ASHAY DHARWADKER
DISTINGUISHED PROFESSOR OF
MATHEMATICS & NATURAL SCIENCES
INSTITUTE OF MATHEMATICS
H501 PALAM VIHAR
DISTRICT GURGAON
HARYANA 1 2 2 0 1 7
INDIA
Abstract 
We show that the mathematical proof of
the four colour theorem [1] directly implies the
existence of the standard model, together with quantum gravity, in its physical
interpretation. Conversely, the experimentally observable standard model and quantum
gravity show that nature applies the mathematical proof of the four colour theorem, at the
most fundamental level. We preserve all the established working theories of physics:
Quantum Mechanics, Special and General Relativity, Quantum Electrodynamics (QED), the
Electroweak model and Quantum Chromodynamics (QCD). We build upon these theories,
unifying all of them with Einstein's law of gravity. Quantum gravity is a direct and
unavoidable consequence of the theory. The main construction of the Steiner system in the
proof of the four colour theorem already defines the gravitational fields of all the particles
of the standard model. Our first goal is to construct all the particles constituting the classic
standard model, in exact agreement with 't Hooft's table
[8]. We are able to predict the
exact mass of the Higgs particle and the CP violation and mixing angle of weak
interactions. Our second goal is to construct the gauge groups and explicitly calculate the
gauge coupling constants of the force fields. We show how the gauge groups are
embedded in a sequence along the cosmological timeline in the grand unification. Finally,
we calculate the mass ratios of the particles of the standard model. Thus, the mathematical
proof of the four colour theorem shows that the grand unification of the standard model with
quantum gravity is complete, and rules out the possibility of finding any other kinds of
particles.


Acknowledgements 
Recent collaborative work with Vladimir Khachatryan shows how to use the grand unified theory to calculate the values of the Cabibbo angle and CKM matrix and also predict the Higgs Boson Mass [arXiv:0912.5189] with precision. We are pleased to announce that The Grand Unification has
been published by Amazon in 2011. 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.




Introduction 

We show that the mathematical proof of
the four colour theorem [1] directly implies
the existence of the standard model, together with quantum gravity, in
its physical interpretation. Conversely, the experimentally observable
standard model and quantum gravity show that nature applies the mathematical
proof of the four colour theorem, at the most fundamental level. We should
emphasize that we preserve all the established working theories of physics:
Planck's Quantum Mechanics, Einstein's Special and General Relativity,
Maxwell's Electromagnetism, Feynman's Quantum Electrodynamics (QED), the
WeinbergSalamWard Electroweak model and GlashowIliopoulosMaiani'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 tRiemann surface. All the particles of the standard model will be obtained by arranging copies of oriented Schrödinger discs on the labeled tRiemann surface, as dictated by the proof of the four colour theorem. In Section 4, we define the Particle Frame in terms of the labeled tRiemann surface. Particle frames associated with spacetime points constitute a vector bundle in mathematical terminology, and a section of the vector bundle i.e. a particle frame at a spacetime 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 CPTransformation 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:
In Section 6, we explicitly define all the Bosons
and their antiparticles, following all the above rules:
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: electronelectron scattering.
In Section 7.1.4, we define the electromagnetic
gauge group
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 YangMills
Weak Field Equations. In Section 7.2.2, we
show how the Z^{0} acts as the neutral carrier of the weak force.
In Section 7.2.3, we give an example of a typical
weak Z^{0} interaction: muonic neutrinoelectron 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
We explicitly define the observable gauge vector bosons [Z^{0}], [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 Z^{0} masses, as well as the ratio of the weak Z^{0} mediated interaction, called its mixing. In Section 7.3, we define the Strong Force
Field. In Section 7.3.1, we define the YangMills
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 quarkantiquark pair, called a meson. In Section
7.3.4, we define the strong gauge group
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 spacetime. 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 spacetime 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 spacetime without selfintersections.
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
We explicitly define the twentyfour 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
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:
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:
We have now calculated all the parameters that define the standard model and its associated force fields, according to 't Hooft's specification [8]. Thus, the mathematical proof of the four colour theorem shows that the grand unification of the standard model with quantum gravity is complete, and rules out the possibility of finding any other kinds of particles. 
1. Special Relativity 

The spacetime 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 forcefree 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 spacetime 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 spacetime 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
The equation (1.1) may be rewritten as

2. The Schrödinger Wave Equation 

Several ingenious physical experiments
performed during the early 1900's irrefutably demonstrated the dual nature
of light waves (electromagnetic radiation) and particles of matter:
light waves behave like material particles and the particles of matter
behave like light waves. Furthermore, physical quantities like energy always
occur in discrete packets, called quanta. The theory of quantum mechanics
[9]
was developed to describe these phenomena, and one of its fundamental rules
is the uncertainty principle: it is impossible to specify precisely and
simultaneously both the position and the momentum of a physical particle.
The main idea was to use concentrated bunches of waves (called wave packets)
to describe all particles (localized particles of matter and quanta of
electromagnetic radiation). A particle is described by a wave function
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
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 spacetime, 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 spacetime. 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 lefthand or the righthand rule. We align this normal vector with the momentum vector p of the particle in the case that p is precisely specified. A lefthanded orientation of a Schrödinger disc D will represent a particle of lefthanded helicity and a righthanded orientation will represent a particle of righthanded 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 oneparticle Schrödinger wave equation is modified to its manyparticle version and its solution Ψ is construed as having many copies that describe the associated fields of the particles. In particular, the copies of the wave function Ψ represent the processes of creation and destruction of particles and the interacting particles of the associated force fields. For example, if the particle is a photon, then we can interpret Ψ to be either the electric field or the magnetic field and equation 2.19 describes the propagation of electromagnetic waves in vacuum. We shall build the standard model of particle physics from copies of oriented Schrödinger discs, arranged in a certain way as dictated by the mathematical proof of the four colour theorem. 
3. The Four Colour Theorem 

We shall now follow the main construction
in the proof of the four colour theorem [1].
Let us briefly state the theorem and sketch the proof, referring to [1]
for details. A map on the sphere or plane is a subdivision of the surface
into finitely many regions. Two regions in a map are said to be adjacent
if they share a whole segment of their boundaries in common. A proper colouring
of the regions of a map is an assignment of a colour to each region such
that no two adjacent regions receive the same colour.
3.1. The Four Colour Theorem. Given any map on the sphere or plane, it is always possible to properly colour the regions of the map using at most four colours 0, 1, 2, 3. Sketch of the proof [1]: In section I on map colouring, we define maps on the sphere and their proper colouring. For purposes of proper colouring it is equivalent to consider maps on the plane and furthermore, only maps which have exactly three edges meeting at each vertex. Lemma 1 proves the six colour theorem using Euler's formula, showing that any map on the plane may be properly coloured by using at most six colours. We may then make the following basic definitions.
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:
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.
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.
We now follow the construction of the tRiemann surface in [1].
Consider the composition of the functions C → C;
z → t
= z^{2} and C → C;
t → w
= t^{12}. The composite is given by the assignment z → t
= z^{2} → w = t^{12
}=
z^{24}.
Take twentyfour identical copies of the map m(4) on the
disc with a cut, labeled k = 1, ..., 24.
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 wRiemann surface. The point w = 0 connects all the discs and is called the branch point. There are twentyfour superposed copies of the map m(4) on the wRiemann surface corresponding to the twentyfour sectors
We shall now label the regions of the maps on the tRiemann surface. Referring to [1] for details, let S_{3} = <σ, ρ> = {1, ρ, ρ^{2}, σρ^{2}, σρ, σ} denote the dihedral group of order 6, abstractly isomorphic to the symmetric group on 3 letters. Corresponding to the trivial representation of S_{3}, we regard Z_{4} as its Eilenberg module and form the split extension Z_{4}]S_{3} that is abstractly isomorphic to the direct product Z_{4}×S_{3}. Next, form the integral group algebras Z(Z_{4}]S_{3}) and ZS_{3}. Again, corresponding to the trivial representation of ZS_{3}, we regard Z(Z_{4}]S_{3}) as its Eilenberg module and form the split extension Z(Z_{4}]S_{3})]ZS_{3} that is abstractly isomorphic to the direct product Z(Z_{4}]S_{3})×ZS_{3}. Let Sym(Z_{4}]S_{3}) denote the symmetric group of order 24! on Z_{4}]S_{3}=24 letters. Then S_{3} embeds in Sym(Z_{4}]S_{3}) 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 S_{3} in the group Sym(Z_{4}]S_{3}). Fix a common coset representative φ_{i} and a pair of elements β, γ of S_{3}. The regions of the maps on the tRiemann surface are labeled by elements of the split extension Z(Z_{4}]S_{3})]ZS_{3} according to the following scheme. 
Labeling Scheme 
The branch point at the center which connects all the sheets is labeled (0, β+γ). 

