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. Google Scholar Citations © 2008

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. The Endowed Chair of the Institute of Mathematics was bestowed upon Distinguished Professor Ashay Dharwadker in 2012 to honour his fundamental contributions to Mathematics and Natural Sciences.

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 θ_w = 30 degrees (this is a running value). The Weinberg angle θ_w 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.

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  

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  

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   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.

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   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  

and its momentum vector is written as   The kinetic energy of the particle is given by the equation   On the other hand, we have Planck's equation for the energy of the wave packet:   We also have de Broglie's equation for the momentum of the wave packet:   To obtain agreement with physical observations we select a wave function (2.1) of the form:   Computing partial derivatives               Introducing the Laplace operator, from (2.7), (2.8) and (2.9) we have   From (2.2), (2.4) and (2.10) we have   On the other hand, from (2.3) and (2.6) we have   Equations (2.11) and (2.12) yield the Schrödinger wave equation   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:     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   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   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   The h² in equation 2.18 cancels and we have   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) = e^i(kX^X+kY^Y+kZ^Z-ωT) 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.

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 Z_N = {0, 1, ..., N-1} under addition modulo N. We regard Z_N as an Eilenberg module for the symmetric group S₃ on three letters and consider the split extension Z_N]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 |Z_N]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 Z_N]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:  

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 

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   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   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]S₃)×ZS₃. 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 {φ_i | 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 φ_i 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.

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].

 
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.

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.

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.

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.

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.

We first associate each colour with a unique absolute value of electric charge according to the following scheme:   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.

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 N_c. 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 N_c = 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 N_c = 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, N_c = 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 N_c = 1
• The W⁺ boson has N_c = 1
• The W^- boson has N_c = 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, N_c = 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, N_c = 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.

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 ψ_S, 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 ψ_S as measuring the degree of disorder of the system defining the particle, so that ψ_S 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 ψ_S, as follows. Write ψ_S = R(δ)φ_j as the unique expression in terms of the common coset representatives and, by definition ψ_S^μ = φ_jR(δ), as described in [1]. Then ψ_S and ψ_S^μ 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)ψ_S = (m, α)ψ_S^μ for any selected region. The given particle is represented on the particle frame corresponding to the t-Riemann surface with φ_i, 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)ψ_S = (m, α)ψ_S^μ. 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 ψ_S = R(δ)φ_j has φ_j equal to the identity permutation.
• For the massive bosons S, we assume that the uniquely associated permutation ψ_S = R(δ)φ_j has δ = 1.
• Furthermore, for the massive Higgs boson, we assume that the permutation φ_j 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 ψ_S, 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 ψ_S 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).

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.

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.

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.

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. 

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.

 

Section 5.1	Leptons
Section 5.2	Quarks

 

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

Let ψ_e-neutrino be the unique permutation associated with the e-neutrino particle according to the mass rule 4.7. Select φ_i and β according to the unique expression ψ_e-neutrino = φ_iR(β) and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φ_i, 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)φ_iR(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 N_c= 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

The e-neutrino antiparticle is equivalent to the e-neutrino particle, as shown by the following construction. Let ψ_e-neutrino be the unique permutation associated with the e-neutrino particle according to the mass rule 4.7. Select φ_i and β according to the unique expression ψ_e-neutrino = φ_iR(β) and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φ_i, 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)φ_i, 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 N_c = 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

Let ψ_electron be the unique permutation associated with the electron particle according to the mass rule 4.7. Select φ_i and γ according to the unique expression ψ_electron = R(γ)φ_i and β = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φ_i, 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)φ_i, 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 N_c = 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

Let ψ_electron be the unique permutation associated with the electron particle according to the mass rule 4.7. Select φ_i and γ according to the unique expression ψ_electron = R(γ)φ_i and β = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φ_i, 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)φ_iR(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 N_c = 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)

Let ψ_μ-neutrino be the unique permutation associated with the μ-neutrino particle according to the mass rule 4.7. Select φ_i and β according to the unique expression ψ_μ-neutrino = φ_iR(β) and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φ_i, 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)φ_iR(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 N_c = 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

The μ-neutrino antiparticle is equivalent to the μ-neutrino particle, as shown by the following construction. Let ψ_μ-neutrino be the unique permutation associated with the μ-neutrino particle according to the mass rule 4.7. Select φ_i and β according to the unique expression ψ_μ-neutrino = φ_iR(β) and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φ_i, 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)φ_i, 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 N_c = 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

