
0
0
0
0
- 3
0
0
1
0
0
Γ20
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
0
- σ
0
0
0
0
- i
- 0
0
0
0
0
0
- 3
0
0
i
0
0
Γ21
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
0
- σ
0
0
0
0
0
- 0
0
0
0
0
1
- 3
0
0
0
1
0
Γ22
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
0
- σ
0
0
0
0
0
- 0
0
0
0
0
- i
- 3
0
0
0
i
0
Γ23
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
0
- σ
0
0
0
0
0
- 0
0
0
0
1
0
- 3
0
0
0
0
- 1
Γ24
+ 1
+ 2
+ σ
+ 0
+ 3
- 1
2/√15
0
0
0
0
- 2
0
2/√15
0
0
0
- σ
0
0
2/√15
0
0
- 0
0
0
0
-3/√15
0
- 3
0
0
0
0
-3/√15
Figure 7.4.6.1. SU(5)g generators
  • SU(5)e = SU(5) is the gauge group for the gravitational force during the Planck epoch in the cosmological timeline. We label the row and the column of each generator of SU(5)e as follows:
Γ1
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
1
0
0
0
- 2
1
0
0
0
0
- 1
0
0
0
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ2
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
- i
0
0
0
- 2
i
0
0
0
0
- 1
0
0
0
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ3
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
1
0
0
0
0
- 2
0
- 1
0
0
0
- 1
0
0
0
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ4
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
1
0
0
- 2
0
0
0
0
0
- 1
1
0
0
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ5
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
- i
0
0
- 2
0
0
0
0
0
- 1
i
0
0
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ6
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
1
0
0
- 1
0
1
0
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ7
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
- i
0
0
- 1
0
i
0
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ8
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
1/√3
0
0
0
0
- 2
0
1/√3
0
0
0
- 1
0
0
-2/√3
0
0
- 0
0
0
0
0
0
- 3
0
0
0
0
0
Γ9
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
1
0
- 2
0
0
0
0
0
- 1
0
0
0
0
0
- 0
1
0
0
0
0
- 3
0
0
0
0
0
Γ10
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
- i
0
- 2
0
0
0
0
0
- 1
0
0
0
0
0
- 0
i
0
0
0
0
- 3
0
0
0
0
0
Γ11
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
0
1
- 2
0
0
0
0
0
- 1
0
0
0
0
0
- 0
0
0
0
0
0
- 3
1
0
0
0
0
Γ12
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
0
- i
- 2
0
0
0
0
0
- 1
0
0
0
0
0
- 0
0
0
0
0
0
- 3
i
0
0
0
0
Γ13
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
1
0
- 1
0
0
0
0
0
- 0
0
1
0
0
0
- 3
0
0
0
0
0
Γ14
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
- i
0
- 1
0
0
0
0
0
- 0
0
i
0
0
0
- 3
0
0
0
0
0
Γ15
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
1
- 1
0
0
0
0
0
- 0
0
0
0
0
0
- 3
0
1
0
0
0
Γ16
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
- i
- 1
0
0
0
0
0
- 0
0
0
0
0
0
- 3
0
i
0
0
0
Γ17
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
0
- 1
0
0
0
1
0
- 0
0
0
1
0
0
- 3
0
0
0
0
0
Γ18
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
0
- 1
0
0
0
- i
0
- 0
0
0
i
0
0
- 3
0
0
0
0
0
Γ19
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
0
- 1
0
0
0
0
1
- 0
0
0
0
0
0
- 3
0
0
1
0
0
Γ20
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
0
- 1
0
0
0
0
- i
- 0
0
0
0
0
0
- 3
0
0
i
0
0
Γ21
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
0
- 1
0
0
0
0
0
- 0
0
0
0
0
1
- 3
0
0
0
1
0
Γ22
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
0
- 1
0
0
0
0
0
- 0
0
0
0
0
- i
- 3
0
0
0
i
0
Γ23
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
0
0
0
0
0
- 2
0
0
0
0
0
- 1
0
0
0
0
0
- 0
0
0
0
1
0
- 3
0
0
0
0
- 1
Γ24
+ 1
+ 2
+ 1
+ 0
+ 3
- 1
2/√15
0
0
0
0
- 2
0
2/√15
0
0
0
- 1
0
0
2/√15
0
0
- 0
0
0
0
-3/√15
0
- 3
0
0
0
0
-3/√15
Figure 7.4.6.2. SU(5)e generators
  • Let SU(5)g×SU(5)e = {(a, b) | aSU(5)g , bSU(5)e} denote the group direct product and define the gravitational gauge group
 