The regions of the maps on the upper half of the upper sheet of the tRiemann surface are labeled as follows (all labels have a positive sign):


The regions of the maps on the lower half of the upper sheet of the tRiemann surface are labeled as follows (all labels have a negative sign):


The regions of the maps on the upper half of the lower sheet of the tRiemann surface are labeled as follows (all labels have a negative sign):


The regions of the maps on the lower half of the lower sheet of the tRiemann surface are labeled as follows (all labels have a positive sign):

Figure 3.5. Labeling scheme for the regions of the maps on the tRiemann surface 
The main construction in the proof [1]
now defines the Steiner system S(5, 8, 24). The 24 points of the
Steiner system are the elements of the underlying set Z_{4}]S_{3}
The rest of the main construction [1] builds the blocks of the Steiner system. Each block consists of 8 points such that any set of 5 points is contained in a unique block [2]. 
4. The Particle Frame 



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 tRiemann 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 spacetime point (X, Y, Z, T), as described in section 2. Particle frames associated with spacetime points constitute a vector bundle in mathematical terminology, and a section of the vector bundle i.e. a particle frame at a spacetime 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 tRiemann 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 spacetime point in vacuum, is shown below.
For the purposes of defining the particles of the standard model, it is convenient to draw the particle frame embedded in flat Euclidean threedimensional 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 fourdimensional spacetime, 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]:

4.1. The Fermion Selection Rule 

Particles of the standard model that obey
the FermiDirac statistics [9] are called
fermions. Hence, distinct particle frames with fermions defined on them
cannot be superposed at a point in spacetime 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.
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 ρ^{2} (generation III) of the tRiemann surface. Type 2 fermions, called quarks, consist of a disc corresponding to a label σ (generation I), σρ (generation II) or σρ^{2} (generation III) of the tRiemann surface. Each generation I, II and III consists of one lepton doublet and one quark doublet. In section 5, we shall see how to use this rule to define all the fermions in the standard model. 
4.2. The Boson Selection Rule 

Particles of the standard model that obey
the BoseEinstein statistics
[9] are called
bosons. Hence, many distinct particle frames with bosons defined on them
can be superposed at a point in spacetime. 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.
In section 6, we shall see how to define all the bosons of the standard model using this rule. In particular, the two pairs of fermion type particles that define a boson are interpreted as creation and destruction operators during interactions in which the boson is exchanged. 
4.4. The Spin Rule 
The particle frame consists of four halfsurfaces:

4.5. The Electric Charge Rule 

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

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

When we speak of charge without any further
specification, we always mean the electric charge. However, in addition
to an electric charge, a particle also has electromagnetic, weak, strong
and gravitational charges according to the following scheme:
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.

4.7. The Mass Rule 
The rest mass of a particle in the standard
model is usually determined from experimental observations. However, we
shall show in 8.3 that the rest masses of all the particles in the standard
model cannot be independent and most of the mass ratios must be fixed quite
precisely due to the structure of the particle frame. We define the assignment
of the rest mass of a particle of the standard model in the following way.
Referring to section 3, recall that each element
ψ
of Sym(Z_{4}]S_{3}) is a permutation
of the underlying set Z_{4}]S_{3}.
Each permutation ψ may be thought of as representing
the entropy or disorder of the set Z_{4}]S_{3},
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_{j}R(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 tRiemann 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:

4.8. The Equivalence Rule 
Given two different selections S_{1}
and S_{2} 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 S_{1} and S_{2} as representing
the same kind of particle in the standard model.

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

4.10. The Helicity Rule 
The helicity of the particle is defined
by selecting one of two possible orientations (lefthanded or righthanded)
for all the active Schrödinger discs on the particle frame. This orientation
carries over to neighboring discs and defines an orientation of the tRiemann
surface. Note that by section 2, the helicity is welldefined and conserved
for massless particles but not for massive particles.

4.11. The CP Transformation Rule 
Given a particle defined on the particle
frame at a spacetime point (X,
Y,
Z,
T) with
momentum vector p, we define the transformations C
and P as follows:

4.12. The Standard Model Completion Rule 

If all the particle frames corresponding
to all the particles in the universe were to be superimposed (hypothetically,
of course), then the fermions and bosons should fit together perfectly
according to the above rules, forming the complete standard model.
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.
Each of the 24 pairs of rays of the particle frame represents the four Schrödinger discs of a unique boson in the standard model, respecting all the above rules. There cannot be any other bosons in the standard model. 
5. Fermions 


5.1. Leptons 


5.1.1. The eneutrino particle 

Let ψ_{eneutrino}
be the unique permutation associated with the eneutrino particle
according to the mass rule 4.7. Select φ_{i}
and β according to the unique expression ψ_{eneutrino}
= φ_{i}R(β)
and γ = 1. Then the particle frame corresponds
to the tRiemann 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_{i}R(β),
β+γ),
according to figures 3.4 and 3.5. This represents the Schrödinger
disc of the eneutrino particle on the particle frame, according
to figures 2.1 and 4.1. By the fermion selection rule 4.1, the eneutrino
particle is a lepton of generation I. By the spin rule 4.4, the spin of
the eneutrino particle is 1/2. By the electric charge rule 4.5,
the electric charge of the eneutrino particle is 0 and by the strong
charge rule 4.6, its strong charge is neutral with N_{c}=
1. The SuperKamiokande experiment [10]
demonstrated that the
eneutrino particle has a small positive rest
mass (not precisely determined as yet), which can be attributed by the
mass rule 4.7 to the HiggsKibble mechanism. This experiment also demonstrated
that the eneutrino particle can be observed with both righthanded
and lefthanded helicities, in agreement with the helicity rule 4.10. Note
that by the equivalence rule 4.8, the
eneutrino particle and antiparticle
5.1.2 represent the same kind of particle in the standard model. Thus,
the eneutrino particle is a Majorana fermion. By the CP
transformation rule 4.11, a righthanded eneutrino particle is
transformed into a lefthanded eneutrino particle and viceversa.

5.1.2. The eneutrino antiparticle 

The eneutrino antiparticle is
equivalent to the eneutrino particle, as shown by the following
construction. Let ψ_{eneutrino}
be the unique permutation associated with the eneutrino particle
according to the mass rule 4.7. Select φ_{i}
and β according to the unique expression ψ_{eneutrino}
= φ_{i}R(β)
and γ = 1. Then the particle frame corresponds
to the tRiemann 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 eneutrino antiparticle on the particle frame, according
to figures 2.1 and 4.1. By the fermion selection rule 4.1, the eneutrino
antiparticle is a lepton of generation I. By the spin rule 4.4, the spin
of the eneutrino antiparticle is 1/2. By the electric charge rule
4.5, the electric charge of the eneutrino antiparticle is 0 and
by the strong charge rule 4.6, its strong charge is neutral with N_{c}
= 1. The SuperKamiokande experiment [10]
demonstrated that the
eneutrino antiparticle has a small positive
rest mass (not precisely determined as yet), which can be attributed by
the mass rule 4.7 to the HiggsKibble mechanism. This experiment also demonstrated
that the eneutrino antiparticle can be observed with both righthanded
and lefthanded helicities, in agreement with the helicity rule 4.10. Note
that by the equivalence rule 4.8, the eneutrino antiparticle and
particle 5.1.1 represent the same kind of particle in the standard model.
Thus, the eneutrino antiparticle is a Majorana fermion. By the
CP
transformation rule 4.11, a righthanded eneutrino antiparticle
is transformed into a lefthanded eneutrino antiparticle and viceversa.

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 tRiemann 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 HiggsKibble mechanism. The electron particle
can be observed with both righthanded and lefthanded helicities, in agreement
with the helicity rule 4.10. By the CP transformation rule 4.11,
a righthanded electron particle is transformed into a lefthanded electron
antiparticle (positron) and a lefthanded electron particle is transformed
into a righthanded electron antiparticle (positron).

5.1.4. The electron antiparticle (positron) 

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 tRiemann 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_{i}R(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 HiggsKibble
mechanism. The electron antiparticle can be observed with both righthanded
and lefthanded helicities, in agreement with the helicity rule 4.10. By
the CP transformation rule 4.11, a righthanded electron antiparticle
is transformed into a lefthanded electron particle and a lefthanded electron
antiparticle is transformed into a righthanded electron particle.

5.1.5. The μneutrino particle 

Let ψ_{μ}_{neutrino}
be the unique permutation associated with the μneutrino
particle according to the mass rule 4.7. Select φ_{i}
and β according to the unique expression ψ_{μ}_{neutrino}
= φ_{i}R(β)
and γ = 1. Then the particle frame corresponds
to the tRiemann 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_{i}R(β),
β+γ),
according to figures 3.4 and 3.5. This represents the Schrödinger
disc of the μneutrino particle on the particle
frame, according to figures 2.1 and 4.1. By the fermion selection rule
4.1, the μneutrino particle is a lepton of
generation II. By the spin rule 4.4, the spin of the μneutrino
particle is 1/2. By the electric charge rule 4.5, the electric charge of
the μneutrino particle is 0 and by the strong
charge rule 4.6, its strong charge is neutral with N_{c}
= 1. The SuperKamiokande experiment [10]
demonstrated that the
μneutrino particle has
a small positive rest mass (not precisely determined as yet), which can
be attributed by the mass rule 4.7 to the HiggsKibble mechanism. This
experiment also demonstrated that the μneutrino
particle can be observed with both righthanded and lefthanded helicities,
in agreement with the helicity rule 4.10. Note that by the equivalence
rule 4.8, the μneutrino particle and antiparticle
5.1.6 represent the same kind of particle in the standard model. Thus,
the μneutrino particle is a Majorana fermion.
By the CP transformation rule 4.11, a righthanded μneutrino
particle is transformed into a lefthanded μneutrino
particle and viceversa.

5.1.6. The μneutrino antiparticle 

The μneutrino
antiparticle is equivalent to the μneutrino
particle, as shown by the following construction. Let ψ_{μ}_{neutrino}
be the unique permutation associated with the μneutrino
particle according to the mass rule 4.7. Select φ_{i}
and β according to the unique expression ψ_{μ}_{neutrino}
= φ_{i}R(β)
and γ = 1. Then the particle frame corresponds
to the tRiemann 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 μneutrino antiparticle on the particle
frame, according to figures 2.1 and 4.1. By the fermion selection rule
4.1, the μneutrino antiparticle is a lepton
of generation II. By the spin rule 4.4, the spin of the μneutrino
antiparticle is 1/2. By the electric charge rule 4.5, the electric charge
of the μneutrino antiparticle is 0 and by the
colour charge rule 4.6, its colour charge is neutral with N_{c}
= 1. The SuperKamiokande experiment [10]
demonstrated that the
μneutrino antiparticle
has a small positive rest mass (not precisely determined as yet), which
can be attributed by the mass rule 4.7 to the HiggsKibble mechanism. This
experiment also demonstrated that the μneutrino
antiparticle can be observed with both righthanded and lefthanded helicities,
in agreement with the helicity rule 4.10. Note that by the equivalence
rule 4.8, the μneutrino antiparticle and particle
5.1.5 represent the same kind of particle in the standard model. Thus,
the μneutrino antiparticle is a Majorana fermion.
By the CP transformation rule 4.11, a righthanded μneutrino
antiparticle is transformed into a lefthanded μneutrino
antiparticle and viceversa.