Let ψ_muon be the unique permutation associated with the muon particle according to the mass rule 4.7. Select φ_i and γ according to the unique expression ψ_muon = R(γ)φ_i and β = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φ_i, 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)φ_i, 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 N_c = 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

Let ψ_muon be the unique permutation associated with the muon particle according to the mass rule 4.7. Select φ_i and γ according to the unique expression ψ_muon = R(γ)φ_i and β = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φ_i, 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)φ_iR(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 N_c = 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

Let ψ_τ-neutrino be the unique permutation associated with the τ-neutrino particle according to the mass rule 4.7. Select φ_i and β according to the unique expression ψ_τ-neutrino = φ_iR(β) and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φ_i, β, γ. We select disc 3 on the upper sheet of the particle frame and its blue region 0, that has the label (+(0, ρ²)φ_iR(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 N_c = 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

The τ-neutrino antiparticle is equivalent to the τ-neutrino particle, as shown by the following construction. Let ψ_τ-neutrino be the unique permutation associated with the τ-neutrino particle according to the mass rule 4.7. Select φ_i and β according to the unique expression ψ_τ-neutrino = φ_iR(β) and γ = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φ_i, 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)φ_i, 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 N_c = 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

Let ψ_tau be the unique permutation associated with the tau particle according to the mass rule 4.7. Select φ_i and γ according to the unique expression ψ_tau = R(γ)φ_i and β = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φ_i, 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)φ_i, 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 N_c = 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

Let ψ_tau be the unique permutation associated with the tau particle according to the mass rule 4.7. Select φ_i and γ according to the unique expression ψ_tau = R(γ)φ_i and β = 1. Then the particle frame corresponds to the t-Riemann surface with this choice of φ_i, 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, ρ²)φ_iR(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 N_c = 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

 

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

Let ψ_up-quark be the unique permutation associated with the up quark particle according to the mass rule 4.7. Select φ_i and β according to the unique expression ψ_up-quark = φ_iR(β) and γ = σ, σρ or σρ². Then the particle frame corresponds to the t-Riemann surface with these choices of φ_i, 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)φ_iR(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 N_c = 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

Let ψ_up-quark be the unique permutation associated with the up quark particle according to the mass rule 4.7. Select φ_i and β according to the unique expression ψ_up-quark = φ_iR(β) and γ = σ, σρ or σρ². Then the particle frame corresponds to the t-Riemann surface with these choices of φ_i, 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)φ_i,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 N_c = 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

Let ψ_down-quark be the unique permutation associated with the down quark particle according to the mass rule 4.7. Select φ_i and γ according to the unique expression ψ_down-quark = R(γ)φ_i and β = σ, σρ or σρ². Then the particle frame corresponds to the t-Riemann surface with these choices of φ_i, 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)φ_i, 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 N_c = 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

Let ψ_down-quark be the unique permutation associated with the down quark particle according to the mass rule 4.7. Select φ_i and γ according to the unique expression ψ_down-quark = R(γ)φ_i and β = σ, σρ or σρ². Then the particle frame corresponds to the t-Riemann surface with these choices of φ_i, 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)φ_iR(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 N_c = 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

Let ψ_charm-quark be the unique permutation associated with the charm quark particle according to the mass rule 4.7. Select φ_i and β according to the unique expression ψ_charm-quark = φ_iR(β) and γ = σ, σρ or σρ². Then the particle frame corresponds to the t-Riemann surface with these choices of φ_i, 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)φ_iR(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 N_c = 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

Let ψ_charm-quark be the unique permutation associated with the charm quark particle according to the mass rule 4.7. Select φ_i and β according to the unique expression ψ_charm-quark = φ_iR(β) and γ = σ, σρ or σρ². Then the particle frame corresponds to the t-Riemann surface with these choices of φ_i, 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)φ_i, 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 N_c = 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

Let ψ_{strange-quark} be the unique permutation associated with the strange quark particle according to the mass rule 4.7. Select φ_i and γ according to the unique expression ψ_{strange-quark} = R(γ)φ_i and β = σ, σρ or σρ². Then the particle frame corresponds to the t-Riemann surface with these choices of φ_i, 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)φ_i, 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 N_c = 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

Let ψ_{strange-quark} be the unique permutation associated with the strange quark particle according to the mass rule 4.7. Select φ_i and γ according to the unique expression ψ_{strange-quark} = R(γ)φ_i and β = σ, σρ or σρ². Then the particle frame corresponds to the t-Riemann surface with these choices of φ_i, 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)φ_iR(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 N_c = 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