Gg = (SU(5)g×SU(5)e)diagonal = {(a, a) | aSU(5)g , aSU(5)e} = SU(5)
(7.4.6.1)

 
Γ1
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
1
0
0
0
- 2
- 2
1
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ2
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
- i
0
0
0
- 2
- 2
i
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ3
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
1
0
0
0
0
- 2
- 2
0
- 1
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ4
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
1
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
1
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ5
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
- i
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
i
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ6
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
1
0
0
- σ
- 1
0
1
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ7
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
- i
0
0
- σ
- 1
0
i
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ8
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
1/√3
0
0
0
0
- 2
- 2
0
1/√3
0
0
0
- σ
- 1
0
0
-2/√3
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ9
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
1
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
1
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ10
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
- i
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
i
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ11
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
1
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
1
0
0
0
0
Γ12
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
- i
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
i
0
0
0
0
Γ13
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
1
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
1
0
0
0
- 3
- 3
0
0
0
0
0
Γ14
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
- i
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
i
0
0
0
- 3
- 3
0
0
0
0
0
Γ15
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
1
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
1
0
0
0
Γ16
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
- i
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
i
0
0
0
Γ17
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
1
0
- 0
- 0
0
0
1
0
0
- 3
- 3
0
0
0
0
0
Γ18
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
- i
0
- 0
- 0
0
0
i
0
0
- 3
- 3
0
0
0
0
0
Γ19
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
1
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
1
0
0
Γ20
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
- i
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
i
0
0
Γ21
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
1
- 3
- 3
0
0
0
1
0
Γ22
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
- i
- 3
- 3
0
0
0
i
0
Γ23
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
1
0
- 3
- 3
0
0
0
0
-1
Γ24
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
2/√15
0
0
0
0
- 2
- 2
0
2/√15
0
0
0
- σ
- 1
0
0
2/√15
0
0
- 0
- 0
0
0
0
-3/√15
0
- 3
- 3
0
0
0
0
-3/√15
Figure 7.4.6.3. The gravitational gauge group Gg= SU(5) generators

The row and column labels
 

± 1
± 1
,
± 2
± 2
,
± σ
± 1
,
± 0
± 0
,
± 3
± 3

in figure 7.4.6.3 specify the Schrödinger discs of the graviton and the gravitational force as defined by equations 7.4.4 and figure 7.4.4. The first component of the row and column labels in figure 7.4.6.3 consists of gravitational charge; the second component specifies how the electric and electromagnetic charges are embedded in the gravitational gauge group during the Planck epoch. The twenty-four observable graviton species are defined as
 

[g(i)] = Γi for i = 1, ..., 24

Each observable graviton [g(i)] is regarded as a superposition of the gravitational charge - gravitational anticharge labels of its row and column multiplied by a complex number viewed on the z-plane. Since t = z2, a rotation by an angle θ around the origin of the z-plane corresponds to a rotation by an angle 2θ around the branch point of the t-Riemann surface. In particular, note that multiplication by ±i, ±1 on the z-plane correspond to rotations of the t-Riemann surface by ±180, ±360 degrees around the branch point, respectively. This means that for the twenty-four observable gravitons [g(i)], for i = 1, ..., 24, the rays defining the twenty-four gravitons g(i), for i = 1, ..., 24, are permuted amongst themselves (and not any other rays) on the particle frame. Thus, the twenty-four observable gravitons [g(i)], for i = 1, ..., 24, correspond to superpositions of particle frames of the twenty-four gravitons g(i), for i = 1, ..., 24.
 

Figure 7.4.6.4. The observable gravitons

The gravitational gauge group acts by conjugation on the twenty-four observable gravitons [g(i)], for i = 1, ..., 24, viewed on the z-plane:
 

Γi-1 Γj Γi for i, j = 1, ..., 24

Again, since t = z2, a rotation by an angle θ around the origin of the z-plane corresponds to a rotation by an angle 2θ around the branch point of the t-Riemann surface and multiplication by ±i, ±1 on the z-plane correspond to rotations of the t-Riemann surface by ±180, ±360 degrees around the branch point, respectively. This means that for any gravitational gauge transformation, the rays defining the twenty-four gravitons g(i), for i = 1, ..., 24, are permuted amongst themselves (and not any other rays) on the particle frame. The strong gauge transformations will always transform superpositions of the twenty-four gravitons g(i), for i = 1, ..., 24, to other superpositions of the twenty-four gravitons g(i), for i = 1, ..., 24.