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 tRiemann 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 HiggsKibble
mechanism. The muon particle can be observed with both righthanded and
lefthanded helicities, in agreement with the helicity rule 4.10. By the
CP
transformation rule 4.11, a righthanded muon particle is transformed into
a lefthanded muon antiparticle and a lefthanded muon particle is transformed
into a righthanded muon antiparticle.

5.1.8. The muon 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 tRiemann 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_{i}R(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 HiggsKibble mechanism. The muon antiparticle
can be observed with both righthanded and lefthanded helicities, in agreement
with the helicity rule 4.10. By the CP transformation rule 4.11,
a righthanded muon antiparticle is transformed into a lefthanded muon
particle and a lefthanded muon antiparticle is transformed into a righthanded
muon particle.

5.1.9. The τneutrino particle 

Let ψ_{τ}_{neutrino}
be the unique permutation associated with the τneutrino
particle according to the mass rule 4.7. Select φ_{i}
and β according to the unique expression ψ_{τ}_{neutrino}
= φ_{i}R(β)
and γ = 1. Then the particle frame corresponds
to the tRiemann 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,
ρ^{2})φ_{i}R(β),
β+γ),
according to figures 3.4 and 3.5. This represents the Schrödinger
disc of the τneutrino particle on the particle
frame, according to figures 2.1 and 4.1. By the fermion selection rule
4.1, the τneutrino particle is a lepton of
generation III. By the spin rule 4.4, the spin of the τneutrino
particle is 1/2. By the electric charge rule 4.5, the electric charge of
the τneutrino particle is 0 and by the strong
charge rule 4.6, its strong charge is neutral with N_{c}
= 1. The SuperKamiokande experiment [10]
demonstrated that the
τneutrino particle has
a small positive rest mass (not precisely determined as yet), which can
be attributed by the mass rule 4.7 to the HiggsKibble mechanism. This
experiment also demonstrated that the τneutrino
particle can be observed with both righthanded and lefthanded helicities,
in agreement with the helicity rule 4.10. Note that by the equivalence
rule 4.8, the τneutrino particle and antiparticle
5.1.10 represent the same kind of particle in the standard model. Thus,
the τneutrino particle is a Majorana fermion.
By the CP transformation rule 4.11, a righthanded τneutrino
particle is transformed into a lefthanded τneutrino
particle and viceversa.

5.1.10. The τneutrino antiparticle 

The τneutrino
antiparticle is equivalent to the τneutrino
particle, as shown by the following construction. Let ψ_{τ}_{neutrino} be
the unique permutation associated with the τneutrino
particle according to the mass rule 4.7. Select φ_{i}
and β according to the unique expression ψ_{τ}_{neutrino}
= φ_{i}R(β)
and γ = 1. Then the particle frame corresponds
to the tRiemann 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,
ρ^{2})R(β)φ_{i},
β+γ),
according to figures 3.4 and 3.5. This represents the Schrödinger
disc of the τneutrino antiparticle on the particle
frame, according to figures 2.1 and 4.1. By the fermion selection rule
4.1, the τneutrino antiparticle is a lepton
of generation III. By the spin rule 4.4, the spin of the τneutrino
antiparticle is 1/2. By the electric charge rule 4.5, the electric charge
of the τneutrino antiparticle is 0 and by the
strong charge rule 4.6, its strong charge is neutral with
N_{c}
= 1. The SuperKamiokande experiment [10]
demonstrated that the
τneutrino antiparticle
has a small positive rest mass (not precisely determined as yet), which
can be attributed by the mass rule 4.7 to the HiggsKibble mechanism. This
experiment also demonstrated that the τneutrino
antiparticle can be observed with both righthanded and lefthanded helicities,
in agreement with the helicity rule 4.10. Note that by the equivalence
rule 4.8, the
τneutrino antiparticle and particle
5.1.9 represent the same kind of particle in the standard model. Thus,
the τneutrino antiparticle is a Majorana fermion.
By the CP transformation rule 4.11, a righthanded τneutrino
antiparticle is transformed into a lefthanded τneutrino
antiparticle and viceversa.

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 tRiemann 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,
ρ^{2})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 HiggsKibble
mechanism. The tau particle can be observed with both righthanded and
lefthanded helicities, in agreement with the helicity rule 4.10. By the
CP
transformation rule 4.11, a righthanded tau particle is transformed into
a lefthanded tau antiparticle and a lefthanded tau particle is transformed
into a righthanded tau antiparticle.

5.1.12. The tau 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 tRiemann 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,
ρ^{2})φ_{i}R(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 HiggsKibble mechanism. The tau antiparticle
can be observed with both righthanded and lefthanded helicities, in agreement
with the helicity rule 4.10. By the CP transformation rule 4.11,
a righthanded tau antiparticle is transformed into a lefthanded tau particle
and a lefthanded tau antiparticle is transformed into a righthanded tau
particle.

5.2. Quarks 


5.2.1. The up quark particle 

Let ψ_{upquark}
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_{upquark}
= φ_{i}R(β)
and γ = σ,
σρ
or
σρ^{2}. Then the particle frame corresponds
to the tRiemann 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_{i}R(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 +σρ^{2},
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 HiggsKibble mechanism. The up quark particle can
theoretically be observed with both righthanded and lefthanded helicities,
in agreement with the helicity rule 4.10. By the CP transformation
rule 4.11, a righthanded up quark particle is transformed into a lefthanded
up quark antiparticle and a lefthanded up quark particle is transformed
into a righthanded up quark antiparticle.

5.2.2. The up quark antiparticle 

Let ψ_{upquark}
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_{upquark} =
φ_{i}R(β)
and γ = σ,
σρ
or
σρ^{2}. Then the particle frame corresponds
to the tRiemann 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 σρ^{2},
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 HiggsKibble mechanism. The up quark antiparticle
can theoretically be observed with both righthanded and lefthanded helicities,
in agreement with the helicity rule 4.10. By the CP transformation
rule 4.11, a righthanded up quark antiparticle is transformed into a lefthanded
up quark particle and a lefthanded up quark antiparticle is transformed
into a righthanded up quark particle.

5.2.3. The down quark particle 

Let ψ_{downquark}
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_{downquark}
= R(γ)φ_{i}
and β = σ,
σρ
or
σρ^{2}. Then the particle frame corresponds
to the tRiemann 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 σρ^{2},
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 HiggsKibble mechanism. The down quark particle can
theoretically be observed with both righthanded and lefthanded helicities,
in agreement with the helicity rule 4.10. By the CP transformation
rule 4.11, a righthanded down quark particle is transformed into a lefthanded
down quark antiparticle and a lefthanded down quark particle is transformed
into a righthanded down quark antiparticle.

5.2.4. The down quark antiparticle 

Let ψ_{downquark}
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_{downquark}
= R(γ)φ_{i}
and β = σ,
σρ
or
σρ^{2}. Then the particle frame corresponds
to the tRiemann 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_{i}R(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 +σρ^{2}, 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 HiggsKibble
mechanism. The down quark antiparticle can theoretically be observed with
both righthanded and lefthanded helicities, in agreement with the helicity
rule 4.10. By the CP transformation rule 4.11, a righthanded down
quark antiparticle is transformed into a lefthanded down quark particle
and a lefthanded down quark antiparticle is transformed into a righthanded
down quark particle.

5.2.5. The charm quark particle 

Let ψ_{charmquark}
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_{charmquark}
= φ_{i}R(β)
and γ = σ,
σρ
or
σρ^{2}. Then the particle frame corresponds
to the tRiemann 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_{i}R(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 +σρ^{2},
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 HiggsKibble mechanism. The charm quark particle
can theoretically be observed with both righthanded and lefthanded helicities,
in agreement with the helicity rule 4.10. By the CP transformation
rule 4.11, a righthanded charm quark particle is transformed into a lefthanded
charm quark antiparticle and a lefthanded charm quark particle is transformed
into a righthanded charm quark antiparticle.

5.2.6. The charm quark antiparticle 

Let ψ_{charmquark}
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_{charmquark}
= φ_{i}R(β)
and γ = σ,
σρ
or
σρ^{2}. Then the particle frame corresponds
to the tRiemann 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 σρ^{2}, 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 HiggsKibble
mechanism. The charm quark antiparticle can theoretically be observed with
both righthanded and lefthanded helicities, in agreement with the helicity
rule 4.10. By the CP transformation rule 4.11, a righthanded charm
quark antiparticle is transformed into a lefthanded charm quark particle
and a lefthanded charm quark antiparticle is transformed into a righthanded
charm quark particle.

5.2.7. The strange quark particle 

Let ψ_{strangequark}
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_{strangequark}
= R(γ)φ_{i}
and β = σ,
σρ
or
σρ^{2}. Then the particle frame corresponds
to the tRiemann 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 σρ^{2},
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 HiggsKibble mechanism. The strange quark particle
can theoretically be observed with both righthanded and lefthanded helicities,
in agreement with the helicity rule 4.10. By the CP transformation
rule 4.11, a righthanded strange quark particle is transformed into a
lefthanded strange quark antiparticle and a lefthanded strange quark
particle is transformed into a righthanded strange quark antiparticle.

5.2.8. The strange quark antiparticle 

Let ψ_{strangequark}
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_{strangequark}
= R(γ)φ_{i}
and β = σ,
σρ
or
σρ^{2}. Then the particle frame corresponds
to the tRiemann 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_{i}R(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 +σρ^{2}, 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 HiggsKibble mechanism. The strange quark antiparticle can theoretically
be observed with both righthanded and lefthanded helicities, in agreement
with the helicity rule 4.10. By the CP transformation rule 4.11,
a righthanded strange quark antiparticle is transformed into a lefthanded
strange quark particle and a lefthanded strange quark antiparticle is
transformed into a righthanded strange quark particle.