Let ψ_top-quark be the unique permutation associated with the top quark particle according to the mass rule 4.7. Select φ_i and β according to the unique expression ψ_top-quark = φ_iR(β) and γ = σ, σρ or σρ². Then the particle frame corresponds to the t-Riemann surface with these choices of φ_i, β, γ. We select disc 4 on the upper sheet of the particle frame and its green region 2, that has the label (+(2, σρ²)φ_iR(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 N_c = 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

Let ψ_top-quark be the unique permutation associated with the top quark particle according to the mass rule 4.7. Select φ_i and β according to the unique expression ψ_top-quark = φ_iR(β) and γ = σ, σρ or σρ². Then the particle frame corresponds to the t-Riemann surface with these choices of φ_i, 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)φ_i, 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 N_c = 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

Let ψ_bottom-quark be the unique permutation associated with the bottom quark particle according to the mass rule 4.7. Select φ_i and γ according to the unique expression ψ_bottom-quark = R(γ)φ_i and β = σ, σρ or σρ². Then the particle frame corresponds to the t-Riemann surface with these choices of φ_i, 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)φ_i, 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 N_c = 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

Let ψ_bottom-quark be the unique permutation associated with the bottom quark particle according to the mass rule 4.7. Select φ_i and γ according to the unique expression ψ_bottom-quark = R(γ)φ_i and β = σ, σρ or σρ². Then the particle frame corresponds to the t-Riemann surface with these choices of φ_i, 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, σρ²)φ_iR(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 N_c = 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

 

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

The photon is the carrier of the electromagnetic force. The unique permutation ψ_photon = R(β)φ_i associated with the massless photon particle must have φ_i 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 φ_i, 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)φ_iR(β), β+γ), (+(0, ρ)φ_iR(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)φ_i, β+γ), (-(0, ρ)R(γ)φ_i, 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 N_c = 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

The photon antiparticle is equivalent to the photon particle, as shown by the following construction. The unique permutation ψ_photon = R(β)φ_i associated with the massless photon particle must have φ_i 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 φ_i, 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)φ_i, β+γ), (-(0, ρ)R(β)φ_i, 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)φ_iR(γ), β+γ), (+(0, ρ)φ_iR(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 N_c = 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

The vector boson Z⁰ particle is the neutral carrier of the weak force. The unique permutation ψ_Z⁰ = R(β)φ_i 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 φ_i, 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, ρ²)φ_iR(β), β+γ), (+(0, σρ²)φ_iR(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)φ_i, β+γ), (-(0, σρ²)R(γ)φ_i, 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 N_c = 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

The Z⁰ antiparticle is equivalent to the Z⁰ particle, as shown by the following construction. The unique permutation ψ_Z⁰ = R(β)φ_i 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 φ_i, 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)φ_i, β+γ), (-(0, σρ²)R(β)φ_i, 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, ρ²)φ_iR(γ), β+γ), (+(0, σρ²)φ_iR(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 N_c = 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

The vector boson W⁺ particle is the positive carrier of the weak force. The unique permutation ψ_W⁺ = R(β)φ_i 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 φ_i, 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)φ_iR(β), β+γ), (+(3, ρ²)φ_iR(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)φ_iR(γ), β+γ), (+(3, ρ²)φ_iR(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 N_c = 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

The W⁺ antiparticle is equivalent to the W^- particle, as shown by the following construction. The unique permutation ψ_W⁺ = R(β)φ_i 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 φ_i, 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)φ_i, β+γ), (-(3, ρ²)R(β)φ_i, 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)φ_i, β+γ), (-(3, ρ²)R(γ)φ_i, 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 N_c = 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

The vector boson W^- particle is the negative carrier of the weak force. The unique permutation ψ_W^- = R(β)φ_i 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 φ_i, 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)φ_i, β+γ), (-(3, σρ)R(β)φ_i, 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)φ_i, β+γ), (-(3, σρ)R(γ)φ_i, 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 N_c = 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

The W^- antiparticle is equivalent to the W⁺ particle, as shown by the following construction. The unique permutation ψ_W^- = R(β)φ_i 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 φ_i, 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, σρ²)φ_iR(β), β+γ), (+(3, σρ)φ_iR(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, σρ²)φ_iR(γ), β+γ), (+(3, σρ)φ_iR(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 N_c = 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

The gluon is the carrier of the strong force. The unique permutation ψ_gluon = R(β)φ_i associated with the massless gluon particle must have φ_i 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 φ_i, 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)φ_iR(β), β+γ), (+(0, σ)φ_iR(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)φ_i, β+γ), (-(0, σ)R(γ)φ_i, 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 N_c = 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