 

8. Grand Unification

We achieve our second goal, the Grand Unification of all the forces: electromagnetic, weak, strong and gravitational. In Section 8.1, we follow the cosmological timeline into the past, from the present to the Big Bang (or equivalent energy scales), showing how the gauge groups are embedded in a sequence
 
U(1) → SU(2) → SU(3) → SU(5)

during the unification and also showing how the Schrödinger discs are identified on the particle frame during the unification. In Section 8.2, we explicitly calculate all the coupling constants: the electromagnetic coupling constant, the weak coupling constant, the strong coupling constant and the gravitational coupling constant. Finally, in Section 8.3, we explicitly calculate the mass ratios of the particles in the standard model. 

 

8.1. The Gauge Groups

If we follow the cosmological timeline forward in time, we can see how the forces separated and how the particle frame evolved to its present form. Viewed backwards in time, we obtain the unification of the forces, showing how the gauge groups are embedded in a sequence U(1) → SU(2) → SU(3) → SU(5) and also how the Schrödinger discs are identified on the particle frame.
 
Figure 8.1. Separation and Unification
↓  Unification
Cosmological Timeline : The Present
Separation  ↑

8.1.1. The Later Epochs, upto the Present

  • Time T = 10-12 seconds upto the present
  • Average temperature 1015 K to 3 K
  • All the four forces are distinct: the gravitational force, the strong force, the weak force and the electromagnetic force
The electromagnetic gauge group 7.1.4 generator with rows and columns labeled, is shown below.
 
Γ1
+ 1
+ ρ
- 1
- ρ
1
Figure 8.1.1.1. The electromagnetic gauge group Ge= U(1) generator

The particle frame, labeled as in figures 3.5 and 4.1, with all the bosons describing the four forces in their present form, is shown below.
 

Figure 8.1.1.2. The particle frame and the force carrier bosons in the present epoch

At this time the particle frame assumes its present form as shown in figure 4.1, represented by the t-Riemann surface given by the composition of the functions C  → C;z → t = z2 and C → C; t → w = t12. The composite is given by the assignment z → t = z2 → w = t12 = z24.
 

↓  Unification
Cosmological Timeline : The Inflationary Epoch
Separation  ↑

8.1.2. The Inflationary Epoch

  • Time T = 10-35 seconds to T = 10-12 seconds
  • Average temperature 1027 K to 1015 K
  • The electromagnetic force and the weak force are unified
The weak gauge group 7.2.8 generators with rows and columns labeled, are shown below. The  electromagnetic gauge group is embedded in the top left-hand 1×1 corner of the weak gauge group, according to the row and column labeling.
 
Γ1
+ ρ
+ σρ
+ ρ2
+ σρ2
- ρ
- σρ
0
1
- ρ2
- σρ2
1
0
Γ2
+ ρ
+ σρ
+ ρ2
+ σρ2
- ρ
- σρ
0
- i
- ρ2
- σρ2
i
0
Γ3
+ ρ
+ σρ
+ ρ2
+ σρ2
- ρ
- σρ
1
0
- ρ2
- σρ2
0
- 1
Figure 8.1.2.1. The weak gauge group Gw= SU(2) generators

The particle frame, labeled as in figures 3.5 and 4.1, now has the Schrödinger discs with labels 1 and ρ identified, corresponding to the unification of the electromagnetic force with the weak force, as shown below.
 

Figure 8.1.2.2. The particle frame and the force carrier bosons in the inflationary epoch

During this epoch the particle frame may be represented by the t-Riemann surface given by the composition of the functions C  → C; z → t = z2 and C → C; t → w = t6. The composite is given by the assignment z → t = z2 → w = t6 = z12.
 

↓  Unification
Cosmological Timeline : The Grand Unification Epoch
Separation  ↑

8.1.3. The Grand Unification Epoch

  • Time T = 10-43 seconds to T = 10-35 seconds
  • Average temperature 1032 K to 1027 K
  • The weak force and the strong force are unified
The strong gauge group 7.3.4 generators with rows and columns labeled, are shown below. The weak gauge group is embedded in the top left-hand 2×2 corner of the strong gauge group, according to the row and column labeling.
 