5.2.9. The top (truth) quark particle 

Let ψ_{topquark}
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_{topquark}
= φ_{i}R(β)
and γ = σ,
σρ
or
σρ^{2}. Then the particle frame corresponds
to the tRiemann 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,
σρ^{2})φ_{i}R(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 +σρ^{2},
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 HiggsKibble mechanism. The top quark particle
can theoretically be observed with both righthanded and lefthanded helicities,
in agreement with the helicity rule 4.10. By the CP transformation
rule 4.11, a righthanded top quark particle is transformed into a lefthanded
top quark antiparticle and a lefthanded top quark particle is transformed
into a righthanded top quark antiparticle.

5.2.10. The top (truth) quark antiparticle 

Let ψ_{topquark}
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_{topquark}
= φ_{i}R(β)
and γ = σ,
σρ
or
σρ^{2}. Then the particle frame corresponds
to the tRiemann 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,
σρ^{2})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
σρ^{2}, 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 HiggsKibble
mechanism. The top quark antiparticle can theoretically be observed with
both righthanded and lefthanded helicities, in agreement with the helicity
rule 4.10. By the CP transformation rule 4.11, a righthanded top
quark antiparticle is transformed into a lefthanded top quark particle
and a lefthanded top quark antiparticle is transformed into a righthanded
top quark particle.

5.2.11. The bottom (beauty) quark particle 

Let ψ_{bottomquark}
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_{bottomquark}
= R(γ)φ_{i}
and β = σ,
σρ
or
σρ^{2}. Then the particle frame corresponds
to the tRiemann 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,
σρ^{2})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 σρ^{2},
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 HiggsKibble mechanism. The bottom quark particle
can theoretically be observed with both righthanded and lefthanded helicities,
in agreement with the helicity rule 4.10. By the CP transformation
rule 4.11, a righthanded bottom quark particle is transformed into a lefthanded
bottom quark antiparticle and a lefthanded bottom quark particle is transformed
into a righthanded bottom quark antiparticle.

5.2.12. The bottom (beauty) quark antiparticle 

Let ψ_{bottomquark}
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_{bottomquark}
= R(γ)φ_{i}
and β = σ,
σρ
or
σρ^{2}. Then the particle frame corresponds
to the tRiemann 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,
σρ^{2})φ_{i}R(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 +σρ^{2}, 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 HiggsKibble
mechanism. The bottom quark antiparticle can theoretically be observed
with both righthanded and lefthanded helicities, in agreement with the
helicity rule 4.10. By the CP transformation rule 4.11, a righthanded
bottom quark antiparticle is transformed into a lefthanded bottom quark
particle and a lefthanded bottom quark antiparticle is transformed into
a righthanded bottom quark particle.

6. Bosons 


6.1.1. The photon particle 

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 tRiemann 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_{i}R(β),
β+γ),
(+(0, ρ)φ_{i}R(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 HiggsKibble
mechanism must assign it zero mass. The photon particle can be observed
with both righthanded and lefthanded 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 righthanded photon
is transformed into a lefthanded photon and viceversa. Thus, CP
symmetry is preserved in particle interactions involving the photon, i.e.
electromagnetic interactions.

6.1.2. The photon antiparticle 

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 tRiemann 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_{i}R(γ),
β+γ),
(+(0, ρ)φ_{i}R(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 HiggsKibble mechanism must assign
it zero mass. The photon antiparticle can be observed with both righthanded
and lefthanded 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 righthanded photon antiparticle is transformed
into a lefthanded photon antiparticle and viceversa. Thus,
CP
symmetry is preserved in particle interactions involving the photon, i.e.
electromagnetic interactions.

6.2.1. Vector boson Z^{0} particle 

The vector boson Z^{0} particle
is the neutral carrier of the weak force. The unique permutation
ψ_{Z}^{0}
= R(β)φ_{i}
associated with the massive Z^{0} particle must have β
= 1 according to the mass rule 4.7 and we select γ
= 1. Then the particle frame corresponds to the tRiemann 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,
ρ^{2})φ_{i}R(β),
β+γ),
(+(0, σρ^{2})φ_{i}R(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, ρ^{2})R(γ)φ_{i},
β+γ),
((0, σρ^{2})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 Z^{0} particle on
the particle frame, according to figures 2.1 and 4.1. By the boson selection
rule 4.2, the Z^{0} particle is a boson. By the spin rule 4.4,
the spin of the Z^{0} particle is 1. By the electric charge rule
4.5, the electric charge of the Z^{0} 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 Z^{0}
particle is 91188 MeV, which can be attributed by the mass rule 4.7 to
the HiggsKibble mechanism. The Z^{0} particle can be observed
with both righthanded and lefthanded helicities, in agreement with the
helicity rule 4.10. Note that by the equivalence rule 4.8, the Z^{0}
particle and antiparticle 6.2.2 represent the same kind of particle in
the standard model. By the CP transformation rule 4.11, a righthanded
Z^{0} particle is transformed into a lefthanded Z^{0}
particle and viceversa. Thus, CP symmetry is preserved in particle
interactions involving the Z^{0} particle, i.e. neutral weak interactions.

6.2.2. Vector boson Z^{0} antiparticle 

The Z^{0} antiparticle is equivalent
to the Z^{0} particle, as shown by the following construction.
The unique permutation
ψ_{Z}^{0}
= R(β)φ_{i}
associated with the massive Z^{0} particle must have β
= 1 according to the mass rule 4.7 and we select γ
= 1. Then the particle frame corresponds to the tRiemann 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, ρ^{2})R(β)φ_{i},
β+γ),
((0, σρ^{2})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, ρ^{2})φ_{i}R(γ),
β+γ),
(+(0, σρ^{2})φ_{i}R(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 Z^{0} antiparticle
on the particle frame, according to figures 2.1 and 4.1. By the boson selection
rule 4.2, the Z^{0} antiparticle is a boson. By the spin rule 4.4,
the spin of the Z^{0} antiparticle is 1. By the electric charge
rule 4.5, the electric charge of the Z^{0} 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 Z^{0}
antiparticle is 91188 MeV, which can be attributed by the mass rule 4.7
to the HiggsKibble mechanism. The Z^{0} antiparticle can be observed
with both righthanded and lefthanded helicities, in agreement with the
helicity rule 4.10. Note that by the equivalence rule 4.8, the Z^{0}
particle 6.1.1 and antiparticle represent the same kind of particle in
the standard model. By the CP transformation rule 4.11, a righthanded
Z^{0} antiparticle is transformed into a lefthanded Z^{0}
antiparticle and viceversa. Thus, CP symmetry is preserved in particle
interactions involving the Z^{0} particle, i.e. neutral weak interactions.

6.3.1. Vector boson W^{+} particle 

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 tRiemann 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_{i}R(β),
β+γ),
(+(3, ρ^{2})φ_{i}R(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_{i}R(γ),
β+γ),
(+(3, ρ^{2})φ_{i}R(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 HiggsKibble mechanism. The W^{+} particle can be observed
with both righthanded and lefthanded 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 righthanded W^{+} particle is transformed into a lefthanded
W^{} particle and a lefthanded W^{+} particle is transformed
into a righthanded 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.

6.3.2. Vector boson W^{+} antiparticle 

The W^{+} antiparticle is equivalent
to the W^{} particle, as shown by the following construction.
The unique permutation 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 tRiemann 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, ρ^{2})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, ρ^{2})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 HiggsKibble mechanism. The W^{+} antiparticle can be observed
with both righthanded and lefthanded 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 righthanded W^{+} antiparticle is transformed into a lefthanded
W^{} antiparticle and a lefthanded W^{+} antiparticle
is transformed into a righthanded 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.

6.4.1. Vector boson W^{} particle 

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 tRiemann 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,
σρ^{2})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, σρ^{2})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 HiggsKibble mechanism. The W^{} particle can be observed
with both righthanded and lefthanded 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 righthanded W^{} particle is transformed into a lefthanded
W^{+} particle and a lefthanded W^{} particle is transformed
into a righthanded 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

6.4.2. Vector boson W^{} antiparticle 

The W^{} antiparticle is equivalent
to the W^{+} particle, as shown by the following construction.
The unique permutation
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 tRiemann 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, σρ^{2})φ_{i}R(β),
β+γ),
(+(3, σρ)φ_{i}R(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, σρ^{2})φ_{i}R(γ),
β+γ),
(+(3, σρ)φ_{i}R(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 HiggsKibble mechanism. The W^{} antiparticle can be observed
with both righthanded and lefthanded 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 righthanded W^{} antiparticle is transformed into a lefthanded
W^{+} antiparticle and a lefthanded W^{} antiparticle
is transformed into a righthanded 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.

6.5.1. The gluon particle 

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
σρ^{2}
and γ = σ,
σρ
or
σρ^{2}. Then the particle frame corresponds
to the tRiemann 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_{i}R(β),
β+γ),
(+(0, σ)φ_{i}R(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 HiggsKibble mechanism must assign it effective
zero mass. The gluon particle can theoretically be observed with both righthanded
and lefthanded 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 righthanded gluon is transformed into a lefthanded gluon and
viceversa. Thus, CP symmetry is preserved in particle interactions
involving the gluon, i.e. strong interactions.

6.5.2. The gluon antiparticle 

The gluon antiparticle is equivalent to
the gluon particle, as shown by the following construction. 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
σρ^{2}
and γ = σ,
σρ
or
σρ^{2}. Then the particle frame corresponds
to the tRiemann surface with this choice of φ_{i},
β,
γ.
By the antiparticle rule 4.9, for the first ray we select discs 17, 18
on the lower sheet of the particle frame and their blue regions 0,
that have the labels ((0, σρ)R(β)φ_{i},
β+γ),
((0, σ)R(β)φ_{i},
β+γ)
respectively, according to figures 3.4 and 3.5. For the second ray we select
discs 23, 24 on the lower sheet of the particle frame and their blue regions
0,
that have the labels (+(0, σρ)φ_{i}R(γ),
β+γ),
(+(0, σ)φ_{i}R(γ),
β+γ)
respectively, according to figures 3.4 and 3.5. These together represent
the Schrödinger discs corresponding to the strong field and the pair
of rays represent the gluon antiparticle on the particle frame, according
to figures 2.1 and 4.1. By the boson selection rule 4.2, the gluon antiparticle
is a boson. By the spin rule 4.4, the spin of the gluon antiparticle is
1. By the electric charge rule 4.5, the electric charge of the gluon antiparticle
is 0. By the strong charge rule 4.6, the gluon antiparticle carries 8 possible
strong anticharge/charge pairs as superpositions of particle frames (there
are 8 antigluon species), so N_{c} = 8. Due to quark confinement,
no free antigluons can be observed. From indirect experimental observations
the gluon antiparticle is massless and moves with the velocity of light,
so by the mass rule 4.7, the HiggsKibble mechanism must assign it effective
zero mass. The gluon antiparticle can theoretically be observed with both
righthanded and lefthanded helicities, in agreement with the helicity
rule 4.10. Note that by the equivalence rule 4.8, the gluon antiparticle
of a species and the particle 6.5.1 of the corresponding species represent
the same kind of particle in the standard model. By the
CP transformation
rule 4.11, a righthanded gluon antiparticle is transformed into a lefthanded
gluon antiparticle and viceversa. Thus,
CP symmetry is preserved
in particle interactions involving the gluon, i.e. strong interactions.

6.6.1. The graviton particle 

The graviton is the carrier of the gravitational
force. The unique permutation
ψ_{graviton}
= R(β)φ_{i}
associated with the massless graviton 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 tRiemann surface with this choice of φ_{i},
β,
γ.
Let m = 0, 1, 2, or 3, corresponding
to the gravitational charge rule 4.6. For the first ray we select discs
1, 12 on the upper sheet of the particle frame and their regions m,
that have the labels (+(m, 1)φ_{i}R(β),
β+γ),
((m, σ)R(γ)φ_{i},
β+γ)
respectively, according to figures 3.4 and 3.5. For the second ray we select
discs 6, 7 on the upper sheet of the particle frame and their regions
m,
that have the labels (+(m, σ)φ_{i}R(β),
β+γ),
((m, 1)R(γ)φ_{i},
β+γ)
respectively, according to figures 3.4 and 3.5. These together represent
the Schrödinger discs corresponding to the gravitational field and
the pair of rays represent the graviton particle on the particle frame,
according to figures 2.1 and 4.1. By the boson selection rule 4.2, the
graviton particle is a boson. By the spin rule 4.4, the spin of the graviton
particle is 2. By the electric charge rule 4.5, the electric charge of
the graviton particle is 0 and by the strong charge rule 4.6, its strong
charge is neutral with N_{c} = 1. By the gravitational charge
rule 4.6, the graviton particle carries 24 possible gravitational charge/anticharge
pairs as superpositions of particle frames (there are 24 graviton species).
The graviton particle has not been observed yet, but it is an inevitable
consequence of the gravitational force and quantum mechanics [8].
It is believed
[8]
that the graviton particle is massless and moves
with the velocity of light, so by the mass rule 4.7, the HiggsKibble mechanism
must assign it zero mass. The graviton particle can theoretically be observed
with both righthanded and lefthanded helicities, in agreement with the
helicity rule 4.10. Note that by the equivalence rule 4.8, the graviton
particle and antiparticle 6.6.2 represent the same kind of particle in
the standard model. By the
CP transformation rule 4.11, a righthanded
graviton is transformed into a lefthanded graviton and viceversa. Thus,
CP
symmetry is preserved in particle interactions involving the graviton,
i.e. gravitational interactions.

6.6.2. The graviton antiparticle 

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

6.7.1. The Higgs particle 

The Higgs particle attributes mass to
all the particles in the standard model, including itself. The unique permutation
ψ_{Higgs}
= R(β)φ_{i} associated
with the massive Higgs particle must have β
= 1 according to the mass rule 4.7. Furthermore, by the mass rule 4.7,
the permutation 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. We select
γ
= 1. Then the particle frame corresponds to the tRiemann surface
with this choice of φ_{i},
β,
γ.
By the Higgs selection rule 4.3, the Higgs particle is given as the intersection
of all the 24 discs of the tRiemann surface. We may regard the
Higgs particle as the intersection of the discs 1 ,..., 12 of the upper
sheet (the origin of the upper sheet) and the Higgs antiparticle as the
intersection of the discs 13, ..., 24 of the lower sheet (the origin of
the lower sheet). However, the particle and antiparticle 6.7.2 are identified
as the branch point (0,
β+γ)
of the tRiemann surface. These 24 discs together represent the
Schrödinger discs corresponding to the Higgs field and the branch
point represents the Higgs particle on the particle frame, according to
figures 2.1 and 4.1. By the Higgs selection rule 4.2, the Higgs particle
is a scalar boson. By the spin rule 4.4, the spin of the Higgs particle
is 0. By the electric charge rule 4.5, the electric charge of the Higgs
particle is 0 and by the strong charge rule 4.6, its strong charge is neutral
with N_{c} = 1. The Higgs particle has not been observed
yet, but it is an inevitable consequence of the HiggsKibble mechanism
[8].
By the mass rule 4.7, we can predict the mass of the Higgs particle (in
MeV) as
(91188 + 80280 + 80280)/2 = 125874

6.7.2. The Higgs antiparticle 

The Higgs antiparticle is equivalent to
the Higgs particle (they are actually identified on the particle frame),
as shown by the following construction. The Higgs particle attributes mass
to all the particles in the standard model, including itself. The unique
permutation
ψ_{Higgs} = R(β)φ_{i}
associated with the massive Higgs particle must have β
= 1 according to the mass rule 4.7. Furthermore, by the mass rule 4.7,
the permutation 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. We select
γ
= 1. Then the particle frame corresponds to the tRiemann surface
with this choice of φ_{i},
β,
γ.
By the Higgs selection rule 4.3 and the antiparticle rule 4.9, the Higgs
antiparticle is given as the intersection of all the 24 discs of the tRiemann
surface. We may regard the Higgs particle as the intersection of the discs
1 ,..., 12 of the upper sheet (the origin of the upper sheet) and the Higgs
antiparticle as the intersection of the discs 13, ..., 24 of the lower
sheet (the origin of the lower sheet). However, the particle 6.7.1 and
antiparticle are identified as the branch point (0,
β+γ)
of the tRiemann surface. These 24 discs together represent the
Schrödinger discs corresponding to the Higgs field and the branch
point represents the Higgs antiparticle on the particle frame, according
to figures 2.1 and 4.1. By the Higgs selection rule 4.2, the Higgs antiparticle
is a scalar boson. By the spin rule 4.4, the spin of the Higgs antiparticle
is 0. By the electric charge rule 4.5, the electric charge of the Higgs
antiparticle is 0 and by the strong charge rule 4.6, its strong charge
is neutral with N_{c} = 1. The Higgs antiparticle has not
been observed yet, but it is an inevitable consequence of the HiggsKibble
mechanism [8]. By the mass rule 4.7, we
can predict the mass of the Higgs antiparticle (in MeV) as
(91188 + 80280 + 80280)/2 = 125874

7. Force Fields 


7.1. The Electromagnetic Force Field 

In Section 7.1.1,
we review Maxwell's Electromagnetic Field Equations. In
Section
7.1.2, we show how the photon acts as the carrier of the electromagnetic
force. In Section 7.1.3, we give an example
of a typical electromagnetic interaction: electronelectron scattering.
In Section 7.1.4, we define the electromagnetic
gauge group
We explicitly define the observable gauge photon and show how the electromagnetic gauge group acts on it by means of the electromagnetic gauge transformations. 
7.1.1. Maxwell's Electromagnetic Field Equations 

Maxwell's equations for electromagnetism
remain unchanged if the spacetime coordinates are subjected to Lorentz
transformations given by equation 1.3. To demonstrate Lorentz invariance,
we must put Maxwell's equations into the fourdimensional form required
by special relativity. We use Einstein's notation for tensors
[5],
with superscripts for components of contravariant fourvectors, subscripts
for components of covariant fourvectors and subscript commas for partial
derivatives. We choose units of distance and time such that the velocity
of light c = 1. Maxwell's equations are usually written as:

7.1.2. The photon as the carrier of the electromagnetic force 

We now show how the photon 6.1.1 can be
regarded as the carrier of the electromagnetic force. The unique permutation
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 tRiemann 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_{i}R(β),
β+γ),
(+(0, ρ)φ_{i}R(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 of the photon 6.1.1 and the electromagnetic
field via the following correspondence:

7.1.3. Electromagnetic Interactions 

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

7.1.4. The Electromagnetic Gauge Group 

The electromagnetic field described by
the tensor F_{μν} is a special
case of a YangMills gauge field, corresponding to the photon. The electromagnetic
YangMills gauge field is specified by its gauge group
G_{e}
and a constant of interaction α_{e},
called its coupling constant. The coupling constant will be calculated
explicitly in 8.2. We shall now construct the gauge group G_{e}
for the electromagnetic field. Consider the unitary group U(1) consisting
of 1×1 complex unitary matrices under
matrix multiplication. The unitary group
U(1) is generated (as a
Lie group) by a single 1×1 matrix, whose
only entry can be assumed to be the real number 1 (for the unitary group
with n parameters, any set of n^{2} linearly independent
n×n
Hermitian matrices is a set of generators). We first define two copies
of U(1), called
U(1)_{e} and U(1)_{w},
as follows:
The row and column labels
in figure 7.1.4.3 specify the Schrödinger discs of the photon and
the electromagnetic force as defined by equations 7.1.2 and figure 7.1.2.
The first component of the row and column labels consists of electromagnetic
charge; the second component specifies how the electromagnetic gauge group
will embed in the weak gauge group 7.2.9. The observable photon
is defined as Γ_{1}. The observable
photon is regarded as a superposition of the electromagnetic charge  electromagnetic
anticharge labels of its row and column multiplied by a complex number
(just 1, in this case) viewed on the zplane. Since t = z^{2},
a rotation by an angle θ around the origin of
the zplane corresponds to a rotation by an angle 2θ
around the branch point of the tRiemann surface. In this case,
multiplication by 1 on the zplane corresponds to a trivial rotation
of the tRiemann surface by 0 degrees around the branch point. This
means that for the observable photon, the rays defining the photon (and
its equivalent antiparticle) are permuted trivially amongst themselves
(and not any other rays) on the particle frame. Thus, the observable photon
corresponds to (trivial) superpositions of particle frames for the photon.
The electromagnetic gauge group acts (trivially) on the observable photon
by conjugation, viewed on the zplane:
Again, since t = z^{2}, a rotation by an angle θ around the origin of the zplane corresponds to a rotation by an angle 2θ around the branch point of the tRiemann surface and multiplication by 1 on the zplane corresponds to a trivial rotation of the tRiemann surface by 0 degrees around the branch point. This means that for any electromagnetic gauge transformation, the rays defining the photon (and its equivalent antiparticle) are permuted amongst themselves (and not any other rays) on the particle frame. The electromagnetic gauge transformations will always transform superpositions of photons to other superpositions of photons. 
7.2. The Weak Force Field 