Γ1
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
0
1
0
- σρ2
- 2
1
0
0
- σ
- σ
0
0
0
Γ2
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
0
- i
0
- σρ2
- 2
i
0
0
- σ
- σ
0
0
0
Γ3
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
1
0
0
- σρ2
- 2
0
- 1
0
- σ
- σ
0
0
0
Γ4
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
0
0
1
- σρ2
- 2
0
0
0
- σ
- σ
1
0
0
Γ5
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
0
0
- i
- σρ2
- 2
0
0
0
- σ
- σ
i
0
0
Γ6
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
0
0
0
- σρ2
- 2
0
0
1
- σ
- σ
0
1
0
Γ7
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
0
0
0
- σρ2
- 2
0
0
- i
- σ
- σ
0
i
0
Γ8
+ σρ
+ 1
+ σρ2
+ 2
+ σ
+ σ
- σρ
- 1
1/√3
0
0
- σρ2
- 2
0
1/√3
0
- σ
- σ
0
0
-2/√3
Figure 8.1.3.1. The strong gauge group Gs= SU(3) generators

The particle frame, labeled as in figures 3.5 and 4.1, now has the Schrödinger discs with labels 1, ρ, ρ2, σρ, σρ2 identified, corresponding to the unification of the weak force with the strong force, as shown below.
 

Figure 8.1.3.2. The particle frame and the force carrier bosons in the grand unification epoch

During this epoch the particle frame may be represented by the t-Riemann surface given by the composition of the functions C  → C; z → t = z2 and C → C; t → w = t4. The composite is given by the assignment z → t = z2 → w = t4 = z8.
 

↓  Unification
Cosmological Timeline : The Planck Epoch
Separation  ↑

8.1.4. The Planck Epoch

  • Time T = 0 seconds to T = 10-43 seconds
  • Average temperature more than 1032 K
  • The strong force and the gravitational force are unified
The gravitational gauge group 7.4.6 generators with rows and columns labeled, are shown below. The strong gauge group is embedded in the top left-hand 3×3 corner of the gravitational gauge group, according to the row and column labeling.
 
Γ1
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
1
0
0
0
- 2
- 2
1
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ2
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
- i
0
0
0
- 2
- 2
i
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ3
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
1
0
0
0
0
- 2
- 2
0
- 1
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ4
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
1
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
1
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ5
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
- i
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
i
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ6
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
1
0
0
- σ
- 1
0
1
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ7
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
- i
0
0
- σ
- 1
0
i
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ8
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
1/√3
0
0
0
0
- 2
- 2
0
1/√3
0
0
0
- σ
- 1
0
0
-2/√3
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ9
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
1
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
1
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ10
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
- i
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
i
0
0
0
0
- 3
- 3
0
0
0
0
0
Γ11
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
1
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
1
0
0
0
0
Γ12
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
- i
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
i
0
0
0
0
Γ13
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
1
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
1
0
0
0
- 3
- 3
0
0
0
0
0
Γ14
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
- i
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
i
0
0
0
- 3
- 3
0
0
0
0
0
Γ15
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
1
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
1
0
0
0
Γ16
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
- i
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
0
- 3
- 3
0
i
0
0
0
Γ17
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
1
0
- 0
- 0
0
0
1
0
0
- 3
- 3
0
0
0
0
0
Γ18
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
- i
0
- 0
- 0
0
0
i
0
0
- 3
- 3
0
0
0
0
0
Γ19
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
1
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
1
0
0
Γ20
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
- i
- 0
- 0
0
0
0
0
0
- 3
- 3
0
0
i
0
0
Γ21
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
1
- 3
- 3
0
0
0
1
0
Γ22
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
0
- i
- 3
- 3
0
0
0
i
0
Γ23
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
0
0
0
0
0
- 2
- 2
0
0
0
0
0
- σ
- 1
0
0
0
0
0
- 0
- 0
0
0
0
1
0
- 3
- 3
0
0
0
0
-1
Γ24
+ 1
+ 1
+ 2
+ 2
+ σ
+ 1
+ 0
+ 0
+ 3
+ 3
- 1
- 1
2/√15
0
0
0
0
- 2
- 2
0
2/√15
0
0
0
- σ
- 1
0
0
2/√15
0
0
- 0
- 0
0
0
0
-3/√15
0
- 3
- 3
0
0
0
0
-3/√15
Figure 8.1.4.1. The gravitational gauge group Gg= SU(5) generators

The particle frame, labeled as in figures 3.5 and 4.1, now has the Schrödinger discs with labels 1, ρ, ρ2, σρ, σρ2, σ identified, corresponding to the unification of the strong force with the gravitational force, as shown below.
 