In Section 7.2.1,
we define the YangMills Weak Field Equations. In Section
7.2.2, we show how the Z^{0} acts as the neutral carrier
of the weak force. In Section 7.2.3, we give
an example of a typical weak Z^{0} interaction: muonic neutrinoelectron
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
We explicitly define the observable gauge vector bosons [Z^{0}], [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 Z^{0} masses, as well as the ratio of the weak Z^{0} mediated interaction, called its mixing. 
7.2.1. YangMills Weak Field Equations 

Corresponding to the three vector bosons
Z^{0}, W^{+}, W^{} we have three electromagnetic
type fields defined by the tensors F_{μν}^{(0)},
F_{μν}^{(+)},
F_{μν}^{()}
respectively. Thus, we have three covariant 4vectors

7.2.2. The vector boson Z^{0} as the neutral carrier of the weak force 

We now show how the vector boson Z^{0}
particle 6.2.1 can be regarded as the neutral carrier of the weak force.
The unique permutation
ψ_{Z}^{0}
= R(β)φ_{i}
associated with the massive Z^{0} particle must have β
= 1 according to the mass rule 4.7 and we select γ
= 1. Then the particle frame corresponds to the tRiemann 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,
ρ^{2})φ_{i}R(β),
β+γ),
(+(0, σρ^{2})φ_{i}R(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, ρ^{2})R(γ)φ_{i},
β+γ),
((0, σρ^{2})R(γ)φ_{i},
β+γ)
respectively, according to figures 3.4 and 3.5. These together represent
the Schrödinger discs of the vector boson Z^{0} particle and
the neutral component of the weak field via the following correspondence:

7.2.3. Weak Z^{0} Interactions 

A typical weak Z^{0} interaction
is shown in the following Feynman diagram for a muonic neutrino  electron
collision: a μneutrino and an electron collide
(elastically) and a virtual Z^{0} (the neutral carrier of the weak
force) is exchanged.
We can visualize this weak Z^{0} interaction as shown below
in figure 7.2.3.2: (a) a μneutrino and an electron
approach each other; (b) two of the Schrödinger discs of the weak
Z^{0} field destroy the old μneutrino
and the old electron (the corresponding wave functions are now interpreted
as destruction operators); (c) the other two Schrödinger discs
of the weak Z^{0} field create a new μneutrino
and a new electron (the corresponding wave functions are now interpreted
as creation operators); (d) together, the four Schrödinger
discs of the weak Z^{0} field constitute the virtual Z^{0}that
is exchanged while transmitting the neutral weak force; (e) the μneutrino
and electron have collided (elastically).

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

We now show how the vector boson W^{+}
particle 6.3.1 can be regarded as the positive carrier of the weak force.
The unique permutation
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 tRiemann 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_{i}R(β),
β+γ),
(+(3, ρ^{2})φ_{i}R(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_{i}R(γ),
β+γ),
(+(3, ρ^{2})φ_{i}R(γ),
β+γ)
respectively, according to figures 3.4 and 3.5. These together represent
the Schrödinger discs of the vector boson W^{+} particle and
the positive component of the weak field via the following correspondence:

7.2.5. Weak W^{+} Interactions 

When a neutron interacts with a neutrino,
a W^{+} can be exchanged, transforming the neutron into a proton
and producing an electron. A down quark in the neutron changes into an
up quark due to an intermediate interaction with a virtual W^{+},
transforming the neutron into a proton. Although quarks are not directly
observed due to confinement, we may still represent this interaction by
the following Feynman diagram.
We can visualize this weak W^{+} interaction as shown below
in figure 7.2.5.2: (a) a down quark and a neutrino approach each other;
(b) two of the Schrödinger discs of the weak W^{+} field destroy
the old down quark and the old neutrino (the corresponding wave functions
are now interpreted as destruction operators); (c) the other two
Schrödinger discs of the weak W^{+} field create a new up
quark and a new electron (the corresponding wave functions are now interpreted
as creation operators); (d) together, the four Schrödinger
discs of the weak W^{+} field constitute the virtual W^{+}
that is exchanged while transmitting the positive weak force; (e) an up
quark and an electron have been created.

7.2.6. The vector boson W^{} as the negative carrier of the weak force 

We now show how the vector boson W^{}
particle 6.4.1 can be regarded as the negative carrier of the weak force.
The unique permutation
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 tRiemann 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,
σρ^{2})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, σρ^{2})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 of the vector boson W^{} particle and
the negative component of the weak field via the following correspondence:

7.2.7. Weak W^{} Interactions 

When a neutron interacts with a neutrino,
a W^{} can be exchanged, transforming the neutron into a proton
and producing an electron. A down quark in the neutron changes into an
up quark due to an intermediate interaction with a virtual W^{},
transforming the neutron into a proton. Although quarks are not directly
observed due to confinement, we may still represent this interaction by
the following Feynman diagram.
We can visualize this weak W^{} interaction as shown below
in figure 7.2.7.2: (a) a down quark and a neutrino approach each other;
(b) two of the Schrödinger discs of the weak W^{} field destroy
the old down quark and the old neutrino (the corresponding wave functions
are now interpreted as destruction operators); (c) the other two
Schrödinger discs of the weak W^{} field create a new up
quark and a new electron (the corresponding wave functions are now interpreted
as creation operators); (d) together, the four Schrödinger
discs of the weak W^{} field constitute the virtual W^{}
that is exchanged while transmitting the negative weak force; (e) an up
quark and an electron have been created.

7.2.8. The Weak Gauge Group 

The weak YangMills field 7.2.1 consists
of three electromagnetic type fields defined by the tensors F_{μν}^{(0)},
F_{μν}^{(+)},
F_{μν}^{()}
corresponding to the three vector bosons Z^{0}, W^{+},
W^{} respectively. The weak YangMills gauge field is specified
by its gauge group G_{w} and a constant of interaction α_{w},
called its coupling constant. The coupling constant will be calculated
explicitly in 8.2. We shall now construct the gauge group G_{w}
for the weak field. Consider the special unitary group SU(2) consisting
of 2×2 complex unitary matrices of determinant
1, under matrix multiplication. The special unitary group
SU(2)
is generated (as a Lie group) by the three 2×2
unitary matrices of determinant 1 (called Pauli generators). We
first define two copies of SU(2), called SU(2)_{w}
and SU(2)_{s}, as follows:
The row and column labels
in figure 7.2.8.3 specify the Schrödinger discs of the three vector
bosons Z^{0}, W^{+}, W^{} and the weak force as
defined by equations 7.2.2, 7.2.4 and 7.2.6, respectively. The first component
of the row and column labels in figure 7.2.8.3 consists of weak charge;
the second component specifies how the weak gauge group will embed in the
strong gauge group 7.3.4. The three observable vector bosons are
defined as
Each observable vector boson [Z^{0}], [W^{+}] or [W^{}]
is regarded as a superposition of the weak charge  weak anticharge labels
of its row and column multiplied by a complex number viewed on the zplane.
Since t = z^{2}, a rotation by an angle θ
around the origin of the zplane corresponds to a rotation by an
angle 2θ around the branch point of the tRiemann
surface. In particular, notice that multiplication by ±i,
±1
on the zplane correspond to rotations of the tRiemann surface
by ±180,
±360
degrees around the branch point, respectively. This means that for the
three observable vector bosons [Z^{0}], [W^{+}], [W^{}],
the rays defining the three vector bosons Z^{0}, W^{+},
W^{} are permuted amongst themselves (and not any other rays)
on the particle frame. Thus, the three observable vector bosons [Z^{0}],
[W^{+}], [W^{}] correspond to superpositions of particle
frames for the three vector bosons Z^{0}, W^{+}, W^{}.
The weak gauge group acts on the three observable vector bosons [Z^{0}],
[W^{+}], [W^{}] by conjugation, viewed on the
zplane:
Again, since t = z^{2}, a rotation by an angle θ around the origin of the zplane corresponds to a rotation by an angle 2θ around the branch point of the tRiemann surface and multiplication by ±i, ±1 on the zplane correspond to rotations of the tRiemann surface by ±180, ±360 degrees around the branch point, respectively. This means that for any weak gauge transformation, the rays defining the three vector bosons Z^{0}, W^{+}, W^{} are permuted amongst themselves (and not any other rays) on the particle frame. The weak gauge transformations will always transform superpositions of the three vector bosons Z^{0}, W^{+}, W^{} to other superpositions of the three vector bosons Z^{0}, W^{+}, W^{}. 
7.2.9. The Weinberg Angle 

The Weinberg angle θ_{w}
is a parameter that gives a relationship between the W^{+}, W^{}
and Z^{0} masses, as well as the ratio of the weak Z^{0}
mediated interaction, called its mixing. Indeed, from figures 7.2.2,
7.2.4, 7.2.6 and 7.2.8.4, it is apparent that the components of the weak
Z^{0} field mix with the components of the weak W^{+},
W^{} fields and the angle subtended by the mixing Schrödinger
discs on the particle frame is exactly π/6 radians
or 30 degrees. Hence, we predict that θ_{w}
= 30 degrees. This is in good agreement with the SLAC experiment [12]
which estimates sin^{2}(θ_{w})
= 0.2397, i.e. θ_{w} =
29.3137 degrees (this is a running value, depending on the momentum
at which it is measured, with a significance of 6 standard deviations).