Figure 8.1.4.2. The particle frame and the force carrier boson in the Planck epoch

During this epoch the particle frame may be represented by the t-Riemann surface given by the composition of the functions C  → C; z → t = z2 and C → C; t → w = t2. The composite is given by the assignment z → t = z2 → w = t2 = z4. A boson must be present as the carrier of the force and for the creation and destruction of particles. Since this is the minimum configuration required to define a boson, according to the boson selection rule 4.2, we have reached a limit with the grand unification of all the four forces immediately after the Big Bang at time T = 0.
 

Figure 8.1.4.3. The Big Bang at time T = 0
 

8.2. The Gauge Coupling Constants

We shall explicitly define the coupling constants by counting boson configurations on the particle frame, shown in figure 4.1. By the boson selection rule 4.2, a boson is defined on the particle frame by selecting a pair of rays. The rays defining the graviton are fixed, since it is the unique spin 2 boson, by the spin rule 4.4. Define B to be the number of ways to form a boson on the particle frame, respecting the equivalence rule 4.8, and excluding the graviton which is fixed. Then  
B =
1

2
(
24
2
) - 1
= 137
(8.2.1)
since, there are 24 rays on the particle frame and we must choose 2 rays to form a boson; but we must divide the total number by 2 to respect the equivalence rule, and then subtract 1 for the excluded graviton configuration. From experimental observations concerning strong interactions among neutrons and protons, the coupling constant for the strong force, after it separates from the gravitational force and the quarks and gluons become confined, is given by  
αstrong = 1
(8.2.2)
The coupling constants for the other fields must be given in comparison to αstrong. Define the coupling constants for the other fields as  
αfield =
1

Bn
/ αstrong
(8.2.3)
where n is the number of carrier bosons for the field. For the electromagnetic field, by 8.2.3 we have
  
αelectromagnetic =
1

1371
/ 1 = 7.29 × 10-3
(8.2.4)
since n = 1 corresponding to the only carrier of the electromagnetic force, the photon. For the weak field, by 8.2.3 we have
  
αweak =
1

1373
/ 1 = 3.89 × 10-7
(8.2.5)
since n = 3 corresponding to the three carriers of the weak force: Z0, W+ and W-. The only remaining field is the gravitational field that becomes dominant only at very high energy scales, near the Planck epoch.  The gravitational coupling constant according to 8.2.3 must be different before and after the strong force separates from the gravitational force. We find that the coupling constant here must be regarded as a running coupling (the theory of running couplings is called the renormalization group). However, we can give explicit bounds on the running gravitational coupling constant. Before the strong force separates from the gravitational force, we have αstrong = 1/ 1378 since the quarks and the 8 gluons that are the carriers of the strong force are free at this energy scale. Then by 8.2.3 we have
  
αgravitational =
1

13724
/
1

1378
= 6.49 × 10-35
(8.2.6)
since n = 24 corresponding to the 24 species of gravitons that act as carriers of the garvitational force. After the strong force separates from the gravitational force, we have αstrong = 1 according to 8.2.2 and then by 8.2.3 we have
  
αgravitational =
1

13724
/ 1 = 5.23 × 10-52
(8.2.7)
since n = 24 corresponding to the 24 species of gravitons that act as carriers of the garvitational force. Thus, the running gravitational coupling constant is given by
  
5.23 × 10-52 < αgravitational < 6.49 × 10-35
(8.2.8)
The gravitational coupling constant must have been a very small number, practically zero, immediately after the Big Bang at time T = 0.
 

8.3. The Mass Ratios

We shall explicitly define the particle mass ratios by counting the number of regions in the subdivision of the upper surface of the particle frame. These numbers are symmetric across the lower surface of the particle frame, giving the antiparticle mass ratios. Define nfield to be the number of regions in the subdivision of the upper surface of the particle frame during the separation of the corresponding force in the cosmological timeline. Then  
ngravitational = 8
nstrong = 16
nweak = 40
nelectromagnetic = 48
(8.3.1)
Figure 8.3.1. Number of regions

We define the mass ratio constants in direct analogy with the coupling constants. The mass ratio constant corresponding to the strong force, after it separates from the gravitational force and the quarks become confined, is given by  

βstrong = 1
(8.3.2)
The mass ratio constants corresponding to the other fields must be given in comparison to βstrong. Define the mass ratio constants corresponding to the other fields as  
βfield = nfield / βstrong
(8.3.3)
Corresponding to the electromagnetic field, by 8.3.3 we have  
βelectromagnetic = 48 / 1 = 48
(8.3.4)
Corresponding to the weak field, by 8.3.3 we have  
βweak = 40 / 1 = 40
(8.3.5)
The only remaining field is the gravitational field that becomes dominant only at very high energy scales, near the Planck epoch.  The mass ratio constant corresponding to the gravitational field according to 8.3.3 must be different before and after the strong force separates from the gravitational force. Before the strong force separates from the gravitational force, we have βstrong = 16, since the quarks are free at this energy scale. Then by 8.3.3 we have  
βgravitational = 8 / 16 = 1/2
(8.3.6)
After the strong force separates from the gravitational force, we have βstrong = 1 according to 8.3.2 and then by 8.3.3 we have  
βgravitational = 8 / 1 = 8
(8.3.7)
To this list we add the mass ratio constant cos(30) = √3/2, given by the Weinberg angle θw = 30 degrees obtained in 7.2.9, and zero, to account for massless particles. Thus, we obtain the eight mass ratio constants  
0, 1/2, √3/2, 1, 8, 16, 40, 48
(8.3.8)
Let us select a unit of mass such that the electron mass is 1. Then, we can compare the mass ratio constants 8.3.8 with the experimentally observed mass ratios, as follows.
 
up quark mass / down quark mass = 5 / 10 = 1/2
charm quark mass / strange quark mass = 1600 / 180  8
top quark mass / bottom quark mass = 180000 / 4500 = 40
tau mass / muon mass = 1771 / 105.658 16
electron mass / unit mass = 1 / 1 = 1
Z0 mass / top quark mass = 91188 / 180000 1/2
Average Mixed Z0, W±mass / tau mass = (91188+80280) / 2 / 1771 48
W± mass / Z0mass = 80280 / 91188 √3/2
Higgs mass / Sum of Z0, W+, W- masses = 125874 / (91188+80280+80280) = 1/2

The exact values of the neutrino masses are not yet known, but we know that they are not zero, and perhaps the neutrino mass ratios will also be found amongst the mass ratio constants 8.3.8. We have now calculated all the parameters that define the standard model and its associated force fields, according to 't Hooft's specification [8]. Thus, the mathematical proof of the four colour theorem shows that the grand unification of the standard model with quantum gravity is complete, and rules out the possibility of finding any other kinds of particles.

 

References

 
[1] Ashay Dharwadker, A New Proof of the Four Colour Theorem, http://www.dharwadker.org , 2000
[2] Ashay Dharwadker, The Witt Design, http://www.dharwadker.org/witt.html , 2001
[3] Ashay Dharwadker, Riemann Surfaces, http://www.eg-models.de/models/Surfaces/Riemann_Surfaces/2002.05.001 , 2002
[4] P. A. M. Dirac, The Evolution of the Physicist's Picture of Nature, Scientific American, 1963
[5] P. A. M. Dirac, General Theory of Relativity, Princeton University Press, 1996
[6] Albert Einstein, On the Generalized Theory of Gravitation, Scientific American, 1950
[7] Albert Einstein, The Meaning of Relativity, Princeton University Press, 1956
[8] Gerard 't Hooft, In search of the ultimate building blocks, Cambridge University Press, 1997
[9] Leonard I. Schiff, Quantum Mechanics, McGraw-Hill Inc, 1968
[10] Official Super-Kamiokande Press Release, Evidence for massive neutrinos, http://neutrino.phys.washington.edu/~superk/sk_release.html , 1998
[11] Steven Weinberg, The First Three Minutes: A Modern View Of The Origin Of The Universe, Basic Books, 1993
[12] Weak Mixing Angle Measurement, SLAC E-158, http://www-project.slac.stanford.edu/e158/ , 2003

 
 


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