7.3. The Strong Force Field 

In Section 7.3.1,
we define the YangMills 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 quarkantiquark pair, called
a meson. In Section 7.3.4, we define the strong
gauge group
We explicitly define the eight species of observable gauge gluons and show how the strong gauge group acts on them by means of the strong gauge transformations. 
7.3.1. YangMills Strong Field Equations 

Corresponding to the eight gluon species
described in 6.5.1, we have eight electromagnetic type fields defined by
the eight tensors F_{μν}^{(s)},
s
= 1, ..., 8. Thus, we have eight covariant 4vectors

7.3.2. The gluon as the carrier of the strong force 

We now show how the gluon 6.5.1 can be
regarded as the carrier of the strong force. The unique permutation
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
σρ^{2}
and γ = σ,
σρ
or
σρ^{2}. Then the particle frame corresponds
to the tRiemann 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_{i}R(β),
β+γ),
(+(0, σ)φ_{i}R(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. By the strong charge rule
4.6, the gluon particle carries eight possible strong charge/anticharge
pairs as superpositions of particle frames (there are eight gluon species
indexed by s = 1, ..., 8). These together represent the Schrödinger
discs of the gluon 6.5.1 and the strong field via the following correspondence,
for s = 1, ..., 8:

7.3.3. Strong Interactions 

No free quarks are observed because of
the phenomenon of quark confinement. Quarks are bound together by
the strong force forming neutrons, protons and mesons. If one were to try
and isolate a quark at a distance greater than the proton volume, the energy
required would be greater than the energy required to form a quarkantiquark
pair (a meson) and the lower energy process is favoured in nature. The
strong force obeys the law of asymptotic freedom: at distances comparable
to the proton volume the strong force effectively vanishes, so that quarks
are essentially free to move about within this confined volume. A typical
strong interaction is shown in the following Feynman diagram for the formation
of an up quark  strange antiquark pair (a meson), called the K^{+}
kaon. An up quark and a strange antiquark experience an attractive force
(which is asymptotically free) and a virtual gluon (the carrier of the
strong force) is exchanged. Suppose that the incoming up quark has a strong
charge σ and the incoming strange antiquark
has a strong charge σρ. The exchanged virtual
gluon A_{s} has a pair of strong charges
σ
 σρ and causes the strong charge of the outgoing
up quark to change to σρ and the strong charge
of the outgoing strange antiquark to change to σ.
We can visualize this strong interaction as shown below in figure 7.3.3.2:
(a) the up quark (with strong charge σ) and
the strange antiquark (with strong charge σρ)
within the confinement volume ; (b) two of the Schrödinger discs of
the strong field destroy the up quark and the strange antiquark (the corresponding
wave functions are now interpreted as destruction operators); (c)
the other two Schrödinger discs of the strong field create a new up
quark (with strong charge σρ) and a new strange
antiquark (with strong charge σ) (the corresponding
wave functions are now interpreted as creation operators); (d) together,
the four Schrödinger discs of the strong field constitute the virtual
gluon that is exchanged while transmitting the strong force; (e) the up
quark (with strong charge σρ) and the strange
antiquark (with strong charge σ) stay within
the confinement volume, forming a meson.

7.3.4. The Strong Gauge Group 

The strong YangMills field 7.3.2 consists
of eight electromagnetic type fields defined by the tensors F_{μν}^{(s)}
for
s = 1, ..., 8, corresponding to the eight gluon species. The
strong YangMills gauge field is specified by its gauge group G_{s}
and a constant of interaction α_{s},
called its coupling constant. The coupling constant will be explicitly
calculated in 8.2. We shall now construct the gauge group G_{s}
for the strong field. Consider the special unitary group SU(3) consisting
of 3×3 complex unitary matrices of determinant
1, under matrix multiplication. The special unitary group
SU(3)
is generated (as a Lie group) by the eight 3×3
unitary matrices of determinant 1 (called GellMann generators).
We first define two copies of SU(3), called SU(3)_{s}
and SU(3)_{g}, as follows:
The row and column labels
in figure 7.3.4.3 specify the Schrödinger discs of the gluon and
the strong force as defined by equations 7.3.2 and figure 7.3.2. The first
component of the row and column labels in figure 7.3.4.3 consists of strong
charges; the second component specifies how the strong gauge group will
embed in the gravitational gauge group 7.4.6. The eight observable
gluon species are defined as
Each observable gluon [A^{(s)}] is regarded as a superposition
of the strong charge  strong anticharge labels of its row and column multiplied
by a complex number viewed on the zplane. Since t = z^{2},
a rotation by an angle θ around the origin of
the zplane corresponds to a rotation by an angle 2θ
around the branch point of the tRiemann surface. In particular,
note that multiplication by ±i,
±1
on the zplane correspond to rotations of the tRiemann surface
by ±180,
±360
degrees around the branch point, respectively. This means that for the
eight observable gluons [A^{(s)}], for s = 1, ...,
8, the rays defining the eight gluons A^{(s)}, for s
= 1, ..., 8, are permuted amongst themselves (and not any other rays) on
the particle frame. Thus, the eight observable gluons [A^{(s)}],
for s = 1, ..., 8, correspond to superpositions of particle frames
of the eight gluons A^{(s)}, for s = 1, ..., 8.
The strong gauge group acts by conjugation on the eight observable gluons
[A^{(s)}], for s = 1, ..., 8, viewed on the
zplane:
Again, since t = z^{2}, a rotation by an angle θ around the origin of the zplane corresponds to a rotation by an angle 2θ around the branch point of the tRiemann surface and multiplication by ±i, ±1 on the zplane correspond to rotations of the tRiemann surface by ±180, ±360 degrees around the branch point, respectively. This means that for any strong gauge transformation, the rays defining the eight gluons A^{(s)}, for s = 1, ..., 8, are permuted amongst themselves (and not any other rays) on the particle frame. The strong gauge transformations will always transform superpositions of the eight gluons A^{(s)}, for s = 1, ..., 8, to other superpositions of the eight gluons A^{(s)}, for s = 1, ..., 8. 
7.4. The Gravitational Force Field 

In Section 7.4.1,
we review General Relativity and curved spacetime. 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 spacetime
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 spacetime without selfintersections.
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
We explicitly define the twentyfour species of observable gauge gravitons and show how the gravitational gauge group acts on them by means of the gravitational gauge transformations. 
7.4.1. General Relativity 

To account for the gravitational force,
Einstein [7] assumed that physical spacetime
forms a curved Riemann space and thereby laid the foundation for his theory
of gravitation. Generalizing equation 1.3 of special relativity to general
relativity, the invariant distance dS between a point (with spacetime
coordinates written as a contravariant fourvector) X
^{μ}
and a neighbouring point X
^{μ}
+ dX ^{μ} is given by

7.4.2. Embedding the particle frame in spacetime 

So far, we have drawn the particle frame
(figure 4.1) embedded in flat Euclidean threedimensional space (X,
Y,
Z)
and associated with the drawing an independent time dimension
T.
Each of the discs in the complex plane have been drawn as discs embedded
in flat Euclidean threedimensional space (X,
Y,
Z).
Such an embedding of the Riemann surface (figure 3.4) in flat Euclidean
threedimensional space has selfintersections. We shall show how to embed
the particle frame in curved fourdimensional spacetime without selfintersections.
Let us first review the construction of the particle frame as discs embedded
in flat Euclidean threedimensional space. Consider the composition of
the functions C → C;
z → t
= z^{2} and C → C;
t → w
= t^{12}. The composite is given by the assignment z → t
= z^{2} → w = t^{12
}=
z^{24}.
Take twentyfour identical copies of the map m(4) on the
disc with a cut, labeled k = 1, ..., 24.
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 wRiemann surface embedded in Euclidean
threedimensional space (X, Y, Z).
The point w = 0 connects all the discs and is called the branch point. There are twentyfour superposed copies of the map m(4) on the wRiemann surface corresponding to the twentyfour sectors
Note that the ray shown in figure 7.4.2.3 will go once around each of
the 24 Schrödinger discs of the particle frame represented by the
wRiemann
surface, as T goes continuously from 0 to 2π.
This embedding will also carry the metric g_{μν}of
spacetime, since the spacetime distance for any two points on the
wRiemann
surface is given via the embedding. If we take an infinitesimal contravariant
fourvector on the ray arg z = 0 and compute its change in one circuit
given by arg z = T, as T goes continuously from 0
to 2π, we obtain precisely the curvature tensor
given by equation 7.4.1.5. Hence, each of the 24 Schrödinger discs
of the particle frame represented by the wRiemann surface carry
the curvature tensor R _{μντλ}
and the Ricci tensor R _{ντ}.
Referring to the mass rule 4.7, the inertial mass of each particle S in the standard model is associated with a unique permutation ψ_{S}. According to general relativity 7.4.1, 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. 
7.4.3. The Gravitational Field Equations 

In analogy with the definition of the
electromagnetic field by Maxwell's equations 7.1.1, we can now define the
gravitational field as follows. The contravariant fourvector (X ^{0},
X
^{1},
X
^{2},
X
^{3})
represents spacetime and K_{μν} = g_{μν}
represents the gravitational potential. Since the g_{μν}
are symmetric, only 10 out of the 16 are independent, hence the gravitational
potential
K may be represented by a tenvector (instead of a fourvector
as in the case of the electromagnetic potential). Define the contravariant
and covariant fourvectors
Corresponding to the twentyfour graviton species described in 6.6.1, we have twentyfour gravitational type fields defined by the twentyfour tensors F_{μν}^{(g)}, g = 1, ..., 24. Thus, we have twentyfour covariant 4vectors

7.4.4. The graviton as the carrier of the gravitational force 

We now show how the graviton 6.6.1 can
be regarded as the carrier of the gravitational force. The unique permutation
ψ_{graviton}
= R(β)φ_{i}
associated with the massless graviton 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 tRiemann surface with this choice of φ_{i},
β,
γ.
Let m = 0, 1, 2, or 3, corresponding
to the gravitational charge rule 4.6. For the first ray we select discs
1, 12 on the upper sheet of the particle frame and their regions m,
that have the labels (+(m, 1)φ_{i}R(β),
β+γ),
((m, σ)R(γ)φ_{i},
β+γ)
respectively, according to figures 3.4 and 3.5. For the second ray we select
discs 6, 7 on the upper sheet of the particle frame and their regions
m,
that have the labels (+(m, σ)φ_{i}R(β),
β+γ),
((m, 1)R(γ)φ_{i},
β+γ)
respectively, according to figures 3.4 and 3.5. By the gravitational charge
rule 4.6, the graviton particle carries 24 possible gravitational charge/anticharge
pairs as superpositions of particle frames (there are 24 graviton species).
These together represent the Schrödinger discs of the graviton 6.6.1
and the gravitational field via the following correspondence, for g
= 1, ..., 24:

7.4.5. Gravitational Interactions 

The gravitational force is extremely weak
compared to the other forces. In most cases, it would be very difficult
to detect a gravitational interaction in an experiment, where it would
be masked by an electromagnetic, weak or strong interaction. However, there
is a possibility that such an interaction has already been observed, albeit
indirectly, in the SuperKamiokande experiment
[10],
where evidence for neutrino oscillations was found. A possible gravitational
interaction is shown in the following Feynman diagram that may explain
neutrino oscillations. A μneutrino is converted
into a τneutrino, and vice versa, by the exchange
of a virtual graviton. Suppose that the incoming μneutrino
has a gravitational charge + 0 and the incoming τneutrino
has a gravitational charge  0. The exchanged virtual graviton g
has a pair of gravitational charges 0  0 and causes the
gravitational charge of the outgoing τneutrino
to change to  0 and the gravitational charge of the outgoing μneutrino
to change to + 0.
We can visualize this gravitational interaction as shown below in figure
7.4.5.2: (a) the μneutrino (with gravitational
charge + 0 ) and the τneutrino (with
gravitational charge  0); (b) two of the Schrödinger discs
of the gravitational field destroy the μneutrino
and the τneutrino (the corresponding wave functions
are now interpreted as destruction operators); (c) the other two
Schrödinger discs of the gravitational field create a new τneutrino
(with gravitational charge  0) and a new μneutrino
(with gravitational charge + 0 ) (the corresponding wave functions
are now interpreted as creation operators); (d) together, the four
Schrödinger discs of the gravitational field constitute the virtual
graviton that is exchanged while transmitting the gravitational force;
(e) the μneutrino (with gravitational charge
+ 0 ) and the τneutrino (with gravitational
charge  0) have oscillated.

7.4.6. The Gravitational Gauge Group 

The gravitational field 7.4.4 consists
of twentyfour gravitational type fields defined by the tensors F_{μν}^{(g)}
for
g = 1, ..., 24, corresponding to the twentyfour graviton species.
The gravitational gauge field is specified by its gauge group G_{g}
and a constant of interaction α_{g},
called its coupling constant. The coupling constant will be calculated
explicitly in 8.2. We shall now construct the gauge group G_{g}
for the gravitational field. Consider the special unitary group SU(5)
consisting of 5×5 complex unitary matrices
of determinant 1, under matrix multiplication. The special unitary group
SU(5)
is generated (as a Lie group) by the twentyfour 5×5
unitary matrices of determinant 1. We first define two copies of SU(5),
called SU(5)_{g} and SU(5)_{e},
as follows:
The row and column labels
in figure 7.4.6.3 specify the Schrödinger discs of the graviton
and the gravitational force as defined by equations 7.4.4 and figure 7.4.4.
The first component of the row and column labels in figure 7.4.6.3 consists
of gravitational charge; the second component specifies how the electric
and electromagnetic charges are embedded in the gravitational gauge group
during the Planck epoch. The twentyfour observable graviton species
are defined as
Each observable graviton [g^{(i)}] is regarded as a superposition
of the gravitational charge  gravitational anticharge labels of its row
and column multiplied by a complex number viewed on the zplane.
Since t = z^{2}, a rotation by an angle θ
around the origin of the zplane corresponds to a rotation by an
angle 2θ around the branch point of the tRiemann
surface. In particular, note that multiplication by ±i,
±1
on the zplane correspond to rotations of the tRiemann surface
by ±180,
±360
degrees around the branch point, respectively. This means that for the
twentyfour observable gravitons [g^{(i)}], for i
= 1, ..., 24, the rays defining the twentyfour gravitons g^{(i)},
for i = 1, ..., 24, are permuted amongst themselves (and not any
other rays) on the particle frame. Thus, the twentyfour observable gravitons
[g^{(i)}], for i = 1, ..., 24, correspond to superpositions
of particle frames of the twentyfour gravitons g^{(i)},
for i = 1, ..., 24.
The gravitational gauge group acts by conjugation on the twentyfour
observable gravitons [g^{(i)}], for i = 1, ..., 24,
viewed on the
zplane:
Again, since t = z^{2}, a rotation by an angle θ around the origin of the zplane corresponds to a rotation by an angle 2θ around the branch point of the tRiemann surface and multiplication by ±i, ±1 on the zplane correspond to rotations of the tRiemann surface by ±180, ±360 degrees around the branch point, respectively. This means that for any gravitational gauge transformation, the rays defining the twentyfour gravitons g^{(i)}, for i = 1, ..., 24, are permuted amongst themselves (and not any other rays) on the particle frame. The strong gauge transformations will always transform superpositions of the twentyfour gravitons g^{(i)}, for i = 1, ..., 24, to other superpositions of the twentyfour gravitons g^{(i)}, for i = 1, ..., 24. 
8. Grand Unification 

We achieve our second goal, the Grand
Unification of all the forces: electromagnetic, weak, strong and gravitational.
In Section 8.1, we follow the cosmological timeline
into the past, from the present to the Big Bang (or equivalent energy scales),
showing how the gauge groups are embedded in a sequence
during the unification and also showing how the Schrödinger discs are identified on the particle frame during the unification. In Section 8.2, we explicitly calculate all the coupling constants: the electromagnetic coupling constant, the weak coupling constant, the strong coupling constant and the gravitational coupling constant. Finally, in Section 8.3, we explicitly calculate the mass ratios of the particles in the standard model. 
8.1. The Gauge Groups 

If we follow the cosmological timeline
forward in time, we can see how the forces separated and how the particle
frame evolved to its present form. Viewed backwards in time, we obtain
the unification of the forces, showing how the gauge groups are embedded
in a sequence U(1) → SU(2) → SU(3) → SU(5)
and also how the Schrödinger discs are identified on the particle
frame.
8.1.1. The Later Epochs, upto the Present
The particle frame, labeled as in figures 3.5 and 4.1, with all the
bosons describing the four forces in their present form, is shown below.
At this time the particle frame assumes its present form as shown in
figure 4.1, represented by the tRiemann surface given by the composition
of the functions C → C;z → t
= z^{2} and C → C;
t → w
= t^{12}. The composite is given by the assignment z → t
= z^{2} → w = t^{12
}=
z^{24}.
8.1.2. The Inflationary Epoch
The particle frame, labeled as in figures 3.5 and 4.1, now has the Schrödinger
discs with labels 1 and ρ identified, corresponding
to the unification of the electromagnetic force with the weak force, as
shown below.
During this epoch the particle frame may be represented by the tRiemann
surface given by the composition of the functions C → C;
z → t
= z^{2} and C → C;
t → w
= t^{6}. The composite is given by the assignment z → t
= z^{2} → w = t^{6
}=
z^{12}.
8.1.3. The Grand Unification Epoch
The particle frame, labeled as in figures 3.5 and 4.1, now has the Schrödinger
discs with labels 1, ρ, ρ^{2},
σρ,
σρ^{2}
identified, corresponding to the unification of the weak force with the
strong force, as shown below.
During this epoch the particle frame may be represented by the tRiemann
surface given by the composition of the functions C → C;
z → t
= z^{2} and C → C;
t → w
= t^{4}. The composite is given by the assignment z → t
= z^{2} → w = t^{4
}=
z^{8}.
8.1.4. The Planck Epoch
The particle frame, labeled as in figures 3.5 and 4.1, now has the Schrödinger
discs with labels 1, ρ, ρ^{2},
σρ,
σρ^{2},
σ
identified, corresponding to the unification of the strong force with the
gravitational force, as shown below.
During this epoch the particle frame may be represented by the tRiemann
surface given by the composition of the functions C → C;
z → t
= z^{2} and C → C;
t → w
= t^{2}. The composite is given by the assignment z → t
= z^{2} → w = t^{2
}=
z^{4}.
A boson must be present as the carrier of the force and for the creation
and destruction of particles. Since this is the minimum configuration required
to define a boson, according to the boson selection rule 4.2, we have reached
a limit with the grand unification of all the four forces immediately after
the Big Bang at time T = 0.

8.2. The Gauge Coupling Constants 

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

8.3. The Mass Ratios 

We shall explicitly define the particle
mass ratios by counting the number of regions in the subdivision of the
upper surface of the particle frame. These numbers are symmetric across
the lower surface of the particle frame, giving the antiparticle mass ratios.
Define n_{field} to be the number of regions in the subdivision
of the upper surface of the particle frame during the separation of the
corresponding force in the cosmological timeline. Then
We define the mass ratio constants in direct analogy with the coupling constants. The mass ratio constant corresponding to the strong force, after it separates from the gravitational force and the quarks become confined, is given by
The exact values of the neutrino masses are not yet known, but we know that they are not zero, and perhaps the neutrino mass ratios will also be found amongst the mass ratio constants 8.3.8. We have now calculated all the parameters that define the standard model and its associated force fields, according to 't Hooft's specification [8]. Thus, the mathematical proof of the four colour theorem shows that the grand unification of the standard model with quantum gravity is complete, and rules out the possibility of finding any other kinds of particles. 
References 


Copyright © 2008 by Ashay Dharwadker. All rights reserved. 